ABSTRACT
Title of dissertation: OPTICAL AND CASIMIR EFFECTS IN
TOPOLOGICAL MATERIALS
Justin H. Wilson, Doctor of Philosophy, 2015
Dissertation directed by: Professor Victor M. Galitski
Department of Physics
Two major electromagnetic phenomena, magneto-optical effects and the Casimir
effect, have seen much theoretical and experimental use for many years. On the
other hand, recently there has been an explosion of theoretical and experimental
work on so-called topological materials, and a natural question to ask is how such
electromagnetic phenomena change with these novel materials. Specifically, we will
consider are topological insulators and Weyl semimetals.
When Dirac electrons on the surface of a topological insulator are gapped or
Weyl fermions in the bulk of a Weyl semimetal appear due to time-reversal symmetry
breaking, there is a resulting quantum anomalous Hall effect (2D in one case and
bulk 3D in the other, respectively). For topological insulators, we investigate the
role of localized in-gap states which can leave their own fingerprints on the magneto-
optics and can therefore be probed. We have shown that these states resonantly
contribute to the Hall conductivity and are magneto-optically active. For Weyl
semimetals we investigate the Casimir force and show that with thickness, chemical
potential, and magnetic field, a repulsive and tunable Casimir force can be obtained.
Additionally, various values of the parameters can give various combinations of traps
and antitraps.
We additionally probe the topological transition called a Lifshitz transition in
the band structure of a material and show that in a Casimir experiment, one can
observe a non-analytic “kink” in the Casimir force across such a transition. The
material we propose is a spin-orbit coupled semiconductor with large g-factor that
can be magnetically tuned through such a transition. Additionally, we propose an
experiment with a two-dimensional metal where weak localization is tuned with
an applied field in order to definitively test the effect of diffusive electrons on the
Casimir force—an issue that is surprisingly unresolved to this day.
Lastly, we show how the time-continuous coherent state path integral breaks
down for both the single-site Bose-Hubbard model and the spin path integral. Specif-
ically, when the Hamiltonian is quadratic in a generator of the algebra used to
construct coherent states, the path integral fails to produce correct results follow-
ing from an operator approach. We note that the problems do not arise in the
time-discretized version of the path integral, as expected.
OPTICAL AND CASIMIR EFFECTS IN
TOPOLOGICAL MATERIALS
by
Justin H. Wilson
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2015
Advisory Committee:
Professor Victor M. Galitski, Advisor & Chair
Professor Sankar Das Sarma
Professor Christopher Jarzynski
Professor Jeremy N. Munday
Professor Jay Deep Sau
c© Copyright by
Justin H. Wilson
2015
Dedication
I dedicate this dissertation to my sister, Megan Walker, for her unwavering
love and support.
ii
Acknowledgments
There are many people to thank that have helped me on my journey: Friends
and family, colleagues and mentors.
First, I thank my advisor Victor Galitski for a great many things. As my
advisor he has patiently been there while I have sometimes struggled or been stuck
on a research problem. Additionally, he provided many opportunities to do what
many starting in science have a hard time grasping: collaborating, getting out there
and making yourself known. Naturally, this entire dissertation would not exist
without him, and I am very thankful he has been my advisor.
I also need to thank my fellow graduate student Andrew Allocca. When he
joined Victor’s group, we immediately began collaborating on a project idea that
Victor and I had formulated, and we’ve since seen the fruits of this labor in three
publications, all of which are presented in this dissertation. Andrew did the bulk of
the technical work for two of these publications, and he deserves credit for that. He
has also become a fast friend.
My other collaborators have also been amazing and helpful, not just scientifi-
cally, but with advise and oftentimes friendships outside of the work environment:
Benjamin Fregoso, Dmitry Efimkin, and Joe Mitchell. Dmitry was the genesis of
the idea in Chapter 2 and has been a pleasure to work with. We have a large group
and everyone in that group has been amazing and also has my thanks.
Outside of collaboration, I thank Sankar Das Sarma for helping me make the
transition from quantum gravity to condensed matter theory, and providing a bit
iii
of the philosophical underpinnings of condensed matter which is presented herein. I
have also been helped by conversations with Michael Levin on the project regarding
path integrals presented in Chapter 5, and Liang Wu for the initial conversation
that led to Chapter 4.
Oh, and I have to thank Jed Pixley. But mostly for not tearing my head off
for my procrastinating on our research collaboration by working on my dissertation.
iv
Table of Contents
List of Figures vii
List of Abbreviations xii
1 Introduction 1
1.1 Probing a material with magneto-optics . . . . . . . . . . . . . . . . . 3
1.2 The Casimir effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Topological materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Weyl semimetals . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Lifshitz Transitions . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Coherent State Path Integrals . . . . . . . . . . . . . . . . . . . . . . 20
2 Resonant Faraday and Kerr effects due to in-gap states on the surface of a
topological insulator 25
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Localized in-gap states . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Optical conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Faraday and Kerr effects . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Observation of material phenomena with the Casimir effect 47
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Nonanalytic behavior of the Casimir force across a Lifshitz transition
in a spin-orbit-coupled material . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Clean spin-orbit coupled materials response function . . . . . 49
3.2.2 Casimir effect results . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Quantum Interference phenomenon in the Casimir effect . . . . . . . 62
3.3.1 Casimir effect results . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Mesoscopic disorder fluctuations . . . . . . . . . . . . . . . . . 70
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
v
4 The repulsive Casimir effect between Weyl semimetals 83
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Electrodynamics of a bulk-Hall effect . . . . . . . . . . . . . . . . . . 85
4.3 Reflection off a bulk-Hall system . . . . . . . . . . . . . . . . . . . . . 88
4.4 The Casimir effect between two idealized Weyl semimetals . . . . . . 92
4.5 Conductivity for a clean Weyl semimetal . . . . . . . . . . . . . . . . 98
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 The breakdown of the coherent state path integral 105
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Breakdown of the spin-coherent state path integral . . . . . . . . . . 106
5.3 Breakdown of the harmonic-oscillator coherent state path integral . . 111
5.4 (Lack of) Relation to the semiclassical anomaly . . . . . . . . . . . . 114
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 Conclusion 125
A Optical reflection from a suspended thin film 128
B Long distance behavior of the Casimir force 131
vi
List of Figures
1.1 The diagrammatic expansion of Π up to the leading correction to
the Drude result. Solid lines represent disorder averaged electron
Green’s functions, dashed lines represent interactions with the dis-
order potential, the shaded regions represent diffusons (labeled with
D) or cooperons (labeled with C) and the circles represent current
vertices. The first three diagrams of the third line together give the
Drude result, while the last term gives the leading correction. The
last line defines the renormalized vertex. . . . . . . . . . . . . . . . . 21
2.1 Here we plot the optical Hall conductivity in units of half the gap (2∆
is the magnetically induced gap) taking into account both the effect
of localized impurity states (the subject of this paper) and chiral ex-
citons – which can be clearly distinguished – and compare it to the
pure, noninteracting optical conductivities (dashed line). The chem-
ical potential is at µ = −∆ and there is a density N/S = 0.035a−20
(see Eq. (2.8)) of Coulomb impurity states with dimensionless cou-
pling to electrons of α = 0.3. The exciton contribution is calculated
with dimensionless Coulomb coupling between electron and holes of
αc = 0.18 and is calculated in Ref. [1]. (See Section 2.2 for discussion
of α and αc = e2/~vF.) . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Energies of the first six states localized on a charged impurity. The
vertical line represents the α we consider for our numerical results. . . 30
2.3 The four cases for the chemical potential. The solid line in the middle
of the gap is the bound state. . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Given the four positions of the chemical potential illustrated in Fig. 2.3,
Coulomb coupling α = 0.3, and a density of N/S = 0.035a20; (a) and
(b) show the longitudinal and Hall conductivities, respectively. Note
that the largest features are at 2|µ|, when the electromagnetic waves
excite electrons from the valence to the conduction band. The lower-
frequency features occur when electromagnetic waves excite electrons
from the valence to the bound states (µ ≤ −∆) or when electromag-
netic waves excite electrons from the bound state to the conduction
band (µ ≥ ∆). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vii
2.5 The transmittance of the electromagnetic wave for a thin film of
Bi2Se3 in the case of a filled valence band and an unoccupied bound
state (µ = −∆) and an occupied bound state (µ = ∆). . . . . . . . . 40
2.6 The measurable optical quantities of (a) the Faraday angle and (b)
the ellipticity of the transmitted wave for Bi2Se3 in the case of a
filled valence band with an unoccupied bound state (µ = −∆) and
an occupied bound state (µ = ∆). The features correspond to the
features in the optical conductivities. . . . . . . . . . . . . . . . . . . 41
2.7 The measurable optical quantities of (a) the Kerr angle and (b) the
ellipticity of the reflected wave for Bi2Se3 in the case of a filled valence
band with an unoccupied bound state (µ = −∆) and an occupied
bound state (µ = ∆). The features correspond to the features in the
optical conductivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 The geometry typically used in experimental measurements of the
Casimir force is a gold-coated sphere suspended above a planar plate
from a cantilever. We consider a lower plate of indium antimonide
with an applied magnetic field. . . . . . . . . . . . . . . . . . . . . . 48
3.2 The Casimir force Fc normalized by the ideal conductor value between
one semiconductor plate and one metallic plate separated by a =
50 nm as a function of applied magnetic field. The red plot (left axis)
corresponds to µ > 0, and the blue plot (right axis) corresponds
to µ < 0. The upper plot uses µ = ±6 meV and the lower uses
µ = ±10 meV. The insets show the band structure above and below
the transition point (marked with a dashed line) along with the two
fixed values of the Fermi energy. . . . . . . . . . . . . . . . . . . . . . 54
3.3 The Casimir force Fc normalized by the ideal conductor value between
two metallic plates at a fixed separation as a function of the applied
magnetic field. The insets show the band structure above and below
the transition along with the fixed value of the Fermi energy. With a
large Fermi energy and small electronic g factor (≈ 2), a prohibitively
large magnetic field is needed to reach the transition. . . . . . . . . . 56
3.4 The Casimir force Fc normalized by the ideal conductor value F0
between two semiconductor plates separated by a = 50 nm as a func-
tion of applied magnetic field. The red plot (left axis) corresponds
to µ > 0, and the blue plot (right axis) corresponds to µ < 0. The
upper plot uses µ = ±6 meV and the lower plot uses µ = ±10 meV.
The insets show the band structure above and below the transition
point along with the two fixed values of the Fermi energy. . . . . . . . 60
3.5 The imaginary frequency longitudinal conductivity of the InSb plate
at iω = 2iµ for µ = 10 meV as a function of the applied magnetic
field. The Lifshitz transition point is indicated with a dashed line. . . 61
viii
3.6 The geometry typically used in experimental measurements of the
Casimir force is a gold coated sphere above a planar plate. Here we
show the sphere suspended from a cantilever. We consider a lower
plate of very thin metal with a weak applied perpendicular magnetic
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 The dependence of the Casimir pressure on the applied magnetic field
between two disordered plates (one 3D and one 2D) at a separation
of a = 250 nm. The 2D plate is described by the Drude model with
the weak localization correction. The force is normalized by the ideal
conductor result and is plotted for three temperatures–3, 1, and 0.1 K,
from top to bottom. The Casimir pressure is normalized by the ideal
result, P0 = − ~cpi
2
240a4 . The inset shows the conductivity of the 2D
plate with WL correction as a function of the applied magnetic field,
normalized by the uncorrected Drude conductivity, at the same three
temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.8 The dependence of the Casimir pressure on temperature between
two Drude model plates (one 3D and one 2D) at a separation of
a = 250 nm. The force is normalized by the ideal conductor result,
and there is no applied magnetic field. The solid line is obtained from
including the WL correction in the 2D plate, and the dashed line is
the result obtained if the effect of WL is ignored. The inset shows the
dependence of the conductivity of the 2D plate as a function of tem-
perature normalized by the uncorrected Drude model conductivity.
The solid curve is obtained from the Drude model with WL correc-
tion and the dashed line at 1 is for comparison to the uncorrected
Drude model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.9 The lowest order approximation to the Casimir energy given in Eq. (3.17).
The grey ovals represent the RPA screened linear response functions,
and the wavy lines represent photon propagators. . . . . . . . . . . . 73
3.10 The primary diagram giving the correlation of disorder fluctuations,
as defined in Eq. (3.21). The components of the diagrams have the
same meanings as given in Fig. 1.1. Diagrams containing more diffu-
sons are found either not to contribute to the correlator or are found
to have a contribution O(1/F τ) smaller. . . . . . . . . . . . . . . . . 75
3.11 The diagrams giving the width of the distribution, explicitly given in
Eq. (3.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.12 The fit of numerical data for the quantity W/EDrude0 to the expected
functional dependence given in Eq. (3.24). The black dots are the
numerical data, the dashed blue line is the fit function, and the dotted
red line is the asymptotic value W/Eplasma0 , which has no dependence
on τ in the leading approximation. . . . . . . . . . . . . . . . . . . . 77
ix
3.13 A plot of the distribution Eq. (3.22) given for several values of a for a
constant value of τ = 4.5× 10−14 s, corresponding to l = 60 nm. The
values of a are 250 (solid blue), 400 (dashed green), 800 (dash-dotted
yellow), and 1600 nm (dotted red). The average EDrude0 and width
W are calculated numerically using Eq. (3.17) and Eq. (3.23). The
plots are scaled so the distributions are all the same height, and so
|EDrude0 −Eplasma0 | is always the same width. Also indicated is the value
of Eplasma0 . One sees that as a is increased the distribution becomes
sharply peaked even compared to the small energy scale set by the
difference from the plasma model. . . . . . . . . . . . . . . . . . . . . 80
4.1 The setup we will consider here is two Weyl semimetals separated by
a distance a in vacuum, and with distance between Weyl cones 2b in
k-space (split in the z-direction). . . . . . . . . . . . . . . . . . . . . 84
4.2 The normalized Casimir force between two semi-infinite bulk Hall
materials. Repulsion is seen for σxya/c . 4.00. P0 = − ~cpi
2
240a4 is the
distant-dependent ideal Casimir Force [2]. For σxya/c→∞, Pc/P0 → 1. 94
4.3 The normalized Casimir force for a thin film Hall plate. This is the
value as a function of the Hall conductivity σ2Dxy and it is inherently
independent of distance (i.e. the pressure goes as 1/a4). . . . . . . . . 96
4.4 A plot of the normalized Casimir force for various thicknesses of a bulk
Hall material (idealized Weyl semimetal). It begins slightly repulsive
for small σxyd/c, and as this increases, it becomes more repulsive until
it reaches the maximum for a thin film material (the dashed line) at
which point it increases to the semi-infinite limit. P0 = − ~cpi
2
240a4 and
σxy = e2b/2pi2~ is the bulk Hall response. Figs. 4.5, 4.6, 4.7, and 4.8
takes into account more material properties. . . . . . . . . . . . . . . 97
4.5 The Casimir force for a thin film Weyl semimetal taking into account
low-energy virtual transitions in the band structure. An anti-trap
clearly develops when the longitudinal conductivity overwhelms the
Hall conductivity. We compare different values of b (or equivalently,
changing the Hall effect). For this plot, a0 = 1 nm, d = 20 nm,
Λ = 2pi/a0, vF = 6× 105 m/s, and µ = 0. . . . . . . . . . . . . . . . 101
4.6 The Casimir force for a thin film Weyl semimetal taking into account
low-energy virtual transitions in the band structure. Here we compare
different vF (the larger vF the smaller σxx is). For this plot,a0 = 1 nm,
d = 20 nm, b = 0.3(2pi/a0), Λ = 2pi/a0, and µ = 0. . . . . . . . . . . 101
4.7 The Casimir force for a thin film Weyl semimetal taking into account
low-energy virtual transitions in the band structure. Here we turn
on a finite chemical potential which causes attraction at very large
distances (and hence a trap). Even small chemical potentials have
this property but the trap is quite far out. For this plot, a0 = 1 nm,
d = 20 nm, b = 0.3(2pi/a0), Λ = 2pi/a0, and vF = 6× 105 m/s. . . . . 102
x
4.8 Here, we vary the cutoff to see how it depends. It is very similar
to varying vF as one can see in Fig. 4.6. For this plot, a0 = 1 nm,
d = 20 nm, b = 0.3(2pi/a0), vF = 6× 105 m/s, and µ = 0 . . . . . . . . 102
xi
List of Abbreviations
APS American Physical Society
AQHE Anomalous quantum Hall effect
RPA Random phase approxmation
TE Transverse electric
TI Topological insulator
TM Transverse magnetic
UCF Universal conductance fluctuations
WL Weak localization
xii
Chapter 1: Introduction
In the low-energy physics regime commonly studied by condensed matter the-
ory and cold atom physics, electromagnetism is the dominant force at play. Elec-
trons are bound to atoms by the Coulomb attraction, the electrostatic attraction
between charges, and in semiconducting solids described by a band structure, a
Fermi sea of electrons (usually described with the outermost shell of electrons hop-
ping amongst atoms) has observable properties mainly through its electromagnetic
coupling. Additionally, electrons interact with one another, again through electro-
magnetism, producing interesting phenomena.
Many interesting things are still being studied within the context of non-
interacting band theory. Due to material properties, sometimes the band structure of
a material has a topological character—usually a topological number such as a Chern
number [8]. Two examples of this considered in this dissertation are topological
insulators and Weyl semimetals [9] (see Sec. 1.3.1 and Sec. 1.3.2 respectively for
more details).
The basic reductionist theory for a material is fully described by many-particles
(usually electrons in a solid) interacting with electromagnetism. However, there are
This introduction borrows from the introductions of the papers covered herein: [3–7], copyright
APS.
1
still very interesting and unsolved problems that emerge when one puts a lot of
charges together in something as commonplace as a solid. And in these cases, the
problem as expressed by the full “wave-function” of all electrons and photons is
both intractable and useless.
In this dissertation, the electromagnetic force is dealt with directly. In simple
terms, a lot of what is considered here is related to the linear response of a material to
an electromagnetic field captured by the conductivity σ defined as j = σE where j is
the current density and E is the applied electric field. While the conductivity is easily
calculated with the Kubo formula [10], interactions and disorder can complicate the
picture.
This linear response is important not only for transport, but also when one
has an incident electromagnetic wave incident on the material. In fact, due to
Maxwell’s equations, the current and charge induced by an incident wave has an
effect on the reflected and transmitted waves. These waves can be measured directly
(optically) or indirectly (e.g. with a Casimir experiment) to observe properties of
the conductivity that transport misses at higher frequencies.
These phenomena can be used to probe the higher frequency response of the
material to electromagnetism which in turn can tell us useful information about the
band structure or other material properties. This is the backdrop under which we
begin. First, in Chapter 2 we investigate the optics with a thin film 3D topological
insulator when we include the effects of states localized to charged impurities. In
Chapter 3 we shift gears to the Casimir effect and see first how the Casimir effect can
investigate phenomenon like topological Lifshitz transitions in the band structure;
2
we also investigate how weak localization could be used to discern the effects of
diffusion on the Casimir effect. In Chapter 4 we look at the Casimir response
between two identical Weyl semimetals to find that they can be repulsive. Finally,
we move to the more theoretical in Chapter 5 to discuss a breakdown of the path
integral and how it is unrelated to previously identified problems. Before we move
into the technical details, we provide an introduction and overview of the material
presented here.
1.1 Probing a material with magneto-optics
As one can observe from just looking around, different materials have different
optical properties. Mirrors are good reflectors of visible light, pure water is a good
at transmitting visible light, and more obscure objects absorb some visible light and
diffusively reflect others. But we can even see that the way light interacts is already
telling us more than just if it reflects: In water the refractive index is greater than
one, causing light rays to “bend”.
Light has another property that can change upon interaction with a material:
polarization. When the polarization changes in a time-reversal symmetry broken
way (due to internal magnetization or an applied magnetic field), these phenomena
are known as magneto-optical phenomena.
The first observation of this kind of phenomenon was discovered by Faraday
in 1845 [11]. In the first experiment, Faraday sent light through glass with traces
of lead in the direction of an applied magnetic field. The angle through which the
3
polarization rotates through, as verified in a number of materials, takes the form
ϑF = V`B, (1.1)
where ` is the length traveled by the light, B is the magnetic field, and V is the Verdet
constant of the material [12]. We already see that this is probing material properties:
The Verdet constant is a property of the material, subject to how electrons in the
material are behaving.
The Faraday effect has found use in many areas of physics from astronomy
in the characterization of interstellar media [13] and planetary science [14] to semi-
conductor physics (such as [15] in the case of GaAs). In as early as 1898 [16], the
resonant behavior that occurs when the frequency of light incident on the object
(in this case, a metallic vapor) is commensurate with transitions in the object was
seen. It was not until the advent of atomic physics and the quantum mechanical
description of the interaction of light and matter that these observations were put
on a firm theoretical base. Now, these experimental observations are powerful tools
to probe material properties.
Similar to the Faraday effect is the Kerr magneto-optical effect (referred here
as the Kerr effect, not to be confused with the nonlinear electro-optical effect of the
same name) discovered by Kerr in 1877 [17]. While the Faraday effect refers to the
transmitted light’s polarization, the Kerr effect concerns rotation of the reflected
light.
For a material in the modern context, the optical conductivity σ(ω) is calcu-
lated with Kubo formula which crucially depends on the band structure (or generally,
4
the ground state and excitations of the material’s Hamiltonian). Then when an inci-
dent light hits a material it causes a current j(ω) = σ(ω)E(ω). This current in turn
creates its own field, and causes a reflected and transmitted wave. The resonant
behavior of this reflection and transmission is all in the optical conductivity σ(ω)
since if there are resonant transitions at ω, we will observe a large current response
as a result (see Chapter 2 for more details on a particular case).
These effects have been around for over a hundred years, but we still see active
use to identify resonances and time-reversal symmetry breaking nature of materials
such as for topological insulators (e.g. [18] and [19]) and chiral superconductors (e.g.
[20]). As we describe in detail in Chapter 2, we will show how a localized in-gap
state will give a resonant Faraday and Kerr effect.
1.2 The Casimir effect
Another, more subtle, electromagnetic phenomenon resulting from quantiza-
tion of the electromagnetic field was found in 1948, when Casimir predicted at-
traction between two neutral, perfectly conducting materials [2], The origin of this
phenomenon is the differences in vacuum energies from infinite free space. Schemat-
ically, when one quantizes a field operator such as the electromagnetic field given
by the vector field Aµ, every classically permitted solution can be occupied by an
integer number of “photons”. Mathematically, this usually means for bosonic fields
that quantum harmonic oscillators describe the excitations of the field (photons for
5
QED). The Hamiltonian for such a system is just
H =
∑
k,σ
~ω(k)
(
a†kσakσ + 12
)
. (1.2)
where [akσ, a†k′σ′ ] = δkk′δσσ′ , ω(k) is the dispersion, σ labels the internal degrees of
freedom (e.g. polarization), and k labels the eigenvectors (e.g. k is the wave vector
in free space).
Usually the constant term in the Hamiltonian that remains when 〈a†kσakσ〉 = 0
(zero photons) is thrown out as a “arbitrary constant” energy. This discarded term
is known as the vacuum energy
E0 =
~
2
∑
kσ
ω(k) (1.3)
It is convenient to throw it out because such a term is formally divergent, but in some
cases, it can change in well-defined finite ways (when the eigenenergies given by ω(k)
change). This is the origin of the Casimir energy: defined as the energy difference
between the vacuum energy of one configuration and what is defined as “free-space”
(in the two-plate geometry this corresponds to letting the plate separation a→∞).
Ec(a) = E0(a)− E0(a→∞). (1.4)
The Casimir force is then defined by the derivative of this object Fc = −∂Ec/∂a.
The divergent term can be understood quite simply two ways. One way in-
volves vacuum fluctuations in free space that do not involve the plates and the
self-energies of each plate individually; both quantities do not depend on distance.
The next, more mathematical way, is to understand that E0 is the first moment
of the density of states E0 = 12
∫
dE E ρ(E). This diverges for unbounded spectra,
6
but as Weyl showed [21], the asymptotics (for the Laplacian operating on a scalar
potential, which with some caveats can be extended to the Maxwell operator) go
as ρ(E) ∝ V
√
E where V is the volume of the system (the next order term is a
constant surface term and has also been derived [22], accounting for surface self-
energies). Since we keep the volume of the system constant (and the surfaces do not
deform), these divergent terms are the same in every configuration and can safely be
discarded. The remaining terms of ρ(E) after these are discarded give a convergent
integral for the first moment, otherwise known as the Casimir energy.
Bringing this back to electromagnetic fields, we can generally get a finite
Casimir Free energy for two distinct objects A and B by considering
Ec = EAB0 − EA0 − EB0 + EFree space0 , (1.5)
(we have over-counted free space by subtracting off each object individually). This
amounts to summing up the vacuum diagrams that only connect objects A and B,
Ec =
A
B
+
A
B
A
B
+ · · · (1.6)
where X ≡ Π˜X accounts for all interactions taking place on plate X. Or math-
ematically,
Ec =
~
β
∑
iωn
′Tr log[I− Π˜ADABΠ˜BDBA]. (1.7)
where the trace is a full operator trace (over position and polarization), ωn are the
Matsubara frequencies with the
∑ ′ representing a sum over positive frequencies
with the zero mode taken at half-weight, and DXY is photon propagator connecting
7
X and Y . This quantity has been derived many times [23] with a particular instance
for compact dielectric objects appearing in [24] derived via path integration and for
two-dimensional sheets in the Appendix of [4]. One can see that reflections off of the
two objects are important. In fact, if we specialize to two parallel plates (labeled
now 1 and 2), we can obtain the Casimir energy density [25, 26] formula known as
the Lifshitz formula with just reflection matrices (R1 and R2 respectively)
Ec(a) =
~
β
∑
iωn
′
∫ d2k
(2pi)2 tr log[I−R1R2e
−2qza], (1.8)
where the trace is now just a matrix trace, and qz ≡
√
ω2n + q2. This discussion
highlights the strange place Casimir energy and force occupies: It is calculated
with purely classical objects (either the classical eigenenergies or reflection matrices
discussed here), but all of the photons experiencing these are virtual and hence
making it a purely quantum effect.
It will be useful to relate Eq. (1.8) to the linear response of the material.
In the two-dimensional limit [4], we have that the reflection matrix description is
equivalent to the random phase approximation (RPA) applied to the linear response
ΠX defined by jµ = ΠX,µνAν for vector potential field Aν ,
Π˜X = (I− ΠXD)−1ΠX , (1.9)
or diagrammatically the RPA is given by
X = X (1 + X ). (1.10)
is the dressed current-current correlation function for plate X while ΠX ≡ X
Π˜X is the usual current-current correlator derived in linear response theory – a
8
clearly material-dependent quantity related to conductivity at real frequencies by
σ(ω) = Π(ω)iω .
It is a general theorem that mirror symmetric objects without time-reversal
symmetry breaking can only attract one another with the Casimir effect [27]. Nonethe-
less, Casimir repulsion between two materials in vacuum is a long sought after phe-
nomenon [26, 28]. There are principally four categories in which repulsion can be
achieved: (i) Modifying the dielectric of the intervening medium [26, 29, 30]; (ii)
Pairing a dielectric object and a permeable object [28] (such as with metamaterials
[31–33]); (iii) Using different geometries [34–36]; and (iv) Breaking time-reversal
symmetry [37–39]. As one can see from Eq. (1.8), the integral generally yields
an attractive force (eigenvalues of the square of the reflection matrix R2 are usually
positive); however, if we break time reversal symmetry, obtaining antisymmetric off-
diagonal terms in the reflection matrix Rxy = −Ryx there is the possibility of Casimir
repulsion [40]. One candidate is a two-dimensional Hall material [38], and similarly,
another is a topological insulator where the surface states have been gapped by a
magnetic field [37, 39]. A Hall conductance does not guarantee repulsion; longitu-
dinal conductance can overwhelm any repulsion from the Hall effect (though the
magnetic field can lead to interesting transitions [4]), and a Hall effect that is too
strong can suppress Kerr rotation and hence lead to attraction. The latter case is an
interesting phenomenon where “more” of a repulsive material can lead to attraction.
In the realm of Casimir experiments, after nearly 50 years of theory [41],
experimental evidence was presented by Lamoreaux [42]. Following this discovery
there was a flurry of theory [23, 43] and experiment [30, 44–48] which led to an
9
astounding amount of theoretical and experimental machinery. With this machinery,
others have observed that the Casimir force can have a nontrivial dependence on
material parameters [49–55], some of which may be tunable. Being able to tune the
Casimir force by modifying the frequency-dependent conductivity [56–58] could have
important applications for precision gravity experiments [59–63] and applications to
nanotechnology [64].
From the other direction, any change of the Casimir force would be an indica-
tion of a change in the material’s properties. Just as a repulsive effect would be a
signature of some time-reversal symmetry breaking , other changes in the Casimir
force can be attributed to other material properties. For instance, Bimonte and
coauthors showed that one can in principle measure the change in Casimir energy
between a normal and a superconducting state [51, 52]. Additionally, it has been
demonstrated that both the Casimir effect and the thermal Casimir effect [53] are
capable of probing phase transitions [54, 55].
Unfortunately, some details of the theory describing experiments are still un-
resolved. When applying the Casimir effect to a real experiment, it is theoretically
calculated by modeling metallic plates with one of two models: the Drude or plasma
model. These simplified models describe the linear response of electrons in the plates
to an electromagnetic field at low frequencies. While the Drude model describes dif-
fusive electrons subject to a random disorder potential, the plasma model describes
ballistic electrons unhindered by disorder. These two models typically provide sim-
ilar predictions of the Casimir force as a function of plate separation, with the
plasma model predicting a slightly stronger attraction than the Drude model in
10
non-magnetic metals.
Quantitative results from many experiments [65–68] seem to favor the plasma
model over a naive Drude model—in some ways, arguably, the more physical of
the two. Many experiments attempt to account for the effect of electrostatic patch
potentials in the plates, expecting the effect to be relevant for agreement with one
model or the other. Several of these [67, 68] find that the correction due to patches
would make agreement with Drude worse while others [53, 69] see agreement with the
Drude model once the effect of patches is minimized. There is recent theoretical and
experimental work specifically to account for the contribution of patch potentials [70,
71]. While the initial theoretical results seemed to weaken the case for the plasma
model, the comparison of calculation and experiment shows the contribution to the
force from patch potentials to be approximately an order of magnitude smaller than
the difference between the Drude and plasma models. However, the authors caution
that the analysis is preliminary and acknowledge future work may be needed. In the
same vein as the work done on patch potentials, there has also been investigation
into the effect of charge disorder[72], finding that quenched and annealed disorder
contribute to the Casimir force in markedly different ways, with quenched disorder
completely overwhelming the Casimir force at large distances. Additionally, another
method that may be able to distinguish between the Drude and plasma models based
on the thermal Casimir force has also been proposed [73, 74], and recent preliminary
experimental observations based on this proposal seem to, again, favor a plasma
model [75].
In this dissertation we will show a material phenomenon that the Casimir effect
11
actually can capture with (Lifshitz transition, see Sec. 1.8), propose an experiment to
see what role diffusion has (or does not have) in the Casimir effect, and demonstrate
a material that could see repulsion (Weyl semimetals, see Sec. 1.3.2).
1.3 Topological materials
Ever since Thouless, Kohmoto, Nightingale, and Den Nijs demonstrated topol-
ogy in the quantum Hall effect [76], topology has found its way into many parts of
condensed matter theory. Topological order has been identified by authors, connect-
ing various gapped ground states [77] distinguishing long-range entangled (LRE)
states and short-range entangled (SRE) states. Once a symmetry is added into
this scheme, the SRE states can be further distinguished in to symmetry protected
topological (SPT) phases and non-topological phases. These SPT phases include
topological insulators (Sec. 1.3.1). For gapless Fermionic states, topology can still
be assigned in terms of the Fermi points (or the undergoing of a Lifshitz transition
as described later) [78]. These features will be important in Weyl semimetals [79].
1.3.1 Topological Insulators
Topological insulators (TIs) represent a new class of solids whose band struc-
ture can be characterized by a Z2 topological invariant [80–83]. As with normal
“non-topological” insulators, their bulk has a filled valence band with an empty
conduction band separated by a gap. But unlike usual insulators, TIs have very
unconventional, symmetry-protected surface states. Interest in TIs has grown con-
12
siderably since the discovery of two-dimensional (HgTe [84, 85]) and subsequently
three-dimensional TIs (Bi2Se3, Bi2Te3 and other Bismuth based materials [86–89]).
The surface states of three-dimensional TIs are described by a Dirac equation for
massless particles, but unlike two-dimensional systems like graphene, there is only
one Dirac “cone” (in general, an odd number)—something that can only be realized
at the surface of a bulk three-dimensional system [90, 91]. Thus, the surface of a TI
is a veritable experimental and theoretical playground for many interesting phenom-
ena including but not limited to both topological superconductivity, which gives rise
to exotic Majorana fermions [92] (these could potentially be used as building blocks
for quantum computation [93]) and the anomalous half-integer quantum Hall effect
(AQHE) [94] (See also Refs. [8, 95] for a review).
The AQHE occurs when time-reversal symmetry is broken, opening up a gap
in the Dirac surface states. Without an external magnetic field, this effect can be
realized by an exchange field that couples to the spins of the electrons on the surface.
The exchange field can be induced either by the proximity effect with an insulating
ferromagnet [96] or by the ordering of magnetic impurities introduced to the bulk
or surface of a TI [94, 97, 98]. Recently, both methods of inducing an exchange field
have been realized experimentally [99–102] and the AQHE has been experimentally
confirmed by transport measurements [103]. The AQHE is the origin of the “image
monopole effect” for an electron in the vicinity of a TI surface [104] as well as
reflectionless chiral electronic states localized on domain walls that separate regions
with the opposite exchange field [105].
Another way to probe the AQHE is with the magneto-optical Faraday and
13
Kerr effects discussed in Sec. 1.1 where the polarization of the transmitted and
reflected electromagnetic waves rotates relative to the wave incident on the TI’s
surface. At low frequencies—when dispersion effects can be neglected—the optics
of the TI nanostructures can be described macroscopically with an additional axionic
θ-term in the action ∆SAE, which is insensitive to microscopical details [96, 106] and
is given by
∆SAE = θ
e2
2pih
∫
dt d3r E ·B. (1.11)
Here θ = 0 for ordinary insulators and θ = pi for topological ones. Moreover for
thin film TIs, the Faraday angle tanϑF = α0 and the Kerr angle tanϑK = 1/α0 are
predicted to be universal [107–109] and depend only on the fine-structure constant
α0 = e2/~c ≈ 1/137.
The theoretical investigation of the Faraday and Kerr effect beyond the low
frequency regime is important not only because real optical experiments occur at
finite frequency but also because single-particle and collective excitations on the
surface of TI start to leave their own fingerprint on optical quantities. In particular,
chiral excitons, which are collective in-gap excitations in the gapped Dirac electron
liquid, reveal their chiral nature [110] via prominent resonances seen in the frequency
dependence of the Faraday and Kerr angles [1]. In Chapter 2 we consider other
in-gap excitations, localized electronic states, which are present due to inevitable
impurities occurring in the TI bulk or on its surface. In usual semiconductors, in-
gap states dominate absorption and magneto-optical effects do not appear without a
magnetic field; the exceptions are magnetic semiconductors where there are similar
14
effects [111].
1.3.2 Weyl semimetals
The topological nature of the Weyl semimetals lie at the Weyl nodes where
for low energies and chemical potential at the Weyl node [9, 78]
HW = ±~vFk · σ, (1.12)
where k and σ are the 3D wave vector and vector of Pauli matrices respectively.
Each node has a chirality (+ or - in the above Hamiltonian) and has a chiral anomaly
[112]. Furthermore, they come in pairs requiring either time-reversal or inversion
symmetry to be broken [79] and they can annihilate each other when brought to
the same place in k space. One theoretically easy way to study these semimetals
is at the phase transition between a topological insulator and a normal insulator.
The topology here lies around the nodes themselves [9]. If you consider the Bloch
states |u(k)〉, and compute the effective vector potential A(k) = i 〈u(k)|∂k|u(k)〉
and Berry curvature F(k) = ∂k ×A(k), then the chirality χ (a quantized number
χ = ±1) can be calculated
χ = 1
2pi
∮
F(k) · dn(k), (1.13)
where the integral is over a Fermi surface that encloses the Weyl node. This is
directly analogous to calculating the flux through a surface enclosing a magnetic
monopole.
Weyl semimetals have a weak longitudinal conductance and a bulk Hall effect
[9, 113]. Clean Weyl semimetals at zero temperature have a zero DC longitudinal
15
conductivity and optical conductivity Re[σxx] ∝ ω [114]. Additionally, they exhibit
a bulk Hall effect exemplified in the DC limit by an axionic field theory [115] where
in addition to the Maxwell action, we have, much akin to Eq. (1.11),
SA =
e2
32pi2~c
∫
d3r dt θ(r, t)µναβFµνFαβ, (1.14)
where θ(r, t) = 2b · r − 2b0t, and 2b is the distance between Weyl nodes in k-
space while 2b0 is their energy offset (resulting from the Chiral anomaly); e is the
charge of an electron; ~ is Planck’s constant; c is the speed of light; Fµν is the
electromagnetic field strength tensor; and µναβ is the fully antisymmetric 4-tensor.
Inversion symmetry breaking Weyl semimetals, on the other hand, do not exhibit a
DC Hall effect [116] and therefore will not see the effects described in this letter. The
electrodynamics of this were investigated in [117] where the authors even comment
on the possibility for a repulsive Casimir effect.
The marginal nature of Weyl semimetals makes them prime candidates for
tuning the Casimir force between attractive and repulsive regimes. In constructed
Weyl semimetals made of heterostructures of normal and topological insulators [118]
an external magnetic field can control the Hall effect [119] and hence the repulsive
effects. Additionally, some of the first materials that have been predicted were
pyrochlore iridates [120–122]; these could also see a repulsion tunable with carrier
doping or an additional magnetic field.
In a real material and experiment at finite temperature, disorder and interac-
tions should be taken into account and in Weyl semimetals they lead to a finite DC
conductivity [113, 114, 118].
16
1.4 Material properties
We investigate in Chapter 3 how some material properties could change the
Casimir effect. Here, we provide a brief overview of these phenomena (Lifshitz
transitions and weak localization).
1.4.1 Lifshitz Transitions
A Lifshitz transition, or second-and-half order transition, occurs when a mate-
rial’s Fermi surface undergoes a topological change [78] – such as the emergence or
collapse of an electron or hole pocket [123, 124]. This is a sharp transition only at
zero temperature, and any finite temperature smooths out what is seen, but as long
as T/µ is small (in a good metal, for instance), relatively sharp effects should still
be seen. One typical way of seeing this effect is through thermopower [124] where
maxima correspond to where the Lifshitz transition occurs.
Mathematically, the effect manifests itself as nonanalytic kinks in the conduc-
tivity and the carrier density as the system is tuned through a transition, so anything
that uses the conductivity should also see a kink, and anything that depends on the
derivative of the conductivity (or density of carriers).
Various models are suspected to undergo some type of Lifshitz transition [125–
127] including the cuprates [128], and experimental evidence of a Lifshitz transition
has been recently observed in iron arsenic superconductors [129].
17
1.4.2 Weak localization
Weak localization (WL) is a well known and greatly studied effect [130–134],
most easily observed in low-dimensional disordered systems at low temperatures
where quantum interference logarithmically decreases the conductivity of a sample
with decreasing temperature. This is most easily understood via the simple Einstein
relation, σ = 2e2νD, where ν is the electronic density of states per spin in the
material and D is the diffusion constant. Weak localization provides a quantum
correction to the diffusion constant, D → D + δD, that is strongly dependent on
both temperature and an applied magnetic field at very low temperatures.
In a semiclassical picture, every electron’s path has some amplitude Ap asso-
ciated with it, and the propagation probability for an electron is represented by a
sum over these trajectories
∑
p,q A∗pAq =
∑
p |Ap|2 +
∑
p6=q A∗pAq. When considering
self-intersecting paths, the interference term
∑
p 6=q A∗pAq has a constructively inter-
fering term as long as electrons maintain their phase coherence. Thus, one direction
of propagation around a loop and its time-reversed counterpart interfere construc-
tively, leading to a higher probability of finding the electron in the loop, and a lower
total probability for the electron to be transmitted through the sample, resulting in
a decrease in conductivity.
This effect occurs at temperatures low enough that electrons will not lose
phase information in the time taken to propagate around a loop. As the tempera-
ture decreases, larger loops can be traversed before phase information is lost, and
the suppression of conductivity becomes more pronounced. The effect can be re-
18
moved by either increasing temperature, so that electron dephasing will occur due
to interactions, or by applying a magnetic field, causing an additional phase for
each loop path proportional to the area it encloses. Even for weak fields, this phase
is enough to ensure the paths will not interfere perfectly constructively anymore,
and the WL effect will be suppressed. In the semiclassical picture illustrated above,
the magnetic field effect can be introduced with a phase Ap → Apeiφp due to flux
enclosed by a closed path. Since paths vary in flux they enclose, the interference
term will be suppressed – suppressing the effect of WL on conductivity.
The response function of a non-interacting disordered electron gas can be
represented diagrammatically as in Fig. 1.1. The simplest approximation considers
only the diagrams shown in the first line of the figure—those without impurity lines
and all diagrams with impurity interaction ladders. In the long wavelength (i.e.
local) limit, q → 0, these terms combine to give the response function ΠDrude =
−ne2m ωnωn+1/τ , from which the well known Drude result for DC conductivity can be
found, σDrude = − limωn→0 ΠDrude/ωn = ne
2τ
m . This approximation is valid for the
one thick disordered plate we consider.
The leading correction to the Drude result comes from diagrams with maxi-
mally crossed impurity lines, shown on the second line of Fig. 1.1, which cannot be
ignored in an accurate treatment of the two dimensional plate at low temperatures.
These diagrams can be represented as a single diagram containing a cooperon. This
approximation to the response function can be written as
Π = ΠDrude + δΠ, (1.15)
19
where δΠ gives the WL correction. An explicit calculation of δΠ in two dimensions,
at low but finite temperature T , and with an external magnetic field H, gives the
Hikami-Larkin-Nagaoka formula [130, 131],
δΠ(iωn;T,H) = −
e2|ωn|
2pi2~
[
ψ
(
1
2
+
~
4eHDτφ(T )
)
− ln
( ~
4eHDτ
)]
, (1.16)
where ψ is the digamma function, D = v
2
F τ
2 is the diffusion constant, and τφ is the
electron dephasing time. This expression diverges logarithmically as T → 0 with a
sign opposite that of the Drude result, leading to a suppression of conductivity. It
also has a very sensitive dependence on an applied magnetic field, becoming very
small at moderate values of H (∼ 100 gauss) even at very low temperatures when
the effect would be large in the absence of such a field.
In two dimensions the primary dephasing mechanism at very low temperatures
is Nyquist electron-electron scattering, and the dephasing time is given by [135]
1
τφ
=
kBT
2piDν~2 ln (piDν~) , (1.17)
where ν = m/2pi~2 is the density of states per spin at the Fermi level for a two
dimensional system.
1.5 Coherent State Path Integrals
Finally, we end our introduction and review with a brief account of the theo-
retical object: Path integrals. This subject bares no direct relation with the previous
topics of magneto-optics and Casimir effect.
Path integrals are widely known for being an alternate formulation of quan-
tum mechanics, and appear in many textbooks as a useful calculational tool for
20
= + + + + ...
+ + + ...
+ +
k
k+q
k'
k'+q
k"
k"+q
k k k
k+q k+q
k k
k+q k+q
k
k+q
k k
k+q
k
k+q
k'
k'+q
k'
k'+q
k'
k'+q
k'
k'+q
k'
k'+q
+=k
k+q
k
k+q
k
k+q
= +
Figure 1.1: The diagrammatic expansion of Π up to the leading correction to the
Drude result. Solid lines represent disorder averaged electron Green’s functions,
dashed lines represent interactions with the disorder potential, the shaded regions
represent diffusons (labeled with D) or cooperons (labeled with C) and the circles
represent current vertices. The first three diagrams of the third line together give
the Drude result, while the last term gives the leading correction. The last line
defines the renormalized vertex.
21
various quantum and statistical mechanical problems (e.g., perturbative expansions,
non-perturbative techniques including the instanton method, and effective theories
[136–138]). From their inception, there has been the problem of writing down a path
integral for any system that can be described by a Hilbert space equipped with a
Hamiltonian. One way to approach this problem is with what is known now as the
generalized coherent state path integral [139, 140] which generalizes the coherent
state path integral for a harmonic oscillator. The key observation with path inte-
gration [137] is that, given a Hamiltonian H, the propagator e−itH at some time t
can be broken up into N slices (e−itH/N)N and in between each multiplicative term
one inserts an (over-)complete set of states parametrized by a continuous parame-
ter. If we take N →∞ we get the time-continuous formulation. This formulation of
path integrals, applied to coherent states, has become widely and routinely used in
many areas of physics (see the many papers collected in [141]), yet despite the many
successes of path integrals, they have been on very shaky mathematical grounds (for
a small “slice” of this history, see [142]).
Glauber coherent states [143] are usually understood as the most classical
states associated with the harmonic oscillator. They obey the classical equations of
motion for a harmonic oscillator and are minimal uncertainty states. Perelomov and
Gilmore [144, 145] extended the definition of coherent states to Lie algebras other
than the Heisenberg algebra (i.e., the harmonic oscillator algebra). Since then,
these “generalized” coherent states have been used in a number of applications (see
[146, 147]). In particular, the coherent states form an overcomplete basis (with
a continuous label) which is a necessary ingredient for the construction of a path
22
integral. For the harmonic oscillator, coherent states are represented by a complex
number, but for coherent states constructed with su(2) (spin), they are points on
the Bloch sphere, S2.
For the case of the harmonic oscillator, it is commonly known that one can
easily go between the normal-ordered Hamiltonian (all annihilation operators com-
muted to the right) and the coherent state path integral [136]; this is due to the
fact that coherent states are eigenvectors of the annihilation operator. For the gen-
eral coherent state path integral, the “classical” Hamiltonian in the path integral
is just the expectation value of the quantum Hamiltonian with a coherent state.
This prescription results in some notable exactly solvable cases, but all such cases
involve non-interacting terms which are essentially linear in the algebra generators
used to construct the coherent-states. When the Hamiltonian involves terms that
are non-linear in generators (interactions), this prescription fails, as we demonstrate
in Chapter 5.
In previous literature, the spin coherent state path integral has sometimes pro-
duced (quantitatively) incorrect results [148–152] unless the time-discretized version
is employed [151, 153]. These problems with the time-continuous path integral were
mostly solved by authors including Stone et al. [154] by identifying an anomaly in
the fluctuation determinant which added an extra phase to the semi-classical prop-
agator. Kochetov had also found this phase earlier than Stone in a general context
[155]. Furthermore, Pletyukhov [156] related the extra phase in the spin path inte-
gral back to Weyl ordering the Hamiltonian in the case of the harmonic oscillator
(in the simplest case, Weyl ordering corresponds to symmetrically ordering annihi-
23
lation and creation operators). Additionally, Weyl ordering has been considered in
the Bose-Hubbard case in [157]. Unfortunately, this solution does not explain the
present breakdown under consideration in Chapter 5 as we will demonstrate.
24
Chapter 2: Resonant Faraday and Kerr effects due to in-gap states
on the surface of a topological insulator
2.1 Overview
Beginning our study into electromagnetic effects of topological materials, we
look at the optical effects of topological insulators. This is a well-studied effect
beginning with calculation of the so-call “giant Kerr effect” [108]. A non-zero Kerr
effect requires time-reversal symmetry breaking: such as an applied magnetic field or
ordered magnetic impurities in such a material as described in the Sec. 1.3.1. Such
impurities can have excess charge, contributing to localized states on the surface of
the topological insulator. These states are magneto-optically active, as we show in
this chapter. This chapter is largely taken with slight modification from the author’s
article [3], copyright APS.
We have shown that localized in-gap states on the magnetically gapped sur-
face of TI are magneto-optically active and lead to peculiar resonant features in
frequency dependence of the Faraday and Kerr angles. In this case, the time rever-
sal symmetry is broken internally by the exchange field – leading to a nonzero Hall
conductivity. The shapes of the resonant features differ considerably from the case
25
0 1 2 3 4
−2
0
Impurity resonance
Exciton resonance
Pure, noninteracting
ω (∆/ℏ)
Re
[σ
xy
] (
e2
/h
)
Figure 2.1: Here we plot the optical Hall conductivity in units of half the gap (2∆
is the magnetically induced gap) taking into account both the effect of localized
impurity states (the subject of this paper) and chiral excitons – which can be clearly
distinguished – and compare it to the pure, noninteracting optical conductivities
(dashed line). The chemical potential is at µ = −∆ and there is a density N/S =
0.035a−20 (see Eq. (2.8)) of Coulomb impurity states with dimensionless coupling
to electrons of α = 0.3. The exciton contribution is calculated with dimensionless
Coulomb coupling between electron and holes of αc = 0.18 and is calculated in
Ref. [1]. (See Section 2.2 for discussion of α and αc = e2/~vF.)
26
of chiral excitons, so they can be easily distinguished, as can be seen in Fig. 2.1 by
the total Hall conductivity taking into account both effects. The magneto-optical
effects are controlled by the appearance of a nonzero Hall conductivity and sim-
ilarly, they inherit frequency dependence from the optical Hall conductivity – in
this manner, Fig 2.1 represents the crucial finding of this chapter. These localized
in-gap states also lead to prominent resonances in frequency dependence of the el-
lipticities of transmitted and reflected waves; thus, they can be effectively probed
in ellipsometry measurements.
2.2 Localized in-gap states
The single-particle Hamiltonian for Dirac electrons interacting with charged
impurities scattered over the surface of a TI is given by
H0 = vF[p× σ]z + ∆σz −
∑
i
Ze2
¯|r− ri|
. (2.1)
Here p is the momentum operator; σ is the vector of Pauli matrices with components
σi; vF is Fermi velocity of Dirac electrons; ri is position of of ith impurity and Ze is
their charge; ¯ is the effective dielectric permitivity on the surface of the TI1; And ∆
1 In the thin film geometry, the screened potential between two charges on the top of the film
has the following form
φexact(r) =
2
+ 1
Ze
r +
4
2 − 1
∞∑
n=1
(− 1
+ 1
)2n Ze
√
r2 + (2dn)2
. (2.2)
In the text, we introduce the effective dielectric constant ¯ so that the resulting Coulomb potential
matches the exact potential in the vicinity of the effective radius a0 of the lowest energy state.
The bulk dielectric constant [158] for Bi2Se3 is  ∼ 100.
27
parametrizes the out-of-plane component of exchange field which gaps the surface
spectrum. The in-plane component can be gauged away and is unimportant for the
phenomena with which this paper is concerned.
In the absence of impurities, the surface spectrum is p = ±
√
(vFp)2 + ∆2 (+
for the conduction band; − for the valence band, separated by a gap 2|∆|). The
wave functions of Dirac states can be presented as |p±〉 = eip·r/~ |ϕp±〉, where the
spinor part is given by
|ϕp±〉 =
1
√
2p(p ±∆)




∆± p
ivFpeiθp



 , (2.3)
where θp is the polar angle of the wave vector p.
If impurities are dilute enough—the case we consider below—they can be con-
sidered independently. The dimensionless effective structure constant α = Ze2/~vF¯
measures their coupling to Dirac states. Further, we assume positively charged im-
purities (Z > 0), and the generalization to Z < 0 is straightforward. Each Coulomb
impurity creates numerous localized states with energies labeled by the quantum
numbers n and total angular momentum j
nj =
∆|n+ γ|
√
(n+ γ)2 + α2
, (2.4)
where γ =
√
j2 − α2, n = 0, 1, 2, . . . for j = 1/2, 3/2, . . . and n = 1, 2, . . . for
j = −1/2,−3/2, . . . (note that for n = 0, the states are not doubly degenerate).
The wave functions of the localized states take the form
|Ψx;nj〉 =
1√
2pi




F+nj(r)ei(j−1/2)θr
−F−nj(r)ei(j+1/2)θr



 , (2.5)
28
where θr is the polar angle in real space, and the functions F±nj(r) are given by [159]
F±nj(r) =
(−1)nλ3/2
∆Γ(1 + 2γ)
√
Γ(1 + 2γ + n)(∆± nj)
(~vFj + ∆α/λ)αn!
× (2λr)γ−1/2e−λr[(~vFj + ∆α/λ)F(−n, 1 + 2γ; 2λr)
∓ ~vFnF(1− n, 1 + 2γ; 2λr)]. (2.6)
where λ =
√
∆2 − 2nj/~vF and F(a, b; z) = 1 + ab z + a(a+1)b(b+1) z
2
2! + · · · is the confluent
hypergeometric function. It should be noted that the state with the lowest energy
(which we refer to as the “lowest state” to differentiate it from the many-body ground
state) is well separated from excited states that lay in the vicinity of continuum of
delocalized electronic states, as seen in Fig. (2.2). Thus, we focus on the lowest
state with energy 0 ≡ 0,1/2 = ∆
√
1− 4α2 and wave functions given by Eq. (2.5)
with j = 1/2 and
F±0,1/2(r) =
2α
~vF
√
2∆(∆± 0)
Γ(1 + 2γ)
(
4α∆r~vF
)γ−1/2
e−2α
∆r
~vF , (2.7)
with an effective radius
a0 =
√
〈r2〉 = ~vF
∆
√
3− 4α2 + 3
√
1− 4α2
4α
∼
√
3
8
~vF
∆
1
α +O(α). (2.8)
For α ≥ 0.5, the j = 1/2 bound states become unstable and classically these
bound electrons collapse into the “nucleus”; this has been extensively discussed in
the case of Dirac fermions in graphene [160–162].
We are interested in the resonant contribution of the localized states to the
optical conductivity, so we approximate the delocalized scattering states by the
29
.0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8
1
Lowest state Excited states
α
En
er
gy
 (∆
)
Figure 2.2: Energies of the first six states localized on a charged impurity. The
vertical line represents the α we consider for our numerical results.
unperturbed delocalized ones as written in Eq. (2.3). While this approximation is
not exact – the delocalized states will be modified due to the potential – it does not
affect the resonant feature, which is due to the difference in energies.
2.3 Optical conductivities
The electromagnetic response on the surface of a TI is described by the optical
conductivity tensor. For noninteracting electrons, this tensor can be written in the
Kubo-Greenwood formulation as
σµν(ω) =
~e2
iS
∑
αβ
fα − fβ
α − β
〈α|jµ|β〉 〈β|jν |α〉
~ω + α − β + iδ . (2.9)
Here ω is the frequency of the incident electromagnetic wave, S is the surface area,
and j = −∂H/∂k = vF[σ × zˆ] is the single-particle current operator. The sum is
over all single-particle states α, including the valence band, the conduction band,
and localized states with their corresponding energies α and occupation numbers
30
fα. The conductivity can be broken up into transitions between (i) surface bands,
denoted by σcv; (ii) a surface band and the localized states, denoted by σimp; and (iii)
localized states, denoted by σimp-imp. One can separate each of these contributions
to the conductivity tensor as σ = σcv +σimp +σimp-imp. In this paper, the impurities
contribute independently; this works well when the sample is dilute enough, i.e.,
given N impurities, (N/S)a20  1.
The contribution between bands can be presented as
σcvµν(ω) = −i~e2
∑
p,γ,γ′
fp,γ − fp,γ′
p,γ − p,γ′
〈ϕp,γ|jµ|ϕp,γ′〉 〈ϕp,γ′|jν |ϕp,γ〉
~ω + p,γ − p,γ′ + iδ . (2.10)
This quantity was evaluated previously and is given by [108, 109]. We also give a
similar derivation at imaginary frequencies in Eqs. (4.50) and (4.51) later and these
can be found from those by a rotation onto the real axis,
Re[σcvxx] =
e2
h
pi
8
[
1 +
(
2∆
~ω
)2
]
Θ(~|ω| − 2|µ|),
Im[σcvxx] =
e2
8h
{
4∆2
~ω|µ| +
[
1 +
(
2∆
~ω
)2
]
log
∣
∣
∣
∣
~ω − 2|µ|
~ω + 2|µ|
∣
∣
∣
∣
}
,
Re[σcvxy] =
e2
4h
2∆
~ω log
∣
∣
∣
∣
~ω − 2|µ|
~ω + 2|µ|
∣
∣
∣
∣ ,
Im[σcvxy] = −
e2
h
pi
4
2∆
~ωΘ(~|ω| − 2|µ|),
(2.11)
assuming |µ| ≥ ∆ (if |µ| < ∆, let µ→ ∆ in these expressions).
The localized states on the Coulomb impurities are labeled by λ = (ri, n, j),
and their matrix elements can be presented in the following form:
〈p± |jµ|λ〉 =
∫
d2x 〈ϕp,±|jµ|Ψx−ri;nj〉 eip·x/~
= eip·ri 〈ϕp±|jµ|Ψp;nj〉 , (2.12)
31
where |Ψp;nj〉 is the Fourier transform of |Ψx;nj〉. The expression for σimpµν can be
further split,
σimpµν = σimp+µν + σimp−µν . (2.13)
Here σimp+µν (σimp−µν ) denotes the contribution due to excitations from localized states
to the conduction band (from the valence band to localized states), which is nonzero
if the localized state is filled (empty). They can both be presented in the form
σimp±µν =
N~e2
iS
∑
λ≶µ
∑
p
f±λ − fp±
λ ∓ p
×
[〈λ|jµ|p±〉 〈p± |jν |λ〉
~ω ± p − λ + iδ +
〈p± |jµ|λ〉 〈λ|jν |p±〉
~ω + λ ∓ p + iδ
]
. (2.14)
Here we have summed over all Coulomb impurities. The phase factor in Eq. (2.12)
depends on the position of the impurity and is canceled in the product of matrix
elements in Eq. (2.14). Integrating the matrix elements over the angle of p and
changing variables from p to p ≡  while taking into account the occupation of the
bands, we obtain
σimp±xx =
iN~e2
S
∑
λ≶µ
∫ ∞
max{|µ|,∆}
 d
(2pi~vF)2
Mλ±xx ()
∓ λ
×
[
1
~ω ± − λ + iδ +
1
~ω + λ ∓ + iδ
]
, (2.15)
σimp±xy =
N~e2
S
∑
λ≶µ
∫ ∞
max{|µ|,∆}
 d
(2pi~vF)2
Mλ±xy ()
∓ λ
×
[
1
~ω ± − λ + iδ −
1
~ω + λ ∓ + iδ
]
, (2.16)
32
where we defined
Mλ±xx () ≡
∫ 2pi
0
dϑp | 〈p± |jx|λ〉 |2, (2.17)
Mλ±xy () ≡ i
∫ 2pi
0
dϑp 〈p± |jx|λ〉 〈λ|jy|p±〉 , (2.18)
and we used the fact that these are real functions.
To evaluate these, we use Eq. (2.12). In position space, the bound state |λ〉 is
of the form shown in Eq. (2.5) (centered around ri), so the Fourier transform takes
the corresponding form
|Ψp;nj〉 =
√
2pi




F˜+nj(p)eiθp(j−1/2)
iF˜−nj(p)eiθp(j+1/2)



 (2.19)
where F˜±nj(p) =
∫∞
0 dr rF
±
nj(r)Jj∓1/2(pr/~) are Hankel transforms of their real-space
counterparts (which can be analytically evaluated given Eq. (2.7)).
In terms of these objects, we can evaluate the integrated matrix elements
Mλ±xx () =
(2pivF)2
2 [(∓∆)|F˜
+
nj(p)|2 + (±∆)|F˜−nj(p)|2]
Mλ±xy () =
(2pivF)2
2 [(∓∆)|F˜
+
nj(p)|2 − (±∆)|F˜−nj(p)|2]
with p =
√
2 −∆2/vF.
Considering just the lowest state with n = 0 and j = 1/2 (labeled with λ = 0),
we obtain
M0±xx () +M0±xy () =
2γ(2pivF)2
4α2
(~vF
∆
)2 Γ
(
γ + 32
)2
Γ(2γ + 1)
(∓∆)(∆ + 0)
∆
× 2F1
(
aγ, aγ + 12 ; 1;
∆2 − 2
4α2∆2
)2
(2.20)
33
M0±xx ()−M0±xy () =
2γ(2pivF)2
64α4
(~vF
∆
)2 Γ
(
γ + 52
)2
Γ(2γ + 1)
(±∆)(2 −∆2)(∆− 0)
∆3
× 2F1
(
aγ + 12 , aγ + 1; 2;
∆2 − 2
4α2∆2
)2
(2.21)
where aγ = (2γ + 3)/4 and 2F1 is the (analytic continuation of the) hypergeometric
function 2F1(a, b; c;x) = 1 + abc x1! +
a(a+1)b(b+1)
c(c+1)
x2
2! + · · · .
If two bound states are at different positions, then by our diluteness assump-
tion (insignificant wave-function overlap) transitions between them will not con-
tribute to the conductivity significantly. However, if the chemical potential is in be-
tween two bound states that live at the same position (e.g. the ground and excited
states of a single impurity), then transitions between those states can contribute to
the conductivity; this contribution is given by
σimp-impxx = N
~e2
iS
∑
nj<µ
mj′>µ
(Φnjmj′)
2δj+1,j′ + (Φmj
′
nj )
2δj,j′+1
nj − mj′
×
[
1
~ω + nj − mj′ + iδ +
1
~ω + mj′ − nj + iδ
]
, (2.22)
and
σimp-impxy = N
~e2
S
∑
nj<µ
mj′>µ
(Φnjmj′)
2δj+1,j′ − (Φmj
′
nj )
2δj,j′+1
nj − mj′
×
[
1
~ω + nj − mj′ + iδ −
1
~ω + mj′ − nj + iδ
]
, (2.23)
where we have defined
Φnjmj′ ≡ v2F
∫ ∞
0
rdr F+mj′(r)F−nj(r). (2.24)
The integral in Eq. (2.24) can be calculated analytically for the functions given
in Eq. (2.6); the result is in terms of Appell hypergeometric functions and can
34
be calculated with the use of an integral identity2. Notice that transitions can
only occur between states that only differ by a quantum of angular momentum as
expected from the form of the single-particle current operator. For our calculations,
we do not consider these transitions since the higher excited states merge with the
continuum – leading to at most a decreasing and smoothing of the threshold.
For the calculation of the frequency-dependent conductivities shown in Fig. 2.4,
we use the dimensionless parameters α = 0.3, (N/S)a20 = 0.035. For a charge on
the surface of a bulk TI, α ∼ 0.09Z; however, in a thin film geometry where the
localized state has a radius a0 & d, where d is the thickness of the thin film, the
situation is more complicated. If a0  d, then we expect α ∼ 3.5Z, but we are in an
intermediate region where the energy level due to the more complicated potential
(see Eq. (2.2)) is more accurately captured by α ∼ 0.3Z. Also, we use four values
of the chemical potential corresponding to four different occupation situations, il-
lustrated in Fig. 2.3. In-gap states correspond to resonance features in Figs. 2.4a
and 2.4b which are well below the threshold of 2max(∆, |µ|). If the in-gap states
are empty, an additional peak appears at 0 + max{∆, |µ|}. If they are occupied, it
appears at frequency max{∆, |µ|}− 0. The shape of the resonance depends weakly
2 The relevant integral is
∫ ∞
0
dr e−brrγ−1F(α1;β1; a1r)F(α2;β2; a2r) =
Γ(γ)
bγ F2(γ;α1, α2;β1, β2;
a1
b ,
a2
b ), (2.25)
where F2 is the Appell hypergeometric function defined by the series
F2(γ;α1, α2;β1, β2;x1, x2) =
∞∑
n,m=0
(γ)n+m(α1)n(α2)m
(β1)n(β2)m
xn1
n!
xm2
m! , (2.26)
with (a)n = a(a+ 1) · · · (a+ n− 1) being the Pochhammer symbol.
35
µ = −1.4∆ µ = −∆
µ = ∆ µ = 1.3∆
Figure 2.3: The four cases for the chemical potential. The solid line in the middle
of the gap is the bound state.
on the value of the chemical potential, and its height disappears at µ/|∆|  1.
Thus, the main role of the chemical potential, if it is outside the gap, is the shifting
of resonant frequencies, and in the next section, which discusses the magneto-optical
effects of a topological insulator film, we exclusively consider the chemical potential
to be situated inside the gap.
It should be noted that in the above calculation, we neglected the Drude
contribution, which appears if the chemical potential lies outside the gap. The
Drude contribution dominates transport, but it is not as important for the optical
conductivity at frequencies ω  1/τ . We further assume that we have only one
localized state, namely the lowest bound state in Eq. (2.7). The excited in-gap
states violate the diluteness criterion: electrons can hop between these states due
to significant wave-function overlap, hence these states will merge with continuum
of delocalized states.
36
.0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
µ = ∆
µ = 1.3∆
µ = −∆
µ = −1.4∆
ω (∆/ℏ)
Re
[σ
xx
] (
e2
/h
)
(a) Longitudinal conductivities.
.
.
0 1 2 3 4
−1.5
−1
−0.5
0
µ = ∆
µ = 1.3∆
µ = −∆
µ = −1.4∆
ω (∆/ℏ)
Re
[σ
xy
] (
e2
/h
)
(b) Hall conductivities.
Figure 2.4: Given the four positions of the chemical potential illustrated in Fig. 2.3,
Coulomb coupling α = 0.3, and a density of N/S = 0.035a20; (a) and (b) show the
longitudinal and Hall conductivities, respectively. Note that the largest features
are at 2|µ|, when the electromagnetic waves excite electrons from the valence to
the conduction band. The lower-frequency features occur when electromagnetic
waves excite electrons from the valence to the bound states (µ ≤ −∆) or when
electromagnetic waves excite electrons from the bound state to the conduction band
(µ ≥ ∆).
37
2.4 Faraday and Kerr effects
We consider the Faraday and Kerr effects at normal incidence and in a thin
film geometry. These conditions are the most favorable for observing the effects of
surface states on the optics. While the Faraday and Kerr effects are quite insensitive
to oblique incidence [163], they decrease considerably (especially the Kerr effect) in
the presence of a mismatch of dielectric constants on the TI film surfaces and due
to longitudinal conductivity [109, 164]. This mismatch—of bulk dielectric constant
to surface effects—can be neglected only if the film thickness is considerably smaller
than the optical wavelength in it, d λ/TI. In real samples, the bulk contributes
considerably to the longitudinal conductivity, which could be reduced in TI films.
Further, we assume that the direction of the exchange field (sign of ∆) is the same
on each surface of the TI; in the opposite case, the effects of both plates on the
optics cancel one another.
In experiments, the incident wave is usually linearly polarized, E = E0xˆ.
Looking in the Appendix A, for the calculation of the reflection matrix, we see that
for ω = ckz, σtotxx = σtotyy , and σtotxy = −σtotyx that
Er = E0
[
σtotxx
2c +
(
σtotxx
2c
)2
+
(
σtotxy
2c
)2
]
xˆ +
σtotxy
2c yˆ
1 + 2σ
tot
xx
2c +
(
σtotxx
2c
)2
+
(
σtotxy
2c
)2 ,
Et = E0
[
1 + σ
tot
xx
2c
]
xˆ +
σtotxy
2c yˆ
1 + 2σ
tot
xx
2c +
(
σtotxx
2c
)2
+
(
σtotxy
2c
)2 .
(2.27)
For calculational purposes, it is convenient to present the incident wave as a combi-
nation of two circularly polarized waves and calculate their reflection r± = |r±|eiΦr±
38
and transmission t± = |t±|eiΦt± amplitudes. In this basis, the reflected and trans-
mitted waves are, respectively, Er = E0(r+e+ + r−e−) and Et = E0(t+e+ + t−e−),
where e± = xˆ± iyˆ represent the two directions of circular polarization. The trans-
mittance through the film is given by T = (|t+|2 + |t−|2)/2; the transmitted wave’s
polarization rotates through an angle ϑF = (Φt+ − Φt−)/2 (the Faraday angle) and
has ellipticity δF = (|t+| − |t−|)/(|t+| + |t−|); and the reflected wave’s polarization
rotates through an angle ϑK = (Φr+ − Φr−)/2 (the Kerr angle) and has ellipticity
δK = (|r+| − |r−|)/(|r+|+ |r−|).
Using the form of Er and Et calculated in Eq. (2.27), we can find the values
for the reflected and transmitted circular polarizations as described above. They
are given by
t± =
e2/h
e2/h+ α0σtot±
, r± = −
α0σtot±
e2/h+ α0σtot±
, (2.28)
where σtot± = σtotxx ∓ iσtotxy , e2/h is the quantum of conductance, and α0 ≈ 1/137
is the fine-structure constant. Additionally, both sides of the thin film contribute
to the optical conductivity, so σtotµν = 2σµν . If we expand in the fine-structure
constant, we have ϑF ∼ 2α0 Reσxy/(e2/h), δF ∼ 2α0 Imσxy/(e2/h), and T ∼ 1 −
4α0 Reσxx/(e2/h). Thus, these quantities track the respective optical conductivities
quite well.
For the numerical calculations, we have used the following parameters, in
addition to the dimensionless parameters taken previously [α = 0.3 and (N/S)a20 =
0.035]. We take the parameters for Bi2Se3 for the gap to be the maximum achievable
by magnetic doping [100], ∆ = 25 meV, and Fermi velocity, vF = 6.2 × 105 m/s.
39
..
0 5 10 15 20
0.97
0.98
0.99
1
µ = ∆
µ = −∆
frequency (THz)
Tr
an
sm
itt
an
ce
Figure 2.5: The transmittance of the electromagnetic wave for a thin film of Bi2Se3
in the case of a filled valence band and an unoccupied bound state (µ = −∆) and
an occupied bound state (µ = ∆).
With these numbers, our density is N/S = 38µm−2 and a0 = 30 nm. It should be
noted that N/S is not the total concentration of impurities, but the concentration
of impurity states with a definite energy 0 inside the gap. The generalization to
the realistic case is discussed in the Conclusions.
The dependence of transmittance on frequency is presented in Fig. 2.5. As
one can easily see, the in-gap states lead to absorption below the threshold (i.e.
when ~ω ∼ 2|µ|), but it is small, not impeding the observation of transmission.
The decrease can be understood from the relation of the longitudinal conducitiv-
ity to transmission, and hence why even magneto-optically inactive states affect
transmission.
The Faraday and Kerr angles’ dependence on frequency is presented in Figs. 2.6a
and 2.7a respectively. As with transmittance, the largest feature is at the threshold.
Since the Faraday angle strongly depends on the real part of the Hall conductivity,
40
..
0 5 10 15 20
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
µ = ∆
µ = −∆
frequency (THz)
ϑ F
(◦
)
(a)
.
0 5 10 15 20
0
0.4
0.8
1.2
1.6
µ = ∆
µ = −∆
frequency (THz)
δ F
(1
0−
2 )
(b)
Figure 2.6: The measurable optical quantities of (a) the Faraday angle and (b) the
ellipticity of the transmitted wave for Bi2Se3 in the case of a filled valence band with
an unoccupied bound state (µ = −∆) and an occupied bound state (µ = ∆). The
features correspond to the features in the optical conductivities.
41
.0 5 10 15 20
0
15
30
45
60
75
90
µ = ∆ µ = −∆
frequency (THz)
ϑ K
(◦
)
(a)
.
0 5 10 15 20
0
0.25
0.5
0.75
1
µ = ∆
µ = −∆
frequency (THz)
δ K
(b)
Figure 2.7: The measurable optical quantities of (a) the Kerr angle and (b) the
ellipticity of the reflected wave for Bi2Se3 in the case of a filled valence band with
an unoccupied bound state (µ = −∆) and an occupied bound state (µ = ∆). The
features correspond to the features in the optical conductivities.
42
we see that it matches it and has a similar resonant structure. The Kerr angle is
more sensitive to the real part of the longitudinal conductivity though and we see
corresponding features at these points – decreasing the Kerr angle from its large 90◦
rotation at zero frequency when the frequency is on resonance with the localized
state. In both cases, the effect due to impurities is similar in nature to the resonant
feature at threshold ω = 2∆.
Lastly, we show the frequency dependence of ellipticities of transmitted and
reflected waves in Figs. 2.6b and 2.7b respectively. Again, we see features when the
incident electromagnetic wave is on resonance with the impurity state. The elliptic-
ity of the transmitted wave follows the imaginary part of the Hall conductivity, and
the reflected wave again is quite sensitive to resonant effects. Thus, in-gap states
can be probed effectively with ellipsometry.
2.5 Conclusions
We have shown that in-gap localized states dominate both the absorption and
magneto-optics for TI films with magnetically gapped surfaces. In particular, they
lead to peculiar resonances in the frequency dependencies of the Faraday and Kerr
effects. This is similar to magnetic semiconductors [111], though in non-magnetic
semiconductors in-gap states usually require a magnetic field to become magneto-
optically active. In the system considered in this paper, the surface spectrum does
not respect time-reversal symmetry due to the gap induced by exchange field. Hence,
we can conclude that the effect we observe is insensitive to details such as electron-
43
hole asymmetry [165], hexagonal corrections [166] to the Dirac spectrum, or to the
profile of impurity potential which we assumed to be the Coulomb potential. We
also assumed that all Coulomb impurities have the same charge and are located
on the surface of TI; they can also be in the bulk of the TI, and their coupling to
the electronic Dirac states will depend on their distance to the surface. If they are
dilute enough, they will also contribute to the optical conductivity, which can be
represented as
σimpµν (ω) =
∫ |∆|
−|∆|
d0P (0)σimpµν (ω, 0), (2.29)
where σimpµν (ω, 0) is the contribution of a single impurity bound sates with energy 0
and P (0) is the concentration of the corresponding states. The finite distribution of
levels, originating from different coupling of impurities with Dirac states, can make
the calculated resonance features shallower and considerably wider. Additionally,
there are variations of the chemical potential δµ which correspond to electron and
hole puddles for δµ > 2∆ [167, 168]. For δµ < 2∆, the variations can bring about
variation of the occupation numbers of impurity states in different regions which
does not qualitatively modify our results. For our results to qualitatively still make
sense, we require the variations in the chemical potential δµ < 2∆.
There have been multiple optical experiments probing topological insulators
that measure the Kerr and Faraday effects. Jenkins et al. studied the Kerr effect and
reflectivity for a fixed frequency and varying the magnetic field [19]. Time-domain
spectroscopy has been utilized on strained HgTe [169, 170]. The large Kerr effect
and thickness independent Drude peaks have also been observed [171]. Additionally,
44
the quantized Faraday angle has been seen with passivated Bi2Se3 in a terahertz
experiment as well as observation of a shifted Dirac cone [172, 173]. Time-domain
spectroscopy was also used in BSTS to see both the surface state and a bulk impurity
band [174]. Recently, the same technique was used on (Bi1−xInx)2Se3 to observe a
topological phase transition as x is varied [175]. At present, all observed features
originate from the bulk physics, but recently new ultrathin magnetically gapped TI
films have been grown [176, 177], and for these samples all conditions necessary for
the observation of magneto-optical effects are satisfied.
In these ultrathin films, the tunneling between opposite surfaces can become
important. The tunneling splits the bands and “splits” the threshold, leading to
features [178] similar to impurity states.
Resonant features from localized in-gap states and from chiral excitons appear
below the threshold 2|∆|, but their shapes have completely different characters.
The localized impurity states are single-particle excitations while excitons are two-
particle excitations. Continuous transitions from a valence band to a localized state
(or from the localized state to a conduction band) contribute to optical conductivity,
hence the additional peak can be interpreted as a splitting of the threshold 2|∆| →
|∆|+ 0 (or to |∆|− 0 if the state is occupied). On the other hand, excitons lead to
a sharp feature in the two-particle spectrum, corresponding to their dispersion law
Eex(q). Since only excitons with zero momentum are optically active, they lead to
features of a single, resonant shape in the magneto-optics [1].
To conclude, we have investigated the role of localized in-gap states on the
surface of a topological insulators in the magneto-optical Faraday and Kerr effects.
45
These in-gap states resonantly contribute to both the longitudinal and Hall conduc-
tivities which in turn leads to peculiar resonances in the frequency dependence of the
Faraday and Kerr angles as well as to the ellipticities of transmitted and reflected
waves. These resonant features that we have predicted can be directly measured by
optical experiments. In fact, their specific shape of these resonant features allows
them to be easily separated from other in-gap excitations.
46
Chapter 3: Observation of material phenomena with the Casimir ef-
fect
3.1 Overview
In this chapter we mainly concern ourselves with materials tangentially related
to the topological materials present in Chapter 2 and Chapter 4. We show how
specific material phenomena, specifically topological Lifshitz transitions and weak
localization in two-dimensions, can appear as a distinct signature in a Casimir force
experiment. This chapter is largely taken from the papers1 [4] and [5], copyright
APS.
3.2 Nonanalytic behavior of the Casimir force across a Lifshitz tran-
sition in a spin-orbit-coupled material
Within the paradigm of observing material phenomena with the Casimir ef-
fect (see discussion in Sec. 1.2), we consider how the Casimir force changes as a
parameter tunes a system across a Lifshitz transition—an extreme case of Fermi
1In this chapter, the technical results were obtained in large part by my coauthor on these
papers and fellow graduate student Andrew Allocca.
47
B Metal(Au)
Semiconductor 
(InSb)
Figure 3.1: The geometry typically used in experimental measurements of the
Casimir force is a gold-coated sphere suspended above a planar plate from a can-
tilever. We consider a lower plate of indium antimonide with an applied magnetic
field.
surface reconstruction in an electronic material (see Sec. 1.4.1). We find that as the
system goes through this transition by tuning a magnetic field, the Casimir force is
both nonanalytic and nonmonotonic as a function of the field. Our model involves a
thin layer of indium antimonide (or another semiconductor with a large g factor, as
discussed below) and could be experimentally realized in the common experimental
setup for Casimir measurements as shown in Fig. 3.1. In this section, we consider
how the Casimir force changes as we tune a two-dimensional spin-orbit-coupled ma-
terial through a Lifshitz transition.
Others have considered the consequences on the Casimir effect of considering
two-dimensional plates instead of thick slabs [179–181], but similar to the particular
case of graphene [182–184], our model requires a more microscopic approach. We
consider the Casimir force at zero temperature between two parallel plates where at
48
least one is modeled as a two-band spin-orbit-coupled material (sufficiently thin to
be considered quasi-two-dimensional) with a fixed chemical potential and tunable
Zeeman splitting due to an external magnetic field. (When considering only one spin-
orbit-coupled plate, the other is a metallic plate, modeled as a clean free electron
gas.) The Zeeman field tunes a gap in this two-band material and causes one of the
Fermi surfaces to form or collapse. This is the simplest realistic model exhibiting
a Lifshitz transition. At these transition points, the Casimir force between the two
plates experiences a kink, as seen in Fig. 3.2.
This could be experimentally measured with the usual plate and sphere geom-
etry as seen in Fig. 3.1. The plate would be a thin layer of InSb while the sphere
would be the usual Au-coated sphere. While we consider the parallel plate scenario,
our calculations can be generalized to the sphere-plate geometry by using the prox-
imity force approximation [23] without damage to the nonanalyticity we observe in
the Casimir force.
3.2.1 Clean spin-orbit coupled materials response function
We consider the single-particle effective Hamiltonian for the conduction bands
of the semiconductor,
H = k
2
2m∗ − µ+ β(σxkx − σyky) + Vzσz, (3.1)
which has eigenvalues
ξ±(k) = k
2
2m∗ − µ±
√
V 2z + β2k2, (3.2)
49
where m∗ and µ are the conduction band effective mass of the electron and chemical
potential. The coefficient β is the strength of the Dresselhaus spin-orbit coupling,
and σi are the Pauli matrices. The factor Vz is the induced Zeeman splitting, given
by Vz = µBg∗B, where µB is the Bohr magneton, g∗ is the material’s g factor, and B
is an applied magnetic field. For all calculations we assume that this Hamiltonian is
a simple model of the relevant bands of the material indium antimonide, for which
m∗ = 0.014m0 , where m0 is the free electron mass, and β = γ〈k2z〉 ' γ
(pi
d
)2
[185],
where d is the thickness of the plate and γ = 760.1 eVA˚3 is the intrinsic Dresselhaus
parameter for the material. We consider InSb plates that are six lattice constants
thick, d = 6 × 0.6479 nm = 3.89 nm. The plates may still be considered effectively
two-dimensional (2D) as long as the energy needed to excite higher electron modes
in the confined direction is much larger than the energy required to excite the two
lowest bands modeled here. Additionally, since the g factor of InSb is g∗ = −51.6
we can also neglect the orbital coupling of the electrons directly to the external
magnetic field as well as the effect of the magnetic field on the metallic plate when
it is considered [186].
For µ > |Vz| there are two bands crossing the Fermi energy. With fixed µ, as
|Vz| is increased the occupation of the upper band decreases until the Fermi surface
disappears entirely when |µ| = |Vz| – the electron pocket defined by that Fermi
surface disappears. Increasing the Zeeman splitting further, the Fermi energy lies
within the gap and only the lower band crosses the Fermi level, giving a single Fermi
surface. This represents the Lifshitz transition for µ > 0, and is shown with the red
dashed line in the insets of Fig. 3.2.
50
If m∗β2 > |Vz| the lower band has a local maximum at k = 0 and a similar
scenario can be considered for min < µ < −|Vz|, where min is the lowest energy of
the lower band. In this case, the lower band crosses the Fermi energy for two distinct
values of k, producing two Fermi surfaces – the inner one enclosing a hole pocket.
Again, increasing |Vz| for fixed µ leads to a shrinking of the inner Fermi surface until
it disappears completely at the point when |µ| = |Vz|. For larger Zeeman splitting,
the Fermi energy again lies within the gap and there is a single Fermi surface. This
senario for µ < 0 is shown with the blue dashed line in the insets of Fig. 3.2. The
disappearance of a Fermi surface by changing Vz in these two scenarios are simple
examples of a Zeeman-driven Lifshitz transition.
Since these transitions occur at a specific value of |Vz|, regardless of the sign
of Vz, the direction of the applied magnetic field is unimportant. For this reason,
we always assume Vz > 0 for simplicity. We also denote the magnetic field strength
needed to reach the Lifshitz transition point as BL = |µ|gµB .
We determine the bare correlation function using the current operator, ji(x) =
ψ†(x) ∂Hˆ[A]∂Ai(x)ψ(x), where Hˆ[A] is the Hamiltonian given in Eq. (3.1) after minimal
coupling. The correlation function is then expressed in terms of the current as
Πij(x, x′) = 〈−δ(x− x′)δij∂Aiji(x) + ji(x)jj(x′)〉
∣
∣
∣
A=0
, (3.3)
where 〈· · · 〉 represents averaging over the ground state [136]. In the case of a weakly
correlated system we can use the approximation that the Casimir effect is determined
by the local current-current response functions; i.e., we need to consider only the
q = 0 limit of Πˆ since nonlocal behavior is screened out. Equivalently, this is a simple
51
extension of the usual plasma model to a spin-orbit-coupled Hamiltonian, which
describes the plates. Furthermore, coupling of the spin to the magnetic fluctuations
of the vacuum field do not need to be considered. In this limit, the correlation
function for the spin-orbit-coupled plates has the form
Πˆ(iω) = −e
2
h




ΠL(iω) ΠH(iω)
−ΠH(iω) ΠL(iω)



 , (3.4)
where e2/h is the quantum of conductance,
ΠH(iω) = Vz
[
cot−1
( ω
2+
)
− cot−1
( ω
2−
)]
(3.5)
ΠL(iω) = µ [Θ(µ− |Vz|) + Θ(µ+ |Vz|)] +
+ − −
2
+
ω2 − 4V 2z
4Vzω
ΠH(iω)
and ± are the positive square roots of
(±)2 = V 2z + max
{
0, 2m∗β2(µ+m∗β2)
[
1±
√
1− µ2−V 2z(µ+m∗β2)2
]}
. (3.6)
3.2.2 Casimir effect results
We use a microscopic quantum field theoretic method to calculate the Casimir
energy at zero temperature in terms of the current-current correlation functions
of the two electron systems under consideration and virtual photons in the three-
dimensional (3D) vacuum between them. Summing up the diagrams in Eq. (1.6), the
Casimir energy at zero temperature for parallel 2D plates separated by a distance
a is given by
Ec(a) =
1
4pi2
∫ ∞
0
dq⊥ q⊥
∫ q⊥
0
dω tr ln
[
1ˆ− ˆ˜ΠADˆ(a) ˆ˜ΠBDˆ(a)
]
, (3.7)
52
where Dˆ(a) = Dˆ(q⊥, iω, a) is the photon propagator and ˆ˜Πi = ˆ˜Πi(q⊥, iω) is the
current-current correlation function for plate i, dressed by interactions with 3D
photons. We choose the gauge with no scalar potential, φ = 0, so the relevant
components of the photon propagator have the form
Dˆ(q⊥, iω, z) =
~
2




q⊥
ω2 0
0 1q⊥



 e
−q⊥|z|.
The dressed current-current correlation function can be expressed in terms of the
bare correlation function, Πˆ, as
ˆ˜Π =
[
1ˆ− ΠˆDˆ(z = 0)
]−1
Πˆ,
which accounts for dynamical screening of photons in the random-phase approxima-
tion (RPA).
We take the derivative of Eq. (3.7) with respect to the plate separation, a,
to obtain an expression for the Casimir force. We then integrate this expression
numerically for fixed separation a = 50 nm and Fermi energy µ, while varying |Vz|,
i.e., varying the magnetic field in an actual experiment. We consider two Fermi
energies, µ = ±6 and ±10 meV, which give that the magnetic fields needed to reach
the transition are BL = 2 and 3.35 T. For all numerical results, we give the Casimir
force in our considered system, Fc, normalized by the Casimir force between ideal
conducting plates, F0 = −~cpi2/240a4, calculated for the same plate separation.
The dependence on plate separation closely follows the usual dependence for the
Casimir force with the magnitude of the force increasing at shorter separations.
Furthermore, the qualitative nature of the effect we find is not affected by the plate
53
0 1 2 3
6.800
6.805
6.810
6.815
6.820
6.825
6.830
B !T"
F
c#F 0!
10
!3
"
5.890
5.892
5.894
5.896
5.898
F
c#F 0!
10
!3
"
0 1 2 3 4 5 6
7.06
7.07
7.08
7.09
7.10
7.11
B !T"
F
c#F 0!
10
!3
"
5.48
5.50
5.52
5.54
5.56
5.58
5.60
F
c#F 0!
10
!3
"
Figure 3.2: The Casimir force Fc normalized by the ideal conductor value between
one semiconductor plate and one metallic plate separated by a = 50 nm as a function
of applied magnetic field. The red plot (left axis) corresponds to µ > 0, and the
blue plot (right axis) corresponds to µ < 0. The upper plot uses µ = ±6 meV and
the lower uses µ = ±10 meV. The insets show the band structure above and below
the transition point (marked with a dashed line) along with the two fixed values of
the Fermi energy.
54
separation.
For the simple system with no spin-orbit coupling (β = 0), i.e., two metallic
plates, the Casimir force as a function of magnetic field is shown in Fig. 3.3. The
magnetic fields needed to drive a Lifshitz transition in a typical system here is
prohibitively large.
As the magnetic field is tuned and the chemical potential is kept fixed and
positive, the Fermi surface changes by the removal of an electron pocket:
B < BL
B < BL
B > BL
, µ > 0 fixed, (3.8)
where the shaded circles represent filled electron states in the 2D k-space. In Fig. 3.3,
we see that for B < BL the Casimir force is constant with varying B, since the carrier
density of the material, which in this case is the only free parameter determining the
value of Πˆ = − e2h [2µΘ(µ− |Vz|) + (µ+ |Vz|) Θ(|Vz| − µ)], is constant in this region.
As the upper band is raised above the Fermi level, the closing of the upper band
Fermi surface is indicated by a kink in the Casimir force, above which the magnitude
of the force increases with B, consistent with the increase in the carrier density in
this region. Unlike spin-orbit-coupled materials – the subject of this paper – this
simple system has a critical field BL = |µ|gµB which is unreasonably large (on the
order of 10 000 T) due to large Fermi energies and small g factors. However, the
spin-orbit-coupled semiconductors have small Fermi energies and large g factors,
leading to a more reasonable value of BL.
For the semiconductors under consideration, the Casimir force as a function
of the magnetic field is presented in Fig. 3.2 for the case of one metallic plate and
55
B !a.u."
F
c
#F 0!
a.
u
."
Figure 3.3: The Casimir force Fc normalized by the ideal conductor value between
two metallic plates at a fixed separation as a function of the applied magnetic field.
The insets show the band structure above and below the transition along with the
fixed value of the Fermi energy. With a large Fermi energy and small electronic g
factor (≈ 2), a prohibitively large magnetic field is needed to reach the transition.
56
one InSb plate and in Fig. 3.4 for the case of two InSb plates. Additionally, relevant
numerical quantities (chemical potential, value of the force, change of force from the
zero magnetic field value, and change of the slope characterizing the kink) associated
with these plots are given in Tables 3.1 and 3.2. As the Zeeman energy changes,
the value of the chemical potential has a strong influence on the behavior of the
Casimir force. For positive values of µ (red curves), the Fermi surface sees behavior
similar to that seen in Eq. (3.8). The behavior of the Casimir force above and below
the transition is similar in both systems we consider, with the force decreasing in
magnitude as the magnetic field strength is increased towards the transition and the
force increasing above the transition for sufficiently large values of B. This increase
at large B is irrespective of Fermi energy or case, and Vz  β
√
2m∗|µ|, leading to
a suppression of the spin-orbit-coupling term and a crossover to the simple metallic
behavior.
For negative values of µ (blue curves), the InSb Fermi surface changes begins
with a hole pocket that disappears at the critical magnetic field
B < BL
B < BL
B > BL
, µ < 0 fixed, (3.9)
where now the empty hole is the hole pocket. The behavior of the Casimir force
is different in the two systems. When considering one InSb plate and one metallic
plate, the force increases with increasing B below the Lifshitz transition for all values
of µ considered and then increases above the transition as well for a sufficiently
strong magnetic field (again, in a crossover to the simple metal case). In the system
composed of two InSb plates, there is no common trend seen in the Casimir force
57
µ (meV) Fc(BL)F0
Fc(BL)−Fc(0)
F0
dFc
dB |B=B+L −
dFc
dB |B=B−L (F0/T)
6 6.806 -0.02756 0.0212
10 7.054 -0.0589 0.0326
-6 5.896 0.0061 -0.0097
-10 5.516 0.0458 -0.0125
Table 3.1: Some important numerical results from the case of the Casimir force
between one metallic plate and one InSb plate, all in units of F0 = −~cpi2/240a4 ×
10−3. The first column gives the value of the force at the transition. The second
column gives the change in the force from B = 0 to the transition. The last column
gives the jump in the derivative of the force with respect to the applied magnetic
field across the transition, giving a measure of the severity of the kink.
µ (meV) Fc(BL)F0
Fc(BL)−Fc(0)
F0
dFc
dB |B=B+L −
dFc
dB |B=B−L (F0/T)
6 5.138 -0.0291 0.0200
10 5.346 -0.0619 0.0310
-6 4.367 -0.0020 -0.0056
-10 4.041 0.0192 -0.0035
Table 3.2: The same as Table 3.1 but for the case of two identical InSb plates.
58
for the negative values of the Fermi energy we consider, except that, again, above a
certain magnetic field strength the force increases with increasing B. For the lowest
of the Fermi energies considered, we see that the Casimir force decreases with B
below the transition and then even more quickly directly above the transition.
The main feature of all of these plots is the sharp kink seen at the Lifshitz
transition point, and this feature should be discernible even considering the effects
of temperature and a substrate. We expect the features to remain for tempera-
tures much less than the energy of the gap at the transition point (i.e. the chemical
potential): 70 K and 116 K for chemical potentials of 6 and 10 meV, respectively.
Additionally, as long as the substrate for either the InSb or Au is a poor conductor,
nonmagnetic, and does not experience an electronic transition in the range of mag-
netic fields needed to reach the Lifshitz transition, then we would expect it to have
at most a small effect on our results, and not to change the nature of the features
we find.
These features can be understood by examining the imaginary frequency AC
conductivities of the InSb plates as a function of magnetic field at a fixed nonzero
frequency; since the plates have no disorder there is no dissipation and the longitu-
dinal DC conductivity is infinite. Both the longitudinal and Hall conductivities at
finite frequency have a discontinuity in their derivatives with respect to B at the
point where |µ| = Vz, just as we find with the Casimir force. The overall trend in
the longitudinal conductivity, shown in Fig. 3.5, mimics the behavior of the Casimir
force we find for positive Fermi energies—decreasing in magnitude below the tran-
sition, then decreasing less drastically directly above the transition until reaching a
59
0 1 2 3
5.13
5.14
5.15
5.16
B !T"
F
c#F 0!
10
!3
"
4.360
4.362
4.364
4.366
4.368
F
c#F 0!
10
!3
"
0 1 2 3 4 5 6
5.35
5.36
5.37
5.38
5.39
5.40
B !T"
F
c#F 0!
10
!3
"
4.03
4.04
4.05
4.06
4.07
4.08
4.09
F
c#F 0!
10
!3
"
Figure 3.4: The Casimir force Fc normalized by the ideal conductor value F0 be-
tween two semiconductor plates separated by a = 50 nm as a function of applied
magnetic field. The red plot (left axis) corresponds to µ > 0, and the blue plot
(right axis) corresponds to µ < 0. The upper plot uses µ = ±6 meV and the lower
plot uses µ = ±10 meV. The insets show the band structure above and below the
transition point along with the two fixed values of the Fermi energy.
60
0 1 2 3 4 5 6
B !T"
Σ x
x!iΩ#
2i
Μ"!a.
u.
"
Figure 3.5: The imaginary frequency longitudinal conductivity of the InSb plate at
iω = 2iµ for µ = 10 meV as a function of the applied magnetic field. The Lifshitz
transition point is indicated with a dashed line.
minimum and increasing with B. All of these results taken together suggest that
the Hall contribution to the Casimir effect from interband spin-orbit interactions,
which are stronger when the bands are closer in energy (i.e., small Vz), works to
suppress the strength of the Casimir force. Since the Lifshitz transition occurs pre-
cisely when Vz = |µ|, for smaller values of the Fermi energy the transition occurs
for smaller values of Vz, meaning that the bands are not so far removed from each
other and interband effects are stronger. Additionally, these effects are stronger in
the system with two InSb plates, as would be expected if they were the result of
spin-orbit coupling.
61
3.3 Quantum Interference phenomenon in the Casimir effect
In this section, we provide a new way to experimentally test the validity of the
diffusive electron model by tuning an external magnetic field (or temperature from
a less practical standpoint) in a Casimir system with a two-dimensional plate. The
proposed experiment would be the typical experiment seen in Fig. 3.6 where the
plate would be quasi-two-dimensional. We find a dramatic change in the Casimir
effect between Drude model plates due to weak localization, shown in Fig. 3.7, that
is just not seen with the plasma model.
Additionally, we test if particular disorder realizations can have a strong im-
pact on our model. The theory behind the use of a diffusive models relies upon
performing an average over all possible realization of a disorder potential in the
material. However, if this disorder average is done at the level of linear response
instead of on the Casimir energy itself, then all effects from, e.g., the nonuniform
nature of physical disorder realizations are neglected. While exact calculation of
these neglected effects is prohibitively difficult, it is possible to estimate whether
ignoring them gives a valid approximation to the Casimir energy.
A fundamental assumption of the WL effect is that electronic motion is dif-
fusive in nature, and its contribution to conductivity is calculated as a correction
to the Drude model. Therefore, any impact found on the nature of the Casimir
effect due to WL would apply only to a diffusive model of metallic plates and not
a ballistic model; a sensitive experimental test of the effects of WL on the Casimir
effect would provide a clear indication of whether a diffusive picture of electronic
62
H Metal(Au)
Quasi-2D Metal (Au)
Figure 3.6: The geometry typically used in experimental measurements of the
Casimir force is a gold coated sphere above a planar plate. Here we show the sphere
suspended from a cantilever. We consider a lower plate of very thin metal with a
weak applied perpendicular magnetic field.
motion correctly describes the physics of the electrons in the experiment.
3.3.1 Casimir effect results
The basic inputs into Eq. (3.10) are the electromagnetic linear response func-
tions for the two plates under consideration which contain all the electromagnetic
properties necessary for a calculation of the Casimir effect. These are in Eqs. (1.15)
and (1.16) in Sec. 1.4.2.
To explore the effects of WL on the Casimir effect, we consider a system
consisting of two flat parallel plates: one thick plate described by the Drude model
and one two-dimensional plate described by the Drude model with an additional
term giving the weak localization effect. With an experimental setup in the typical
plate-sphere geometry, as shown in Fig. 3.6, the gold layer on the sphere is thick
63
enough to be most accurately described as a three dimensional material. Since the
effect of weak localization in 3D materials is much weaker than in 2D films, we only
consider the WL effect in the 2D plate. In addition to this system of primary interest,
we also consider the Casimir pressure between two plasma plates and between two
Drude plates without weak localization for points of comparison. The latter of these
will also give the expected behavior of the system including the 2D plate with WL
correction at sufficiently high magnetic fields to completely suppress the WL effect
(H & 100 gauss). In these cases we consider the same geometry, with one thick semi-
infinite plate and a parallel 2D plate. For a calculation of the Casimir pressure in
these systems we start from the well-known Lifshitz equation for the Casimir energy
density at finite temperature [40] without mixing of polarizations upon reflection,
Ec(a) = kBT
∑
{ωn}
′
∫ d2q
(2pi)2
[
ln
(
1− r(1)TMr
(2)
TMe−2
√
q2+ω2na
)
+ ln
(
1− r(1)TEr
(2)
TEe−2
√
q2+ω2na
)]
. (3.10)
In this expression,
∑ ′
denotes a sum over positive Matsubara frequencies, counting
the n = 0 term with half weight. The functions r(i)TM and r
(i)
TE are the reflection
coefficients of plate i for the two polarizations of light. The subscript TM refers to
the polarization where the magnetic field is perpendicular to the plane of incidence,
and similarly for TE with the electric field. The reflection coefficients depend on both
q and the Matsubara frequency ωn = 2pinkBT , and may also depend on the other
parameters of the system under consideration, such as the applied magnetic field H
and additional temperature dependence. The reflection coefficients can be written
explicitly in terms of the dielectric functions of the plates, i(iωn), or alternatively in
64
terms of the electromagnetic linear response functions of the plates, Πi(iωn), which
are related to the dielectric functions as
(iωn) =



1− Π2D(iωn)δ(z)/ω2n for 2D systems
1− Π(iωn)/ω2n for 3D systems.
(3.11)
The reflection coefficients have the form
r2DTM(q, iωn) =
Π2D(iωn)
Π2D(iωn)− 2ω
2
n
q⊥
r2DTE(q, iωn) =
Π2D(iωn)
Π2D(iωn)− 2q⊥
(3.12)
for two-dimensional plates, where we have defined q⊥ =
√
q2 + ω2n. For very thick
three-dimensional metallic plates (thickness d→∞), the reflection coefficients have
the form
r3DTM(q, iωn) = −
√
q2⊥ − Π(iωn)− q⊥ + q⊥ω2nΠ(iωn)√
q2⊥ − Π(iωn) + q⊥ − q⊥ω2nΠ(iωn)
r3DTE(q, iωn) = −
√
q2⊥ − Π(iωn)− q⊥√
q2⊥ − Π(iωn) + q⊥
.
(3.13)
The Casimir pressure is found from Eq. (3.10) by taking its derivative with respect
to plate separation, a.
We consider three systems of one thick plate and one 2D plate: both plasma
plates, both Drude plates, and most importantly, a thick Drude plate with a 2D
Drude plate including the weak localization correction term given in Eq. (1.16).
In all of these systems, we fix the plate separation at a = 250 nm and calculate
the Casimir pressure as a function of either an externally applied magnetic field
or temperature, staying in the low temperature regime where Eq. (1.17) is valid.
Additionally, we set the elastic mean free path of the electrons in disordered plates
to be l = 15 nm and the Fermi energy and effective electron mass to be those of
65
-40 -20 0 20 400.145
0.146
0.147
0.148
0.149
0.150
0.151
0.152
H (gauss)
P cW
L /P 0
-40 -20 0 20 400.980
0.982
0.984
0.986
H (gauss)
σ WL/σ
Dru
de
Figure 3.7: The dependence of the Casimir pressure on the applied magnetic field
between two disordered plates (one 3D and one 2D) at a separation of a = 250 nm.
The 2D plate is described by the Drude model with the weak localization correc-
tion. The force is normalized by the ideal conductor result and is plotted for three
temperatures–3, 1, and 0.1 K, from top to bottom. The Casimir pressure is normal-
ized by the ideal result, P0 = − ~cpi
2
240a4 . The inset shows the conductivity of the 2D
plate with WL correction as a function of the applied magnetic field, normalized by
the uncorrected Drude conductivity, at the same three temperatures.
66
gold: F = 5.53 eV and m∗ = 1.10m0 where m0 is the free electron mass [187]. We
normalize all Casimir pressures by the ideal conductor result, P0 = − ~cpi
2
240a4 , with
a = 250 nm.
In addition to the disagreement on the magnitude of the effect between Drude
and plasma models, we find that there is qualitatively different behavior when ac-
counting for the effect of weak localization. The Casimir pressure between plasma
plates has no dependence on the strength of the applied magnetic field, at least for
such weak fields as we consider here, and only a very weak dependence on temper-
ature in this low temperature regime—the change of the normalized pressure from
10 K to 0.1 K is a decrease of 1.7 × 10−4. In stark contrast, the Casimir pressure
when considering a Drude plate with WL effects shows both a highly nontrivial
dependence on even a weak applied magnetic field (at low temperatures), shown in
Fig. 3.7, and also a sharp decrease with decreasing temperature (with no applied
magnetic field), shown in Fig. 3.8. Both the temperature and magnetic field effects
are expected when considering the Casimir pressure as a function of the conductivity
of the plates. The sharp drop in the Casimir pressure with decreasing temperatures
matches the drop in conductivity of the 2D plate obtained from theory, shown in
the inset of Fig. 3.8, and the strong dependence of the Casimir pressure on a weak
magnetic field closely follows the dependence of the conductivity of the 2D plate as
obtained from theory, shown in the inset of Fig. 3.7, and seen in magnetoresistance
experiments with 2D thin films [133, 134]. Indeed, we find that applying a mag-
netic field of only H = 40 gauss perpendicular to the plates is enough to reduce the
suppression of the pressure by approximately 40%.
67
0 1 2 3 4 5
0.145
0.150
0.155
0.160
T (K)
P cW
L /P 0
0 1 2 3 4 5
0.980
0.985
0.990
0.995
1.000
T (K)
σ WL/σ
Dru
de
Figure 3.8: The dependence of the Casimir pressure on temperature between two
Drude model plates (one 3D and one 2D) at a separation of a = 250 nm. The
force is normalized by the ideal conductor result, and there is no applied magnetic
field. The solid line is obtained from including the WL correction in the 2D plate,
and the dashed line is the result obtained if the effect of WL is ignored. The
inset shows the dependence of the conductivity of the 2D plate as a function of
temperature normalized by the uncorrected Drude model conductivity. The solid
curve is obtained from the Drude model with WL correction and the dashed line at
1 is for comparison to the uncorrected Drude model.
68
At T = 0.1 K and H = 0 gauss we find that by correctly accounting for the
effect of WL in the 2D plate the Casimir pressure is 11% less than if the 2D plate
were described by a simple Drude model without the WL correction. At this tem-
perature and magnetic field, the change in the Casimir pressure from including the
WL correction is larger in magnitude than the difference in the Casimir pressures
predicted by the plasma model and naive Drude model, i.e.,
PDrudec − PWLc
P plasmac − PDrudec
= 1.14 for T = 0.1 K, (3.14)
so the effect is large enough to be measurable for a low enough temperature.
There are several ways to increase the size of the effect even beyond this,
the most straightforward being to lower the temperature even further. We also
find that the effect can be increased by decreasing the electron mean free path, l,
equivalent to increasing the impurity concentration, which can be seen by examining
the dependence of Eq. (1.16) on the mean free path, given partially through the
dephasing time in Eq. (1.17). When considering smaller values of l, however, one
must be sure that the impurity concentration is still below the limit of complete
Anderson localization, or else this model of diffusion breaks down. Alternatively,
when considering much larger values of l, which would make the effect smaller, one
must be sure that the mean free path is much smaller than the sample dimension
L or else the model of a disordered system breaks down. In an actual experimental
system neither of these issues is likely to arise.
Since a controlled smooth variation of temperature in Casimir effect experi-
ments is almost an impossibility, especially at low temperatures where vibrational
69
noise is difficult to remove due to the boiling of cryogenic liquids [188], an experi-
mental test of the effects of weak localization on the Casimir effect could more easily
be performed at fixed low temperature with varying magnetic field, looking for the
effect shown in Fig. 3.7. Only weak magnetic fields would be necessary for such
an experiment, as applying a magnetic field as weak as tens of gauss perpendicular
to the plates would be enough to reduce the effect by a significant percentage. We
propose that an experimental test of these effects could be performed in the nor-
mal plate-sphere geometry with a very thin metallic film at a fixed separation, at
a fixed low temperature, and with a varying weak magnetic field. While the exact
numerical values the forces measured in this geometry are almost guaranteed to
differ from the results we find, the general trends in the temperature and magnetic
field dependence of the force are expected to remain.
3.3.2 Mesoscopic disorder fluctuations
When considering disordered systems one must determine when to perform
averaging over disorder potential realizations. Different realizations of the disorder
potential will give different Casimir energies, and local fluctuations in the disorder
away from this average will cause certain patches on each plate to vary in how attrac-
tive they are–very similar to the phenomenon of universal conductance fluctuations
(UCF) [189, 190]—leading to a self-averaging of the Casimir energy between two
macroscopic plates. This argument would imply that instead of carrying out the
averaging procedure on the linear response Π, which gives the Lifshitz formula with
70
the Drude model, we should perform averaging over the entire Casimir energy itself.
In practice, however, it is not possible to consider an exact disorder potential or to
perform the averaging procedure over the entire Lifshitz formula, and it is unknown
if the simplification of using the disorder averaged linear response (i.e. the Drude
model) in the Lifshitz formula is still a legitimate approximation.Another way of
phrasing this issue is that the approximation
〈Ec[Π]〉 = Ec[〈Π〉] + δEc ≈ Ec[〈Π〉] (3.15)
leads to a naive violation of the Nernst theorem [191, 192] (though this seems to
actually be an order of limits issue that has since been resolved [193]), but it is
unclear if it nonetheless closely approximates the exact expression for the Casimir
energy one would obtain if disorder were to be treated exactly or if averaging were
done at the appropriate stage of the calculation.
Here we calculate what effect fluctuations from the average in any particular
realization of a disorder potential have on the Casimir energy at low temperature,
where conductance fluctuations are strongest. We start from a microscopic version
of the Lifshitz formula in position space directly written down from Eq. (1.7),
Ec [Π1,Π2] = kBT
∑
{ωn}
′
Tr ln (1−M) , (3.16)
where
M =
∫
dr1dr2dr3 ˆ˜Π1(r, r1)Dˆ(r1, r2) ˆ˜Π2(r2, r′3)Dˆ(r3, r′).
Here, Dˆ is the photon propagator, which in M connect the screened response of
one plate to the other, and ˆ˜Πi is the RPA screened electromagnetic linear response
71
functions for plate i, schematically given by (1ˆ − ΠˆiDˆ(0))−1Πˆi, where Πˆ is the
unscreened linear response function and Dˆ(0) is the photon propagator along the
plate. From this point on we will consider plate 1 to be disordered with a particular
disorder realization and for simplicity we will assume that plate 2 is homogeneous
and not disordered. We will further take both plates to be two-dimensional. For
two-dimensional plates, the linear response function and photon propagator are 2×2
matrices, with the components of Π being proportional to the AC conductivity of
the plates.
We are interested in the case of metallic plates without a Hall effect so both
response functions are proportional to the identity matrix. The photon propagator
is diagonal as well. The matrix trace in Eq. (3.16) becomes trivial, leaving us with
a sum over photon polarizations. We make the further approximation that the
Casimir energy is well described by just the first term obtained from expanding the
logarithm,
Ec [Π1,Π2] ≈ −kBT
∑
{ωn}
′
∫ 4∏
i=1
dri
∑
X=TE,TM
Π˜X1 (r4, r1)DX(r1, r2)Π˜X2 (r2, r3)DX(r3, r4),
(3.17)
where now we label the two photon polarization with the superscript X. Diagram-
matically, this approximation can be represented as in Fig. 3.9.
To ensure the following procedures are analytically tractable, we must make
one further simplifying approximation. We assume that the response function for
plate 1, Π1, which depends on an exact disorder realization, can be written as
Π1 = 〈Π1〉 + δΠ1, i.e. the exact response function can be written as the disorder
72
...
Figure 3.9: The lowest order approximation to the Casimir energy given in
Eq. (3.17). The grey ovals represent the RPA screened linear response functions,
and the wavy lines represent photon propagators.
averaged (Drude) response plus another small term to account for the particular
disorder realization, here called δΠ1. (Note that this δΠ1 is distinct from the function
of similar name given in Eq. (3.3) and Eq. (1.16) which define a correction to the
correlation function obtained after disorder averaging.) With this notation, we now
expand the RPA on plate 1 to first order in δΠ1 as,
Π˜1 ≈
〈Π1〉
1− 〈Π1〉D
︸ ︷︷ ︸
=Π˜D1
+
[
1
1− 〈Π1〉D
+
〈Π1〉D
(1− 〈Π1〉D)2
]
δΠ1 (3.18)
≈ Π˜D1 +
1
1− 〈Π1〉D(0)
(
1 + Π˜D1 D(0)
)
︸ ︷︷ ︸
=Γ1
δΠ1. (3.19)
This is the form of Π˜1 that is used in Eq. (3.17).
We now look to the probability distribution of the Casimir energy due to
fluctuations in the disorder realization in plate 1, now contained entirely within the
73
function δΠ1, which is given by
PEc [E ] = 〈δ (Ec − E)〉
=
∫
D(δΠ1)
∫ dx
2pi e
ix(Ec−E)× (3.20)
× exp
[
−1
2
∫ 4∏
i=1
driδΠ1(r1, r2)K1(r1, · · · , r4)δΠ1(r3, r4)
]
.
This expression makes use of the disorder averaged correlator of two unaveraged
response functions for the disordered plate, which can be written as,
K−11 (r1, r2, r3, r4) = 〈δΠ1(r1, r2)δΠ1(r3, r4)〉 , (3.21)
and is also given diagrammatically in Fig. 3.10. The function K−11 is very similar
to the central object of interest considered in the context of UCF [189, 190, 194],
and it is calculated in the same manner. It is related to the size of the fluctuations
of the conductivity, δσ2, (or equivalently in 2D, the conductance) in a similar way
to how the linear response function Π is related to the conductivity. The only
difference between the calculation of this function here and in the context of UCF
is that the latter is primarily concerned with conduction of electrons through a
system with attached leads, usually at zero temperature, while we consider a system
with no leads at finite temperature. As such, most of the qualitative properties
of conductance fluctuations apply in our analysis of fluctuations in the Casimir
energy as well, though the exact form of K−11 differs by small numerical factors.
With this insight, we can already draw several conclusions about the nature of the
distribution we will obtain from Eq. (3.20). Most importantly, for weak disorder
we can expect fluctuations of the Casimir energy around the average value to be
74
Figure 3.10: The primary diagram giving the correlation of disorder fluctuations,
as defined in Eq. (3.21). The components of the diagrams have the same meanings
as given in Fig. 1.1. Diagrams containing more diffusons are found either not to
contribute to the correlator or are found to have a contribution O(1/F τ) smaller.
small since conductance fluctuations are small in good metals: δσ2/σ2 ∼ 1/(F τ)2.
Additionally, we could expect the size of the fluctuations to be reduced by a factor of
2 if a magnetic field were applied to the sample. This is because the diagram for K−11
given in Fig. 3.10 gives the same contribution at zero magnetic field if all diffusons are
replaced with cooperons, but the cooperon contribution is suppressed in magnetic
fields in the same way as the weak localization correction to the conductivity.
In order to evaluate Eq. (3.20), we perform a saddle point approximation on the
functional integral over the disorder fluctuation, δΠ1, which after a straightforward
calculation gives,
PEc [E ] =
1√
2piW 2
exp
[
−(E − E
Drude
0 )
2
2W 2
]
, (3.22)
where we have defined the quantities EDrude0 , the average, and W , the width of
the energy distribution. We find that the average energy is given by the same
expression as in Eq. (3.17), but with the substitution Π˜1 → Π˜D1 , i.e. replacing the
exact unaveraged response function Π1 with the disorder averaged (Drude) response.
Therefore, the average EDrude0 is simply an approximation of the exact Drude result.
75
Figure 3.11: The diagrams giving the width of the distribution, explicitly given in
Eq. (3.23).
Additionally, we find that the square of the width of the distribution can be written
explicitly as,
W 2 = (kBT )2
∑
{ωn,ωn′}
′
∫ 4∏
i=1
dridr′i
∑
X,Y
DX(r1, r2)Π˜X2 (r2, r3)DX(r3, r4)×
×DY (r′1, r′2)Π˜Y2 (r′2, r′3)DY (r′3, r′4)ΓX1 ΓY1 K−11 (r1, r4, r′1r′4), (3.23)
which can be represented diagrammatically as in Fig. 3.11. The multiple diagrams
in this figure result from an expansion of the Γ1 factors of Eq. (3.23),
ΓX1 Γ
Y
1 K−11 =
(
1 + Π˜D1 DX(0)
)(
1 + Π˜D1 DY (0)
)
K−11
(1− 〈Π1〉DX(0)) (1− 〈Π1〉DY (0))
≡
(
1 + Π˜D1 DX(0)
)(
1 + Π˜D1 DY (0)
)
K˜−11 .
We compute these expressions numerically in the same way that we calculate
the Casimir pressure in Sec. 3.3.1. For both plates we use the Fermi energy and
electron mass of gold, and we consider plate 1 to be disordered while plate 2 is a
76
50 100 150 200 250 300 350
0.0145
0.0150
0.0155
0.0160
0.0165
0.0170
ϵFτ/ℏ
W/ℰ 0W/ℰ 0
Figure 3.12: The fit of numerical data for the quantity W/EDrude0 to the expected
functional dependence given in Eq. (3.24). The black dots are the numerical data,
the dashed blue line is the fit function, and the dotted red line is the asymptotic
value W/Eplasma0 , which has no dependence on τ in the leading approximation.
clean plasma plate. We use the material parameters for gold in plate 2 so that in
the limit of weakening disorder we are left with identical plasma plates. We vary the
parameter τ to determine the dependence of W and E0, and the numerical results
for their ratio are fit to the expected functional dependence, as shown in Fig. 3.12.
In the parameter range we are interested in, we find the result,
W
EDrude0
≈ W
Eplasmac
+ C1
~
F τ
. (3.24)
In this expression C1 ≈ 0.096 is a distance independent constant and Eplasma0 is
the Casimir energy between two clean plasma model plates calculated in the same
approximation as EDrude0 , given in Eq. (3.17). In the same way, we also find this
77
ratio’s dependence on the distance between the plates. It suffices to consider only
the first term for this purpose, since the second term in Eq. (3.24) has no dependence
on a. We find,
W
Eplasmac
≈ C2
√ ~c
Fa
, (3.25)
where C2 ≈ 0.038 is another constant independent of both τ and a. The form of
the disorder dependence in Eq. (3.24) is expected since a weakening of disorder,
interpreted as an increase of the scattering time τ , will make a disordered plate
more like a plasma plate. Therefore, a very large scattering time should give a
very good approximation to the plasma result. Note, however, that the complete
removal of disorder through the limit τ →∞ has no physical meaning at this point
in the calculation, since W has already necessarily been calculated in the presence
of disorder. We can see from these two expressions that the distribution will be
relatively sharply peaked, in the sense that W/EDrude0  1, for plates that are not
too close together and are in the disorder regime 1/F τ  1, as we have considered
thus far.
We can get a better understanding of how peaked the energy distribution is
around its average value by comparing its width W to a smaller relevant energy
scale, EDrude0 − Eplasma0 , by combining Eq. (3.24) and Eq. (3.25). We obtain,
W
|EDrude0 − Eplasma0 |
≈ C2
(√ ~c
Fa
+
C2
C1
cτ
a
)
. (3.26)
We see that the nature of the energy distribution Eq. (3.22) depends on the two
dimensionless quantities ~c/(Fa) and cτ/a. Both of these dependencies can be
understood intuitively. The dependence on ~c/(Fa) can be understood as arising
78
from the relevant photonic energy scale. The most important photons are those with
wavelength equal to twice the distance between the plates, and when this distance
is large, these long wavelength photons are able to average over larger areas of the
plates, reducing the effect of local fluctuations. The dependence on cτ/a is similarly
straightforward. It is a comparison of two time scales: the impurity scattering
time, τ , and the time for photons to traverse the distance between the plates, a/c.
When the ratio is small, electrons will have many impurity scattering events before
interacting with a photon, so any effects due to impurities will be very important.
There are several regimes we can now explore. Here, we will always consider
plates of the same material, so the Fermi energy is a fixed parameter and we can only
vary a and τ . First, if the plates are very close, meaning ~c/(Fa) is large, then the
distribution is very wide regardless of the size of τ . Second, if τ is large compared
to a/c, meaning that photons interact with any given electron many times between
impurity scattering events, then the distribution is again very wide, regardless of the
size of ~c/(Fa). The only regime in which the distribution is very sharply peaked
is when both dimensionless parameters are small. This requires that the plates are
much farther apart than both length scales ~c/F and cτ , so that each electron
undergoes many impurity scattering events between photon interactions and effect
of disorder is more pronounced, but also so that long wavelength photons are most
important, averaging out the disorder fluctuations.
Ultimately, this result means that for a given level of disorder, we can always
go to large enough plate separations so that relatively small local fluctuations in
the disorder potential of metallic plates are not likely to greatly affect the Casimir
79
ℰ[ℰ]
ℰ0 ℰ0plasma
Figure 3.13: A plot of the distribution Eq. (3.22) given for several values of a for a
constant value of τ = 4.5× 10−14 s, corresponding to l = 60 nm. The values of a are
250 (solid blue), 400 (dashed green), 800 (dash-dotted yellow), and 1600 nm (dotted
red). The average EDrude0 and width W are calculated numerically using Eq. (3.17)
and Eq. (3.23). The plots are scaled so the distributions are all the same height,
and so |EDrude0 − Eplasma0 | is always the same width. Also indicated is the value of
Eplasma0 . One sees that as a is increased the distribution becomes sharply peaked even
compared to the small energy scale set by the difference from the plasma model.
80
energy, as shown in Fig. 3.13. The difficulty here is that for large values of the inelas-
tic scattering time τ , the distance at which the distribution becomes very sharply
peaked may be so large that the Casimir effect itself will become unmeasurably
small. We note that the values of the parameters τ and a used to get the results
in Sec. 3.3.1 give a distribution very sharply peaked around its average, so we are
justified in our use of the Drude model despite any of the stated concerns over the
disorder averaging procedure.
3.4 Conclusions
As we have shown in the first part of this chapter, tuning through a Lifshitz
transition in this material causes a kink in the Casimir force while the microscopics
control the nature and severity of the kink. We expect similar features to be found in
other materials with such transitions – particularly due to the change in the carrier
concentration across such a transition. This is one way in which precision Casimir
force experiments could be used as a probe of nontrivial electronic properties or
transitions. This is not exclusive to the particular semiconductor considered here;
not only could the Casimir effect be used to probe Lifshitz transitions in other
materials, but it could conceivably be used to detect other phenomena such as the
Fermi surface reconstruction and the superconducting transition in cuprates and
disorder-driven phenomena such as localization.
In the second part of this chapter, we showed that the weak localization cor-
rection to the Drude model at low temperatures may give the Casimir pressure
81
a nontrivial dependence on both temperature and applied perpendicular magnetic
field. Moreover, we find that, for low enough temperatures, WL effects changes
the Casimir pressure from the expected value without WL by an amount greater
than the difference between the Drude and plasma model predictions. Since these
effects are not applicable in a model of a 2D plate without disorder, i.e. the plasma
model, a high precision experimental test measuring this temperature or magnetic
field dependence would give a definitive indication of whether a diffusive model truly
describes the behavior of electrons in Casimir experiments.
Lastly, we explored the effect that fluctuations in the disorder potential can
have on the Casimir energy and the validity of using the Drude model considering
that the correct averaging procedure would give a result that differs from the Drude
model by the inclusion of nonlocal disorder fluctuation contributions. We find that
for a given level of disorder, one can always overcome the effects of fluctuations
by holding the plates far enough apart, which justifies the use of the Drude model
theoretically.
82
Chapter 4: The repulsive Casimir effect between Weyl semimetals
4.1 Overview
In this chapter, we calculate the Casimir effect in a topological material not
considered before with the Casimir force: Weyl semimetals. This is work based on
and largely taken from the work done in the paper [6].
In the Sec. 1.3.2, we gave a brief overview of what Weyl semimetals are and
the possibilities for experimental realizations. At the time of this writing, there has
not been an experimental realization of the time-reversal symmetry breaking variety
(but there has been some that break inversion symmetry [195]), so we proceed with
a discussion including the pure band structure. We begin first with the electrody-
namics in the Weyl semimetal resulting from an axionic action (which results in a
bulk Hall effect). Using the reflection coefficients, we calculate the Casimir effect,
Fig. 4.2. From there, we connect this to the known thin-film result for Chern insu-
lators [38, 39] by making the object a thickness d as seen in Fig. 4.1 and results in
Fig. 4.4.
Taking d a justifies a thin film limit where the two-dimensional conductiv-
ities are σ2Dµν = σµνd. In this case, we investigate the role that the full frequency
response of the conductivities has on the Casimir effect—finding that at shorter
83
Figure 4.1: The setup we will consider here is two Weyl semimetals separated by a
distance a in vacuum, and with distance between Weyl cones 2b in k-space (split in
the z-direction).
distances, attraction prevails, creating an “anti-trap” Figs. 4.5, 4.6, and 4.8.
From here, we tune the chemical potential in order to guarantee long distance
attraction simulating the effect of a finite DC conductivity (see the appendix B).
This reveals a finite region where the plates remain repulsive.
In this letter, we are concerned with Casimir repulsion in identical time-
reversal broken systems. Specifically, we will study how Weyl semimetals with
time-reversal symmetry breaking can exhibit Casimir repulsion. The key ingredient
to Casimir repulsion in this letter is the existence of a nonzero bulk Hall conductance
σxy 6= 0, σxy = −σyx [9].
The material we are interested in is marginal in both the case of longitudinal
conductance and an overwhelming Hall effect: Weyl semimetals [9] with the Casimir
setup seen in Fig. 4.1 and resulting normalized Casimir pressure in Fig. 4.4.
We simulate this effect in the latter part of this letter by raising the chemical
84
potential in our clean system, leading to intraband transitions that contribute to
the longitudinal conductivity (in the DC limit these are singular contributions).
4.2 Electrodynamics of a bulk-Hall effect
While the reference [117] discusses the electrodynamics we discuss here, it only
gives the dispersion relation. So, we present as part of this work the derivation of
the electrodynamics inside this material.
First, as we discussed in Sec. 1.3.2, the action that modifies the usual Maxwell
action S0 = −14
∫
d4xFµνF µν is
SA =
e2
32pi2~c
∫
d3r dt θ(r, t)µναβFµνFαβ, (4.1)
where θ(r, t) = 2b · r− 2b0t, and 2b is the distance between Weyl nodes in k-space
while 2b0 is their energy offset; e is the charge of an electron; ~ is Planck’s constant;
c is the speed of light; Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength
tensor; and µναβ is the fully antisymmetric 4-tensor. From here on out we will use
the fine-structure constant α = e2/4pi~c ≈ 1/137 where convenient.
These yield the following Maxwell equations [117]
∇ · E = − 2piαb ·B,
∇×B = 1c
∂E
∂t +
2
piα(b0B + b× E),
∇ ·B = 0,
∇× E = −1c
∂B
∂t
(4.2)
Consider this material in the semi-infinite space z > 0. First, we confirm that
this does not produce any surface currents.
85
The action can be written simply as
SA =
e2
8pi2~c
∫
z>0
d3r dt θ(r, t)µναβ∂µAν∂αAβ. (4.3)
If we take the functional derivative, we obtain the 4-current jµ(x) = δSAδAµ(x) with the
form
jµ(x) = αpi [Θ(z)
µναβ∂νθ(x) ∂αAβ + δ(z)θ(x)µαβz∂αAβ]. (4.4)
The term proportional to the Heaviside theta function Θ(z) is just the bulk response
that we have already written down in Eq. (4.2), but the other term is a boundary
term (proportional to the Dirac delta-function δ(z)), and it diverges at large x since
θ(x) is not bounded in our case.
Thus, in order to simplify things, we let b0 = 0 and b = bzˆ. This ensures that
there is no surface current since θ(z = 0) = 0; we are purely looking at the bulk Hall
effect in this material. The other surfaces with a surface current have the so-called
Fermi-arcs [9], and for this work we won’t consider them.
In the material itself Maxwell’s equations (4.2) are satisfied. To keep track of
the Hall response, σxy = e
2b
2pi2~ so that Maxwell’s equations take the form in k-space
k · E = −iσxyc zˆ ·B (4.5)
k×B = −ωcE + i
σxy
c zˆ× E (4.6)
k ·B = 0 (4.7)
k× E = ωcB (4.8)
If we take Eq. (4.8) and substitute it into Eq. (4.5), we obtain a modified equation
k ·
(
E− iσxyω zˆ× E
)
= 0, (4.9)
86
as well as the rewriting of Eq. (4.6)
k×B = −ωc
(
E− iσxyω zˆ × E
)
. (4.10)
This suggests the displacement field D = E − iσxyω zˆ × E which corresponds to a
frequency-dependent dielectric constant
(ω) =








1 iσxy/ω 0
−iσxy/ω 1 0
0 0 1








. (4.11)
With this, we can combine Maxwell’s equations into (setting c = 1 for ease of
calculations)
[k⊗ k− k2I]−1(ω)D(ω) = ω2D(ω). (4.12)
If we assume we have a wave vector k propagating in this material, then we can
determine the two polarizations by taking the determinant of the matrix in Eq. (4.12)
det[k⊗ k− k2I+ ω2(ω)] = 0. (4.13)
Solving for ω = ωk reveals two frequencies
(ω±k )2 = k2 + 12σ
2
xy ± σxy
√
k2z + 14σ2xy. (4.14)
Now, to figure out what the polarizations are, we use the fact that we still have
rotational symmetry in the xy-plane, and that k · D = 0. Thus, as a basis for
polarizations, we choose e1 = 1k (−kz, 0, kx) and e2 = (0, 1, 0) with k = (kx, 0, kz).
The resulting equations for D±(k) can be solved by usual linear algebra, re-
sulting in the unnormalized vector
D˜±(k) = ω
±
k
k (
√
k2z + 14σ2xy ∓ 12σxy)e1 ± ikze2. (4.15)
87
Notice how the polarizations D˜±(k) are elliptical in opposite ways, and oscillate
at different frequencies. The former property will lead to interesting reflection and
transmission properties—essential ingredients to the calculating of the Casimir ef-
fect.
4.3 Reflection off a bulk-Hall system
Now consider a wave in the vacuum coming towards the material (z < 0)
incident on the material with wave vector k (kz > 0). Due to time translational
invariance and spatial invariance in the x and y directions, kx, ky, and ω remain
the same between materials, but kz changes across the boundary. To assess this, we
match ω2 = k2x + k2y + k2z to the dispersion in Eq. (4.14) to obtain the two new k±z .
We find (restoring c)
(k±z )2 = kz(kz ± σxy/c). (4.16)
Along with the polarization vectors (these are not the same as D˜±, those kept k
constant, these keep everything but kz constant)
D± = kk±e1 ∓ i kzk±z e2. (4.17)
Notice that for kz < |σxy/c|, one of the elliptical polarizations does not propagate
into the material, independent of angle of incidence.
Now, to obtain the reflection matrix, we call our incident wave E0 with
previously defined wave-vector k. Our reflected wave Er has wave-vector kr =
(kx, ky,−kz), and in the material E± = −1(ω)D± are the two polarizations trans-
mitted with wave vectors k±. The relevant Maxwell equations at the interface
88
between vacuum and the bulk-Hall material are then given by matching the electric
and magnetic field parallel to the surface
(E0 + Er − E+ − E−)× zˆ = 0,
(q× E0 + qr × Er − k+ × E+ − k− × E−)× zˆ = 0.
(4.18)
We can break up the polarization of the incident and reflected waves into transverse
electric (TE) and transverse magnetic (TM) where E = ETMe1 + ETEe2, and the
reflection matrix is defined such that




ETMr
ETEr



 = R(ω,k)




ETM0
ETE0



 . (4.19)
Note that R(ω,k) is being written as a function of ω, kx, and ky; kz is defined in
terms of them. Solving for these sets of equations for Er in terms of E0, we obtain
R∞(ω,k) =
c
σxy




σxy/c+ k−z − k+z i(2kz − k−z − k+z )
−i(2kz − k−z − k+z ) σxy/c+ k−z − k+z



 . (4.20)
For the Lifshitz formula (see Eq. (1.8)), it is calculationally useful to put this formula
in terms of imaginary frequency ω → iω and define q2z = ω2/c2 + k2x + k2y = −k2z and
so we can let kz → iqz. Rotating k±z is problematic due to branch cuts in the square
root function, so we rewrite the above expressions before we make the rotations
using
k+z ± k−z =
√
2kz(kz ±
√
k2z − σ2xy/c2). (4.21)
Rotating from the real kz axis to the imaginary kz = iqz axis is now not an issue,
the only relevant branch cut is for kz < α and slightly below the real-axis. Defining
Q± =
√
2qz(
√
q2z + σ2xy/c2 ± qz), (4.22)
89
the rotated version of Eq. (4.20) takes the form
R∞(iqz) =
1
σxy/c




Q− − σxy/c −Q+ + 2qz
Q+ − 2qz Q− − σxy/c



 . (4.23)
But we are also interested in thickness dependence, not just the semi-infinite
case. This requires restricting our action Eq. (4.1) to 0 < z < d. To solve this,
we just need to add another set of matching conditions. In addition to the in-
cident E0 and reflected Er waves, we now have forward moving Weyl polariza-
tions E↑± with k
↑
± = (kx, ky, k±z ), backwards moving Weyl polarizations E↓± with
k↓± = (kx, ky,−k±z ), and a transmitted wave Et with wave-vector the same as the
transmitted k.
The resulting matching conditions are
(E0 + Er − E↑+ − E↑− − E↓+ − E↓−)× zˆ = 0,
(k× E0 + kr × Er − k↑+ × E↑+ − k↑− × E↑− − k↓+ × E↓+ − k↓− × E↓−)× zˆ = 0,
(E↑+eik
+
z d + E↑−eik
−
z d + E↓+e−ik
+
z d + E↓−e−ik
−
z d − Eteikzd)× zˆ = 0,
(k↑+ × E↑+eik
+
z d + k↑− × E↑−eik
−
z d
+k↓+ × E↓+e−ik
+
z d + k↓− × E↓−e−ik
−
z d − k× Eteikzd)× zˆ = 0.
(4.24)
These equations can still be solved for the reflection matrix with the result
Rd(ω,q) =




Rxx(ω,q) Rxy(ω,q)
−Rxy(ω,q) Rxx(ω,q)



 , (4.25)
90
where
Rxx(ω,q) = σxy{sin(k−z d)[ik+z cos(k+z d) + σxy sin(dk+z )]− ik−z cos(k−z d) sin(k+z d)}/D,
(4.26)
Rxy(ω,q) = σxy{k−z cos(k−z d) sin(k+z d) + sin(k−z d)[k+z cos(k+z d)− 2ikz sin(k+z d)]}/D,
(4.27)
D =[2ik−z cos(k−z d) + (2kz − σxy) sin(k−z d)][2ik+z cos(k+z d) + (2kz + σxy) sin(k+z d)].
(4.28)
Again, for analytic continuation purposes, it is useful to get the expression in terms
of k+z ± k−z which can be done and after the whole procedure is carried out and a
lot of algebra done,
Rd(iqz) =




Rxx(iqz) Rxy(iqz)
−Rxy(iqz) Rxx(iqz)



 , (4.29)
with
Rxx(iqz) = −12
σxy
c (Q− sinhQ+d+
σxy
c coshQ+d−Q+ sinQ−d−
σxy
c cosQ−d)/D,
(4.30)
Rxy(iqz) = −12
σxy
c (Q+ sinhQ+d+ 2qz coshQ+d−Q− sinQ−d− 2qz cosQ−d)/D,
(4.31)
where
D = (Q2+ + 12
σ2xy
c2 ) coshQ+d+ (2qzQ+ +
σxy
c Q−) sinhQ+d
+ (Q2− − 12
σ2xy
c2 ) cosQ−d+ (2qzQ− −
σxy
c Q+) sinQ−d. (4.32)
91
There are two limits of this expression to consider. One is the semi-infinite limit
where qzd→∞, and the other is in the thin film limit where qzd→ 0 as σxyd = σ2Dxy
is held constant.
In the limit as qzd→∞, only the hyperbolic sines and cosines remain and we
obtain
lim
qzd→∞
Rxx(iqz) =
−12
σxy
c (Q− +
σxy
c )
Q2+ + 12
σ2xy
c2 + 2qzQ+ +
σxy
c Q−
=
Q− − σxy/c
σxy/c
, (4.33)
lim
qzd→∞
Rxy(iqz) =
−12
σxy
c (Q+ + 2qz)
Q2+ + 12
σ2xy
c2 + 2qzQ+ +
σxy
c Q−
=
2qz −Q+
σxy/c
. (4.34)
We have recovered the semi-infinite limit.
On the other hand, the thin film result as found by [39, 109] and derived for
completion in Appendix A (setting σxx = σyy = 0 as well as σxy = −σyx) is
R0(iqz) =
1
1 + (σ2Dxy /2c)2




−(σ2Dxy /2c)2 −σ2Dxy /2c
σ2Dxy /2c −(σ2Dxy /2c)2



 , (4.35)
And indeed, if σxyd ≡ σ2Dxy is held constant, then
lim
qzd→0
Rd(iqz) = R0. (4.36)
Thus, Rd interpolates between the thin film and semi-infinite case. We will see in the
next section that these limits are both important in the calculation of the Casimir
effect.
4.4 The Casimir effect between two idealized Weyl semimetals
To begin this section we quote again the formula for the Casimir force ob-
tained from the Lifshitz formula Eq. (1.8), the source of the main integral which we
92
compute. If R1 and R2 are the reflection matrices for plates 1 and 2 respectively
and they are separated by a distance a, then the Casimir pressure is
Pc(a) = 2~
∫ d2k
(2pi)2
∫ dω
2pi qz tr([I−R1R2e
−2qza]−1R1R2)e−2qza, , (4.37)
where the trace is a matrix trace and qz =
√
ω2/c2 + k2. This integral generally
yields an attractive force; however, if we break time reversal symmetry as the Weyl
semimetals do, we can obtain antisymmetric off-diagonal terms for the reflection
matrix Rxy = −Ryx there is the possibility of Casimir repulsion [40].
Considering the semi-infinite case first, we see that R∞(iqz) only depends on
the ratio cqz/σxy. This dependence has implications for the Casimir force. After
changing variables to solely qz, we can inspect the Casimir pressure, and we have
an expression
Pc =
2~c
(2pi)2
∫
dqz q3z g
[
qz
σxy/c , 2qza
]
, (4.38)
with a function g( qzσxy/c , 2qza) given by
g( qzσxy/c , 2qza) = tr([I−R∞(iqz)2e−2qza]−1R∞(iqz)2)e−2qza (4.39)
If we then change variables to x = 2aqz and normalize by Casimir’s original result
for perfect conductors P0 = − ~cpi
2
240a4 [2], we can write the equation for the pressure
as
Pc = −P0
15
2pi4
∫ ∞
0
dx x3 g
[
x
2σxya/c , x
]
, (4.40)
so Pc/P0 = f(σxya/c) a function of only σxya/c.
With this formulation, we plot normalized force Pc/P0 as a function of σxya/c
obtaining the single function seen in Fig. 4.2. We see that for σxya/c . 4.00 we
93
0 4 8 12 16 20
−0.020
0.1
0.2
0.27
Attractive
σxya/c
P c
/P
0
Repulsive
Figure 4.2: The normalized Casimir force between two semi-infinite bulk Hall ma-
terials. Repulsion is seen for σxya/c . 4.00. P0 = − ~cpi
2
240a4 is the distant-dependent
ideal Casimir Force [2]. For σxya/c→∞, Pc/P0 → 1.
obtain repulsion while for σxya/c & 4.00 we obtain attraction. Thus, these similar
materials trap each other at a fixed distance simply dependent on the Hall conduc-
tivity,
aTrap ≈
4.00
σxy/c
. (4.41)
If we insert the value of σxy = e2b/2pi2~ into this expression, we find aTrap ≈ 860/b.
This means that if 1/b ∼ O(nm), then aTrap ∼ O(µm) quite reasonable.
As the distance between the materials gets large, Pc/P0 → 1. This behavior
is markedly different from the thin film Hall case obtained by Tse and MacDonald
in [38, 39]. They found a small (two-dimensional) quantum Hall effect implies
a quantized and repulsive Casimir force at large distance. In our case, we get
attraction at large distances for a bulk Hall material independent of the magnitude
of the Hall effect. To resolve this seeming inconsistency, imagine a finite thickness
94
of the bulk Hall material of thickness d, then the two-dimensional conductivity σxyd
diverges as d → ∞, and in the case of a 2D quantum Hall plate with infinite Hall
conductivity, the Casimir effect is attractive and approaches P0.
To make this argument more precise, we consider Weyl plates that are of
thickness d (using Rd given in Eq. (4.29))
Notice that we can writeRd as a function of only two variablesRd = Rd(cqz/σxy, σ2Dxy /c)
where σ2Dxy = σxyd. Thus, we can perform similar transformations as before to see
that the Casimir force for two ideal Weyl semimetals of thickness d,
Pc = −P0
15
2pi4
∫ ∞
0
dx x3 g
[
x
2σxya/c , σxyd/c, x
]
. (4.42)
Thus, we have Pc = P0f(σxya/c, σxyd/c).
The limiting cases can be understood now by considering first Eq. (4.37). The
exponential constrains qz ∼ 1/a and since the “thin-film” limit is limqzd→0Rd(iqz) =
R0(iqz), we have that d/a → 0. In other words, the thin film limit is applicable
when we are considering d a. Notably, it applies when we’re in the long distance
limit independent of thickness. The opposite limit is just when qzd → ∞, and by
similar arguments, that means d  a is when the semi-infinite case applies. Both
limits leave σxya/c and σxyd/c unaffected (though in the thin film case σxya drops
out while in the semi-infinite case σxyd→∞ has the same limit as qzd→∞).
We can thus conclude that no matter what we calculate, for a d, we should
eventually approach the thin film limit. The thin film limit can actually be calculated
exactly [38, 39] and is
PTFc = P0
90
pi4 Re{Li4
[
(σ2Dxy /c)2/(σ2Dxy /c+ 2i)2
]
} (4.43)
95
0 1 2 3
−0.12
0
0.1
0.22
Attractive
Repulsive
σ2Dxy /c
PT
F
c
/P
0
Figure 4.3: The normalized Casimir force for a thin film Hall plate. This is the
value as a function of the Hall conductivity σ2Dxy and it is inherently independent of
distance (i.e. the pressure goes as 1/a4).
where Li4 is the polylogarithm of degree 4, defined by
Lis(z) =
∞∑
k=1
zk
ks .
The resulting PTFc /P0 function is plotted in Fig. 4.3 as a function of σ2Dxy . Note that
this function has a minimum value of PTFc ≈ −0.117P0 representing how repulsive we
can get. For large enough σ2Dxy /c, the force does become attractive—corresponding
roughly to when (σ2Dxy /2c)2 > σ2Dxy /2c (i.e. when Kerr rotation is suppressed as can
be easily seen from Eq. (4.35)).
The cross-over between the thin film case and the semi-infinite case can be seen
in Fig. 4.4. As σxyd/c is increased, the Casimir energy approaches the semi-infinite
case. However, for any finite d, each curve asymptotically approaches its thin-
film value (and never goes lower than the minimum value represented by dashed
horizontal line in Fig. 4.4). Thus, coming from the semi-infinite case, the trap gets
96
0 4 8 12 16 20
−0.12
0
0.1
0.2
0.27
Attractive
Repulsive
σxya/c
P c
/P
0
σxyd/c = 0.25
σxyd/c = 0.75
σxyd/c = 1.25
σxyd/c = 1.75
σxyd/c = 2.25
σxyd/c = 2.75
σxyd/c→∞
Figure 4.4: A plot of the normalized Casimir force for various thicknesses of a
bulk Hall material (idealized Weyl semimetal). It begins slightly repulsive for small
σxyd/c, and as this increases, it becomes more repulsive until it reaches the maximum
for a thin film material (the dashed line) at which point it increases to the semi-
infinite limit. P0 = − ~cpi
2
240a4 and σxy = e2b/2pi2~ is the bulk Hall response. Figs. 4.5,
4.6, 4.7, and 4.8 takes into account more material properties.
97
pushed further and further out for smaller thicknesses until the Casimir pressure
becomes purely repulsive for all distances. This not only clearly connects our case
to the previously known thin-film result, it also provides a theoretical justification
for considering a thin-film limit d a with a two-dimensional conductivity σµνd.
4.5 Conductivity for a clean Weyl semimetal
In order to add in more material effects and see how they might affect the
Casimir force, we need to calculate the conductivity tensor for the band theory
offered by Weyl semimetals.
We will mainly be interested in the effects of virtual vacuum transitions that
are low in energy, which corresponds to plates that are far apart from one another
(in the two-dimensional case, as seen in the Appendix B). Thus, we will use the
low-energy chiral Hamiltonian for a pair of Weyl nodes
HW = ±~vFσ · (k± b), (4.44)
where vF is the Fermi velocity and b is the position of the of Weyl node in k-space.
The exact band structure will be important as the plates get closer though weighting
will still be larger on the lower energy modes.
For calculational ease, we set ~ = 1 = vF. Now, if we calculate the conduc-
tivity for a fixed kz, we obtain basically the same conductivity for the surface of a
topological insulator with a chemical potential both in and out of the gap.
In fact, the eigenstates differ only slightly from that considered in Eq. (2.1).
98
We restrict ourselves to HW with kz − b. If eik·x |fk±〉 diagonalizes HW , then
〈fk+|σx|fk−〉 =
(kz − b)kx + iky

√
2 − (kz − b)2
, (4.45)
〈fk+|σy|fk−〉 =
(kz − b)ky − ikx

√
2 − (kz − b)2
, (4.46)
〈fk±|σx|fk±〉 = ±
kx
 , (4.47)
where  ≡
√
k2x + k2y + (kz − b)2. The inter-band contribution to the conductivity
takes the form
σ˜interµν (iω; kz) =
e2
i~
∑
k,γ 6=γ′
nkγ − nkγ′
kγ − kγ′
〈fkγ|jµ|fkγ′〉 〈fkγ′ |jν |fkγ〉
iω + kγ − kγ′
, (4.48)
where nkγ is the occupation in that band at momentum k and jµ = σµ are the single
particle current operators. For intra-band quantities with nk± = θ(µ∓ k)
σ˜intraµν (iω; kz) = −
e2
i~
∑
k
δ(µ− k,+)
〈fk+|jµ|fk+〉 〈fk+|jν |fk+〉
iω , (4.49)
assuming only the upper-band for simplicity and without loss of generality (due to
particle-hole symmetry).
Putting these together and performing the integrals, we obtain the result at
finite chemical potential
σ˜xx(iω; kz) =
e2
4pi~
[(
1− 4(kz − b)
2
ω2
) i
4
log
(
2∆− iω
2∆ + iω
)
+
∆
ω
]
, (4.50)
σ˜xy(iω; kz) =
e2
2pi~
[kz − b
ω
i
2
log
(
2∆− iω
2∆ + iω
)]
, (4.51)
where ∆ = max{|kz − b|, |µ|}. With these quantities we can then use the cutoff
procedure explained in [196]
σµν(iω) =
∫ Λ
−Λ
dkz
2pi σ˜µν(iω; kz). (4.52)
99
With Eq. (4.52) we obtain (throwing away terms that go to zero as the cutoff
increases to infinity and multiplying by two for the two nodes and bringing back in
the constants ~ and vF)
σxx(iω) =
e2
12pi2~vF
[
5
3ω + 2ω log
(vFΛ
ω
)
+ 4 µ
2
~2ω − ω log
(
1 + 4µ
2
~2ω2
)]
, (4.53)
σxy(iω) =
e2b
2pi2~ . (4.54)
These quantities are what we use in the next section as input for the Casimir Force.
Notice that σxy(iω) is unchanged at this order. Due to the linear dispersion of the
Weyl nodes, we have a logarithmic cutoff dependence. This regulation comes more
naturally in a low energy theory that has parabolic kz such as H = ~vF[kxσx +
kyσy + `(k2z − b2)σz], where ` is a length scale. Note that in Eq. (4.53), rotating to
real frequencies we get the correct result for two Weyl nodes for Re[σxx(ω)] [114],
and a result with the appropriate logarithmic divergence for Im[σxx(ω)] [197]. This
can be understood in terms of charge renormalization due to the band structure,
but for ease of our purposes we let Λ ∼ 1/a0 where a0 is the lattice spacing. For
our plots in Figs. 4.5, 4.6, 4.7, and 4.8 we choose a lattice spacing of a0 = 1 nm, a
thickness of d = 20 nm, b = 0.3(2pi/a0), Λ = 2pi/a0, vF = 6 × 105 m/s, and µ = 0
unless its the parameter we are varying.
One can use one of the two equivalent ways of calculating the Casimir energy:
the reflection matrix as given in [38] (and calculated in Appendix A) along with
Lifshitz formula Eq. (1.8). In order to avoid an unphysical negative σxx(iω) as well
as for consistency, we cutoff the photon energies in the Lifshitz formula to run from
0 to Λ—an approximation valid for a cvF Λ
−1.
100
0.1 1 2 3 4
−2
−1
0
1
2
3
b = 0.2(2pi)/nm
b = 0.3(2pi)/nm
b = 0.4(2pi)/nm
b = 0.5(2pi)/nm
a (µm)
P c
/P
0
(1
0−
2 )
Figure 4.5: The Casimir force for a thin film Weyl semimetal taking into account
low-energy virtual transitions in the band structure. An anti-trap clearly develops
when the longitudinal conductivity overwhelms the Hall conductivity. We compare
different values of b (or equivalently, changing the Hall effect). For this plot, a0 =
1 nm, d = 20 nm, Λ = 2pi/a0, vF = 6× 105 m/s, and µ = 0.
0.1 1 2 3 4
−0.7
0
2
4
vF = 3× 105 m/s
vF = 6× 105 m/s
vF = 12× 105 m/s
vF = 18× 105 m/s
a (µm)
P c
/P
0
(1
0−
2 )
Figure 4.6: The Casimir force for a thin film Weyl semimetal taking into account
low-energy virtual transitions in the band structure. Here we compare different vF
(the larger vF the smaller σxx is). For this plot,a0 = 1 nm, d = 20 nm, b = 0.3(2pi/a0),
Λ = 2pi/a0, and µ = 0.
101
0.1 1 2 3 4
−0.62
0
1
2
µ = 0meV
µ = 10meV
µ = 17meV
µ = 25meV
a (µm)
P c
/P
0
(1
0−
2 )
Figure 4.7: The Casimir force for a thin film Weyl semimetal taking into account low-
energy virtual transitions in the band structure. Here we turn on a finite chemical
potential which causes attraction at very large distances (and hence a trap). Even
small chemical potentials have this property but the trap is quite far out. For this
plot, a0 = 1 nm, d = 20 nm, b = 0.3(2pi/a0), Λ = 2pi/a0, and vF = 6× 105 m/s.
0.1 1 2 3 4
−0.7
0
2
4
Λ = 2pi /nm
Λ = 3pi /nm
Λ = 4pi /nm
Λ = 5pi /nm
a (µm)
P c
/P
0
(1
0−
2 )
Figure 4.8: Here, we vary the cutoff to see how it depends. It is very similar
to varying vF as one can see in Fig. 4.6. For this plot, a0 = 1 nm, d = 20 nm,
b = 0.3(2pi/a0), vF = 6× 105 m/s, and µ = 0
102
First, in all of the plots in Figs. 4.5, 4.6, 4.7, and 4.8, we see that we get an anti-
trap for these at approximately 650 nm, and if we increase b as in Fig. 4.5 (with, say,
an applied magnetic field), it not only moves closer to zero separation, but the overall
repulsive behavior can be enhanced. On the other hand, if we increase vF as we see
in Fig. 4.6, we see the region of attraction is suppressed, but the overall repulsive
behavior at large distances is maintained. Modifying Λ, as we see in Fig. 4.8 will
have effects similar to modifying vF, but since it appears logarithmically, it needs to
change by orders of magnitude to give appreciable changes. The cutoff Λ is actually
a stand-in for higher band effects, and we can see that here it saves the analysis for
short distances. This “anti-trap” effect occurs at short distances when higher order
band effects also play a role, but any other effects will contribute to the longitudinal
conductivity in such a way that an anti-trap will appear.
Interesting effects at larger distances begin to occur when we introduce a finite
chemical potential as we see in Fig. 4.7. In addition to the anti-trap we get at shorter
distances, we start to see a trap at much longer distances. This is not surprising since
at zero frequency there is a divergent longitudinal conductivity, see Appendix B.
Thus, we know that at long distances, the Casimir force must be attractive, but by
modifying the Hall effect, we have an intermediate regime of repulsion.
A similar effect would occur if we took finite temperature or disorder correc-
tions to the longitudinal conductivities—both yield finite DC conductivities.
103
4.6 Conclusions
Considering the form of the conductance in terms of the fine structure constant,
σxyd/c = α 2bdpi , we see that bd controls the strength of the repulsion in the thin film
limit. Without longitudinal conductance, the repulsive regime roughly corresponds
to when (σxyd/c)2 . σxyd/c or equivalently 2bdpi . 1α . The longitudinal conductance
introduces vF into the scheme; relevant photons have ω ≈ c/a and thus it becomes
important for σxxd/c ∼ α cvF
d
a (neglecting constants) which both emphasizes that vF
controls the longitudinal conductance’s contribution to the Casimir effect and that
the term is suppressed at longer distances.
We have shown here how Weyl semimetals can exhibit a tunable repulsive
Casimir force (with, for instance, magnetic field tuning b) and how it can depend on
the thickness of the material. In the thin film limit, we showed how the semimetallic
nature of these materials can work to create attraction at smaller distances scales,
and how a finite longitudinal conductivity will create long distance attraction along
with repulsion at intermediate distances. The marginal nature of these (at present
purely theoretical) materials could be useful for controlling the Casimir force be-
tween attractive and repulsive regimes.
104
Chapter 5: The breakdown of the coherent state path integral
5.1 Overview
In this chapter, we change subjects away from optical and Casimir effects to
a purely theoretical paradigm: path integration. This bares no relation to what we
have previously discussed.
We outline another problem with the time-continuous coherent state path
integral. This problem manifests itself in two simple examples: (i) the spin-coherent
state path integral and (ii) the harmonic oscillator coherent state path integral
(in particular, the single-site Bose-Hubbard model). The single-site Bose-Hubbard
Hamiltonian is a minimal model that demonstrates the problem with the normal-
ordered path integral. However, the problem itself is more general than the toy
model considered here and clearly persists in more complicated models, including
lattice Bose-Hubbard models. We use an exact method of calculating the partition
function mathematically developed by Alekseev et al. [198] (and more recently used
by Cabra et al. [199] for the spin path integral with H = Sz), and demonstrate
that the exact result differs from the correct partition function in the cases of both
normal-ordering of operators (as prescribed by most textbooks) and when using
Weyl ordering (i.e., it cannot be accounted for with the phase anomaly found by
105
Solari and Kochetov [153, 155] and elaborated on by Stone et al. [154]).
This breakdown is yet to be fully resolved, but there has been interesting effort
in this regard by others inspired by this work [200]. The culprit of the problem
seems to be, unsurprisingly to those familiar with coherent state path integrals, the
continuity assumption for paths. However, the path integral works just fine when
one has a Hamiltonian that is a linear combination of generators of some Lie algebra
used to generate the coherent states. We illustrate these points with two examples,
and elaborate on the analysis of these points in what follows.
This chapter is mostly based on the paper [7], copyright APS.
5.2 Breakdown of the spin-coherent state path integral
We begin with the coherent state path integral for spin with the standard
SU(2) algebra defined on the operators {Sx, Sy, Sz} with [Si, Sj] = iijkSk, and we
define our Hilbert space by taking the matrix representation of the SU(2) group in
(2s + 1)-by-(2s + 1) matrices (s being the spin of the system). Irrespective of the
algebra, we can in general define a Hermitian matrix H that acts on states in our
Hilbert space, and this will be our Hamiltonian. Generally, H can be written as a
polynomial of algebra generators.
If |s〉 is the maximal state of Sz in our spin-s system, then we can define
spin-coherent states as
|n〉 = e−iφSze−iθSy |s〉 (5.1)
where (θ, φ) are coordinates on the sphere S2 along the unit vector n (i.e., a point
106
on the standard Bloch sphere). These coherent states are overcomplete such that
2s+ 1
4pi
∫
S2
dn |n〉 〈n| = 1 (5.2)
where dn = dφ d(cos θ) is the standard measure on S2. Using this continuous, over-
complete basis, one can derive the standard path integral for the partition function
for spin from Z = tr e−βH in the standard way [136] discussed in the Sec. 1.5
Z ′ =
∫
Dn(τ) exp
{
−
∫ β
0
dτ [−〈n(τ)|∂τn(τ)〉+ 〈n(τ)|H|n(τ)〉]
}
. (5.3)
We call the partition function as given by the time-continuous path integral Z ′ in
order to distinguish it from Z = tr e−βH since we will find that in general they
may not agree. The path integral is over all closed paths (since it is the parititon
function). The first term in the action for Eq. (5.3), 〈n|∂τn〉, is the Berry phase
term and in (θ, φ) coordinates
−〈n|∂τn〉 = −is(1− cos θ)∂τφ. (5.4)
We assume 〈n|H|n〉 = H(cos θ) for some function H(x) (this is true if and
only if H is diagonal). This puts the φ dependence of the action solely in the Berry
phase term of the action. We then integrate the Berry phase term by parts; the
boundary term becomes
∫ β
0
dτ 〈n(τ)|∂τn(τ)〉 = is(φ(β)− φ(0))(1− cos θ(0)) + is
∫ β
0
dτ φ(τ)dcos θdτ , (5.5)
remembering that the paths considered in Z ′ are closed paths (so cos θ(0) = cos θ(β)).
By the same logic, φ(β) = φ(0) + 2pik for any integer k, and we can break up the
integral into topological sectors given by the integer k (the winding number of a path
107
around the z-axis). Note that this makes the (perfectly reasonable) assumption for
continuous paths that the φ-integral is the universal covering of S1 (simply: the real
numbers).
At this point in calculations our path integral now takes the form
Z ′ =
∑
k∈Z
∫
Dn(τ) exp
{
2piisk(1− cos θ(0))
+ is
∫ β
0
dτφ(τ)dcos θdτ −
∫ β
0
dτH(cos θ)
}
, (5.6)
and we can now consider the measure Dn(τ) = DφD(cos θ). For each time-slice Dφ
integrates over the universal covering of S1, so we can perform this integral. This is
due to the fact that our only φ-dependence is multiplying dcos θdτ from integrating by
parts, and we use the standard identity for functional integrals (derived easily from
a the usual δ-function identity)
∫
Dφ e−i
∫ β
0 dτ φ(τ)f(τ) = δ(f), (5.7)
to get that cos θ must be constant (i.e., dcos θdτ = 0). This δ-function allows us to do the
path integral over D(cos θ), except for the initial value which we call x := cos θ(0)1.
Taking all of this into account, the path integral can then be written as
Z ′ =
∞∑
k=−∞
∫ 1
−1
dx e2piiks(1−x)−βH(x). (5.8)
The sum over k can be evaluated as a sum of delta functions using the formula
∞∑
k=−∞
e2piks(1−x) =
∞∑
n=−∞
δ(s(1− x)− n) (5.9)
1This is even clearer in the case of a time slice picture, where each time slice imposes f(τi) =
f(τi+1) except that one of these remains undefined, which we take to be f(0) for convenience
108
of the form for all integers n. Since x is in the interval −1 to +1, only finitely many
n contribute (n = 0 to n = 2s to be exact). We can rewrite the sum over n as a sum
over (suggestively) m := s− n and we get the answer (dropping overall constants)
Z ′ =
s∑
m=−s
e−βH(m/s). (5.10)
Eq. (5.10) looks very promising, but H(m/s) is not the same as 〈m|H|m〉. First let
us see where it does work. Take the simple Hamiltonian H = Sz, then 〈n|H|n〉 =
s cos θ, and thus H(x) = sx. This immediately yields
Z ′H=Sz =
s∑
m=−s
e−βm, (5.11)
and it is easily calculated (in operator language) that Z ′H=Sz = ZH=Sz . The two
methods agree for the particular Hamiltonian H = Sz (the case considered by Cabra
et al. [199]). On the other hand, if we take H = S2z and s = 1, we can evaluate
〈n|S2z |n〉 = 12 (cos2 θ + 1); from which we have
H(x) = 1
2
(
x2 + 1
)
. (5.12)
Thus, Z ′H=S2z = 2e
−β + e−β/2, but this conflicts with ZH=S2z = 2e−β + 1 by more
than just a multiplicative constant. Thus, we have Z ′H=S2z 6= ZH=S2z for s = 1, and
in fact Z ′H=S2z 6= ZH=S2z for all s > 1/2.
Importantly, the two methods agree for any Hamiltonian when s = 1/2. This
comes from the fact that any (diagonalized) Hamiltonian for a two state sytem
(s = 1/2) can be written as H = a+ bSz after an appropriate rotation (and, in fact,
S2z = 1/4), and the above method gives Z ′ = Z when H = a+ bSz.
109
Further, say we have a Hamiltonian that is a polynomial of Sz/s (any finite-
dimensional Hamiltonian, with an appropriate unitary transformation can be writ-
ten in this way)
H =
N∑
i=1
ai
(Sz
s
)n
. (5.13)
First we note that
〈m|H|m〉 = Em =
2s∑
k=0
ak
(m
s
)k
.. (5.14)
On the other hand,
H(m/s) = 〈θm, φ|H|θm, φ〉 , (5.15)
where θm is defined by cos θm = m/s and this expression is independent of φ.
Equivalently, we can write
H(m/s) = 〈s|eiθmSyHe−iθmSy |s〉 .
The expression we are now interested in is
〈s|eiθmSyHe−iθmSy |s〉 =
N∑
k=0
ak
1
sk 〈s|e
iθmSySkz e−iθmSy |s〉
=
N∑
k=0
ak
1
sk 〈s|(e
iθmSySze−iθmSy)k|s〉
=
N∑
k=0
ak
1
sk 〈s|[Szm/s− Sx(1−m
2/s2)1/2]k|s〉 .
We can do a binomial expansion at this point, but we must note that if we have an
odd amount of Sx’s the expectation value will be zero (recall Sx = (S+ + S−)/2).
110
Thus, the first two terms in this expansion are
hk :=
1
sk 〈s|[Szm/s− Sx(1−m
2/s2)1/2]k|s〉 = 1sk 〈s|S
k
zmk/sk|s〉
+
1
sk
(m
s
)k−2
(
1− m
2
s2
) k−2∑
n=0
k−2−n∑
l=0
〈s|SlzSxSnz SxSk−2−l−nz |s〉+ · · · . (5.16)
Evaluating the right hand side of Eq. (5.16), we can write it as
hk =
(m
s
)k
+
1
s2+n
(m
s
)k−2
(
1− m
2
s2
) k−2∑
n=0
k−2−n∑
l=0
〈s|SxSnz Sx|s〉+ · · · ,
=
(m
s
)k
+
1
2s
(m
s
)k−2
(
1− m
2
s2
) k−2∑
n=0
(k − 1− n)
(
1− 1s
)n
+ · · · .
Recall that k ≤ N so it does not grow with s, then the second term is clearly of the
order 1/s and can be dropped for large enough spin s.
This method clearly works for H being any polynomial of Sz, and after that
polynomial is set, taking s→∞ (or when the polynomial’s degree is much smaller
than two times the spin).
Putting this together with the conditions found
H(m/s) ∼ 〈m|H|m〉+O(1/s).
This general result shows that “semiclassically” (i.e. s tends to infinity), we
will still arrive at sensible results.
5.3 Breakdown of the harmonic-oscillator coherent state path inte-
gral
To motivate looking for this same issue in a system with the Weyl-Heisenberg
algebra (i.e., the harmonic oscillator algebra), it is known [201] that one can contract
111
u(2) (since we constructed our coherent states for spins with su(2)) into the Weyl-
Heisenberg algebra by considering u(2) = span{S0, Sx, Sy, Sz} = u(1)⊕su(2), where
we define [S0, Si] = 0. Then define the operators J0 := S0, J1,2() := Sy,x, and
J3() := S0 + −2Sz to get the commutation relations [J3, J1,2] = ∓iJ2,1, [J1, J2] =
−i2J3 + iJ0, and [J0, Ji] = 0. If we let  → 0, we recover exactly the Weyl-
Heisenberg algebra: h4 = span{1, x, p, a†a} with [x, p] = i, [a†a, x] = −ip, [a†a, p] =
ix. Intuitively, imagine the Bloch sphere that Sx, Sy, and Sz define with a radius
s. As s grows, if we concentrate, say, on the south pole, it looks like a plane in
phase-space2. This is what the contraction captures: It essentially goes to a tangent
plane which asymptotically is exact as s→∞. Observe that Sz is related to a†a in
this contraction, so we might suspect that terms quadratic in a†a give problems like
those found with S2z in the spin-coherent state path integral.
A Hamiltonian that uses the Weyl-Heisenberg algebra to construct its coherent
states is the Bose-Hubbard model. For a single site, we can write
H = −µn+ U
2
n(n− 1), (5.17)
where n = a†a and the a (a†) is the annihilation (creation) operator for the algebra
[a, a†] = 1. The form n(n − 1) = a†a†aa comes from the normal ordering required
from a path integral of the form
Z ′ =
∫
D2z exp
{
−
∫ β
0
dτ
[
1
2
(z∗z˙ − z˙∗z)− µ|z|2 + U
2
|z|4
]}
. (5.18)
We can solve this path integral with the same method used to obtain Eq. (5.10) in
2Technically, the contraction is independent of the representation s defines, but in this large
representation, it operates nearly identically.
112
the spin-coherent state path integral. Let z = √neiθ, so that the measure becomes
D2z = DnDθ and the action becomes
S =
∫
dτ(inθ˙ − µn+ U
2
n2). (5.19)
Integrating by parts on the nθ˙ term then integrating over Dθ will fix n to be constant
and give us a separation into different topological sectors (depending on how many
times a path wraps around zero in phase-space), and the boundary term will fix n
to be an integer. Since n is radial, it can only be positive so we directly obtain
Z ′ =
∞∑
n=0
eµnβ−U2 n2β. (5.20)
But this differs from the partition function that we can easily calculate in operator
language:
Z =
∞∑
n=0
eµnβ−U2 n(n−1)β. (5.21)
We see that a similar problem to that of the spin coherent state path integral
here. To see it explicitly, for U  1, we have Z ′ ∼ 1 + eµ−U/2 + · · · , but Z ∼
1+eµ+e2µ−U + · · · . With different asymptotics, Z and Z ′ are different expressions.
Note that if we let µ→ µ+ U2 in Z ′, that we will get same result. This substitution
for µ corresponds to replacing n in Eq. (5.17) by 〈n〉 = |z|2 when writing down our
action (so instead of 〈n2〉, one gets 〈n〉2—just as in the case of Sz, fluctuations in
the generators are getting in the way).
113
5.4 (Lack of) Relation to the semiclassical anomaly
As discussed in the introduction, others have identified problems in the co-
herent state path integral that are related to the semiclassics of the model. To get
the correct result, an anomaly in the fluctuation determinant modifies the propa-
gator in a non-trivial manner giving the correct result. This is usually discussed
in the literature in context of the SU(2) coherent state path integrals, and here
we do the calculation for the single-site Bose-Hubbard model as we have used in
Sec. 5.3. However, as early as Berezin [140], it was recognized that the semiclassics
of the coherent state path integral was correct if one used Weyl-ordering instead
of normal-ordering. To be specific, if one has a complex function H(z, z∗) in the
Lagrangian of the coherent state path integral, the normal-ordering scheme defines
〈z|H|z〉 = H(z, z∗). This means that we can directly map
H =
∑
nm
αnm(a†)man =⇒ 〈z|H|z〉 =
∑
nm
αnm(z∗)mzn, . (5.22)
On the other hand, the Weyl symbol is defined such that
H =
∑
nm
αnm[(a†)man]S =⇒ HW (z, z∗) =
∑
nm
αnm(z∗)mzn, (5.23)
where [· · · ]S represents the fully symmetric operator product (e.g. [a†a]S = (aa† +
a†a)/2). These two representations can be mapped into one another, such as [140]
〈z|H|z〉 =
∫ d2v
2pi HW (v, v
∗)e− 12 (z∗−v∗)(z−v). (5.24)
However, the semiclassical picture actually tracks HW instead of 〈z|H|z〉.
114
In order to do semiclassics, we need a small parameter. Still considering
Eq. (5.17), let us change our algebra slightly to incorporate this small parame-
ter (akin to the standard ~→∞ for normal semiclassics): h−1 such that [a, a†] = h.
The quantity h−1 is the representation index (called γ in [155]). We note here that
different h’s change the coherent states |z〉 in the following way: if z = 1√2(u + iv),
then u = q/c, v = p/d, and h = ~/(cd). We have used q and p as the standard
position and momentum for the harmonic oscillator. Up until now we have been
considering h = 1.
We first look at the Hamiltonian H = ωn/h where n = a†a. We first find the
propagator between two coherent states. The propagator as defined by the path
integral with the Lagrangian L = iz∗z˙ − 〈z|H|z〉 is then given by
〈zf |e−iHT |zi〉 = e−
1
2h (|zf |2+|zi|2)+
1
h z∗f zie−iωT−
1
2 iωT . (5.25)
Let us try to prove this in the operator language, we can insert a complete set
of states and obtain
〈zf |e−iHT |zi〉 =
∑
n
〈zf |n〉 〈n|zi〉 e−iωTn/h (5.26)
= e− 12h (|zf |2+|zi|2)
∑
n
(
1
hz
∗
fzi
)n 1
n!e
−iωTn (5.27)
= e− 12h (|zf |2+|zi|2)+ 1h z∗f zie−iωT . (5.28)
But we have that Eq. (5.28) is not the same as Eq. (5.25). However, if we use
the Weyl-ordered version of the Hamiltonian though (HW = ω(aa† + a†a)/(2h) =
ωn/h+ ω/2h, then we would have a multiplicative constant term that would go as
e− i2ωT—in agreement with the path integral approach.
115
Now, we move to the more interesting case of the path integral for the single-
site Bose-Hubbard model. We will be interested in the Lagrangian that has
H(z, z∗) = −µh |z|
2 +
U
2h |z|
4. (5.29)
In the algebra described in the previous section, we can write our single-site
Bose-Hubbard Hamiltonian as
H = −µhn+
U
2hn(n− h), (5.30)
which if we consider normal ordering 〈z|H|z〉 = H(z, z∗). If we let µ˜ := µ + Uh2 ,
then we can write
H = − µ˜hn+
U
2hn
2. (5.31)
We will be using this form of H throughout. The propagator looks like
K(z∗f , zi;T ) = 〈zf |eiµ˜n
T
h−
i
2Un2
T
h |zi〉 (5.32)
Now, we want to decouple n2, so we make use of the Hubbard-Stratonvich transfor-
mation
e− i2Un2 Th =
√
iT
2piUh
∫
dω e i2U ω2 Th+niω Th , (5.33)
and this leads directly to an evaluation of the propagator which we can write now
as
K(z∗f , zi;T ) =
√
iT
2piUh
∫
dω e i2Uhω2T 〈zf |ei(µ˜+ω)n
T
h |zi〉 (5.34)
=
√
iT
2piUh
∫
dω e 1h z∗f ziei(ω+µ˜)T− 12h (|zf |2+|zi|2)+ i2Uhω2T . (5.35)
116
We will take the stationary phase approximation of this quantity later to obtain
agreement with (or rather, disagreement with) the semiclassics given by the path
integral.
If we change variables so that ω¯ + µ = ω + µ˜, we get
K(z∗f , zi;T ) =
√
iT
2piU
∫
dω¯ e 1h z∗f ziei(ω¯+µ)T− 12h (|zf |2+|zi|2)+ i2Uh ω¯2T+ i2 ω¯T+ i8UhT . (5.36)
On the other hand, we can Weyl order the Hamiltonian to get
HW = −
1
2hµ
(
a†a+ aa†
)
+
U
2h
[
a†a†aa
]
S (5.37)
= −1hµ
(
n+ h
2
)
+
U
2h
(
n2 + nh+ h
2
2
)
(5.38)
= −1h
(
µ− Uh
2
)
n+ U
2hn
2 +
Uh
4
(5.39)
In order to check the above propagator K with its analogue with the Weyl Hamil-
tonian KW , we can use the previous result in Eq. (5.36) to find the result for this
Weyl symmetric Hamiltonian
KW (z∗f , zi;T ) =
√
iT
2piU
∫
dω¯ e 1h z∗f ziei(ω¯+µ)T− 12h (|zf |2+|zi|2)+ i2Uh ω¯2T− i2 ω¯T− i8UhT . (5.40)
In terms of the path integral we can write out
K(z∗f , zi;T ) =
∫ z∗(T )=z∗f
z(0)=zI
D2z exp {Φ[z, z∗]/h} , (5.41)
where we have defined
Φ = S + Γ, (5.42)
Γ =
1
2
[
z∗fz(T ) + z∗(0)zI − |zf |2 − |zI |2
]
, (5.43)
S = 1
2
∫ T
0
dt (zz˙∗ − z∗z˙)− i
∫ T
0
dt hH(z, z∗), (5.44)
117
and where H(z, z∗) is defined by Eq. (5.29). We solve for the classical state by
varying the action to get separate equations for z and z∗. These look like
z˙cl = −ih
∂H
∂z∗cl
zcl(0) = zi (5.45)
z˙∗cl = ih
∂H
∂zcl
z∗cl(T ) = zf . (5.46)
This allows us to evaluate the semiclassical propagator to be
Ksc(zf , zi;T ) =
[
∂2Φcl
∂z∗f∂zi
]1/2
exp
{
Φcl/h+
i
2
∫ T
0
B dt
}
, (5.47)
where Φcl := Φ[zcl, z∗cl] and
B =
[
h ∂
2H
∂z∂z∗
]
z=zcl
. (5.48)
With our specific Hamiltonian, the classical equations to be satisfied are
z˙cl = −izcl
(
−µ+ U |zcl|2
)
= i(µ+ ω)zcl, (5.49)
z˙∗cl = iz∗cl(−µ+ U |zcl|2) = −i(ω + µ)z∗cl, (5.50)
where we defined
ω(zcl, z∗cl) = −U |zcl|2, (5.51)
and we can see that we get ddtω = 0. This allows us to solve the equations (5.50) to
get
zcl(t) = ei(ω+µ)tzi, (5.52)
z∗cl(t) = ei(ω+µ)(T−t)z∗f . (5.53)
118
And we get the consistency equation
ω = −Uz∗fziei(ω+µ)T . (5.54)
Now, we go about evaluating everything
Φcl =
∫ T
0
dt
[
−i(ω + µ)|zcl|2 − i
(
−µ|zcl|2 +
U
2
|zcl|4
)]
(5.55)
=
∫ T
0
dt
[
i(ω + µ)ωU − i
(
µωU +
U
2
ω2
U2
)]
(5.56)
=
iT
2
ω2
U (5.57)
Γcl = z∗fziei(ω+µ)T −
1
2
(
|zf |2 + |zi|2
)
. (5.58)
This allows us to write
Φcl = z∗fziei(ω+µ)T +
iT
2U ω
2 − 1
2
(
|zf |2 + |zi|2
)
. (5.59)
Now, we define
Φω = z∗fziei(ω+µ)T +
iT
2U ω
2, (5.60)
that independently depends on ω. We notice that
∂Φω
∂ω = 0, (5.61)
in fact this is just the consistency equation Eq. (5.54). We now perform some partial
differentiation
∂2Φcl
∂z∗f∂zi
=
∂
∂z∗f
[∂Φω
∂zi
+
∂Φω
∂ω
∂ω
∂zi
]
(5.62)
=
∂2Φω
∂z∗f∂zi
+
∂2Φω
∂ω∂zi
∂ω
∂z∗f
+
∂2Φω
∂z∗f∂ω
∂ω
∂zi
+
∂2Φω
∂ω2
∂ω
∂zi
∂ω
∂z∗f
. (5.63)
119
Now Eq. (5.61) gives us that
∂2Φω
∂z∗f∂ω
+
∂2Φω
∂ω2
∂ω
∂z∗f
= 0 (5.64)
∂2Φω
∂zi∂ω
+
∂2Φω
∂ω2
∂ω
∂zi
= 0. (5.65)
Eq. (5.64) and Eq. (5.65) can be used to get rid of derivatives of ω in Eq. (5.63) and
we get
∂2Φcl
∂z∗f∂zi
=
∂2Φω
∂z∗f∂zi
− ∂
2Φω
∂ω∂zi
∂2Φω
∂ω∂z∗f
(∂2Φω
∂ω2
)−1
. (5.66)
If we write out things we get
∂2Φω
∂z∗f∂zi
= ei(ω+µ)T , (5.67)
∂2Φω
∂ω∂zi
= iTz∗fei(ω+µ)T , (5.68)
∂2Φω
∂ω∂z∗f
= iTziei(ω+µ)T , (5.69)
∂2Φω
∂ω2 =
iT
U − T
2z∗fziei(ω+µ)T . (5.70)
Putting this together we get
∂2Φcl
∂z∗f∂zi
= ei(ω+µ)T +
T 2ziz∗fe2i(ω+µ)T
iT/U − T 2z∗fziei(ω+µ)T
(5.71)
=
iTei(ω+µ)T/U
iT/U − T 2z∗fziei(ω+µ)T
(5.72)
=
(iT
U
)(∂2Φω
∂ω2
)−1
ei(ω+µ)T . (5.73)
Thus we have
[
∂2Φcl
∂z∗f∂zi
]1/2
=
( iT
hU
)1/2(1
h
∂2Φω
∂ω2
)−1/2
e i2 (ω+µ)T . (5.74)
120
And lastly,
B = −µ+ 2U |zcl|2 = −(µ+ 2ω), (5.75)
so that
i
2
∫ T
0
B dt = − i
2
T (µ+ 2ω). (5.76)
Finally, we have the full semiclassical operator (summing over ω, solutions to the
consistency equation (5.54))
Ksc(z∗f , zi;T ) =
∑
ω
( iT
hU
)1/2(1
h
∂2Φcl
∂ω2
)−1/2
exp
[
1
hΦcl +
i
2
(ω + µ)T − i∆
]
,
(5.77)
=
∑
ω
( iT
hU
)1/2(1
h
∂2Φcl
∂ω2
)−1/2
exp
[
1
hΦcl −
i
2
ωT
]
, (5.78)
where
i∆ = i
2
(µ+ 2ω)T, (5.79)
and the sum is over solutions to the consistency equation ∂ωΦω = 0. The ∆ term
comes from the anomaly in the fluctuation determinant and as evaluated by Ko-
chetov [155]. Notice that this coincides with Eq. (5.40) when we take the steepest
descent of the ω¯ integral as h → 0. Thus, we see that the path integral is giving a
result inconsistent with normal ordering and consistent with Weyl ordering. This
speaks to the fact that we lose operator order when going to the path integral [202].
However, let us see if this was actually the problem with the coherent state
path integral as we had evaluated before. Recall Eq. (5.20) and Eq. (5.21) define Z
121
and Z ′ (exact Hamiltonian and path integral respectively). Now if ZW represents
the exact partition function as given by the Weyl ordering, we obtain
ZW =
∑
n
eβµn−β U2 (n2+n+1/2)+βµ/2 ∼ eβ(U+µ)/2(1 + eβ(µ−U) + · · · ). (5.80)
We see from this that Z 6= Z ′ 6= ZW . This solution, as it exists for the semiclas-
sical analysis, does not apply to the calculational procedure we used for the exact
calculation found in Sec. 5.3.
5.5 Conclusions
While the semiclassical result is not a new one, it shows that the path integral
is not dealing with the same Hamiltonian. Unfortunately, our exact calculation of
Z ′ (see Eq. (5.61)) suggests that the path integral is dealing with H ′ = −µn+ U2 n2
while semiclassics suggests it is dealing with HW = −µn + U2 n(n + 1) (going back
to h = 1). These two methods differ but both are not dealing with the Hamiltonian
under consideration, Eq. (5.30). In the case of the Weyl ordered Hamiltonian, we
can write our original Hamiltonian in Eq. (5.30) as H = HW − Un which is Weyl
ordered (up to a constant). This ordering can be used to modify the path integral
by an extra term: −U |z|2. This correction to the path integral suggested by Weyl
ordering does not fix the exact calculation of Z ′ as can be easily shown, but it does
motivate an ad hoc correction to the path integral to “fix” our exact calculation.
We use the following action:
S =
∫
dt
(
−µ|z|2 + U
2
|z|2(|z|2 − 1)
)
. (5.81)
122
This action is constructed by just changing the operator n to a function |z|2; while
this gives correct results with the method which gives Eq. (5.20), there is no a priori
reason to suspect this of being the action. Similarly, if in the spin-coherent state path
integral, we replace the operator Sz with its expectation value 〈n|Sz|n〉 everywhere,
we will get the correct result. This means, in particular, for H = S2z that instead
of 〈S2z 〉 in the spin-path integral we have 〈Sz〉2. In general, if one substitutes the
generators of the coherent states in the Hamiltonian with their expectation value,
one obtains the correct result for Z with the methods used to derive Eq. (5.54) and
Eq. (5.61).
Corrections aside, a simple way to see what has gone wrong is to return to
Eq. (5.12). This H(x) function can not achieve the value 0, but H = S2z clearly
has such an eigenvalue. This is due to the fact that for higher dimensional repre-
sentations of SU(2) not every eigenvector of Sz can be rotated into another with
a standard SU(2) rotation. On the other hand, the coherent states we used are a
complete set for even higher dimensional representations, so in principle, we should
not lose any information about the m = 0 state. Continuity in n seems to be the
culprit: H(x) came from a time discretized form (between time slices j and j + 1)
〈nj+1|S2z |nj〉, and we have 〈n|S2z | − n〉 = 0, so 〈nj+1|S2z |nj〉 can attain zero, but not
for any paths that are “close” to each other (i.e. nj ≈ nj+1) as the continuous time
path integral assumes. As such, the discrete time path integral (before a continuity
assumption is imposed) can unambiguously give the correct results to a calculation.
To conclude, in the time-continuous formulation of the path integral, neither
the action suggested by Weyl-ordering nor the action constructed by normal ordering
123
gives correct results when evaluating Z via path integrals.
124
Chapter 6: Conclusion
In this dissertation, we have largely been concerned with how electromagnetic
phenomena are modified due to materials considered to be topological (Chapters 2
and 4) or undergoing a topological transition (a Lifshitz transition, Sec. 3.2). Along
the way, we investigated the question of how to test diffusive models of metals in
the Casimir effect theoretically and experimentally (Sec. 3.3), and we looked into
the breakdown of the coherent state path integral (Chapter 5).
The main effects considered here considered both the classical and quantum
aspects of the electromagnetic field. The classical electromagnetic field experiences
magneto-optical effects captured by the Faraday and Kerr rotations, and those see a
dependence on the optical conductivity of a material—which can be a proxy for many
interesting effects such as AQHE and resonant structure of impurities or excitons,
as Chapter 2 discusses. For thin film TIs at finite frequencies, the Kerr and Faraday
rotations differ from their low frequency, universal behavior due to other material
phenomena. We added to this picture of finite frequency optical experiments on
thin film TIs, captured in Fig. 2.1, by including the resonant behavior of localized
impurity states.
The quantum electromagnetic field has virtual vacuum fluctuations of pho-
125
tons that cause a force between materials. Using one object as a probe, one can
look for interesting material phenomenon like those considered in Chapter 3. The
band structure can experience a topological phase transition known as a Lifshitz
transition, causing a “kink” in the optical conductivity that translates directly to
a “kink” in the Casimir force as we found in Fig. 3.2 which could be measured.
Additionally, we shed light on the problem in the Casimir community of how to
model the low frequency behavior of a material: with a diffusive model (Drude)
or a ballistic model (Plasma). In this vein, we proposed an experiment to use an
effect, namely weak localization, to see a qualitative change in the Casimir effect
if and only if the electronic model the Casimir effect is accessing is diffusive (as it
should be in the metals under consideration). The results for how a diffusive model
changes the Casimir effect are captured in Fig. 3.7, where magnetic field tunes this
weak localization effect. Instead of the usual numerical differences in models, this
would get at the effect in a profound way. Theoretically, we also checked to see if
the approximation of disorder averaging our linear response instead of the Casimir
energy itself was accurate, and found that in the parameter ranges usually under
consideration, it is fine provided the plates are not closer to each other than roughly
c/vF × `MF where `MF is the mean free path of electrons in the disordered material
(a photon should not travel between the plates many times before an electron done
scattering).
In search of materials that could experience a Casimir repulsion, we turn
to Weyl semimetals in Chapter 4 and their peculiar electromagnetic properties.
Beginning with idealized case of semi-infinite and thick Weyl plates characterized
126
solely by a low-energy axionic action Eq. (4.1), we find that a region of repulsion
that can be increased for smaller thicknesses (see Fig. 4.4). We then determine how
the actual low-energy band structure could modify these results by calculating the
optical conductivity in Eq. (4.53) and then modifying parameters to see how the
Casimir effect changes. Repulsion is replaced with attraction at shorter distances
scales, but survives at larger scaled, see Figs. 4.6 and 4.5. Even the inclusion of a
technically infinite DC conductivity by means of a finite chemical potential does not
entirely destroy this intermediate repulsive region as Fig. 4.7 demonstrates.
Finally, in Chapter 5 we discussed how a particular exact method for calcu-
lating the path integral exposes a breakdown that does not fit into the usual known
schemes that correct for such problems. The problem seems to be connected to
continuity of paths, since calculational methods used prior to this assumption seem
to produce correct results. Curiously, if the Hamiltonian used to generate the path
integral is linear in the generators of the Lie algebra with which the coherent states
are formed, the procedure goes through smoothly and gives the correct result.
The electromagnetic force—the fundamental force responsible for nearly all
condensed matter phenomenon—and topology are important to probing, classify-
ing, and discovering new materials. The realization of topology in band structure
gives interesting novel physical phenomena, and the use of direct electromagnetic
phenomena in optics and Casimir experiments, as we have considered here, can very
powerfully probe material properties resulting from this topology.
127
Appendix A: Optical reflection from a suspended thin film
In this appendix we consider the reflection and transmission of an electromag-
netic wave incident on a thin-film material characterized by conductance σµν(ω).
Similar calculations have been previously done [109]; we provide this here for completeness—
especially when connecting the classical electrodynamcs to the RPA from quantum
field theory in Eqs. (1.9) and Eqs. (1.10).
To begin say we have a thin film at z = 0 characterized by σµν(ω) such that
J(x, ω) = σ(ω)·E(x, ω) δ(z). Taking continuity into account, if we Fourier transform
in position space, we have, by the continuity equation (defining k‖ = (kx, ky))
ρ = k‖ω · σ · E. (A.1)
Consider an incident plane wave in z < 0: E0eik·x−iωt, the reflected wave
Ereikr·x−iωt with kr = (kx, ky,−kz), and transmitted wave Eteik·x−iωt. Then, the
relevant Maxwell equations are
∇ · E = ρ, (A.2)
∇×B = 1c
∂E
∂t +
1
cσ · E δ(z). (A.3)
∇× E = −1c∂tB (A.4)
By ∇ × E = −1c∂tB, we know that the parallel component of E is parallel across
128
the film, or
zˆ× (E0 + Er − Et) = 0. (A.5)
Now, we can take Eq. (A.3), and integrate around a closed contour illustrated in to
get
e1 · (B0 + Br −Bt) =
1
c (e1 × zˆ) · σ · E
=
1
ce1 · (zˆ× σ · E).
This is true for any e1 perpendicular to zˆ, so we can just write our equations to be
solved as
zˆ× (k× E0 + kr × Er − k× Et) = −
ω
c σ · Et, (A.6)
zˆ× (E0 + Er − Et) = 0. (A.7)
If we satisfy these, then Eq. (A.2) will be automatically satisfied across the bound-
ary. At this point, we specify the components: transverse electric and transverse
magnetic (depending on whether the electric or magnetic field, respectively, is per-
pendicular to the plane of incidence).
Without loss of generality, assume yˆ is perpendicular to the plane of incidence
so that yˆ · k = 0, E0 = ETM0 k× yˆ/k +ETE0 yˆ (similarly for Et), and ETMr k× yˆ/k −
ETEr yˆ.
Solving these linear equations is quite simple. Remembering our designation of
the x-direction in the plane of incidence, and y-direction perpendicular (transverse),
129
we have
σ =




σxx σxy
σyx σyy



 (A.8)
in general. This leads to the reflection matrix that connects




ETMr
ETEr



 = R(ω,k)




ETMr
ETEr



 , (A.9)
and the transmission matrix




ETMt
ETEt



 = T (ω,k)




ETMr
ETEr



 . (A.10)
Appropriately defined, the resulting matrices are
R(ω,k) = 1D




ckz
ω
σxx
2c + det
σ
2c −
σxy
2c
−σyx2c ωckz
σyy
2c + det
σ
2c



 , (A.11)
T (ω,k) = 1D




1 + ωckz
σyy
2c
σxy
2c
σyx
2c 1 +
ckz
ω
σxx
2c



 , (A.12)
where
D = 1 + ckzω
σxx
2c +
ω
ckz
σyy
2c + det
σ
2c. (A.13)
These coefficients can be used for optical (Faraday and Kerr effects) and Casimir
calculations (with the Lifshitz formula Eq. (1.8)).
130
Appendix B: Long distance behavior of the Casimir force
For the thin film geometry, we can easily analyze the long distance behavior
of the Casimir force. To see this we look at Eq. (A.11). Rotating to imaginary
frequencies ω → iω and kz → iqz leaves things relatively unchanged. Now, we
expect ω ∼ O(cqz) and photons between the plates will have qz ∼ c/a. Thus, large
distances correspond to small frenquency/wavenumber (as compared to the length
scales inherent in σµν). To see this, look at the Lifshitz formula Eq. (1.8), rewritten
by changing variables to q2z = ω2 + k2.
Ec(a) = ~
∫ ∞
0
qz dqz
(2pi)2
∫ cqz
0
dω
2pi tr log[I−R
2(ω, qz)e−2qza]. (B.1)
If we change variables to x = ω/cqz and u = 2qza, then we obtain
Ec(a) =
~c
(2a)3
∫ ∞
0
u2 du
(2pi)2
∫ 1
0
dx
2pi tr log
[I−R2 ( cux2a , u2a
)
e−u
]
. (B.2)
The exponential keeps u ∼ O(1) while clearly x ∼ O(1). Thus, when a is large
compared to the length scales in R(ω, qz), we can expand R2(ω, qz) in this expression
and clearly ω = qz = 0 is the very far limit (a→∞) of this expression.
Thus, we consider two limiting cases that are relevant to the text. To simplify
things we take the case σxx = σyy and σxy = −σyx.
131
Case 1 (Diamagnetic response):
σxx
c =
ωp
ω +O(ω
0), σxyc = O(ω
0). (B.3)
A pure system has a diamagnetic response which in a dirty system translates to
a Drude peak in the optics and finite conductivity. This response has this 1/ω
behavior which for photons that go as c/a, will be quite large. The origin of this
might be understood with a free electron system where (at imaginary frequencies)
σbulkxx =
ne2
m
1
ω , (B.4)
where n is the three-dimensional density of electrons, m is their mass, and e is
their charge. As a thin film, we might just naively integrate this over the bulk for
simplicity to obtain
σxx
c =
ne2d
mc
1
ω . (B.5)
Thus, ωp = ne
2d
mc . As an aside, this takes the form ωp = 2ndλvfα where λ is the
de-Broglie wavelength of the electrons, vf their Fermi velocity, and α is the fine-
structure constant.
At large distances (a c/ωp), this will dominate the reflection matrix leaving
us solely with
R(ω, qz) =
1
cqz
2ω
ωp
ω +
ω
2ckz
ωp
ω +
ω2p
4ω2




ckz
2ω
ωp
ω +
1
4
(ωp
ω
)2
0
0 ω2ckz
ωp
ω +
1
4
(ωp
ω
)2



 (B.6)
At low frequencies, this has the expansion
R(ω, qz) ∼ I− 2 ωωp




ω
cqz 0
0 cqzω



 . (B.7)
132
The relevant expression for the Lifshitz formula is then
R2
(
cux
2a ,
u
2a
)
∼ I− 2 cuaωp




x2 0
0 1



 . (B.8)
The rest is just expansion of the Lifshitz integrand
tr log
[I−R2 ( cux2a , u2a
)
e−u
]
∼ 2 log
[
1− e−u
]
− 2 cuaωp
x2 + 1
1− eu . (B.9)
Integrating over x leaves us with
∫ 1
0
dx tr log
[I−R2 ( cux2a , u2a
)
e−u
]
∼ 2 log
[
1− e−u
]
− 2 cuaωp
4/3
1− eu . (B.10)
As can be easily verified, we know that
∫ ∞
0
du u2 log[1− e−u] = 1
3
∫ ∞
0
du u
3
1− eu = −
pi4
45
. (B.11)
Thus, we can immediately write down the Casimir energy
Ec = −
~cpi
1440a3
(
1− 4caωp
+ · · ·
)
. (B.12)
Recall that E0 = − ~cpi1440a3 is the Casimir energy for ideal conductors. And for all
relevant scales, this energy represents an attractive force.
Case 2 (Hall response):
σxx(ω) = o(ω0), σxy(ω) = σxy + o(ω0). (B.13)
In this case, we have assumed σxx(ω = 0) = 0, and as a result, at large distances
the reflection matrix takes the simple form
R( cux2a , u2a)→
1
1 +
σ2xy
4c2




σ2xy
4c2 −
σxy
2c
σxy
2c
σ2xy
4c2



 , as a→∞. (B.14)
133
The integral involving this matrix is done in a straightforward manner using the
integral identity
Li4(λ) = −
1
2
∫ ∞
0
du u2 log[1− λe−u], (B.15)
we then just need the eigenvalues of R, which are σxy/(σxy ± 2ic). Thus,
Ec(a) = E0
45
pi4 Re
{
Li4
[( σxy
σxy + 2ic
)2
]}
, (B.16)
where E0(a) = − ~cpi1440a3 as we have defined previously. This behavior is the long-
distance behavior as we have described the conductivities. If we further assume
σxy/c 1, we get
Ec(a) ∼ −E0
45
4pi4
(σxy
c
)2
. (B.17)
Notice that this is a repulsive force (not because it is positive, but because it has a
negative slope). It remains repulsive until roughly σxy/2c ≈ 1. The force plot can
be seen in Fig. 4.3.
134
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