Casimir Effect between Chern Insulators
Wang-Kong Tse
We %develop a theory for investigate the Casimir effect between two-dimensional electron systems driven to the quantum Hall regime by a strong perpendicular magnetic field. In the large separation ($d$) limit where retardation effects are essential we find i) that the Casimir force is quantized in units of ${3\hbar c}\alpha^2/{8\pi^2}{d^4}$, and ii) that the force is repulsive for mirrors with same type of carrier, and attractive for mirrors with opposite types of carrier. The sign of the Casimir force is therefore electrically tunable in ambipolar materials like graphene. The Casimir force is suppressed when one mirror is a charge-neutral graphene system in a filling factor $\nu=0$ quantum Hall state.
Casimir_CondMat.tex
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\begin{document}

%\twocolumn[
\hsize\textwidth\columnwidth\hsize\csname@twocolumnfalse\endcsname
%\title{Casimir Effect between Chern Insulators}
%\title{Null Casimir Effect between Graphene Mirror in $\nu = 0$
%  Quantum Hall State and Metals}
\title{Quantized Casimir Force} 
%Repulsive and Null Casimir Effect from Two-Dimensional Mirrors 
%  in the Quantum Hall Regime}
%Long-range Repulsive Casimir Effect between Quantum Hall Mirrors} %Liquids}
%\title{Casmir Effect between Quantum Hall Liquids}
%Two-Dimensional Electron Systems in
%  Strong Magnetic Fields}
%\title{Casmir Effect between Two-Dimensional Chern Insulators}
%in the Quantum Hall regime}

\author{Wang-Kong Tse}
\author{A.~H. MacDonald}
\affiliation{Department of Physics, University of Texas, Austin, Texas 78712, USA}

\begin{abstract}
We %develop a theory for 
investigate the Casimir effect between two-dimensional electron 
systems driven to the quantum Hall regime by a strong perpendicular magnetic 
field. In the large separation ($d$) limit where retardation effects are essential
we find i) that the Casimir force is quantized in units of 
${3\hbar c}\alpha^2/{8\pi^2}{d^4}$, and ii) that the 
force is repulsive for mirrors with same type of carrier, 
and attractive for mirrors with opposite types of carrier. 
The sign of the Casimir force is therefore electrically tunable in ambipolar 
materials like graphene.  The Casimir force is 
suppressed when one mirror is a charge-neutral graphene system 
in a filling factor $\nu=0$ quantum Hall state. 
\end{abstract}
%
\pacs{12.20.-m,05.40.-a,78.20.Ls,73.43.-f}
%
\maketitle

\newpage

%\section{Content}

%1. Abstract
%2. Introduction and Motivation, Summary of results
%3. Casimir Energy from Microscopic Consideration of the Correlation Energy
%4. Numerical Results
%5. Discussion
%6. Supplementary Info (Conductivity, Reflection Coefficients)

%\section{Introduction}

\textit{Introduction ---} 
The Casimir effect is an intriguing quantum electrodynamic phenomenon
in which the vacuum energy of the electromagnetic field 
is altered by the presence of two closely-spaced mirrors coupled to
the field,
%interactions between the field and the matter in two
%closely spaced mirrors,
%The Casimir effect is an intriguing quantum electrodynamic phenomenon in
%which the vacuum energy of the electromagnetic field
%is altered by interactions between light and the matter in two
%closely spaced mirrors, 
resulting in a measurable force between 
them. The force between two parallel metal plates, for 
example, is attractive because the plates 
restrict the number of vacuum electromagnetic modes present in the 
space between them. 
%thus creating an inward radiation pressure. 
%is attractive because the plates are optically reflective and 
%restrict the number of 
%vacuum electromagnetic modes present in the space between them. 
%, which therefore reduces the electromagnetic energy density inside compared to outside the
%metal plates cavity.  
%There has been a continuing interest in this subject Continuing 
Growing interest in this facscinating topic has been driven by improving
Casimir force measurement capabilities
%using atomic force microscope \cite{Mohi_exp1} and micromachined oscillator \cite{Decca_exp1} 
at small separations from $1\,\mu$m down to about $10$ nm
\cite{%Lamor_exp1,
Mohi_exp1,Eder_exp1,Capasso_exp2,Decca_exp1} and by new materials.
%In addition to conventional dielectric materials and metals, using semiconductors such as Si as mirrors points to the interesting possibility of 
%controlling the Casimir force by modulating the carrier density \cite{Casimir_RMP}. 
When semiconductors like Si are used as mirrors 
instead of conventional dielectric materials and metals, for example,  
it is possible to control the Casimir force by 
optical or electrical carrier density modulation \cite{Casimir_RMP}.
This property, combined with recent advances in micro-electromechanical systems (MEMS) and
nano-electromechanical systems (NEMS), could enable new
electromechanical applications. 

At submicron distances, the Casimir
force between materials usually dominates the gravitational force. 
Measurement of the Casimir force
has therefore played an important role in the search for the  
%also leads to constraints of the magnitude of 
non-Newtonian gravitational forces suggested by a number of
unification theories \cite{unify}.
Because the Newtonian gravitational law has been poorly tested below $10\,\mu\mathrm{m}$, 
an accurate quantitative understanding of the Casimir effect is key to
%instrumental to 
the identification of any new gravitational force law which might  
prevail at small length scales.
Theories of the Casimir effect have in fact  
been employed \cite{gravityprop,gravityexp}
in establishing constraints on the magnitude of non-Newtonian 
gravitational effects.  In these studies measured force values are
compared with known contributions from Newtonian gravity, 
and Casimir forces computed using a separate set of
measurements of the detailed dielectric properties of 
the test materials \cite{Advances}. The Casimir force can be sensitive to 
unintended variations in impurity density, degree of surface roughness, and
sample thickness.
%, and modeling such as the effect from surface roughness. 
%experimental determination of the detailed dielectric properties of 
%modeling of the optical properties of 
%the materials that make up the mirrors. 
The errors associated with Casimir force estimation are therefore difficult to control 
and this in turn crucially 
%sensitively 
limits the precision of bounds placed on hypothetical 
gravitational forces. 
%This in turn depends crucially on a seperate experimental determination of the detailed dielectric properties of 
%modeling of the optical properties of 
%the materials that make up the mirrors, which sensitively affect 
%the precision of the resulting bounds of the hypothetical gravitational force. 
A way to suppress or even neutralize 
%eliminate 
the Casimir effect at small length 
scales is therefore desirable, enabling a more sensitive direct measurement of gravity. 

In this Letter, we address one possibility for suppressing 
%nullifying 
the Casimir effect %The key is to consider the Casimir effect between 
by developing
a theory of the Casimir force between two-dimensional (2D) electron
systems in the presence of strong perpendicular
magnetic fields. The development of quantized Landau levels and the
associated quantum Hall effect opens up a new regime for investigations of the 
Casimir effect. We show that in ambipolar materials 
like graphene the Casimir force
is electrically tunable between attractive and repulsive values, and that 
in the large-separation relativistic
regime, the Casimir force is quantized. 
Importantly, a %zero 
strongly suppressed Casimir force can be achieved under a high magnetic field by using charge-neutral
graphene sheets as mirrors. 

%
\textit{Theory ---} When spatial dispersion in the mirrors is negligible,
%electronic systems forming the 
the Casimir effect is determined by their local 
(\textit{i.e.}, %in-plane momentum 
$q = 0$) 
charge and current response
functions. 
In this limit, the Casimir energy (per unit area) can be elegantly expressed
in terms of the reflection coefficients of the mirrors \cite{Reynaud} %\cite{Lif_JETP}, 
%It is then convenient to express the Casimir energy in terms 
%of the reflection coefficients of the mirrors \cite{Lif_JETP}, 
%
\begin{eqnarray}
E &=&
\frac{\hbar}{4\pi^2}\int_0^{\infty}\mathrm{d}q_{\perp}q_{\perp}\int_0^{q_{\perp}c}\mathrm{d}\omega
\nonumber \\
&&\mathrm{tr}\,\mathrm{ln}
\left[\mathbb{I}-r'^{\mathrm{L}}(iq_{\perp},i\omega)r^{\mathrm{R}}(iq_{\perp},i\omega)\right], \label{Lif}
%E = \int\frac{\mathrm{d}\omega}{2\pi}\sum_q\mathrm{tr}\,\mathrm{ln}\left[\mathbb{I}-R'^{\mathrm{L}}(\omega)R^{\mathrm{R}}(\omega)\right], \label{Lif}
\end{eqnarray}
%
where $r'^{\mathrm{L}}, r^{\mathrm{R}}$ are the reflection 
coefficients \textit{into} the cavity of the left (L) and right
(R) mirrors, $q_{\perp} = \sqrt{(\omega/c)^2-q^2}$ is the component
of photon momentum perpendicular to the mirror planes, and 
$q$ the in-plane momentum component.  

We apply Eq.~(\ref{Lif}) to mirrors made of ultrathin films 
that can be adequately modeled as a quasi-2D layer; this 
class of systems includes atomically thin materials like 
graphene and bilayer graphene \cite{GraphRev}. Several groups have
studied the Casimir force between graphene sheets in the 
absence of an external
magnetic field \cite{graphene_Casimir}.
%The Casimir effect between graphene sheets without external magnetic
%field was studied in Refs.~
%
When a field is present the optical characteristics of a 2D mirror depend on its
longitudinal $\sigma_{xx}$ and Hall $\sigma_{xy}$
conductivities.  For general 
angle of incidence $\theta$, straightforward calculations yield \cite{TI_Mag} the 
following expressions for the tensor components of the reflection coefficients of a 2D %a 2D 
mirror in a plane with fixed $\hat{z}$ direction coordinate $z$:
%
\begin{eqnarray}
r_{xx} &=& -2\pi e^{i2q_{\perp}%q_0\lambda
z}\left\{\bar{\sigma}_{xx}(\omega)/\lambda+2\pi\left[\bar{\sigma}_{xx}^2 (\omega)+\bar{\sigma}_{xy}^2 (\omega)\right]\right\}/{R},\nonumber \\
r_{xy} &=& -r_{yx} = -2\pi e^{i2q_{\perp} z}\bar{\sigma}_{xy}(\omega)/R, \nonumber \\
r_{yy} &=& -2\pi e^{i2q_{\perp}
z}\left\{\bar{\sigma}_{xx}(\omega)\lambda+2\pi\left[\bar{\sigma}_{xx}^2(\omega)+\bar{\sigma}_{xy}^2(\omega)\right]\right\}/{R},
\nonumber \\
\label{SLRefl}
\end{eqnarray}
%
%$R_{xx} = -2\pi\mathrm{exp}(i2k\lambda z)[\sigma_{xx}/\lambda+2\pi(\sigma_{xx}^2+\sigma_{xy}^2)]$,
%$R_{xy} = -R_{yx} = -2\pi\mathrm{exp}(i2k\lambda z)\sigma_{xy}$, and
%$R_{yy} = -2\pi\mathrm{exp}(i2k\lambda
%z)[\sigma_{xx}\lambda+2\pi(\sigma_{xx}^2+\sigma_{xy}^2)]$,
where $\lambda = \mathrm{cos}\theta = q_{\perp}/(\omega/c)$, ${R} =
1+2\pi\bar{\sigma}_{xx}(\lambda+1/\lambda)+4\pi^2[\bar{\sigma}_{xx}^2+\bar{\sigma}_{xy}^2]$, 
and
$\bar{\sigma}_{xx,xy} = \sigma_{xx,xy}/c$ are \textit{dimensionless}
%longitudinal and transverse optical 
optical conductivities normalized by $c$. The expression for $r'$ differs %only 
by the replacement 
$\mathrm{exp}(i2q_{\perp} z) \to \mathrm{exp}(-i2q_{\perp} z)$. 

We now apply a strong magnetic field normal to two parallel 2D mirrors 
that are separated by a distance $d$. 
%Between mirrors separated by a distance $d$, 
When $d/c$ is larger than  
characteristic electronic time scales the  
retardation effects captured by Eq.~\ref{Lif} are essential.  
In the quantum Hall regime, the large $d$ limit holds for $d/c$ much larger than 
$\mathrm{max}(\hbar/\Delta,\hbar/\Gamma)$ where $\Delta$ describes 
an inter-Landau level transition energy and $\Gamma$ a disorder broadening energy.  
% Allan: Actually does the above apply at all filling factors - including quantum Hall `risers';
%maybe we don't want to say any more than we need to here.
%
Because $r'^{\mathrm{L}}r^{\mathrm{R}} \propto \mathrm{exp}(-2q_{\perp}d)$ in 
Eq.~(\ref{Lif}), electromagnetic correlations between mirrors
separated by large distances are carried by long-wavelength virtual
photons; the integral in Eq.~(\ref{Lif}) is therefore  
dominated by low-frequency contributions. The value of 
the Casimir energy is thus determined at large separations by the static longitudinal
and Hall conductivities
$\sigma_{xx,xy}(\omega = 0)$.
%approximated by the static response 
%using static values in the conductivities. 
If the applied magnetic field is strong enough that the 
%Under strong magnetic fields when 
mirrors sit 
%are sitting 
on well-formed quantum Hall plateaus,
%the electronic band structure of the 2D mirrors is composed of Landau levels, 
%form quantum Hall states, 
then the low-frequency longitudinal conductivity vanishes 
$\bar{\sigma}_{xx}^{\mathrm{L,R}} \simeq 0$ and the 
Hall conductivity $\bar{\sigma}_{xy}^{\mathrm{L,R}} \simeq
\nu_{\mathrm{L,R}}\alpha/2\pi$, where $\alpha = 1/137$ is the fine
structure constant and $\nu_{\mathrm{L,R}} = 2\pi \hbar
n_{\mathrm{L,R}}/(eB)$ are the Landau level filling factors of the left and 
right mirrors with carrier density $n_{\mathrm{L,R}}$. 
Expanding the reflection coefficients Eq.~(\ref{SLRefl}) up to leading order in 
$\alpha$ yields $\mathrm{tr}\,\mathrm{ln}[\mathbb{I}-r'^{\mathrm{L}}r^{\mathrm{R}}]
\simeq
\mathrm{ln}(1+2 e^{-2q_{\perp}d}\nu_{\mathrm{L}}\nu_{\mathrm{R}}\alpha^2)
\simeq 2 e^{-2q_{\perp}d} \nu_{\mathrm{L}}\nu_{\mathrm{R}}\alpha^2$.
It then follows from Eq.~(\ref{Lif}) that the 
%leading contribution to the 
Casimir force per unit area $F = -\partial E/\partial d$ 
between two quantum Hall mirrors is 
%
\begin{equation}
F =
\frac{3\hbar c}{8\pi^2}\alpha^2\frac{\nu_{\mathrm{L}}\nu_{\mathrm{R}}}{d^4}. \label{Main}
\end{equation}
%
Eq.~(\ref{Main}) is %the main 
a central result of this Letter.  Because the quantum Hall effect occurs 
only for discrete integer and small-denominator rational values of $\nu$,
we conclude that the large-separation asymptotic 
Casimir force is quantized in units of ${3\hbar
  c}\alpha^2/{8\pi^2}{d^4}$. Comparing with the well-known result
between perfect metals $F_0 = -\hbar c\pi^2/240 d^4$, 
one notices that
the 
%long-range 
Casimir force is suppressed by a factor $\propto \alpha^2$.
% Allan:  What we are referring to as force is actually force per area.  Should we 
% say force per area? 
%In the following, we provide a microscopic explanation of this result. 

%
\begin{figure}%[!htb]%[h]
  \includegraphics[width=8.5cm,angle=0]{Two_Bubbles_paper_4.pdf}
\caption{(Color online). Feynman diagrams for the ground state
  correlation energy between mirrors, where green wavy lines represent the photon Green
  function $\mathcal{D}$ and filled blue/dark bubbles represent the current-current
  correlation function $\Pi$. (a). Leading-order diagram for the
  longitudinal contribution $E_{{l}}$ to the ground state correlation
  energy. The transverse contribution $E_{{t}}$ is given by  
  replacing $\mathcal{D}_{{l}},\Pi_{l}\,\to\,\mathcal{D}_{{t}},\Pi_{t}$. (b) Leading-order diagram for the 
  Hall contribution $E_{\mathit{H}}$, this diagram is to be counted
  twice because an equivalent diagram results from interchanging
  $\mathcal{D}_{{l}}$ and $\mathcal{D}_{{t}}$ and 
  using $\Pi_{\mathit{H}} \equiv \Pi_{xy} = -\Pi_{yx}$. (c) Diagram for the
  current-current correlation function. The first diagram on the right 
represents the diamagnetic contribution whereas the second represents
the paramagnetic contribution.} 
\label{Fey1}
\end{figure}
%

It is informative to also understand Eq.~(\ref{Main}) from the perspective of
quantum electrodynamic perturbation theory in which the  
%a diagrammatic perturbative analysis. 
%Casimir effect, arising from mutual electromagnetic correlations 
%between two spatially separated systems, contributes to the 
%ground-state \cite{APReview,Pit,Barash} energy of the coupled system. 
Casimir effect arises from mutual electromagnetic correlations 
between two separated systems and therefore contributes to the 
ground-state \cite{APReview,Pit,Barash} energy of the coupled system. 
In the following, we adopt the axial gauge with a zero 
electromagnetic scalar potential $\phi = 0$.  The longitudinal (photon
momentum parallel to vector potential) $\mathcal{D}_{l}$ and transverse
(photon momentum perpendicular to vector potential) $\mathcal{D}_{t}$ 
interlayer photon propagators \cite{Marusic_Propag} are given as a function of mirror separation by 
%between the mirrors %are expressed as 
$\mathcal{D}_{l}(q_{\perp},\omega) = -(2\pi i c 
q_{\perp}/\omega^2)\mathrm{exp}(iq_{\perp}d)$ and $\mathcal{D}_{t}(q_{\perp},\omega) = -[2\pi i
/(q_{\perp}c)]\mathrm{exp}(iq_{\perp}d)$. In the absence of a magnetic
field, the longitudinal contribution to the interaction energy is given to leading order in $\alpha$ by 
the diagram in Fig.~\ref{Fey1}a:
%in the perturbative expansion in $1/c$ 
%
% Allan:  Note that I've changed the discussion of longitudinal and transverse.
% We can discuss if there's something you don't agree with.  The change in 
% terminology requires a change in the figures.
%
\begin{equation}
E_{{l}} =
-\hbar\int\frac{\mathrm{d}\omega}{2\pi}\sum_q\Pi_{l}^{\mathrm{L}}\mathcal{D}_{l}\Pi_{l}^{\mathrm{R}}\mathcal{D}_{l}, \label{El} 
\end{equation}
%
where $\Pi_{l}^{\mathrm{L,R}}$ is the longitudinal component of the
current-current correlation tensor of the mirrors. 
%is determined only by the longitudinal current-current correlation
%function.  
%$\mathcal{D}_{xx}(q_{\perp},\omega) = -(2\pi i c q_{\perp}/\omega^2)\mathrm{exp}(iq_{\perp}d)$ is the longitudinal
%photon Green function \cite{Marusic_Propag}, where we have chosen $x$
%as the longitudinal direction. 
%(The $\Pi_{t}^{\mathrm{L}}\mathcal{D}_{t}\Pi_{t}^{\mathrm{R}}\mathcal{D}_{t}$
%contribution is higher order in $\alpha$.) 
By virtue of the continuity equation for charge density,
%charge conservation expressed in the continuity equation, 
the longitudinal current-current correlation function is related to the 
density-density correlation function $\chi$ by $\Pi_{l}(q,\omega) = 
(e\omega/q)^2\chi(q,\omega)$.
Eq.~(\ref{El}) therefore describes the
leading-order contribution to the Casimir effect due to fluctuations
of charge density. 
%There has been growing interest \cite{Casimir_RMP} in the Casimir 
%effect when one of the mirrors is a semiconductor, where the charge carriers can be
%electrons or holes. 
%We point out that 
Because $\Pi_{l} \propto \sigma_{xx}$ is independent of the sign of 
charge carriers, the Casimir force between two
parallel-plate mirrors placed in vacuum is attractive 
for like or unlike carrier densites. The transverse interaction energy
$E_{t}$ gives a comparable contribution to Eq.~(\ref{El}) with integrand
$\Pi_{t}^{\mathrm{L}}\mathcal{D}_{t}\Pi_{t}^{\mathrm{R}}\mathcal{D}_{t}$
where  $\Pi_{t}^{\mathrm{L,R}}$ is the transverse component of the
current-current correlation tensor.
% Similarly, the transverse
% interaction energy is given by $E_{{t}} = 
% -\hbar\int({\mathrm{d}\omega}/{2\pi})\sum_q\Pi_{t}^{\mathrm{L}}\mathcal{D}_{t}\Pi_{t}^{\mathrm{R}}\mathcal{D}_{t}$
% with $\Pi_{t}^{\mathrm{L,R}}$ the transverse component of the
% current-current correlation tensor. 

% When spatial dispersion can be
% ignored 
% The large separation limit of 
% Eq.~(\ref{El}) implies that $\Pi_{t}^{\mathrm{L,R}} \approx
% \Pi_{l}^{\mathrm{L,R}}$. 

When a magnetic field is present, the Casimir 
% Allan: Are we using longitudinal and transverse in the same sense all the time.
% Let's discuss the best terminology.   
energy acquires a contribution that depends on the Hall current-current correlation function 
$\Pi_{\mathit{H}}(q,\omega)$, for which the leading-order diagram (Fig.~\ref{Fey1}b) yields
% 
\begin{equation}
E_{\mathit{H}} = 
 2\hbar\int\frac{\mathrm{d}\omega}{2\pi}\sum_q\Pi_{\mathit{H}}^{\mathrm{L}}\mathcal{D}_{t}\Pi_{\mathit{H}}^{\mathrm{R}}\mathcal{D}_{l}. \label{Et}
\end{equation} 
%
Expressed in terms of the Hall conductivity, $\Pi_{\mathit{H}} = -i\omega\sigma_{xy}$ in
Eq.~(\ref{Et}).  Performing the complex plane rotations  \cite{Reynaud} $q_{\perp}
\to iq_{\perp}$ and $\omega \to i\omega$, we therefore obtain  
%Eq.~(\ref{Et}) as 
%
\begin{equation}
E_{\mathit{H}} =
\frac{2\hbar}{c^2}\int_0^{\infty}\mathrm{d}q_{\perp}q_{\perp}\int_0^{q_{\perp}c}{\mathrm{d}\omega}\sigma_{xy}^{\mathrm{L}}(iq_{\perp},i\omega)\sigma_{xy}^{\mathrm{R}}(iq_{\perp},i\omega)^{-2q_{\perp}d}, \label{Etcond}
\end{equation}
%
showing that the Hall contribution $E_{\mathit{H}}$ depends on the product of the Hall conductivities of 
the two mirrors, and is therefore sensitive to the signs of their
charge carriers. To leading order in $\alpha$,
%where $\alpha = 1/137$ is the fine structure constant, 
%in the RPA expansion of Figs.~\ref{Fey1}-\ref{Fey2}, 
the Casimir energy in a magnetic field is given by the sum of these
three contributions $E = E_{{l}}+E_{{t}}+E_{\mathit{H}}$.  

When the magnetic field is strong, Landau quantization of the electronic energy
spectrum of the mirror leads to energy gaps $\Delta$. 
%Electromagnetic responses of the mirrors are dictated by \textit{local}
%current-current correlation functions for separations $d$ exceeding the 
%the magnetic length $l_B = \sqrt{\hbar/eB}$ ($l_B \lesssim 10^{-8}\,\rm{m}$ for $B \simeq 10\,\rm{T}$). 
%Current responses of the mirrors are predominantly local for separations $d$ exceeding the 
%electronic mean free path.
%, determined by the magnetic length $l_B =
%\sqrt{\hbar/eB} \lesssim 10^{-8}\,\rm{m}$ for $B \simeq 10\,\rm{T}$. 
We confine our discussion to
low temperatures $k_{\mathrm{B}}T \ll \hbar c/d \ll \Delta$ at which the quantum
Hall effect is established and thermal contributions to the Casimir
force can be ignored. 
% Because the mirror separation is typically much larger than the 
% intra-mirror electronic mean free path, which is determined by the magnetic
% length $l_B = \sqrt{\hbar/eB}$, the mirrors are dictated by local
% current-current responses. 
In this limit density fluctuations and hence the longitudinal and 
transverse contributions to the Casimir energy
%, which are proportional to the diagonal components of the
%conductivity tensors, 
are suppressed because the 2D quantum Hall
systems are incompressible \cite{Note1}.
Hence, Hall-like quantum fluctuations of the current dominates. 
%The Hall-like quantum fluctuations of the current 
%then dominate. 
%suppressed. 
Evaluating the Hall energy in Eq.~(\ref{Etcond}), 
%alone 
we can rederive %easily 
%one recovers
the Casimir force result Eq.~(\ref{Main}) 
obtained earlier from macroscopic electrodynamic considerations. 
%
\begin{figure}%[!htb]%[h]
  \includegraphics[width=8.7cm,angle=0]{Cmr_Force_Graphene_SL_n12_pm3e15_nu12_5em1_15em1_1.pdf}
\caption{(Color online). Casimir force $F$ normalized by the perfect 
  metal value $F_0 = -\hbar c \pi^2/240 d^4$ as a function of
  separation $d$ in a strong magnetic field. Solid lines correspond to numerical results and
  dashed lines to the analytic large $d$ limits in Eq.~(\ref{Main}). The case with
  opposite-sign carrier densities on the two graphene sheets $n_{\mathrm{L}} =
  3\times10^{11}\,\mathrm{cm}^{-2} = -n_{\mathrm{R}}$ is depicted by
  the red (grey) curve whereas the case with the same-sign carrier density 
  $n_{\mathrm{L}}=n_{\mathrm{R}}=3\times10^{11}\,\mathrm{cm}^{-2}$ is depicted by the black (dark)
  curve.  At these carrier densities, the magnetic fields
  $B = 25, 8.3\,\mathrm{T}$ correspond to filling factors $\vert \nu\vert = 2, 6$.
  These calculations are valid for $d \gg l_{B}$ where 
$l_{B} = 257/\sqrt{B}\,{\AA}$ 
%$l_{B} \sim  10^{-8}{\rm m}$ 
is the magnetic length
  and weak disorder.} 
%For graphene, $ v = 10^6\,\mathrm{ms}^{-1}$ and $\varepsilon_{\mathrm{c}} 
%\simeq 2.7\,\mathrm{eV}$.} 
\label{Plot1}
\end{figure}
%

\textit{Repulsive Casimir Effect ---} Repulsive Casimir forces are unusual and have been 
realized for the first time in recent experiments 
involving test bodies immersed in a liquid
medium \cite{Capasso_exp1}. 
An important implication of Eq.~(\ref{Main}) is that repulsive
Casimir forces can be easily obtained in ambipolar 2D quantum 
Hall systems by gating them to like-sign %opposite  
carrier 
densities.  
We demonstrate this repulsive Casimir effect by considering the 
case of two 
graphene sheets, numerically evaluating the Casimir
force $F = -\partial 
E/\partial d$ 
%the Lifshitz formula 
from the Casimir energy $E$ given by Eqs.~(\ref{Lif})-(\ref{SLRefl}).
%The low-energy Hamiltonian of graphene is given 
%by $H = \hbar v\bm{\sigma}\cdot\bm{k}$, where $v =
%10^6\,\mathrm{ms}^{-1}$ is the electron's band velocity, and 
We obtain the graphene sheet's optical conductivity tensor 
from the Kubo formula \cite{Suppl}. 
%
\begin{figure}%[!htb]%[h]
  \includegraphics[width=8.7cm,angle=0]{Cmr_Force_SLGraphene_Slab_SLGraphene_Combined_n12_0_nu12_0_B_10_20_30.pdf}
\caption{(Color online). Casimir force versus separation for magnetic field strengths $B = 10,
  20, 30\,\mathrm{T}$ between (a) two charge-neutral graphene sheets;
  (b) a charge-neutral graphene sheet and a half-space composed of an
  insulator with dielectric constant $\epsilon$. In the large $d$ regime where the 
  suppression effect becomes prevalent, the insulator is characterized
  by its low-frequency dielectric behavior, and we take $\epsilon \simeq 4$ corresponding to
  SiO$_2$ or hexagonal BN.
%  approximately dielectric behavior is approximately frequency independent. 
% the dielectric constant $\epsilon = 4$ corresponding to
%   SiO$_2$ or hexagonal BN). In the large $d$ regime where the
%   suppression becomes prevalent, the dielectric behavior is 
%   approximately frequency independent. 
% Frequency dependence and dissipative effect do 
% not affect suppression of the Casimir force and can be easily
% incorporated with a Drude-Lorentz model. 
} 
\label{nu0}
\end{figure}
%
Fig.~\ref{Plot1} shows the calculated Casimir force 
%using Eq.~(\ref{Lif}) with Eq.~(\ref{SLRefl})
% and Eqs.~(\ref{Cond})-(\ref{function_f})
versus $d$ between (1) an electron-doped graphene sheet and a hole-doped graphene sheet 
(grey/red curve), and (2) between two electron-doped graphene sheets
(dark/black curve). The plot is normalized by the Casimir force
between two perfect metals $F_0 = -\hbar c \pi^2/240 d^3$. 
For small separations the Casimir force is attractive 
%negative 
for both cases.
%This is because for small separations the density-density fluctuations are the dominant 
%contribution to the Casimir effect. 
For separations large enough that retardation effects are important 
%to reach the relativistic large $d$ limit 
however, contributions from charge density fluctuations 
are exponentially suppressed, Hall
current fluctuations gain dominance, and repulsive forces can appear.  
%come to the fore. 
Our numerical results for both cases settle %closely 
onto the values predicted by the analytic result Eq.~(\ref{Main}) beyond a
threshold separation, $d_{\mathrm{QH}}$.
For $d > d_{\mathrm{QH}}$ we find Casimir attraction for 
opposite-sign carrier densities and repulsion for like-sign
carrier densities. 
The crossover length $d_{\mathrm{QH}}$ can be estimated 
by expanding Eq.~(\ref{Lif}) to one higher-order term in 
$1/d$ and $\alpha$ beyond the limit given by Eq.~(\ref{Main}).
%can be obtained. 
We find \cite{Suppl} that $d_{\mathrm{QH}} \gtrsim 
l_B(c/v)f(\nu_{\mathrm{L}},\nu_{\mathrm{R}})$, where $l_B =
\sqrt{\hbar/eB}$ is the magnetic length, $v = 10^6\,\mathrm{ms}^{-1}$ is the band velocity in graphene, 
and $f(\nu_{\mathrm{L}},\nu_{\mathrm{R}})$ is a monotonic increasing
function of $\nu_{\mathrm{L,R}} \propto 1/B$ of order $\mathcal{O}(1)$. 
%Note that $d_{\mathrm{QH}}$ decreases with magnetic field, 
%in agreement with Fig.~\ref{Plot1}.  
In agreement with numerical results in Fig.~\ref{Plot1}, $d_{\mathrm{QH}}$
decreases with magnetic field. 
%, suggesting that the repulsive regime can be 
%$d_{\mathrm{QH}}$ decreases with magnetic field, in agreement with Fig.~\ref{Plot1}.  
%implying that the repulsive regime can be enlarged  
%Note that $d_{\mathrm{QH}}$ decreases with magnetic field.
%increases with magnetic field.
%in agreement with Fig.~\ref{Plot1}.  

% Allan:  Let's discuss the following: 
% Cite E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics Pergamon 
% Press, Oxford, 1981, Pt. II. for 2D results? 
%\textit{Discussion ---} 
\textit{Casimir Force Quenching ---} 
We have shown that the asymptotic Casimir force between 2D quantum Hall insulators follows the same power 
law as the Casimir force for ideal thick metals but is weaker by a 
factor  of $\pi^4 \nu_{L} \nu_{R}  \alpha^2/90$ 
where $\alpha$ is the fine structure constant.   At the same time it is qualitatively stronger than 
the Casimir force between ordinary 2D insulators which falls off especially
rapidly (like $d^{-6}$) at large distances. 
% Allan- OK?  
For the special case of charge-neutral graphene in a $\nu=0$ quantum
Hall state \cite{nuzeroQH}, the leading Hall contribution Eq.~(\ref{Main}) to the Casimir force 
vanishes, 
%it is easy to show that 
and the system behaves like an ordinary 2D insulator, with the Casimir force
given asymptotically at large $d$ by $F = -9\hbar c\,{\sigma_{xx}^{\mathrm{L}}}'(0)
{\sigma_{xx}^{\mathrm{R}}}'(0)/2d^6$ where $'$ denotes a derivative with
 respect to frequency \cite{Suppl}. 
%We find that $F$ is inversely proportional to $B$ 
%$F \propto (\hbar c^3/d^6)\alpha^2(l_B/v)^2 \propto 1/B$ 
%and is smaller than the ideal metal case by $10^{-10}\,-\,10^{-8}$ for $d \sim 10\,-\,100\,\mu\mathrm{m}$.
Fig.~\ref{nu0}(a) shows the 
calculated Casimir forces 
between two parallel charge-neutral graphene sheets as a function of magnetic field strength. 
%As anticipated, 
%As anticipated, 
%Remarkably, 
The force value %reaches zero 
is substantially suppressed 
for 
large 
separations $d \gtrsim 10\,\mu\mathrm{m}$ at a typical 
field strength of $10\,\mathrm{T}$, indicating that the
relativistic-regime Casimir force is quenched by strong magnetic
fields. %In the asymptotic regime, 
In this regime, we find \cite{Suppl} that 
%
\begin{equation}
F = -\frac{9\hbar
  c^3}{4\pi^2d^6}\alpha^2\left(\frac{l_B}{v}\right)^2\left[\frac{1}{2}+h(1)\right]^2, \label{F01}
\end{equation}
%
where $h(n) = \sum_{m = n}^{\infty}{1}/{\left(\sqrt{m}+\sqrt{m+1}\right)^3}$ and $h(1)
\approx 0.25$. For $d \simeq 10-100\,\mu\mathrm{m}$, $F$ is
suppressed by $\sim 10^{-8}-10^{-10}$ compared to the ideal metal value $F_0$.
%force $F_0$ between ideal metals.
%ideal metal value $F_0$.
%$F \propto (\hbar c^3/d^6)\alpha^2(l_B/v)^2 \propto 1/B$ and $F/F_0 \sim
%10^{-8}\,-\,10^{-10}$ for $d \simeq 10\,-\,100\,\mu\mathrm{m}$. 
%is suppressed by $8$ to
%$10$ orders of magnitude compared to the ideal metal case.
%is smaller than the ideal
%metal case by $10^{-10}\,-\,10^{-8}$ 
%for $d = 10\,-\,100\,\mu\mathrm{m}$.

Because of experimental difficulties  
in maintaining precise parallelism between two plates, it is common in
Casimir force measurements to replace one of the plates by a sphere 
%employ a sphere opposite a plate 
to mimic the parallel-plates geometry at the point of closest
separation. It is therefore essential 
%imperative 
to 
%confirm 
ascertain 
%examine 
whether the quenching effect of the Casimir force 
%Casimir force quenching we find above 
also survives under this geometry. 
Fig.~\ref{nu0}(b) shows the full numerical results of the
Casimir force between a charge-neutral graphene sheet and a dielectric
half-space mimicking a large insulating sphere with radius $R \gg d$. 
%the sphere-plate configuration. 
The Casimir force 
again drops rapidly, and does so at a larger value of $d$ due to the
higher opacity of the dielectric region. The
quenching effect of the Casimir force should therefore remain observable in a 
sphere-plate geometry with a charge-neutral graphene sheet in the $\nu = 0$ quantum
Hall state. 

The strong quantum Hall effects of graphene 2D electron systems might be 
attractive for circumstances where a weak Casimir force is desirable,
such as in experimental tests of the short-distance gravitational law. 
The $\nu=0$ state in particular has the exceptionally weak Casimir 
force characteristic of 2D insulators.  The quantum Hall effect can 
also be used to gate the graphene sheet accurately to charge 
neutrality, ensuring that electrostatic forces
are absent. % leaving only gravitational forces.    

In conclusion, we find that the large-separation Casimir force
asymptote is quantized in the quantum Hall regime, and that 
in ambipolar systems it can be tuned electrically between attractive and repulsive 
values.  When one of the mirrors is 
charge neutral, strong suppression of the 
%strongly suppressed 
Casimir effect can be 
achieved. 
%Our findings could have useful implications in
%microscale or nanoscale electromechanical systems and  submillimeter gravitational force measurement. 
Our findings %indicate 
suggest that the use of two-dimensional electronically gapped
materials offer a new strategy for control of the Casimir effect. 

The authors thank Ricardo Decca for helpful comments. This work was supported by Welch Foundation grant TBF1473, NRI-SWAN, and DOE Division of Materials Sciences and Engineering grant DE-FG03-02ER45958.


\begin{thebibliography}{20}

%\bibitem{Lamor_exp1} S. K. Lamoreaux, Phys. Rev. Lett. \textbf{78}, 5 (1997).
\bibitem{Mohi_exp1} 
%U. Mohideen and A. Roy, Phys. Rev. Lett. \textbf{81}, 4549 (1998);
%A. Roy and U. Mohideen,  Phys. Rev. Lett. \textbf{82}, 4380 (1999); 
%A. Roy, C.-Y. Lin, and U. Mohideen, Phys. Rev. D \textbf{60}, 111101(R) (1999);
B. W. Harris, F. Chen, and U. Mohideen, Phys. Rev. A \textbf{62}, 052109 (2000).
\bibitem{Eder_exp1} T. Ederth, Phys. Rev. A \textbf{62}, 62104 (2000).
\bibitem{Capasso_exp2} H. B. Chan \textit{et al}., Science
  \textbf{291}, 1941 (2001). 
\bibitem{Decca_exp1} R. S. Decca \textit{et al}., Phys. Rev. D
  \textbf{75}, 077101 (2007).
\bibitem{Casimir_RMP} G. L. Klimchitskaya, U. Mohideen, and
  V. M. Mostepanenko, Rev. Mod. Phys. \textbf{81}, 1827 (2009), Int. J. Mod. Phys. B \textbf{25}, 171 (2011).
\bibitem{unify} See, for example, E. Fischbach and C. L. Talmadge, \textit{The Search for
    Non-Newtonian gravity} (Springer 1999); J. C. Long, H. W. Chan,
  and J. C. Price, Nucl. Phys. B \textbf{539} 23 (1999); J. C. Long
  and J. C. Price, C. R. Phys. \textbf{4} 337 (2003); 
E. G. Adelberger, B. R. Heckel, and A. E. Nelson,
Annu. Rev. Nucl. Part. Sci. \textbf{53} 77 (2003); R. Onofrio,
  New J. Phys. \textbf{8}, 237 (2006); I. Antoniadis \textit{et al}.,
  C. R. Physique \textbf{12}, 755 (2011).
\bibitem{gravityprop}  V. M. Mostepanenko and Y. Sokolov,
  Phys. Lett. A \textbf{125} 405 (1987); \textit{ibid} \textbf{132}
  313 (1988).
\bibitem{gravityexp} R. S. Decca \textit{et al}., Eur. Phys. J. C
  \textbf{51}, 963 (2007); V. B. Bezerra \textit{et al}., Phys. Rev. D \textbf{81}, 055003 (2010);
V. B. Bezerra \textit{et al}., Phys. Rev. D \textbf{83}, 075004
(2011); A. O. Sushkov \textit{et al}., Phys. Rev. Lett. \textbf{107}, 171101 (2011).
\bibitem{Advances} M. Bordag, G. L. Klimchitskaya, U. Mohideen, and 
 V. M. Mostepanenko, \textit{Advances in the Casimir Effect} (Oxford 2009). 
%See, \textit{e.g.}, the recent review R. Onofrio, New J. Phys. \textbf{8}, 237 (2006) and references therein. 
%\bibitem{gravityexp1} 
\bibitem{Reynaud} M. T. Jaekel and S. Reynaud, J. Phys. I (France)
  \textbf{1}, 1395 (1991).
%even if the slab is perfect insulator. 
%characterized by a real dielectric constant.
\bibitem{GraphRev} A. H. Castro Neto, F. Guinea, N. M. R. Peres, 
K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. \textbf{81}, 109
(2009).
\bibitem{graphene_Casimir} J. F. Dobson, A. White, and A. Rubio,
  Phys. Rev. Lett. \textbf{96}, 073201 (2006); 
G. G\'omez-Santos, Phys. Rev. B \textbf{80}, 245424
  (2009); D. Drosdoff and L. M. Woods, 
 \textit{ibid.} \textbf{82}, 155459 (2010); I. V. Fialkovsky,
  V. N. Marachevsky, and D. V. Vassilevich, \textit{ibid.} \textbf{84}, 035446 
  (2011); B. E. Sernelius, \textit{ibid.} \textbf{85}, 195427 (2012).
\bibitem{TI_Mag} W.-K. Tse and A. H. MacDonald, Phys. Rev. B
  \textbf{82}, 161104 (2010); Phys. Rev. B \textbf{84}, 205327 (2011).   
\bibitem{APReview} I. E. Dzyaloshinskii, E. M. Lifshitz, and
  L. P. Pitaevskii, Adv. Phys. \textbf{10}, 165 (1961).  
%\bibitem{AGD} A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,
%  \textit{Methods of Quantum Field Theory in Statistical Physics}
%  (Dover, Place, 1975). 
%\bibitem{Lif_JETP} E. M. Lifshitz, Sov. Phys. JETP \textbf{2}, 73  (1956). 
%\bibitem{GraphRev} 
\bibitem{Pit} I. E. Dzyaloshinskii and L. P. Pitaevskii,
  Sov. Phys. JETP \textbf{36}, 1282 (1959).
\bibitem{Barash} Y. S. Barash and V. L. Ginzburg,
  Sov. Phys.-Usp. \textbf{18}, 305 (1975). 
%\bibitem{PinesNoz} D. Pines and P. Nozieres, \textit{The Theory of
%    Quantum Liquids}, vol. 1, Westview Press, 1989.
%\bibitem{Advances} M. Bordag, G. L. Klimchitskaya, U. Mohideen, and
%  V. M. Mostepanenko, \textit{Advances in the Casimir Effect}, Oxford 2009. 
%\bibitem{Silvester} J. R. Silvester, Maths Gazette \textbf{84}, 460
%  (2000).  
%\bibitem{FW} A. L. Fetter and J. D. Walecka, \textit{Quantum Theory of
%    Many-Particle Systems}, McGraw-Hill 1971. 
%\bibitem{Lifshitz} E. M. Lifshitz, Sov. Phys. JETP \textbf{2}, 73
%  (1956). 
\bibitem{Marusic_Propag} V. Despoja, M. \v Sunji\' c, and L. Maru\v si\'
  c,
  Phys. Rev. B \textbf{80}, 075410 (2009). 
\bibitem{Note1} %It is worth emphasizing that 
The %absence 
suppression of the longitudinal Casimir energy contribution   
is unique to a \textit{2D} insulating system. For thick insulating slabs with 
thickness $w \gtrsim d$, Eq.~(\ref{Lif}) implies that 
the longitudinal contribution is finite even when $\sigma_{xx} \to 0$. 
\bibitem{Capasso_exp1} J. N. Munday, F. Capasso and V. A. Parsegian,
  Nature \textbf{457}, 170 (2009). 
%\bibitem{Marusic_Temp} V. Despoja, M. \v Sunji\' c, and L. Maru\v si\' c,
%  Phys. Rev. B \textbf{83}, 165421 (2011). 
%\bibitem{MarusicvdW} V. Despoja, M. Sunjic, and L. Marusic,
%  Phys. Rev. B \textbf{75}, 045422 (2007).
%\bibitem{Tomas} M. S. Toma\v s, Phys. Rev. A \textbf{51}, 2545 (1995). 
%\bibitem{QAH} W.-K. Tse, Z. Qiao, Y. Yao, A. H. MacDonald, and Q. Niu,
%  Phys. Rev. B \textbf{83}, 155447 (2011); Z. Qiao, S. A. Yang,
%  W. Feng, W.-K. Tse, J. Ding, Y. Yao, J. Wang, and Q. Niu,
%  Phys. Rev. B \textbf{82}, 161414(R) (2010). 
\bibitem{Suppl} The calculation of the optical conductivity and next
  higher-order correction to the Casimir energy are detailed in the Supplemental
  Material. 
%the derivation of $d_{\mathrm{QH}}$ are detailed in the Supplemental Material. 
% \bibitem{Decca_Review} R. Decca, V. Aksyuk, and D. Lopez,
%   \textit{Casimir force in micro and nano electro mechanical systems},
%   in \textit{Casimir Physics} (Springer 2011), p. 102.
\bibitem{nuzeroQH} J. G. Checkelsky, L. Li, and N. P. Ong,
  Phys. Rev. Lett. \textbf{100}, 206801 (2008); J. G. Checkelsky,
  L. Li, and N. P. Ong, Phys. Rev. B \textbf{79}, 115434 (2009);  
L. Zhang, J. Camacho,
  H. Cao, Y. P. Chen, M. Khodas, D. E. Kharzeev, A. M. Tsvelik,
  T. Valla, and I. A. Zaliznyak, Phys. Rev. B \textbf{80}, 241412(R)
  (2009); Y. Zhao, P. Cadden-Zimansky, F. Ghahari, and Philip Kim,
  Phys. Rev. Lett. \textbf{108}, 106804 (2012).
%\bibitem{PFA} B. Derjaguin, Kolloid Z. \textbf{69}, 155 (1934).
% \bibitem{Remark} %Noting that the suppression effect is not affected,  
% Since we wish to demonstrate the suppression effect in the large $d$
% regime, a frequency-independent dielectric constant
% We use a frequency-independent dielectric constant for
% simplicity. Frequency dependence and dissipation of the dielectric do
% not affect suppression of the Casimir force and can be easily
% incorporated with a Drude-Lorentz model. 
% The suppression effect on the Casimir force is not affected by our
% choice of a constant $\epsilon$. 
%\bibitem{TICasimirpapers} A. G. Grushin and A. Cortijo,
%  Phys. Rev. Lett. \textbf{106}, 020403 (2011); A. G. Grushin, P. Rodriguez-Lopez, and
%A. Cortijo, Phys. Rev. B \textbf{84}, 045119 (2011);
%P. Rodriguez-Lopez, Phys. Rev. B \textbf{84}, 165409 (2011); L. Chen and
%S. Wan, Phys. Rev. B \textbf{84}, 075149 (2011).  


\end{thebibliography}

\end{document}

1208.3786.extracted_bib
\begin{thebibliography}{20}

%\bibitem{Lamor_exp1} S. K. Lamoreaux, Phys. Rev. Lett. \textbf{78}, 5 (1997).
\bibitem{Mohi_exp1} 
%U. Mohideen and A. Roy, Phys. Rev. Lett. \textbf{81}, 4549 (1998);
%A. Roy and U. Mohideen,  Phys. Rev. Lett. \textbf{82}, 4380 (1999); 
%A. Roy, C.-Y. Lin, and U. Mohideen, Phys. Rev. D \textbf{60}, 111101(R) (1999);
B. W. Harris, F. Chen, and U. Mohideen, Phys. Rev. A \textbf{62}, 052109 (2000).
\bibitem{Eder_exp1} T. Ederth, Phys. Rev. A \textbf{62}, 62104 (2000).
\bibitem{Capasso_exp2} H. B. Chan \textit{et al}., Science
  \textbf{291}, 1941 (2001). 
\bibitem{Decca_exp1} R. S. Decca \textit{et al}., Phys. Rev. D
  \textbf{75}, 077101 (2007).
\bibitem{Casimir_RMP} G. L. Klimchitskaya, U. Mohideen, and
  V. M. Mostepanenko, Rev. Mod. Phys. \textbf{81}, 1827 (2009), Int. J. Mod. Phys. B \textbf{25}, 171 (2011).
\bibitem{unify} See, for example, E. Fischbach and C. L. Talmadge, \textit{The Search for
    Non-Newtonian gravity} (Springer 1999); J. C. Long, H. W. Chan,
  and J. C. Price, Nucl. Phys. B \textbf{539} 23 (1999); J. C. Long
  and J. C. Price, C. R. Phys. \textbf{4} 337 (2003); 
E. G. Adelberger, B. R. Heckel, and A. E. Nelson,
Annu. Rev. Nucl. Part. Sci. \textbf{53} 77 (2003); R. Onofrio,
  New J. Phys. \textbf{8}, 237 (2006); I. Antoniadis \textit{et al}.,
  C. R. Physique \textbf{12}, 755 (2011).
\bibitem{gravityprop}  V. M. Mostepanenko and Y. Sokolov,
  Phys. Lett. A \textbf{125} 405 (1987); \textit{ibid} \textbf{132}
  313 (1988).
\bibitem{gravityexp} R. S. Decca \textit{et al}., Eur. Phys. J. C
  \textbf{51}, 963 (2007); V. B. Bezerra \textit{et al}., Phys. Rev. D \textbf{81}, 055003 (2010);
V. B. Bezerra \textit{et al}., Phys. Rev. D \textbf{83}, 075004
(2011); A. O. Sushkov \textit{et al}., Phys. Rev. Lett. \textbf{107}, 171101 (2011).
\bibitem{Advances} M. Bordag, G. L. Klimchitskaya, U. Mohideen, and 
 V. M. Mostepanenko, \textit{Advances in the Casimir Effect} (Oxford 2009). 
%See, \textit{e.g.}, the recent review R. Onofrio, New J. Phys. \textbf{8}, 237 (2006) and references therein. 
%\bibitem{gravityexp1} 
\bibitem{Reynaud} M. T. Jaekel and S. Reynaud, J. Phys. I (France)
  \textbf{1}, 1395 (1991).
%even if the slab is perfect insulator. 
%characterized by a real dielectric constant.
\bibitem{GraphRev} A. H. Castro Neto, F. Guinea, N. M. R. Peres, 
K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. \textbf{81}, 109
(2009).
\bibitem{graphene_Casimir} J. F. Dobson, A. White, and A. Rubio,
  Phys. Rev. Lett. \textbf{96}, 073201 (2006); 
G. G\'omez-Santos, Phys. Rev. B \textbf{80}, 245424
  (2009); D. Drosdoff and L. M. Woods, 
 \textit{ibid.} \textbf{82}, 155459 (2010); I. V. Fialkovsky,
  V. N. Marachevsky, and D. V. Vassilevich, \textit{ibid.} \textbf{84}, 035446 
  (2011); B. E. Sernelius, \textit{ibid.} \textbf{85}, 195427 (2012).
\bibitem{TI_Mag} W.-K. Tse and A. H. MacDonald, Phys. Rev. B
  \textbf{82}, 161104 (2010); Phys. Rev. B \textbf{84}, 205327 (2011).   
\bibitem{APReview} I. E. Dzyaloshinskii, E. M. Lifshitz, and
  L. P. Pitaevskii, Adv. Phys. \textbf{10}, 165 (1961).  
%\bibitem{AGD} A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,
%  \textit{Methods of Quantum Field Theory in Statistical Physics}
%  (Dover, Place, 1975). 
%\bibitem{Lif_JETP} E. M. Lifshitz, Sov. Phys. JETP \textbf{2}, 73  (1956). 
%\bibitem{GraphRev} 
\bibitem{Pit} I. E. Dzyaloshinskii and L. P. Pitaevskii,
  Sov. Phys. JETP \textbf{36}, 1282 (1959).
\bibitem{Barash} Y. S. Barash and V. L. Ginzburg,
  Sov. Phys.-Usp. \textbf{18}, 305 (1975). 
%\bibitem{PinesNoz} D. Pines and P. Nozieres, \textit{The Theory of
%    Quantum Liquids}, vol. 1, Westview Press, 1989.
%\bibitem{Advances} M. Bordag, G. L. Klimchitskaya, U. Mohideen, and
%  V. M. Mostepanenko, \textit{Advances in the Casimir Effect}, Oxford 2009. 
%\bibitem{Silvester} J. R. Silvester, Maths Gazette \textbf{84}, 460
%  (2000).  
%\bibitem{FW} A. L. Fetter and J. D. Walecka, \textit{Quantum Theory of
%    Many-Particle Systems}, McGraw-Hill 1971. 
%\bibitem{Lifshitz} E. M. Lifshitz, Sov. Phys. JETP \textbf{2}, 73
%  (1956). 
\bibitem{Marusic_Propag} V. Despoja, M. \v Sunji\' c, and L. Maru\v si\'
  c,
  Phys. Rev. B \textbf{80}, 075410 (2009). 
\bibitem{Note1} %It is worth emphasizing that 
The %absence 
suppression of the longitudinal Casimir energy contribution   
is unique to a \textit{2D} insulating system. For thick insulating slabs with 
thickness $w \gtrsim d$, Eq.~(\ref{Lif}) implies that 
the longitudinal contribution is finite even when $\sigma_{xx} \to 0$. 
\bibitem{Capasso_exp1} J. N. Munday, F. Capasso and V. A. Parsegian,
  Nature \textbf{457}, 170 (2009). 
%\bibitem{Marusic_Temp} V. Despoja, M. \v Sunji\' c, and L. Maru\v si\' c,
%  Phys. Rev. B \textbf{83}, 165421 (2011). 
%\bibitem{MarusicvdW} V. Despoja, M. Sunjic, and L. Marusic,
%  Phys. Rev. B \textbf{75}, 045422 (2007).
%\bibitem{Tomas} M. S. Toma\v s, Phys. Rev. A \textbf{51}, 2545 (1995). 
%\bibitem{QAH} W.-K. Tse, Z. Qiao, Y. Yao, A. H. MacDonald, and Q. Niu,
%  Phys. Rev. B \textbf{83}, 155447 (2011); Z. Qiao, S. A. Yang,
%  W. Feng, W.-K. Tse, J. Ding, Y. Yao, J. Wang, and Q. Niu,
%  Phys. Rev. B \textbf{82}, 161414(R) (2010). 
\bibitem{Suppl} The calculation of the optical conductivity and next
  higher-order correction to the Casimir energy are detailed in the Supplemental
  Material. 
%the derivation of $d_{\mathrm{QH}}$ are detailed in the Supplemental Material. 
% \bibitem{Decca_Review} R. Decca, V. Aksyuk, and D. Lopez,
%   \textit{Casimir force in micro and nano electro mechanical systems},
%   in \textit{Casimir Physics} (Springer 2011), p. 102.
\bibitem{nuzeroQH} J. G. Checkelsky, L. Li, and N. P. Ong,
  Phys. Rev. Lett. \textbf{100}, 206801 (2008); J. G. Checkelsky,
  L. Li, and N. P. Ong, Phys. Rev. B \textbf{79}, 115434 (2009);  
L. Zhang, J. Camacho,
  H. Cao, Y. P. Chen, M. Khodas, D. E. Kharzeev, A. M. Tsvelik,
  T. Valla, and I. A. Zaliznyak, Phys. Rev. B \textbf{80}, 241412(R)
  (2009); Y. Zhao, P. Cadden-Zimansky, F. Ghahari, and Philip Kim,
  Phys. Rev. Lett. \textbf{108}, 106804 (2012).
%\bibitem{PFA} B. Derjaguin, Kolloid Z. \textbf{69}, 155 (1934).
% \bibitem{Remark} %Noting that the suppression effect is not affected,  
% Since we wish to demonstrate the suppression effect in the large $d$
% regime, a frequency-independent dielectric constant
% We use a frequency-independent dielectric constant for
% simplicity. Frequency dependence and dissipation of the dielectric do
% not affect suppression of the Casimir force and can be easily
% incorporated with a Drude-Lorentz model. 
% The suppression effect on the Casimir force is not affected by our
% choice of a constant $\epsilon$. 
%\bibitem{TICasimirpapers} A. G. Grushin and A. Cortijo,
%  Phys. Rev. Lett. \textbf{106}, 020403 (2011); A. G. Grushin, P. Rodriguez-Lopez, and
%A. Cortijo, Phys. Rev. B \textbf{84}, 045119 (2011);
%P. Rodriguez-Lopez, Phys. Rev. B \textbf{84}, 165409 (2011); L. Chen and
%S. Wan, Phys. Rev. B \textbf{84}, 075149 (2011).  


\end{thebibliography}