# Qualitative chirality effects on the Casimir-Lifshitz torque with liquid
crystals

Wijnand Broer  Institute of Physics, University of Chinese Academy of
Sciences, 100190 Beijing, China  Bing-Sui Lu  Division of Physics and Applied
Physics, Nanyang Technological University, Singapore 637371, Singapore  Rudolf
Podgornik  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Institute of Physics, Chinese Academy of
Sciences, 100190 Beijing, China  Kavli Institute for Theoretical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Wenzhou
Institute of the University of Chinese Academy of Sciences, Wenzhou, Zhejiang
325000, China  Department of theoretical physics, J. Stefan Institute, , 1000
Ljubljana, Slovenia and Department of Physics, Faculty of Mathematics and
Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
<EMAIL_ADDRESS>
###### Abstract

We model a cholesteric liquid crystal as a planar uniaxial multilayer system,
where the orientation of each layer differs slightly from that of the adjacent
one. This allows us to analytically simplify the otherwise acutely complicated
calculation of the Casimir-Lifshitz torque. Numerical results differ
appreciably from the case of nematic liquid crystals, which can be treated
like bloc birefringent media. In particular, we find that the torque deviates
considerably from its usual sinusoidal behavior as a function of the
misalignment angle. In the case of a birefringent crystal faced with a
cholesteric liquid one, the Casimir-Lifshitz torque decreases more slowly as a
function of distance than in the nematic case. In the case of two cholesteric
liquid crystals, either in the homochiral or in the heterochiral
configuration, the angular dependence changes qualitatively as a function of
distance. In all considered cases, finite pitch length effects are most
pronounced at distances of about 10 nm.

##  I Introduction

Casimir-Lifshitz interactions [1, 2, 3] are macroscopic dispersion forces that
arise from quantum mechanical and thermal fluctuations of the electromagnetic
field. The name ‘dispersion forces’ originates from the fact that their
properties are governed by the dielectric and magnetic dispersion of the
interacting materials [4, 5, 6, 7]. Casimir-Lifshitz forces share the same
physical origin as the van der Waals-London forces, which conventionally
denote the interaction in cases where retardation effects are negligible.

These forces are studied for both fundamental and practical reasons. From a
fundamental point of view, the Casimir-Lifshitz force plays an important role
in the search for deviations from Newtonian gravitation in short range
gravitation experiments. Its study can facilitate the precise comparisons
between theoretical predictions and experimental data [8]. More practically,
Casimir-Lifshitz interactions affect the actuation dynamics of nano- and
micro-mechanical systems, such as switches, cantilevers, and actuators at a
sub-micrometer length scale [9, 10, 11, 12, 13, 14]. The Casimir-Lifshitz
interactions also underpin the stability of colloidal and biophysical
macromolecular systems. These forces represent one of the pillars of the
fundamental Deryaguin-Landau-Verwey-Overbeek theory of colloid stability [15].

Precise evaluation of the Casimir-Lifshitz force between real materials
requires a detailed knowledge of the materials’ electromagnetic
susceptibilities as inputs [2, 3]. Because this interaction is mediated by
virtual photons, the frequency of which cannot be controlled directly, the
Casimir-Lifshitz interaction is a broadband phenomenon. Consequently, the
electromagnetic susceptibilities of the materials involved must be known -
either theoretically or experimentally - as a function of frequency in a
sufficiently broad range. Another requirement to evaluate the Casimir-Lifshitz
interaction is solving Maxwell’s equations for the given geometric
configuration of the interacting bodies. This latter aspect of the Casimir-
Lifshitz calculations is the focus of this paper.

The presence of anisotropies, either in the susceptibilities or the
morphological anisotropy induced by the shape of the interacting bodies [16],
implies the existence of not only forces but also Casimir-Lifshitz torques
between the interacting bodies [17]. Specifically, the dielectric anisotropy
in the plane of reflection, which will exclusively concern us in what follows,
creates a dielectric contrast in the azimuthal direction, which gives rise to
the Casimir-Lifshitz torque attempting to align the orientations of both
materials[18]. An exact analytical description of the Casimir-Lifshitz torque
between two planar uniaxially anisotropic half spaces was originally based on
the direct solution of the Maxwell equations and the pertaining dispersion
equation by Barash more than four decades ago [19]. An alternative methodology
applicable to the Casimir-Lifshitz torque can be based on the solutions of the
Maxwell’s equations by interpreting them as an eigenvalue problem. This method
is known as the transfer matrix method. It is especially suited for describing
electromagnetic field propagation through anisotropic materials [20, 21, 22],
and can be used to derive the Casimir torque between semi-infinite
birefringent plates [23, 24]. The results of the transfer matrix methodology
were later shown to be consistent with the direct method of Barash [25]. Some
recent examples of the application of this method in the context of Casimir
physics include biaxial materials, [26] Weyl semi-metals, [27, 28] and
magnetic ferrite slabs [29].

The first experimental observation of Casimir-Lifshitz torques has been
accomplished only recently, by relying on the anisotropic dielectric response
of nematic liquid crystals [30]. It seems that further experiments on Casimir-
Lifshitz torques with liquid crystals are within experimental reach presently
or in the near future. Apart from the Casimir-Lifshitz torques, liquid
crystals exhibit diverse other Casimir fluctuation phenomena and so might
prove to be an abundant source of a variety of distinct Casimir phenomena [31,
32]. The simplest way to theoretically describe the anisotropy of liquid
crystals is by approximating them as uniaxial half spaces with a unique
orientation [33]. This essentially approximates the liquid crystal as a
birefringent half space. In what follows we will refer to this approximation
as ’half space approximation’. This is indeed a valid approach if the liquid
crystal is aligned, such as in the case of nematic liquid crystals [34,
*Kornilovitch2018]. However, a cholesteric liquid crystal actually consists of
a sequence of many thin, planar, uniaxially oriented layers, whose orientation
varies periodically in space with a periodicity given by the cholesteric pitch
[36]. This type of layered system has been shown to be amenable to the
transfer matrix method [20] as electromagnetic waves in layered anisotropic
media can be described by an algebra of 4$\times$4 matrices [37].

However, a major complication arises from the fact that the cholesteric liquid
crystal consists of a large number of layers ($\sim 10^{4}$) which generally
implies a product of an equally large number of 4$\times$4 matrices. Even
though the non-retarded limit reduces the size of the matrices to 2$\times$2,
it does not reduce the number of matrix multiplications [38, 34, 39, 40].
Since the computational complexity of matrix multiplication increases linearly
with the number of multiplications, an explicit analytical expression for the
transfer matrix can reduce this complexity by up to four orders of magnitude.
Hence here we propose a way to analytically simplify this problem and derive
an explicit expression for the composite transfer matrix for a specific planar
multilayer structure that models a cholesteric liquid crystal. We have taken
advantage of the facts that each layer is very thin, and that the difference
in orientation between each successive layer is small [36] . This allows us to
write an analytic result for the transfer matrix based on the Baker-Campbell-
Hausdorff (BCH) formula [41] as a second order expansion in this parameter,
$\delta$, which is inversely proportional to the pitch length of the
cholesteric liquid crystal. The form of the transfer matrix is then obtained
as a function of the ordinary and extraordinary eigenvalues of the Maxwell
equations, averaged over the orientations in the multilayer system.
Consequently, our solutions enable us to analyze the interaction between two
semi-infinite homochiral or heterochiral layers and thus quantify
macroscopically the effects of chirality. Our results point to a realistic
possibility of measuring the effects of chirality in the macroscopic Casimir-
Lifshitz torque for interaction geometries involving oriented cholesteric
liquid crystals. Note that this is different from effects of chirality of an
intervening medium [42, 43, 44].

The paper is organized as follows. After the introduction we will discuss the
solutions of the Maxwell equations for a general planar anisotropic multilayer
in section II. Next we will introduce the simplification for the cholesteric
liquid crystal configuration in section III, the numerical implementation of
which is then covered in section IV. Conclusions can be found in section V.

##  II Planar anisotropic multilayer

###  II.1 Transfer matrix

Here we will formulate the theory of Casimir-Lifshitz interactions in a system
composed of a one dimensional layering of finite thickness anisotropic
dielectric slabs. (See Fig. 1) Later on, we will use this geometry to model a
cholesteric liquid crystal. Our approach is based on an application of the
transfer matrix method for the electromagnetic field modes propagating through
one-dimensional stratifications. This method was introduced as an algebra of 4
$\times$ 4 matrices by Berreman [20] and later reformulated by Yeh [21]. Here
we will follow the latter formulation, which is based on the change of basis
transform in linear algebra. However, both formulations are equivalent [45].
We assume that the anisotropic stratification coincides with the direction of
the cholesteric axis, oriented along the $z$-direction and perpendicular to
the translational $x$-$y$-plane of symmetry of the laboratory coordinate
system. (See Fig. 1)

We start by formulating Maxwell’s equations as an eigenvalue problem, as we
have done before for the case of single interfaces [25]. Assume that the
electromagnetic (EM) wave propagates within one of the anisotropic slabs
labeled as $j$ whose anisotropic plane is facing the surface, which is defined
as the $x$-$y$-plane in the laboratory coordinate system. Furthermore, the
magnetic and electric anisotropy axes are assumed to be identical, implying
that the electric permittivity and the magnetic permeability are given by:

$\underline{\pmb{\varepsilon}}_{j}(\omega)=\left(\begin{smallmatrix}\varepsilon_{jx}\cos^{2}\theta_{j}+\varepsilon_{jy}\sin^{2}\theta_{j}&(\varepsilon_{jx}-\varepsilon_{jy})\sin\theta_{j}\cos\theta_{j}&0\\\
(\varepsilon_{jx}-\varepsilon_{jy})\sin\theta_{j}\cos\theta_{j}&\varepsilon_{jx}\sin^{2}\theta_{j}+\varepsilon_{jy}\cos^{2}\theta_{j}&0\\\
0&0&\varepsilon_{jz}\end{smallmatrix}\right)$ (1)

$\underline{\pmb{\mu}}_{j}(\omega)=\left(\begin{smallmatrix}\mu_{jx}\cos^{2}\theta_{j}+\mu_{jy}\sin^{2}\theta_{j}&(\mu_{jx}-\mu_{jy})\sin\theta_{j}\cos\theta_{j}&0\\\
(\mu_{jx}-\mu_{jy})\sin\theta_{j}\cos\theta_{j}&\mu_{jx}\sin^{2}\theta_{j}+\mu_{jy}\cos^{2}\theta_{j}&0\\\
0&0&\mu_{jz},\end{smallmatrix}\right)$ (2)

where $\theta_{j}$ denotes the angle between the optic axis of layer $j$ and
the $x$-axis of laboratory’s coordinate system. It must be stressed that the
entries of both $\underline{\pmb{\varepsilon}}_{j}$ and
$\underline{\pmb{\mu}}_{j}$ depend on frequency, but that the argument will be
suppressed from now on. Because of this dependence it is convenient to
represent the electromagnetic fields in the frequency domain as well. After
also transforming to Fourier space, the vectorial Maxwell equations are given
by

$\mathbf{k}\times\mathbf{E}=\frac{\omega}{c}\mathbf{B},\qquad\mathbf{k}\times\mathbf{H}=-\frac{\omega}{c}\mathbf{D}$
(3)

where $\mathbf{B}=\underline{\pmb{\mu}}\cdot\mathbf{H}$ and
$\mathbf{D}=\underline{\pmb{\varepsilon}}\cdot\mathbf{E}$. Eq. 3 is a system
of six linear equations with six unknowns, four of which are independent.
Elimination of the $z-$component of $\mathbf{E}$ and $\mathbf{H}$ leads to the
following 4 dimensional eigenvalue equation: [46]

$\underline{\pmb{Q}}_{j}\pmb{\Psi}=q~{}\pmb{\Psi},$ (4)

for the 4 dimensional EM field vector
$\pmb{\Psi}=(E_{x},E_{y},H_{x},H_{y})^{T}$. $\underline{\pmb{Q}}_{j}$ denotes
a 4$\times 4$ anti-diagonal block matrix whose explicit entries are given in
the supplemental material, section I.

The four eigenvector solutions of Eq. 4 represent the orientation dependent
modes, which both can propagate forward and backward. They are denoted by
$\pmb{\Psi}_{jo}^{\pm}$ and $\pmb{\Psi}_{je}^{\pm}$, where the subscripts $e$
and $o$ denote extraordinary and ordinary, respectively (for the case of
birefringent media), and the superscripts $+$ and $-$ indicate forward and
backward propagating modes, respectively. The mode eigenvalues are given by

$\displaystyle
q^{\pm}_{jo}=\pm\sqrt{\varepsilon_{jy}\mu_{jx}\omega^{2}/c^{2}-(\mu_{jx}/\mu_{jz})k^{2}_{\rho}\cos^{2}(\theta_{j}-\eta)-k^{2}_{\rho}\sin^{2}(\theta_{j}-\eta)}$
(5a) $\displaystyle
q^{\pm}_{je}=\pm\sqrt{\varepsilon_{jx}\mu_{jy}\omega^{2}/c^{2}-(\varepsilon_{jx}/\varepsilon_{jz})k_{\rho}^{2}\cos^{2}(\theta_{j}-\eta)-k_{\rho}^{2}\sin^{2}(\theta_{j}-\eta)}$
(5b)

where $k_{\rho}$ denotes the radial component of the wavevector, and $\eta$
represents the azimuthal component of the EM wave propagation direction.

Figure 1: One dimensional stratification of $N$ finite thickness anisotropic
dielectric slabs representing an arbitrary planar multilayer stack. The
stratified media, $i=1,N$, with generally anisotropic layers are embedded in
an isotropic bathing medium. Each layer has an arbitrary material composition,
thickness and orientation.

Here we formulate a 4$\times$4 transfer matrix formalism for a one-dimensional
stratified system composed planar slabs of finite thickness with in-plane
anisotropy (See Fig. 1), based on the methodology of Ref.[21].

The matrix in Eq. 4 has distinct eigenvalues so it is diagonalizable. This
brings us to the so called _propagation matrix_

$\underline{\pmb{P}}_{j}\equiv\mathrm{diag}(\exp(-iq_{je}d_{j}),\exp(iq_{je}d_{j}),\exp(-iq_{jo}d_{j}),\exp(iq_{jo}d_{j})).$
(6)

Multiplication from the left of the matrix $\underline{\pmb{P}}_{j}$ by the EM
field vector represents propagation within an anisotropic slab in the
$z$-direction over a distance $d_{j}$, the thickness of the $j$th layer:

$\pmb{\Psi}(z+d_{j})=\underline{\pmb{P}}_{j}\cdot\pmb{\Psi}(z),$

where $\pmb{\Psi}$ again denotes a 4 dimensional EM wave vector.
$\underline{\pmb{P}}_{j}$ is a valid representation only in the eigenvector
basis where the matrix is diagonal. Hence it is transformed to the $xy$ basis
as follows

$\exp\left({-i\underline{\pmb{Q}}_{j}d_{j}}\right)=\underline{\pmb{S}}_{j}\cdot\underline{\pmb{P}}_{j}\cdot\underline{\pmb{S}}_{j}^{-1},$
(7)

where the columns of $\underline{\pmb{S}}_{j}$ consist of the eigenvectors of
Maxwell’s equations:

$\underline{\pmb{S}}_{j}=\begin{pmatrix}\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}\\\
\pmb{\Psi}_{je}^{+}&\pmb{\Psi}_{je}^{-}&\pmb{\Psi}_{jo}^{+}&\pmb{\Psi}_{jo}^{-}\\\
\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}\end{pmatrix}$
(8)

which is analogous to what is called ‘rotation matrix’ in Ref. [37]. In Eq. 8
the normalization constants of the eigenvectors can be omitted because
$\underline{\pmb{P}}_{j}$ is multiplied by the inverse of
$\underline{\pmb{S}}_{j}$ from the right. The total transfer matrix that
represents the EM wave propagation through all $N$ layers is given by

$\underline{\pmb{T}}_{N}\equiv\mathrm{...}\underline{\pmb{S}}_{j-1}\cdot\underline{\pmb{P}}_{j-1}\cdot\underline{\pmb{S}}_{j-1}^{-1}\cdot\underline{\pmb{S}}_{j}\cdot\underline{\pmb{P}}_{j}\cdot\underline{\pmb{S}}_{j}^{-1}\cdot\underline{\pmb{S}}_{j+1}\cdot\underline{\pmb{P}}_{j+1}\cdot\underline{\pmb{S}}_{j+1}^{-1}\mathrm{...}\equiv\prod_{j=1}^{N}\underline{\pmb{S}}_{j}\cdot\underline{\pmb{P}}_{j}\cdot\underline{\pmb{S}}_{j}^{-1}$
(9)

where the product symbol denotes matrix multiplication from the right for each
subsequent index.

Next we have to consider what happens outside the slab, i.e. the cases $j=0$
and $j=N+1$. The EM field propagating through the isotropic embedding medium
can be written as a linear combination of the $s$\- and $p$-polarized modes.
By the same token as before, since the $s$\- and $p$-polarized modes are the
eigenmodes for isotropic media, a matrix can be constructed with the $s$\- and
$p$-polarized mode eigenvectors as columns for $j=0$:

$\underline{\pmb{S}}_{0}=\begin{pmatrix}\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}\\\
\pmb{\Psi}_{js}^{+}&\pmb{\Psi}_{js}^{-}&\pmb{\Psi}_{jp}^{+}&\pmb{\Psi}_{jp}^{-}\\\
\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}&\rule[-4.30554pt]{0.5pt}{10.76385pt}\end{pmatrix}$
(10)

the explicit form of which can be found in the supplemental material, section
I. However, the case $j=N+1$ is a little different. Here only forward
propagating modes exist, which does not affect the matrix itself but instead
only affects the vector on which the matrix acts. After all, in the $s-$ and
$p-$mode basis, this information can be represented by two nonzero components,
which represent the amplitudes of the forward propagating $s-$ and
$p-$polarized modes, (see Section II.2 for details). Note that the matrices of
Eqs. 8 and 10 are singular at zero frequency. However, physically ’zero
frequency’ corresponds to a static limit where the frequency is small on the
scale of the main absorption frequency of the materials. Hence within this
limit it is possible to obtain the inverses of the matrices of Eqs. 8 and 10.
See also supplemental material, section V for more details on how to take the
non-retarded limit.

Since Eq. 9 is valid in the laboratory $xy$ basis, it must be transformed to
the $sp-$mode basis, which is the proper eigenmode basis for isotropic media.
The total transfer matrix is hence

$\underline{\pmb{M}}=\underline{\pmb{S}}_{0}^{-1}\cdot\underline{\pmb{T}}_{N}\cdot\underline{\pmb{S}}_{0}$
(11)

where $\underline{\pmb{T}}_{N}$ is given by Eq. 9. For the general case
considered here it is not possible to give an explicit closed expression for
the matrix $\underline{\pmb{M}}$. As a test case, let us consider the limit of
an infinitely thick slab. This limit does not directly and algebraically
follow from Eq. 11 (see supplemental material, section III for details). It
can be shown that for a semi-infinite slab, the transfer matrix simplifies to

$\underline{\pmb{M}}=\underline{\pmb{S}}_{0}^{-1}\cdot\underline{\pmb{S}}_{1}.$
(12)

###  II.2 Lifshitz formula

The Casimir-Lifshitz interaction free energy can be obtained from the secular
determinant of the modes, whose zeros give the eigenfrequencies of the bound
states. [37] However, in its turn the secular determinant itself can be
rewritten in terms of the Fresnel reflection coefficients that can be obtained
from the transfer matrix formalism. This route then connects the transfer
matrix formalism with the Casimir-Lifshitz interactions which we elucidate in
what follows. Grosso modo one can follow either the approach based on the
Green function tensors, closest to the original papers by Lifshitz et al [2,
*Lifshitz61]. (See e.g. Refs. [47, 48, 49]) or follow the heuristic derivation
by van Kampen et al. [50, 51], which was later shown to be actually exact
[52], and based on the summation over the allowed EM modes within a given
geometry, more akin to the original paper by Casimir [1]. It can be shown that
both approaches lead to the same result, [53] namely the Lifshitz formula for
dispersion interactions. Here we will take advantage of this result.

We assume that the matrix $\underline{\pmb{M}}$ from Eq. 11 is known and that
its entries are given by $\underline{\pmb{M}}=\Big{(}M_{ik}\Big{)}$ with
$i,k\in\\{1,2,3,4\\}$. In such a case, the Fresnel reflection matrix elements
can be shown to be given by [21, 45]:

$\displaystyle
r_{ss}=\frac{M_{21}M_{33}-M_{23}M_{31}}{M_{11}M_{33}-M_{13}M_{31}}$ (13a)
$\displaystyle
r_{sp}=\frac{M_{33}M_{41}-M_{31}M_{43}}{M_{11}M_{33}-M_{13}M_{31}}$ (13b)
$\displaystyle
r_{ps}=\frac{M_{11}M_{23}-M_{13}M_{21}}{M_{11}M_{33}-M_{13}M_{31}}$ (13c)
$\displaystyle
r_{pp}=\frac{M_{11}M_{43}-M_{13}M_{41}}{M_{11}M_{33}-M_{13}M_{31}}.$ (13d)

We have verified that combining Eq. 12 with Eq. 13 restores the known Fresnel
reflection matrix [22, 25] for a semi-infinite birefringent plate.

To determine the Casimir energy and torque, Eqs. 13a, 13b, 13c and 13d must be
inserted into the Lifshitz formula. The energy per unit area is given by:

$\frac{E_{Cas}}{A}=\frac{k_{b}T}{4\pi^{2}}\sum_{n=0}^{\infty}(1-\tfrac{1}{2}\delta_{0n})\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\ln[D(k_{\rho},\eta,\\{\theta_{j}\\},\varphi,i\zeta_{n})]k_{\rho}\mathrm{d}k_{\rho}\mathrm{d}\eta$
(14)

where [48]

$D=\det(\underline{\pmb{I}}-\underline{\pmb{r}}_{1}(\\{\theta_{j}\\},i\zeta_{n})\cdot\underline{\pmb{r}}_{2}(\theta_{1}+\varphi,i\zeta_{n})e^{-2k_{3}a})$
(15)

and the reflection matrices are given by

$\underline{\pmb{r}}_{q}=\begin{pmatrix}r_{q,ss}&r_{q,sp}\\\
r_{q,ps}&r_{q,pp}\end{pmatrix}\qquad q\in\\{1,2\\}.$ (16)

The subscript $q$ is merely a label for each multilayer stack. The entries of
the matrices are given by Eqs. 13a, 13b, 13c and 13d. All quantities are
evaluated at the imaginary Matsubara frequencies $\zeta_{n}=\frac{2\pi
nk_{b}T}{\hbar},$ so that each contribution to the Casimir energy decreases
monotonically with $n$. The Casimir torque is then given by

$\tau(a,\varphi)=-\frac{\partial E_{Cas}}{\partial\varphi}$ (17)

where $\varphi$ denotes the angle between the optic axes of the layers of each
stack closest to the gap. This result is equivalent to that of Ref. [37] which
uses the formalism of Berreman [20], while here the same formalism has been
used with the basis suggested by Yeh [21].

##  III Modeling a Cholesteric liquid crystal

Figure 2: Illustration of the continuum approximation used here to model a
liquid crystal. It consists of layers with identical thicknesses $d$ but each
with a slightly different orientation, each with identical increments
$\delta\ll 1$. Consequently, the multilayer structure manifests as a
continuous inhomogeneity in the direction perpendicular to the plane of
reflection (defined as the $z$-direction), where the orientation effectively
behaves as a helix with pitch length $L$.

A cholesteric liquid crystal is modeled as a nonmagnetic uniaxial planar
multilayer stack. All anisotropic layers of equal thickness $d$ are made of
identical materials, and only differ by their orientation $\theta_{j}$. We
start by writing Eq. 7 as

$\exp({-i\underline{\pmb{Q}}_{j}d})=\underline{\pmb{S}}_{j}\cdot\underline{\pmb{P}}_{j}\cdot\underline{\pmb{S}}_{j}^{-1}$

for a single slab with label $j$.

Now let us formulate the conditions of validity of this approximation. In
accordance with an actual cholesteric liquid crystal [36], each layer is
assumed to be infinitesimally thin, i.e. $d\rightarrow 0$. Moreover, let the
difference in orientation between each subsequent layer, $\delta$, also be
infinitesimally small, in such a way that this quantity parametrically
describes a helix in space. Consequently, an infinitesimal change in $z$
within the liquid crystal is proportional to an infinitesimal change in the
orientation:

$\mathrm{d}z=d=\frac{L}{2\pi}\delta,$ (18)

where $L$ denotes the pitch length, in which the orientations of the layers
complete one period, and $\delta\ll 1$. Hence there is some freedom of choice,
we can use either $z$ or $\theta$ as a (continuous) variable. In what follows
we will use two variables interchangeably. Another assumption is that the
total thickness $Nd$, where $N$ denotes the total number of layers, is
infinite. This is justifiable because a typical thickness of liquid crystals
is of the order of several microns, and we will limit ourselves to separation
distances of less than one micron ($a\ll Nd$), where the Casimir-Lifshitz
torque is likely to be large enough to be measured [30]. Finally, it will be
assumed that the pitch length of the cholesteric liquid crystal $L$, in which
the orientations of the layers complete one period, is smaller than the total
thickness. Hence at least one period is assumed to fit inside the crystal. We
can summarize the condition for our approximation as follows:

$d\ll L\ll Nd,$ (19)

which means that the difference between the first orientation and last
orientation is at least $2\pi$ radians. This is applicable to a cholesteric
liquid crystal [36]. We also require $d$ to be small on the scale of the
matrix norm of $\underline{\pmb{Q}}_{j}$:

$||\underline{\pmb{Q}}_{j}||d\ll 1\quad\text{for all }1\leq j\leq N.$ (20)

Physically, Eq. 20 can be understood as the thickness $d$ being much smaller
than the typical values of the relevant wavelengths. After all, in a
continuous medium approach, the relevant wavelengths are assumed to be larger
than the typical distances between atoms. Since the physical thickness is
actually of the same order of magnitude, it can be approximated as a numerical
integration stepsize in the context of a continuous medium approach. The
condition Eq. 20 allows us to use the Baker-Campbell-Hausdorff (BCH) formula
[41]. The second condition of Eq. 19 allows us to use Eq. 12 instead of Eq.
11.

In section IV of the supplemental material we derive the following expression
valid up to the second order in the angular deviation in the orientation
between the neighboring uniaxial layers

$\prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\exp\Bigg{(}-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}+O(\delta^{2})\Bigg{)}\longrightarrow\exp\Bigg{(}-i\int\limits_{\min(z)}^{\max(z)}\underline{\pmb{Q}}(\theta)\mathrm{d}z+O(\delta^{2})\Bigg{)},$
(21)

where eventually we assumed that $j$ changes as a continuous variable
associated with $z$. The full derivation is somewhat lengthy so we will
summarize it here. In short, we use the Baker-Campbell-Haussdorff (BCH)
formula [41] up to third order in $\delta$. Then we combine the BCH formula
with finite difference coefficients [54] to determine that, three nearest
neighbors to the right are needed to obtain the same accuracy as the BCH
formula.

In order to evaluate the argument of the exponential matrix in Eq. 21, and we
will try to determine its diagonalized form. Let us focus on the leading order
term: $-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}$. Even though Eq. 7 tells us
that each individual matrix $\underline{\pmb{Q}}_{j}$ is diagonalizable, it
does not follow in general that their sum is also diagonalizable. However,
within this particular approximation it is assumed that the commutators
between the matrices $\underline{\pmb{Q}}_{j}$ are negligible and the matrices
effectively commute, with the error of this approximation being of second
order. Commuting diagonalizable matrices can be shown to have the same
eigenvectors. Consequently, each diagonalizable matrix
$\underline{\pmb{Q}}_{j}$ can be transformed to the same eigenvector basis via
one matrix, say $\underline{\pmb{S}}$. In other words, the matrices
$\underline{\pmb{Q}}_{j}$ can be simultaneously diagonalized:

$\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}=\underline{\pmb{S}}\cdot\underline{\pmb{D}}\cdot\underline{\pmb{S}}^{-1}$

where $\underline{\pmb{D}}$ is a diagonal matrix. This furthermore implies
that:

$\exp\left(-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}\right)=\underline{\pmb{S}}\exp\left(-id\underline{\pmb{D}}+O(d^{2})\right)\underline{\pmb{S}}^{-1}.$
(22)

We know that the matrices $\underline{\pmb{S}}$ and $\underline{\pmb{D}}$
exist but now we want to determine their explicit form. The physical
interpretation of Eq. 22 is that, in this approximation, the orientations of
the different layers are averaged over the entire crystal. Since $d\rightarrow
0$ and $N\rightarrow\infty$, it can be proposed that $d\sim 1/N$, and the left
hand side of Eq. 22 becomes an average over all eigenvalues. Hence it follows
that

$\underline{\pmb{D}}=\frac{1}{N}\sum_{j=1}^{N}\text{diag}(q_{e}((\theta_{j}-\eta)),-q_{e}((\theta_{j}-\eta)),q_{o},-q_{o})$
(23)

which changes to an average with respect to a continuous distribution where
the sum is replaced by an integral. The approximation of Eq. 23 is of course
simple and crude, its error being of second order in the stepsize $d$. It
assumes that the planar multilayered system can be approximated as a single
slab with an averaged orientation. Essentially the effect of the presence of
multiple layers is treated additively in this case.

The transfer matrix for such a configuration is given by (see Eq. 22):

$\exp\left(-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}\right)=\left<\underline{\pmb{S}}\right>\underline{\pmb{P}}_{tot}(\left<q_{e}\right>)\left<\underline{\pmb{S}}\right>^{-1},$
(24)

where

$\underline{\pmb{P}}_{tot}(\left<q_{e}\right>)=\text{diag}(\exp(-id_{tot}\left<q_{e}\right>),\exp(id_{tot}\left<q_{e}\right>,\exp(-id_{tot}q_{o}),\exp(id_{tot}q_{o})).$
(25)

Here $d_{tot}=Nd$ represents the total thickness, and in the limit
$d_{tot}\rightarrow\infty$, $\underline{\pmb{P}}_{tot}$ tends to
${\underline{\pmb{P}}_{tot}\rightarrow\text{diag}(1,0,1,0)}.$ (This does not
follow directly algebraically, but from physical considerations, see
Supplemental Material, section III ). In Eq. 25 the eigenvectors are the ones
corresponding to $\left<q_{e}\right>$:

$\left<\underline{\pmb{S}}\right>=\underline{\pmb{S}}(\left<q_{e}\right>,\left<\theta\right>),$
(26)

where $\underline{\pmb{S}}$ is given by Eq. 8 and $\left<q_{e}\right>$ denotes
the extraordinary eigenvalue, averaged over the orientations:

$\left<q_{e}\right>=\frac{1}{N}\sum_{j=1}^{N}q_{e}((\theta_{j}-\eta))\longrightarrow\frac{1}{N\delta}\int\limits_{0}^{N\delta}q_{e}(\theta-\eta)\mathrm{d}\theta.$
(27)

The rightmost expression follows after the continuity approximation that
allows us to replace the sums over the discrete index $j$ by integrals over
the continuous variable $\theta$. We assume here that the pitch length, the
length of one period, is smaller than the total thickness, see Eq. 19.

Due to the periodicity of $q_{e}$, $N\delta$ can be replaced by $\pi$, so that
$q_{e}$ is averaged over one period and it can take
all possible real values. This is because the thickness of the liquid crystal
is assumed to be larger than the pitch length. (See Eq. 19) Consequently, at
least one period can be fit into the crystal. The integer part of the ratio
between the thickness and the pitch length will dominate the remainder,
because the remainder represents a part of the liquid crystal located at an
infinite distance from the plane of reflection. Setting $N\delta=\pi$ also
allows us to set $\eta=0$, because the integral becomes translation invariant
over one period. In this case, the integral of Eq. 27 can be evaluated
analytically:

$\left<q_{e}\right>=\frac{1}{\pi}\int\limits_{0}^{\pi}q_{e}(\theta)\mathrm{d}\theta=\pm\frac{2}{\pi}\sqrt{\varepsilon_{1x}\left(\tfrac{k_{\rho}^{2}}{\varepsilon_{1y}}+\tfrac{\zeta^{2}}{c^{2}}\right)}E\left(\frac{k_{\rho}^{2}(\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}-1)}{\varepsilon_{1x}\left(\tfrac{k_{\rho}^{2}}{\varepsilon_{1y}}+\tfrac{\zeta^{2}}{c^{2}}\right)}\right),$
(28)

where $E(z)$ denotes the complete elliptic integral of the second kind, given
by [55]

$E(z)=\int_{0}^{\tfrac{\pi}{2}}\sqrt{1-z\sin^{2}t}\mathrm{d}t,$

the value of which is real for arguments $-1\leq z\leq 1$.

To determine the average orientation, we must solve the following equation:

$\left<q_{e}\right>=\pm\sqrt{\varepsilon_{1x}\tfrac{\zeta^{2}}{c^{2}}+\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}k_{\rho}^{2}\cos^{2}\left<\theta\right>+k_{\rho}^{2}\sin^{2}\left<\theta\right>}$
(29)

which has four possible solutions:

$\left<\theta\right>=\pm\arccos\left(\pm\frac{\sqrt{k_{\rho}^{2}+\varepsilon_{1x}\zeta^{2}/c^{2}-\left<q_{e}\right>^{2}}}{k_{\rho}\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right).$
(30)

Here the different signs of $\left<\theta\right>$ represent the different
chiralities of the liquid crystal: a positive $\left<\theta\right>$ is
associated with right handed chirality and the minus sign corresponds to left
handed chirality, if the axial wave vector component is positive. The
different signs in the argument of the arc-cosine in Eq. 30 come from the
different square roots in Eq. 29, associated with the two different
propagation directions of the extraordinary modes in the liquid crystal.
However, since the quantity $\left<q_{e}\right>$ is an average along the
helix, its sign must be interpreted as the direction of the helical spiral.
Alternatively, the different signs of the square root of $\left<q_{e}\right>$
in Eq. 29 can be thought of as the location of the crystal: since the
thicknesses of the crystals are assumed to be infinite, the negative square
root represents the crystal being on the left and the positive square root
represent the crystal being on the right. This distinction depends on the
arbitrary choice of coordinates, and it is hence not physical. Eq. 30 provides
four solutions whereas there are only two physically distinguishable shapes of
the helix, namely left- and right handed. Therefore we need to establish how
they can be physically interpreted. We require that the physical
$\left<\theta\right>$ is real. First we have to verify that such solutions
exist. To this end we distinguish two cases:
$\varepsilon_{1y}\geq\varepsilon_{1x}$ and
$\varepsilon_{1y}\leq\varepsilon_{1x}$. In order for $\left<\theta\right>$ to
be real, we need to check that

$0\leq
k_{\rho}^{2}+\varepsilon_{1x}\tfrac{\zeta^{2}}{c^{2}}-\left<q_{e}\right>^{2}\leq
k_{\rho}^{2}\left(1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right),$

which can be seen to be the case since

$\min\left(1,\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right)\leq\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\cos^{2}\theta+\sin^{2}\theta\leq\max\left(1,\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right)$

for all real $\theta$. Here it is assumed that $|\left<\theta\right>|\leq\pi$.
Hence the real value of the average within the domain
$-\pi\leq\left<\theta\right>\leq\pi$ orientation is

$\left<\theta\right>_{L}=\arccos\left(-\frac{\sqrt{k_{\rho}^{2}+\varepsilon_{1x}\zeta^{2}/c^{2}-\left<q_{e}\right>^{2}}}{k_{\rho}\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\
$ (31a)

$\left<\theta\right>_{R}=-\arccos\left(-\frac{\sqrt{k_{\rho}^{2}+\varepsilon_{1x}\zeta^{2}/c^{2}-\left<q_{e}\right>^{2}}}{k_{\rho}\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\
$ (31b)

if the crystal is on the left, and

$\left<\theta\right>_{L}=-\arccos\left(\frac{\sqrt{k_{\rho}^{2}+\varepsilon_{1x}\zeta^{2}/c^{2}-\left<q_{e}\right>^{2}}}{k_{\rho}\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\
$ (32a)
$\left<\theta\right>_{R}=\arccos\left(\frac{\sqrt{k_{\rho}^{2}+\varepsilon_{1x}\zeta^{2}/c^{2}-\left<q_{e}\right>^{2}}}{k_{\rho}\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\
$ (32b)

if the crystal is on the right. Here $\left<q_{e}\right>$ is given by Eq. 28.
The subscripts $L$ and $R$ in Eqs. 32 and 31 denote the left- and right
handedness of the liquid crystal, respectively. Note that Eqs. 32 and 31
depend on frequency, which in turn affects the frequency dependence of the
reflection matrix. Finally, Eqs. 28, 32 and 31 have to be inserted into Eq. 8
in order to obtain $\left<\underline{\pmb{S}}\right>_{1}$.

Now we move on to the next order term in the BCH expansion. Here the
contribution of nested commutators is neglected, i.e. commutators are assumed
to commute with other matrices. In order to take advantage of Eq. 24 we apply
the BCH formula again to separate the BCH terms as follows (see Supplemental
material, section IV for a detailed derivation):

$\exp\left(-i\frac{L}{\pi}\int\limits_{0}^{N\delta}\underline{\pmb{Q}}(\theta)\mathrm{d}\theta-3\frac{L^{2}}{\pi^{2}}\int\limits_{0}^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial\underline{\pmb{Q}}(\theta)}{\partial\theta}\right]\mathrm{d}\theta+O(\delta^{3})\right)$
(33)
$=\left<\underline{\pmb{S}}\right>_{1}\underline{\pmb{P}}_{tot}(\left<q_{e}\right>)\left<\underline{\pmb{S}}\right>_{1}^{-1}\exp\Bigg{(}-3\frac{L^{2}}{\pi^{2}}\int\limits_{j=1}^{(N-3)\theta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial\underline{\pmb{Q}}(\theta)}{\partial\theta}\right]\mathrm{d}\theta\Bigg{)},$

where we have taken advantage of the transfer matrix for the single interface
case, Eq. 12. The argument of rightmost exponential matrix on the right hand
side of Eq. 33 is not jointly diagonalizable with the first BCH term. 111The
matrices may still be approximately jointly diagonalizable. However at this
point it is not clear how to estimate this approximation, or what the common
eigenvector basis should be. However, this correction is expected to be small
compared to the averaged factor on its left. This allows us to write it as a
Taylor expansion. Hence we obtain

$\
\prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\left<\underline{\pmb{S}}\right>_{1}\underline{\pmb{P}}_{tot}(\left<q_{e}\right>)\left<\underline{\pmb{S}}\right>_{1}^{-1}\left(\underline{\pmb{I}}-3\frac{L^{2}}{\pi^{2}}\int\limits_{0}^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial\underline{\pmb{Q}}(\theta)}{\partial\theta}\right]\mathrm{d}\theta+O(\delta^{3})\right).$
(34)

The leading order term in Eq. 34 can be understood as a slab with infinite
thickness and an averaged orientation. The second order correction describes
the effect due to the finite thicknesses of the layers and their slightly
different orientations. For a liquid crystal of infinite thickness Eq. 34
becomes

$\prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\underline{\pmb{S}}_{0}^{-1}\left<\underline{\pmb{S}}\right>_{1}\left(\underline{\pmb{I}}-3\frac{L^{2}}{\pi^{2}}\int\limits_{0}^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial\underline{\pmb{Q}}(\theta)}{\partial\theta}\right]\mathrm{d}\theta+O(\delta^{3})\right),$
(35)

Next we are faced with the task of writing Eq. 35 as an expansion in $\delta$.
Note that the integral absorbs one factor of $\delta$ and that the integration
from 0 to $N\delta=\pi$ will not contribute to the integral (See supplemental
material, section I for the explicit expressions for the commutator), so the
integral is expected to be small $\sim\delta$. Within the approximations used
here, it suffices to calculate the integral up to first order in $\delta$. (A
more precise calculation would be beyond the approximation and hence be
inconsistent). We will denote this first order coefficient by
$\underline{\pmb{A}}$. Hence the transfer matrix is

$\prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\underline{\pmb{S}}_{0}^{-1}\left<\underline{\pmb{S}}\right>_{1}\left(\underline{\pmb{I}}-3\frac{L^{2}}{\pi^{2}}\underline{\pmb{A}}\delta+O(\delta^{3})\right),$
(36)

where $\underline{\pmb{A}}$ is given by the integral in Eq. 35, and its
explicit expressions can be found in the supplemental material, section I.
Note that the error of the BCH formula is of third order in $\delta$, as the
higher order term Eq. 36 is actually quadratic in $\delta$. So this term does
not have to follow the sign convention of Eqs. 31 and 32. Since the
approximate transfer matrix is known now, Eq. 13 can be applied to it to
obtain the corresponding Fresnel matrix. Note that the factor of $L^{2}$
cancels out of the higher order correction for the Fresnel matrix. However,
it’s worth pointing out that $\delta$ is inversely proportional to $L$, (see
Eq. 18) so that the final result still depends on the pitch length. We again
refer to the supplemental material, section I for the explicit expressions for
the corrections to the Fresnel matrix. It’s worth noting that the Fresnel
matrices are invariant under $\pmb{k}\rightarrow-\pmb{k}$, hence they are
identical whether they are placed on the left or on the right (if they have
the same orientation). However in general this is not necessarily true [57].

##  IV Results

Figure 3: Dielectric functions of pure, nematic and doped, cholesteric 5CB.
The dielectric function of nematic 5CB as proposed in Ref. [34, 35] (indicated
by the black squares and blue triangles) is modified to account for the
presence of a chiral dopant. This affects the static values (see the magenta
crosses and the red circles), which were obtained from Ref. [58]

Here we will use the results derived in section III to calculate the Casimir-
Lifshitz torque given by Eq. 17. The first configuration we consider is
composed of a birefringent half-space, a finite vacuum gap of thickness 10 nm
$<a<$1 $\mu$m, and a cholesteric half-space. Next we will investigate the
combination of two cholesteric liquid crystals. The former in general should
be accessible to experiments [30]. While realistically, surface roughness
generally plays a significant role at this range [59], the roughness effects
have been considered elsewhere in detail [60, 61, 62] and we will ignore them
here. We have verified our calculations by reproducing earlier results
obtained by implementations of the Barash formula [63, 33, 64] (See
supplemental material, section V).

In order to calculate Casimir-Lifshitz interactions, the frequency dependent
dielectric function of the interacting materials must be known. This is why we
introduce it here first. We will ignore the temperature dependence of the
dielectric function and assume a constant temperature of 293 K. This is
because we are interested in the cholesteric phase only.

The dielectric function of a nematic liquid crystal can be described by a
semi-empirical three band model [65]:

$\varepsilon_{N,x,y}(i\zeta)=\left(1+\sum_{k=1}^{3}\frac{C_{k,x,y}}{1+\tfrac{\zeta^{2}}{\omega_{k}^{2}}}\right)^{2}.$
(37)

However,a pure 5CB crystal is nematic and not cholesteric. It needs to be
doped to reach a cholesteric phase, which is typically described by a Debye
model [66]:

$\varepsilon_{D,x,y}(i\zeta)=\varepsilon_{\infty,x,y}+\frac{\varepsilon_{x,y}(0)-\varepsilon_{\infty,x,y}}{1+\zeta\tau}$
(38)

Therefore we propose to model a doped nematic liquid crystal as the sum of
these dielectric functions

$\varepsilon_{x,y}(i\zeta)=\varepsilon_{D,x,y}(i\zeta)+\varepsilon_{N,x,y}(i\zeta)$
(39)

However, the Debye relaxation processes take place on a much smaller frequency
scale than the Matsubara frequencies at room temperature. More precisely, the
smallest Matsubara frequency ($n=1$) at 293 K is six orders of magnitude
larger than the typical values of the Debye relaxation frequencies. [66, 58]
Therefore we propose to approximate it as the static term from Eq. 39:

$\varepsilon_{x,y}(i\zeta)\approx\varepsilon_{D,x,y}(0)+\varepsilon_{5CB,x,y}(i\zeta)$
(40)

in other words, the Debye processes only affect the static permittivity of the
liquid crystal. The different components of the dielectric functions for both
the nematic and cholesteric case in Fig. 3.

(a)

(b)

(c)

(d)

Figure 4: The Casimir torque between a birefringent (BaTiO3) crystal and a
cholesteric liquid crystal. (a): a left handed facing a birefringent one. (b):
a right handed crystal facing a birefringent one. (c): the Casimir torque as a
function of misalignment angle at a distance of $a=10$ nm. The different
curves are: the triangles pointing right show the usual result for a nematic
liquid crystal. Red circles: left handed crystal, $\delta=0$. Blue triangles
pointing left: left handed crystal, $\delta\neq 0$. Magenta crosses: right
handed crystal, $\delta=0$. Black squares: right handed crystal, $\delta\neq
0$. (d): The amplitude of the Casimir torque as a function of separation. The
triangles pointing right show the nematic case. The cholesteric case is shown
by the other two curves, of which the red circles represents $\delta=0$ and
the black squares show the perturbed case. The inset of panel (d) shows the
Casimir torque for the cases (a) (red circles) and (b) (blue crosses) at
distances between 380 and 800 nm.

Now we are in a position to specify the materials, i.e. assign values to the
parameters in Eq. 40. Let the cholesteric liquid crystal consist of 96 mass %
nematic 5CB (4- cyano-4’-pentyl-biphenyl) doped with 4 mass % chiral dopant
S811. [58] The value of the static Debye term for this mixture was taken from
a recent experiment [58]. The dielectric function of 5CB was established in
Ref. [34] based on data from Ref. [67]. The optical data of barium titanate
(BTO $\rm BaTiO_{3}$) [68] will be used to describe the semi-infinite
birefringent half-space. For more details about the parameter values and their
effect on the Casimir-Lifshitz interaction we refer to section VI of the
supplemental material. Throughout this section we choose the value of $\delta$
(if nonzero) to be 0.03, which corresponds to a pitch length of 200 nm. (See
Eq. 18).

(a)

(b)

(c)

(d)

Figure 5: The Casimir torque between two cholesteric liquid crystals of
identical chirality. (a): homochiral configuration of two left handed crystals
(b): homochiral configuration of two right handed crystals. (c): Casimir
torque as a function of misalignment angle at a distance of $a=10$ nm. The
different curves are: triangles pointing right: two nematic liquid crystals.
Black crosses: unperturbed, case (a). blue squares: perturbative contribution
to case (a) . Red circles: unperturbed case (b). orange circles: perturbative
contribution to case (b). (d): The Casimir torque as a function of
misalignment angle for distances between 380 an 800 nm. The blue triangles
represent case (a) and the red circles represent case (b).

(a)

(b)

(c)

(d)

Figure 6: The Casimir torque for the heterochiral case. (a): the left handed
crystal is placed on the left, and the right handed crystal is on the right.
(b): the reverse of panel (a). (c): several different contributions at 10 nm
distance. Black crosses: difference between case (a) and case (b), including
only lowest order terms. Red asterisks: idem, including higher order terms.
Orange circles: perturbative contribution to case (a), blue squares:
perturbative contribution to case (b). (d): Casimir torque at distances
between 380 and 800 nm, both for case (a) (black asterisks), and case (b) (red
circles).

The results are shown in Fig. 4. The upper two panels Figs. 4(a) and 4(b)
display the two cases we distinguish here: the BaTiO3 crystal can be faced
with either a right handed or a left handed crystal. Fig. 4(c) shows the
Casimir torque as a function of the misalignment angle $\varphi$. For
reference, we have included the usual case for a nematic liquid crystal,
indicated by the cyan curve with triangles pointing to the right. The leading
order term in Eq. 36 creates a shift in the angular dependence of the torque
of about $\pi/2$ radians, making the proportionality closer to $\cos 2\varphi$
than the usual $\sin 2\varphi$. The sign of the torque changes depending on
the left- or right handedness of the cholesteric liquid crystal, which is a
qualitative chirality effect. While the leading order term horizontally shifts
the graph, the next order term enhances the amplitude in only one direction,
creating a clear deviation from the usual sinusoidal shape of the curve (even
though it is still periodic with period $\pi$). For this reason we have
verified that

$\int\limits_{0}^{\pi}\tau\mathrm{d}\varphi=0$ (41)

so that no energy is gained or lost during a rotation over a period. Note that
the shape of the curve cannot be attributed to retardation effects, which are
negligible at a distance of 10 nm. The higher order correction actually is
most significant at short distances $\sim$ 10 nm, whereas retardation effects
become more pronounced at larger distances. Finally, Fig. 4(d) shows the
distance dependence of the amplitude of the torque. Not only is the amplitude
enhanced significantly compared to the nematic case, it also decreases more
slowly as a function of distance. Since the configuration of a birefringent
and a semi-infinite liquid crystalline sample is presently experimentally
accessible [30], this could make experimental observation of the torque easier
at larger distances. The inset of Fig. 4(d) shows that the angular dependence
remains the same at large distances (between 380 and 800 nm): the positions of
the extrema and the nodes do not change as a function of distance. The higher
order correction of the BCH formula (Eq. 36 is negligible in this separation
range, contributing significantly only at distances of $\sim$ 10 nm.

Next, we proceed to the case of two liquid crystals facing each other. Here we
distinguish between the so called homo and heterochiral cases, shown in Figs.
5 and 6 respectively. In the former case two right handed or two left handed
crystals face each other, and in the latter case a left handed crystal is
faced with a right handed one.

First we will discuss the homochiral case, as shown in Fig. 5. Figs. 5(a) and
5(b) illustrate the possible configurations: two right- and two left-handed
crystals. Fig. 5(c) shows a plot of the Casimir torque as a function of
misalignment angle at a distance of 10 nm. For reference, the case of two
nematic liquid (light blue triangles) has also been included. As in the
previous case, (see Fig. 4), the leading order term shifts the usual sine to a
cosine. However, unlike the previous case, here the higher order terms
decrease the amplitude of the torque in one direction, which does give rise to
a deviation from the sinusoidal form. This short distance is the only range
where the finite pitch length effects, represented by the higher order term in
Eq. 36, actually contribute significantly. At this range, it seems as though
the torque for the left handed configuration is opposite of that of the right
handed one. However, at larger distances it can be seen that this is not the
case. In Fig. 5(d) the Casimir torque is plotted as a function of the
misalignment angle for distances between 380 and 800 nm. At this distance
range, the higher order term of Eq. 36 contribute negligibly. Still even the
lower order terms give rise to a significant deviation from the sinusoidal
behavior from the previous case. Moreover, the position of the extrema changes
with the distance, and the nodes are not on the x-axis. In other words, the
angular dependence changes with the separation distance, which is
qualitatively different from the previous case or the case of two birefringent
materials. The left- and right-handed configurations do not exhibit opposite
torques, but the phase difference between the curves changes as a function of
distance as well. It is worth pointing out that, if a particular angle is
chosen, near one of the nodes, it becomes possible to see the torque change
sign as a function of distance. This kind of sign change has also been
reported for biaxial materials [64] and Weyl semi-metals [27], for example.

Next we will address the heterochiral case shown in Fig. 6. Figs. 6(a) and
6(b) show the configurations of a left handed crystal on the left and a right
handed crystal on the right, and the reverse configuration, respectively.
These configurations are expected to be physically indistinguishable. To test
this, we plot the different contributions to the torque as a function of
misalignment angle again at 10 nm distance, see Fig. 6(c). It can be seen that
the difference between the torque of Fig. 6(a) and Fig. 6(b) is non-negligible
if the higher order term in Eq. 36 is omitted. (See the black crosses in Fig.
6(c)) However, the inclusion of the higher order terms renders this difference
negligible. (The red asterisks in Fig. 6(c)). So even though the higher order
corrections themselves are different (as indicated by the orange circles and
blue squares in Fig. 6(c)), the total torque is identical for both
configurations. Finally, the torque is plotted as a function of the
misalignment angle for distances of 380 to 800 nm in Fig. 6(d). Here the
torque behaves qualitatively similar to the previous case, with the phase
somewhere in between that of the torque for the two homochiral configurations
of Fig. 5(d). As expected, the torques for the configurations of Figs. 6(a)
and 6(b) are identical.

##  V Conclusions and outlook

In order to understand the effects of cholesteric chirality on the Casimir-
Lifshitz torque, we have modeled a cholesteric liquid crystal as a uniaxial
planar multilayer system. The system consists of identical layers of equal
thicknesses, but the orientation of each layer differs slightly from that of
the adjacent one. This model allows us to derive an analytical simplification
of the Fresnel matrices as an expansion in this small orientation difference,
by means of the Baker-Campbell-Hausdorff (BCH) formula.

Numerically, we have obtained results that are appreciably different from the
case of two semi-infinite birefringent plates, which is a valid approximation
for nematic liquid crystals. [33, 30] In particular, in the case of a
birefringent half space facing a cholesteric liquid crystal, the leading order
term in the BCH expansion shifts the angular dependence of the torque by about
$\pm\pi/2$ radians, depending on the chirality of the liquid crystal. At very
short distances of about 10 nm, the higher order BCH term gives rise to a
significant deviation from the usual sinusoidal shape of the torque as a
function of misalignment angle. Moreover, the amplitude of the torque
decreases more slowly as a function of distance than in the nematic case.
Since the configuration of a birefringent crystal faced with a liquid crystal
is presently experimentally accessible, [30] we believe that especially this
latter result can be useful for detecting the torque at large separation
distances.

The case of two cholesteric liquid crystals consists of three physically
different configurations: two homochiral ones (two left-handed crystals or two
right-handed crystals), and one heterochiral one. In each case, the higher
order term of the BCH expansion, which is associated with finite pitch length,
contributes significantly at about 10 nm distance. We have shown that it must
be included to obtain physically consistent results in the heterochiral case.
At larger distances, the angular dependence of the torque turns out to change
as a function of distance, while again significantly deviating from the known
sinusoidal behavior. Moreover, for certain values of the misalignment angle,
the torque can change sign as a function of distance. Present technology does
not yet make it possible to reach a sufficient degree of parallelism to
measure the Casimir torque between two liquid crystals with satisfactory
accuracy. [14] However, present efforts directed at a resolution of this
problem [8] give us hope that it may be possible in the future to measure the
torque for such a case.

Possible future endeavors include calculations beyond the approximations used
here. For example, one could allow discontinuity in the inhomogeneous
dielectric function. This would make it possible to study the effect of a thin
water layer between two layers of the cholesteric liquid crystal.
Alternatively, one could explore a larger distance range in which case one
needs to investigate the effects of a finite total thickness. Finally, the
approximate solution of the Maxwell equations obtained here may be of interest
for the study of cholesterics in different contexts, or for the understanding
of chiral media in general.

###### Acknowledgements.

We acknowledge funding from the Key project #12034019 of the National Natural
Science Foundation of China (WB and RP) and the Singapore Ministry of
Education Academic Research Fund Tier 1 Grant No. RG160/19(S) (B-SL).

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