,

# Quantifying phases in homogenous twisted birefringent medium

Dipti Banerjee∗ and Srutarshi Banerjee ∗<EMAIL_ADDRESS><EMAIL_ADDRESS>$*$Department of Physics,Vidyasagar College for
Women
39, Sankar Ghosh lane,Kolkata-700006, West Bengal,INDIA  $**$Department of
Electrical Engineering,Indian Institute of Technology
Kharagpur-721302,West Bengal,INDIA

(12.06.12)

###### Abstract

The internal birefringence of an optical medium develops the dynamical phase
through natural rotation of incident polarized light. The uniform twist of the
medium induces an external birefringence in the system.This can be visualized
through the geometric phase by the solid angle in association with the angular
twist per unit thickness of the medium $k$.An equivalent physical analysis in
the $l=1$ orbital angular momentum sphere also has been pointed out.

keywords: Birefringent,geometric phase.
Pacs code:$42.25.Bs$

The theory of twisted birefringent material was developed long back by Gennes
paper1  and Chandrasekhar paper2  in connection with optics of cholesteric and
twisted nematic liquid crystals.In birefringent media three kinds of twists
are studied: (i) the limit of twist pitch is very short with respect to the
optical wavelength paper3  (ii) the twist pitch is comparable to the optical
wavelength that is solved by the previous approach. (iii) the very long twist
pitch is known as Geometric Optics Approximation or the Maguin paper4  limit.
Much later Jones paper5 ,paper6 , attracted our attention in formulating the
transformation through optical elements such as linear and circular polarizer,
retarder, rotator,etc arranged in proper sequence by the method of $(2\times
2)$ matrix. Azzam studied later by differential matrix formalism paper7  the
anisotropic behavior of the optical medium with and without depolarization.

The property of birefringence develops quantum phases in the optical material.
The appeared phases may be either dynamical or geometric or mixture of
both.There are four types of Geometric Phases (GP) that have been reported in
optics so far: i)The first identified GP is Pancharatnam phase paper8
,$\Omega/2$, where $\Omega$ is the solid angle enclosed by the path on the
Poincare sphere. Berry explained the quantal counterpart paper9  of
Pancharatnam’s phase in case of cyclic adiabatic evolution. He also studied
the phase two-form (GP) paper10  in connection with the dielectric tensor of
the optical medium. ii) The second kind of phase was experimentally performed
by Chaio and co-workers paper11  when the light with fixed polarization slowly
rotate around a closed path with varied directions. The developed GP was the
spin-redirection or coiled light phase. iii) The third one was developed by
the squeezed state of light through the cyclic changes by Lorentz
transformation paper12 . iv) The fourth GP was studied by van Enk paper13 , in
case of cyclic change of transverse mode pattern of Gaussian light beam
without affecting the direction of propagation or the polarization of
light.Bhandari paper14  studied the details of geometric phase in various
combination of optical material. Berry et.al. paper15  studied the phase two-
form (GP) of the twisted stack of birefringent plates.

The physical mechanism of these different kind of geometric phases originate
from spin or orbital angular momentum of polarized photon.The first
observation of the angular momentum of light was performed by Beth paper16
through an experiment where a beam of right circularly polarized light was
passed through a birefringent medium (quarter-wave plate) and transformed to
left circularly polarized light.Indeed as pointed out by Enk paper13  that
Pancharatnam phase in mode space is associated with spin angular momentum
transfer of light and optical medium.The orbital angular momentum GP of
polarized photon has been studied experimentally by Galvez et.al.paper17  in
mode space and theoretically by Padgget paper18  in Poincare sphere.In recent
days the OAM beams are generated by a kind of birefringent plate known as
”q-plates”, which are very fruitful application in classical and quantum
regime.In an interesting method the physics of OAM beams by ”q-plates” has
been developed by Santamato et.al. paper19  recently.

All these recent findings indicate that our previous study on the GP of
polarized photon (passing through polarization matrix M and rotator) in
connection with helicity, was an obvious new representation paper20
.Explicitly with the spinorial representation of polarized photon by spherical
harmonics, we consider paper21  that as the light gets a fixed polarization,
its helicity in connection with the spin angular momentum also fixed. It
varies along with the rotation of plane of polarization of light.We have
expanded this idea by finding the dielectric matrix and isolating GP in terms
of helicity of polarized photon paper22 .In this paper the properties of the
birefringent medium are visualized through the dynamical and geometric phases.
Here it has been assumed that the polarized photon is passed through the
uniformly twisted medium having very long twist pitch nearly at the Maguin
limit.In the next section the Jones matrix representation of birefringent
medium is reviewed while in section-2 the dynamical and geometric phase for
slow uniform angle of twist has been evaluated.

##  I The matrix representation of birefringent medium

An optical element could change the polarization of the incident polarized
light. Jones developed the effect by representation of $2\times 2$ matrix
paper5

$\vec{D}^{\prime}=M\vec{D}$ (1)

If the input polarization of light is unaltered after going through an optical
medium, then the state can be identified as the eigenvector $\vec{D}_{i}$ of
the optical component $M_{i}$ following the eigen value equation

$M_{i}\vec{D}_{i}=d_{i}\vec{D}_{i}$ (2)

where $d_{i}$ is the corresponding eigenvalue of a particular polarization
matrix $M_{i}=\pmatrix{m_{1}&m_{4}\cr m_{3}&m_{2}}.$
The optical properties such as birefringence and dichroism of a homogeneous
medium can be described by the differential matrix $N$.Jones paper6  pointed
out that $N$ governs the propagation of the polarized light vector
$\varepsilon$ at a particular wavelength through an infinitesimal distance
within the optical element

$\frac{d\varepsilon}{dz}=\frac{dM}{dz}\varepsilon_{0}=\frac{dM}{dz}M^{-1}\varepsilon=N\varepsilon$
(3)

where it is evident that $N$ is the operator that determines $dM/dz$ from
polarization matrix $M$ as follows

$N=\frac{dM}{dz}M^{-1}=\pmatrix{n_{1}&n_{2}\cr n_{3}&n_{4}}$ (4)

If $N$ is independent of z, it shows its dependence on polarization matrix M
paper6  by

$M=M_{0}\exp(\int{Ndz})$ (5)

In the lamellar representation suggested by Jones paper6 , a thin slab of a
given medium is equivalent to a pile of retardation plates and partial
polarizers. Eight constants are required to specify the real and imaginary
part of the four matrix elements of N matrix ($2\times 2$), each possessing
one of the eight fundamental properties.The eight optical properties can be
paired paper7  for following four properties.
i) Isotropic refraction and absorption
ii) Linear birefringence and linear dichroism along the xy coordinate axis.
iii) Linear birefringence and linear dichroism along the bisector of xy
coordinate axes.
iv) Circular birefringence and circular dichroism.
The optical medium that has circular birefringence and linear birefringence
will be our point of interest paper7 , and could have the following matrix
form

$\displaystyle\theta_{cb}=\tau\pmatrix{0&-1\cr 1&0}$ (6)
$\displaystyle\theta_{lb}=\rho\pmatrix{0&-i\cr i&0}$ (7)

These $\theta_{cb}$ and $\theta_{lb}$ matrices form the required differential
matrix.

$\displaystyle
N=\theta_{cb}+\theta_{lb}=\pmatrix{0&-\tau-i\rho\cr\tau+i\rho&0}=\pmatrix{0&n_{2}\cr
n_{3}&0}$ (8)

where, $\tau$ is the circular birefringence that measures the rotation of the
plane polarized light per unit thickness and $\rho$ is the part of linear
birefringence that measures the difference between the two principal constants
along the coordinate axes.

The evolution of the ray vector
$\varepsilon={\varepsilon_{1}\choose\varepsilon_{2}}$ as in eq.(3) passing
through such medium $N$, could be re-written as

$\displaystyle\frac{d\varepsilon_{1}}{dz}=n_{1}\varepsilon_{1}+n_{2}\varepsilon_{2}$
(9)
$\displaystyle\frac{d\varepsilon_{2}}{dz}=n_{3}\varepsilon_{1}+n_{4}\varepsilon_{2}$
(10)

For pure birefringent medium represented by eq.(8), one may use the evolution
of ray vector

$\frac{d\varepsilon_{1}}{dz}=n_{2}\varepsilon_{2},\\\
\frac{d\varepsilon_{2}}{dz}=n_{3}\varepsilon_{1}$ (11)

which implies that the spatial variation of component of electric vectors in
one direction give the effect in the other perpendicular direction. Thus an
exchange of optical power between the two component states of the polarized
light takes place indicating the rotation of the ray vector after entering the
medium.

Geometrically this state $\varepsilon$ is a point P on the surface of the
Poincare sphere that defines a position vector $\vec{p}$ in three dimensional
space. Huard pointed out paper23  that the evolution of the vector $\vec{p}$
is equivalent to the cyclic change of the state vector during the passage of
infinitesimal distance dz of the optical medium. The spatial change of vector
as it passes through the crystal becomes

$\frac{d\vec{p}}{dz}=\vec{\Omega}\times\vec{p}$ (12)

A natural twist for an elementary angle $d\alpha=\Omega dz$ is experienced by
the instantaneous vector $\vec{p}$ for thickness $dz$. The magnitude and
direction of the rotation vector depends on thickness and inherent property of
the optical medium.

Jones pointed out paper6  that as homogeneous birefringent crystal is
uniformly twisted about the direction of transmission, the $N$ matrices are
transformed upon rotation both for angle of twist dependent and independent of
crystal thickness.In the former case, Jones mentioned the twisted matrix
$N^{\prime}$ in terms of untwisted matrix $N_{0}$ and rotation matrix.

$N^{\prime}=S(\theta)N_{0}S(-\theta)$ (13)

For angle of twist independent on crystal thickness,Jones showed that paper6
the twisted matrix $N^{\prime}$ becomes

$N^{\prime}=N_{0}-kS(\pi/2)$ (14)

where $S(\pi/2)=\pmatrix{0&-1\cr 1&0\cr}$ denotes the rotation matrix for
normal incidence of light. The solution of the above eqs.(13) and (14) is
$\varepsilon^{\prime}=\exp(N^{\prime}z)\varepsilon^{\prime}_{0}$ where
$\varepsilon^{\prime}_{0}$ is the value of the vector $\varepsilon^{\prime}$
at $z=0$. We realize here from the definition of angle of twist per unit
distance that k has similarity with $\Omega$ in eq.(12). The basic difference
lies in their space of appearance where the former $(k)$ exist in the external
and later $(\Omega)$ in the internal space.

In the next section we will point out that the nature of quantum phases will
depend on the kind of angle of twist. Due to the property of inherent
birefringence represented by $\eta$, a natural twist is realized by the
incident polarized light acquiring dynamical phase. External twist of the
optical medium develops further external birefringence of the medium by k that
could be visualized by geometric phase(GP). It may be noted that GP will
differ as and when the angle of twist is independent or dependent on the
crystal thickness. From the work of Santamato paper19  one can realize that in
former case GP is visualized in OAM sphere.

##  II Quantum phases by twisting homogeneous birefringent medium

A polarized light traveling in the z direction can be written as a two
component spinor

$|\psi>={\psi_{+}\choose\psi_{-}}$ (15)

in terms of the electric displacement vector $d_{x},d_{y}$ where
$\psi_{\pm}=(d_{x}\pm id_{y})/\sqrt{2}$ . Berry paper9  pointed out that the
polarization matrix $M$ satisfying $M|\psi>=1/2|\psi>$, can be determined from
the eigenvector $|\psi>$ using the relation $(|\psi><\psi|-1/2)$.

From the spherical harmonics, the eigenvector

$|\psi>={{Y_{1/2}}^{-1/2,1/2}\choose{Y_{1/2}}^{-1/2,-1/2}}={\cos\frac{\theta}{2}\exp
i(\phi+\chi)/2\choose\sin\frac{\theta}{2}\exp-i(\phi-\chi)/2}$ (16)

can be considered here omitting the phase factor $\exp-i(\phi-\chi)/2$, from
above $|\psi>$ as

$|\psi>={\cos\theta/2e^{i\phi}\choose\sin\theta/2}$ (17)

In view of Berry paper9  the polarization matrix could be expressed as

$\left.\begin{array}[]{lcl}M(r)&=&\frac{1}{2}{\psi_{+}\choose\psi_{-}}(\psi_{+}~{}~{}~{}\psi_{)}-\frac{1}{2}\\\
&=&\frac{1}{2}\pmatrix{{\psi_{+}\psi_{+}}&{\psi_{+}\psi_{-}}\cr{\psi_{-}\psi_{+}}&{\psi_{-}\psi_{-}}}-\frac{1}{2}\pmatrix{1&0\cr
0&1}\\\
&=&\frac{1}{2}\pmatrix{{\psi_{+}\psi_{+}-1}&{\psi_{+}\psi_{-}}\cr{\psi_{-}\psi_{+}}&{\psi_{-}\psi_{-}-1}}\end{array}\right\\}$
(18)

Representing each term by spherical harmonics paper21  as

${Y_{1}}^{1}\approx\psi_{-}\psi_{+}\approx{Y_{1/2}}^{-1/2,1/2}.{Y_{1/2}}^{-1/2,-1/2}$
${Y_{1}}^{-1}\approx\psi_{+}\psi_{-}\approx{Y_{1/2}}^{1/2,1/2}.{Y_{1/2}}^{1/2,-1/2}$

and

${Y_{1}}^{0}\approx(\psi_{+}\psi_{+}-1)\\\ or(1-\psi_{-}\psi_{-})\\\ \\\
\approx{Y_{1/2}}^{1/2,1/2}.{Y_{1/2}}^{-1/2,-1/2}$

$-{Y_{1/2}}^{-1/2,1/2}.{Y_{1/2}}^{1/2,-1/2}$
also

${Y_{1/2}}^{1/2,1/2}.{Y_{1/2}}^{-1/2,-1/2}\approx 1$

that results the polarization matrix in eq.(18) of the form

$\left.\begin{array}[]{lcl}M(r)&=&\frac{1}{2}\pmatrix{\cos\theta&\sin\theta
e^{-i\phi}\cr{\sin\theta e^{i\phi}}&{-\cos\theta}}\end{array}\right\\}$ (19)

Every elements of the above polarization matrix eq.(19) can be realized as the
product harmonics ${Y_{1}}^{1}$, ${Y_{1}}^{-1}$ and ${Y_{1}}^{0}$. This enable
one to realize here the polarization matrix for orbital angular momentum $l=1$
paper21 .

$M(r)=\frac{1}{2}\pmatrix{{Y_{1}}^{0}&{Y_{1}}^{1}\cr{Y_{1}}^{-}1&{Y_{1}}^{0}\cr}\\\
$ (20)

The polarization matrix for this case $(l=1)$ parameterized by $(\theta,\phi)$
lies on the conventional Poincare sphere and equivalent as OAM sphere for
$l=1$. Spin angular momentum (SAM) of polarized photon is associated with
optical polarization. Another parameter $\chi$ for helicity is included that
extend the Poincare sphere to $(\theta,\phi,\chi)$ whose picture is seen in
fig.1. SAM space is possible to realize by parameters $(\theta,\chi)$.There
are two eigenvalues of helicity operator $+1$ and $-1$ that correspond to the
right handed state (spin parallel to motion) and left handed state (spin
opposite to motion) respectively. Hence the parameter for helicity, $\chi$
changes with the change of polarization of light For every OAM sphere there
exist two SAM hemi-sphere. Since the eigen values of helicites for polarized
photons is $\pm 1$, the factor $1/2$ from the polarization matrix M has been
omitted. For higher OAM states $l=2,3..$, further study is needed to evaluate
the polarization matrix for a particular orbital angular momentum from the
respective product harmonics ${Y_{l}}^{m}$.

Figure 1: Spinorial representation of polarized photon

To study with helicity of photon the words of Berry paper10 , ”Photons have no
magnetic moment and so cannot be turned with a magnetic field but have the
property of helicity to use” is very supportive. With this view we will study
here further.

The property of birefringence of the optical medium can be represented by the
differential matrix $N$. At a particular position $z$ of optical medium, the
spatial variation of the polarization matrix $M$ becomes,

$N=(\frac{dM}{d\theta})(\frac{d\theta}{dz})M^{-1}$ (21)

Considering $z=cos\theta$, the thickness of the optical medium, the N matrix
can be obtained from M in eq.(19), where $\theta$ is the angular variable of
light after refraction.

$N=\eta\pmatrix{0&-e^{i\phi}\cr e^{-i\phi}&0\cr}$ (22)

This N matrix has complex eigenvalues $\pm i\eta$. Comparing with our previous
work paper21  it may be pointed out that $\eta$ has dependence on $\theta$
through the eigenvalue $1/sin\theta$ of the eigenvector,

$\left(\begin{array}[]{c}\pm ie^{i\phi}\\\ 1\end{array}\right)$ (23)

Thus N matrix will be different for different values of $\theta$. The
birefringence can not be measured if $\theta=0$ that makes $\eta=\infty$ where
for $\theta=\pi/2$ and $\pi/3$ the accepted values of $\eta$ will be $1$ and
$2$. Also its dependence on $\sin\theta$ indicate that $\eta$ might have only
non-negative values. The nature of the optical medium could be identified
comparing the above N matrix in eq.(22) with eq.(8). It is seen that developed
N matrix is homogenous possessing both circular and linear birefringence
represented by $\eta\cos\phi$ and $(-\eta\sin\phi)$ respectively.

When the angle of twist has dependence on the crystal thickness by
$\theta=kz$,following Jones paper6  the twisted matrix $N^{\prime}$ will be

${N}^{\prime}=\pmatrix{0&-\eta e^{i\phi}+k\cr\eta e^{-i\phi}-k&0\cr}$ (24)

The corresponding twisted ray will be obtained as the initial polarized light
suffers rotation in opposite direction of the twisted matrix $N^{\prime}$.

$\varepsilon^{\prime}=\pmatrix{\cos\theta&\sin\theta\cr-\sin\theta&\cos\theta\cr}\left(\begin{array}[]{c}ie^{i\phi}\\\
1\end{array}\right)$ (25)

in other words

$\varepsilon^{\prime}=\left(\begin{array}[]{c}ie^{i\phi}\cos\theta+\sin\theta\\\
-ie^{i\phi}\sin\theta+\cos\theta\end{array}\right)$ (26)

.

Light having fixed polarization and helicity if suffers the slow variation of
path in real space it can be mapped on to the surface of unit sphere in the
wave vector space. Slow twist pitch as considered here is comparable with the
Maguin limit paper4  in connection with the optics of cholesteric and twisted
nematic liquid crystal.

The effect of twist help to achieve the geometric phase by the initial state
$|A>$ of polarized light as it unite with the final $|A^{\prime}>$.

$<A|A^{\prime}>=\pm\exp(i\gamma(C)/2)$ (27)

where $\gamma$ is the solid angle swept out on its unit sphere.

This work is based on the consideration of polarized light passing normally
through a medium having linear and circular birefringence. The incident
polarized light suffers a natural twist due to inherent property of the
medium. As a result the dynamical phase $\gamma$ is developed in the optical
medium $N$. Further external twist with the consideration of
$d\theta=d\theta^{\prime}$, develops the phase $\gamma^{\prime}$ that has
dynamical and geometrical part too.

$\displaystyle\gamma=\varepsilon^{*}\frac{d\varepsilon}{d\theta}\frac{d\theta}{dz}={\varepsilon^{*}}N\varepsilon$
(28)
$\displaystyle\gamma^{\prime}={\varepsilon^{\prime}}^{*}\frac{d\varepsilon^{\prime}}{d\theta}\frac{d\theta}{dz}={\varepsilon^{\prime}}^{*}N^{\prime}{\varepsilon^{\prime}}$
(29)

The dynamical phase $\gamma$ can be obtained using eq.(22) and (23) in (28)

$\displaystyle\gamma={\varepsilon^{*}}N\varepsilon$ $\displaystyle=$
$\displaystyle{\varepsilon_{1}}^{*}n_{2}\varepsilon_{2}+{\varepsilon_{2}}^{*}n_{3}\varepsilon_{1}$
(30) $\displaystyle=$ $\displaystyle(-ie^{-i\phi})(-\eta e^{i\phi})+(\eta
e^{-i\phi})(ie^{i\phi})$ $\displaystyle=$ $\displaystyle 2i\eta$

Comparing this eq.(30) with (27), the developed dynamical phase $\gamma$
appears as imaginary term in exponent. It could be varied between $1\&2$ for
positive $\theta$ values through $1/\sin\theta$. Hence Fig.-2 shows the
uniform variation of this dynamical phase $\gamma$ with the natural
birefringence of the medium $\eta$. If $\theta$ becomes negative, the
corresponding $\eta$ shows negative value also.

Figure 2: Variation of dynamical phase $\gamma$

Any ray passing through $N$ or $N^{\prime}$ will suffer a natural twist due to
the internal dynamics of the birefringent medium. The polarized light passing
through the twisted medium $N^{\prime}$ will acquire the phase
$\gamma^{\prime}$, that has two parts, one from the dynamics and another
through the parametric change of the medium. The phase $\gamma^{\prime}$ will
contain both the dynamical and geometric phase in the exponent. To grasp the
geometric phase due to external twist one could make the difference
$\gamma^{\prime}-\gamma$ to eliminate the dynamical phase developed.

Figure 3: Variation of Net Phase $\gamma^{\prime}$ with $\eta$ and k

To calculate the net quantum phase $\gamma^{\prime}$ appeared after a twist,
the twisted matrix $N^{\prime}$ in eq.(24) is used which intuitively will act
on the twisted light ray $\varepsilon^{\prime}$ in eq.(26).

Figure 4: Variation of $\gamma^{\prime}$ with k and $\cos(\phi)$.

$\displaystyle\gamma^{\prime}={{\varepsilon^{\prime}}_{1}}^{*}{n^{\prime}}_{2}{\varepsilon^{\prime}}_{2}+{\varepsilon_{2}}^{*}{n^{\prime}}_{3}{\varepsilon^{\prime}.}_{1}$
(31) $\displaystyle=$ $\displaystyle(-ie^{-i\phi}\cos\theta+\sin\theta)(k-\eta
e^{i\phi})(-ie^{i\phi}\sin\theta+\cos\theta)$ $\displaystyle+$
$\displaystyle(ie^{-i\phi}\sin\theta+\cos\theta)(\eta
e^{-i\phi}-k)(ie^{i\phi}\cos\theta+\sin\theta)$ $\displaystyle=$
$\displaystyle i[2\eta{\sin}^{2}\theta\cos 2\phi-2k\cos\phi]$

The above phase $\gamma^{\prime}$ at $\theta=\pi/2$ becomes

$\gamma^{\prime}=2i[\eta\cos 2\phi-k\cos\phi]$ (32)

Here the respective variation of net phase $\gamma^{\prime}$ with $\eta$ and k
are shown in fig.3 and in fig.4.

Figure 5: Variation of $\Gamma$ with $\eta$ and k.

As the dynamical phase is independent of twist angle $\theta$,the geometric
phase in case of $\theta=\pi/2$ could be recovered by

$\Gamma=\gamma^{\prime}-\gamma=i[2\eta(\sin^{2}\theta\cos
2\phi-1)-2k\cos\phi]$ (33)

Figure 6: Variation of $\Gamma$ with $\cos\phi$ and k.

in other words

$\Gamma=i\eta[(1-\cos 2\theta)\cos 2\phi-2]-2ik\cos\phi$ (34)

Similar nature of curve is seen for the net phase $\gamma^{\prime}$ and
geometric phase $\Gamma$. Fig.5 and Fig.6 shows the variation of $\Gamma$ with
$\eta$ and k and $\Gamma$ with $\cos\phi$ and k respectively.

Figure 7: Variation of $\Gamma_{1}$ with $\cos\phi$ and k.

It may be noted that $\eta$ and $k$ parameters have dependence on the angle of
incidence $\theta$ and angle of external twist $\theta^{\prime}$ respectively.
Here for simplicity the two angles ($\theta=\theta^{\prime}$) are considered
equal. The angle $\phi$ is associated with the natural(internal) twist of the
light ray inside the medium. The three types of phases
$\gamma,\gamma^{\prime}$ and $\Gamma$ could be studied graphically with
respect to variation of k and $\eta$. At normal incidence $\theta=\pi/2$,the
GP will be

$\Gamma=i[2\eta(\cos 2\phi-1)-2k\cos\phi]$ (35)

At $\theta=0$ angle of twist,GP has been identified as $\Gamma_{1}$

$\Gamma_{1}=-2i[\eta+k\cos\phi]$ (36)

Fig.7 and 8 shows the variation of GP at $\theta=0$ with respective parameters
$\cos\phi$, $\eta$ and k.

Figure 8: Variation of $\Gamma_{1}$ with $\eta$ and k.

The property of circular birefringence of the medium visualized by the
parameter $\eta$ makes a natural twist to the incident light and for appearing
dynamical phase. The external twist of the optical medium though is associated
with $k$, the developed geometric phase has its dependence on $\eta$ also.
Graphical analysis shows that the presence of external birefringence
introduces a spiral behavior of the geometric phase. To realize GP in the OAM
and SAM space the respective parameters $\phi$ and $\chi$ are responsible.

We like to extend our study using circularly polarized light incident on the
birefringent medium.Let us identify the LCP and RCP of polarization of light
by

$|L>={1\choose i},~{}~{}~{}~{}~{}|R>={1\choose-i}$ (37)

The $|L>$ and $|R>$ states gives rise to the dynamical phases for up polarized
photon

$\displaystyle\gamma_{L}={<L|N|L>}=-2i\eta\cos\phi$ (38)
$\displaystyle\gamma_{R}={<R|N|R>}=2i\eta\cos\phi$ (39)

In case of twisted birefringent crystal, incidence of left circularly
polarized light (LCP) on $N^{\prime}$ develop the phase $\gamma^{\prime}$

$\begin{array}[]{lcl}{\gamma^{\prime}}_{L}=<L|N^{\prime}|L>\\\
=(1-i)e^{-i\theta}\pmatrix{0&-\eta e^{i\phi}+k\cr\eta
e^{-i\phi}-k&0\cr}{1\choose i}e^{i\theta}\\\ =2ik-2i\eta\cos\phi\end{array}$
(40)

that consists both the dynamical and geometrical phases. The later(GP) could
be isolated by $\Gamma=\gamma^{\prime}-\gamma$

$\Gamma=2ik-2i\eta\cos\phi+2i\eta\cos\phi=2ik$ (41)

Here it seen that for circularly polarized light the dynamical phase and
geometric phase depend only on the external birefringence k and the internal
birefringence $\eta$ respectively.

Discussion:

In this communication two types of birefringence internal and external are
studied. Due to the property of inherent birefringence represented by $\eta$,
a natural twist is realized by the incident polarized light acquiring
dynamical phase. The dynamical phase $\gamma$ in all cases varies linearly
with internal birefringence $\eta$ of the medium. External twist of the
optical medium develops further external birefringence of the medium by k that
could be visualized by geometric phase(GP). It may be noted that GP will
differ as and when the angle of twist is independent or dependent on the
crystal thickness. GP has dependence on both the internal and external
birefringence when eigen polarized light is passed through the twisted optical
medium. It depends completely on the external birefringence k of the optical
medium for the passage of left or right circularly polarized light. Further
could be noted that if $\phi=0$ the value of geometric phase for the two types
of polarized light becomes identical. In future we wish to study the twisted
optical medium having the property of dichroism.

Acknowledgement: This work had been supported by the Abdus Salam International
Centre for Theoretical Physics (ICTP), Trietse, Italy. Correspondence with
Prof.Santamato,Napoli,Italy is gratefully acknowledged. Also the help from
Mr.S.Bhar, colleague of DB at VCFW (Department of Physics) is acknowledged.

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# Exact Solution to the Quantum and Classical Dimer Models on the Spectre
Aperiodic Monotiling

Shobhna Singh  School of Physics and Astronomy, Cardiff University, The
Parade, Cardiff CF24 3AA, United Kingdom  H. H. Wills Physics Laboratory,
Tyndall Avenue, Bristol, BS8 1TL, United Kingdom  Felix Flicker  H. H. Wills
Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom

###### Abstract

The decades-long search for a shape that tiles the plane only aperiodically
under translations and rotations recently ended with the discovery of the
‘spectre’ aperiodic monotile. In this setting we study the dimer model, in
which dimers are placed along tile edges such that each vertex meets precisely
one dimer. The complexity of the tiling combines with the dimer constraint to
allow an exact solution to the model. The partition function is
$\mathcal{Z}=2^{N_{\textrm{Mystic}}+1}$ where $N_{\textrm{Mystic}}$ is the
number of ‘Mystic’ tiles. We exactly solve the quantum dimer (Rokhsar
Kivelson) model in the same setting by identifying an eigenbasis at all
interaction strengths $V/t$. We find that test monomers, once created, can be
infinitely separated at zero energy cost for all $V/t$, constituting a
deconfined phase in a 2+1D bipartite quantum dimer model.

The dimer model is one of the oldest models in statistical physics. Given a
graph (vertices connected by edges), a ‘perfect dimer matching’ is a set of
edges (dimers) such that each vertex connects to precisely one member of the
set. The dimer model then considers the set of all perfect matchings. The
model characterises a wide range of physical processes including adsorption
[1, 2, 3, 4, 5], zero modes in electronic tight binding models [6, 7, 8], and
magnetism, where dimers are used for example in analytic approaches to the
Ising model [9]. The _quantum_ dimer model (QDM), also called the Rokhsar
Kivelson model, introduces quantum superpositions of dimer placements [10,
11]. Originally introduced to capture the physics of resonating valence bond
states [12] in theories of high-temperature superconductivity [10], QDMs are
now understood to host a range of exotic phenomena such as quantum spin
liquids, topological order, and fractionalisation [11, 13]. They have recently
been realised experimentally in programmable quantum simulators [14, 15].

The utility of classical dimer models derives in part from an efficient method
(the ‘FKT algorithm’) for enumerating perfect matchings developed by Fisher,
Kasteleyn, and Temperley [1, 2, 3, 4, 5]. The result permits an exact solution
to any $N$-vertex planar dimer model in the form of the partition function:

$\displaystyle\mathcal{Z}_{N}\left[w\right]=\sum_{\mathcal{M}_{i}\in\mathcal{M}}\,\,\prod_{e\in\mathcal{M}_{i}}w\left(e\right).$
(1)

Here, $\mathcal{M}_{i}$ is a perfect matching in the set of all perfect
matchings $\mathcal{M}$, and $e$ are the edges, of the graph. Setting weight
$w=1$ on all edges, $\mathcal{Z}\left[1\right]$ counts the number of perfect
matchings. From $\mathcal{Z}$ all thermodynamic functions of state immediately
follow. Of particular interest is the free energy per dimer [16] in an
$N$-vertex graph

$\displaystyle
f_{N}\left[w\right]=\frac{1}{N/2}\ln\left(\mathcal{Z}_{N}\left[w\right]\right).$
(2)

For certain regular graphs admitting periodic embeddings, $\mathcal{Z}$ has
been evaluated analytically [4, 5, 16]. Owing to the importance of graph
connectivity to the behaviour of dimer models, they have recently begun to be
studied on infinite graphs with aperiodically ordered planar embeddings [17,
18, 19, 8]. Such graphs, which capture the symmetries of physical
quasicrystals [20, 21], are irregular, meaning their vertices meet different
numbers of edges, typically leading to a large degree of frustration in dimer
arrangements. They admit planar embeddings which are long-range ordered,
meaning their diffraction patterns feature sharp Bragg peaks [20, 21], despite
lacking a discrete translational symmetry. Examples include the graph version
of the Penrose tiling [22, 17] and the Ammann-Beenker tiling [23, 18, 19]. The
long range order often permits analytical results; for example, in a
modification of the Ammann Beenker tiling an exact solution to the dimer model
can be approximated to arbitrary accuracy using transfer matrices [18].

This year saw a major advance in the study of aperiodic tilings with the
discovery of the ‘spectre’ aperiodic monotile [24]. The spectre positively
answered the decades-old question of whether there exists a shape that tiles
the plane only aperiodically under translations and rotations [24]. Spectre
tilings, either finite or infinite, can be created by the ‘composition rules’
in Fig. 1, reproduced from Ref. 24, in which each tile is replaced with copies
of itself so as to build a larger tiling. Each tile becomes its mirror image
under composition, meaning that all tiles have the same chirality after each
composition.

Here we provide an exact analytical solution to both the classical and quantum
dimer models on spectre tilings. We treat the vertices and edges of the tiles
as those of a graph. Since we are only concerned with graph connectivity, we
straighten the curved edges of the spectre tiles (resulting in what is termed
‘Tile(1,1)’ in Refs. [25, 24]). Each spectre tile, labelled $S_{0}$, can have
either 13 or 14 edges depending on its environment [24]. To ensure that all
tiles are identical at the level of graph connectivity we add a vertex (gold
in Fig. 1) to any 13-edge tiles. This makes the graphs bipartite, meaning
vertices divide into two sets such that edges only connect vertices in
different sets. We discuss the non-bipartite case briefly at the end.

_Results (classical)_ — Starting from a single spectre $S_{0}$, a finite
number of compositions $\mathcal{N}$ generates a finite connected tile set
$S_{\mathcal{N}}$, while an infinite number of compositions results in a
tiling of the Euclidean plane [24]. Even though each tile has 14 vertices, the
total number of vertices can still be odd; in such cases the number of perfect
matchings is zero, since a dimer must connect a pair of vertices. By
construction, any tiling built by composition can also be seen as a
concatenation of the once-composed tiles $S_{1}$ and $M_{1}$ (Fig. 1). The two
green spectres in Fig. 1 together make up a tile called the ‘Mystic’, which we
denote $M_{0}$. We term the dark green tile the ‘Upper Mystic’ $M_{0}^{+}$. It
plays a special role as the only entirely internal tile in $S_{1}$ and
$M_{1}$. It is also marked out as special by appearing $\pi/6$ rotated from
the other tiles, which appear only $\pi/3$ rotated amongst themselves (Fig.
1). The Mystic $M_{0}$ contains four internal edges. Of these four, exactly
one of the two central edges must be covered by a dimer in any perfect
matching. Either choice forces a range of other dimers. Choosing the red dimer
in Fig. 1 forces all the pink and purple dimers; instead choosing the blue
dimer forces all the light blue and purple dimers. The purple dimers are the
same in both cases, and so these edges are always covered in any perfect
matching. In fact the figure demonstrates that every non-boundary edge of
$S_{1}$ and $M_{1}$ either lies on $M_{0}^{+}$, or has a fixed dimer
occupation as shown. The only degrees of freedom are the two dimer matchings
per $M_{0}^{+}$ (and therefore per $M_{0}$), or possibly the boundaries of
$S_{1}$ or $M_{1}$ within larger regions. From now on we focus on
$S_{\mathcal{N}}$ regions unless otherwise stated.

Figure 1: The composition rules for the spectre tiling (after Ref. 24). The
vertices of the spectre $S_{0}$ are indicated; the gold vertex is added
whenever it is not implied by the meeting of tiles. The two once-composed
tiles $S_{1}$ and $M_{1}$ can be pieced together without overlaps to construct
the infinite aperiodic tiling. Note that composition mirrors $S_{0}$ tiles in
such a way that only one chirality appears at any level of composition. The
Mystic $M_{0}$ is the two green tiles (the darker tile being the Upper Mystic
$M_{0}^{+}$). Of the four internal edges of $M_{0}$ either the red or dark
blue dimer must appear in any perfect matching. Choosing red, all pink and
purple dimers are forced. Choosing blue, all light blue and purple dimers are
forced. The only freedom on internal $S_{1}$ and $M_{1}$ edges is therefore
two dimer matchings per Mystic (these cases are shown at the bottom, labelled
$|r\rangle$ (red) and $|b\rangle$ (blue) for convenience in the quantum
model). The gold vertex appears only on the boundaries of $S_{1}$ and $M_{1}$,
so does not affect this argument.

The only dimer within either $S_{1}$ or $M_{1}$ to meet a boundary vertex
appears on $M_{1}$ (vertex circled in green in Fig. 1). Fig. 2 shows the
twice-composition of $S_{0}$, which we term $S_{2}$. The special dimer has
been highlighted in gold. It forces the two closest green dimers, which in
turn force every other green dimer. The result is that all internal edges of
$S_{2}$ not on Upper Mystics are again constrained.

In fact this behaviour is generic for $S_{\mathcal{N}}$ regions. Ref. 24 lists
all possible ways in which $S_{1}$ and $M_{1}$ can meet. The boundaries of
$S_{1}$ and $M_{1}$ consist only of bivalent or trivalent vertices. The bottom
vertex of the Mystic (the rightmost of the two lowest vertices of $M_{0}$ in
Fig. 1) meets the boundary of $S_{\mathcal{N}}$ exactly once. In all other
cases it appears internally. It does so at a trivalent vertex connecting three
regions and touches the (gold) boundary dimer of $M_{1}$. Since a dimer
meeting a trivalent vertex forces the absence of dimers on both other edges,
the existence of even a single forced dimer along the network of $S_{1}$ and
$M_{1}$ boundaries is enough to force all remaining dimer placements. The only
exception is a twofold freedom along the boundary of the entire connected tile
set $S_{\mathcal{N}}$ (only relevant for finite tile patches). Hence, the
total number of dimer matchings is

$\displaystyle\mathcal{Z}_{N}\left[1\right]=2^{N_{\textrm{Mystic}}+1}$ (3)

where $N_{\textrm{Mystic}}$ is the number of Mystic tiles $M_{0}$ (equal to
the number of Upper Mystic tiles $M_{0}^{+}$).

Figure 2: Twice-composition of the spectre, $S_{2}$. The mirror of $M_{1}$
(Fig. 1) is highlighted in pink. The gold dimer (highlighted with an arrow)
reaches the boundary of $M_{1}$, and forces all green dimers. The only freedom
in dimer placements is the twofold choice on Upper Mystic tiles, and on the
boundary. Hence the number of perfect matchings is
$2^{N_{\textrm{Mystic}}+1}$.

In the thermodynamic limit $S_{\mathcal{N}\rightarrow\infty}$, for which the
number of vertices $N\rightarrow\infty$, the free energy per dimer is

$\displaystyle f_{\lim
N\rightarrow\infty}\left[1\right]=\frac{\textrm{ln}\left(2\right)}{3\left(5+\sqrt{15}\right)}\approx
0.02604$ (4)

(see Appendix A). To confirm this result we exactly calculated the free energy
per dimer numerically in finite patches $S_{2}$ to $S_{6}$ using the FKT
algorithm [1, 3]. The results, shown in Fig. 3, converge towards the
analytical result. The convergence is slow owing to the fractal boundary of
$S_{\mathcal{N}\rightarrow\infty}$, so we also show the result of a series
acceleration method [26] (Appendix B) which gives a rapid convergence.

Figure 3: The free energy per dimer $f_{N}$ for patches of spectres containing
$N$ vertices. Square points represent compositions $S_{2}$ to $S_{6}$. The
dashed line shows the analytical result of Eq. 4, valid for
$N\rightarrow\infty$. Convergence is slow owing to the fractal boundary of
$S_{\mathcal{N}\rightarrow\infty}$; green circles represent a series
acceleration (see Appendix B).

The free energy per dimer in the spectre tiling, Eq. (4), is significantly
smaller than values obtained in periodic lattices [16], e.g. the square
(0.583), honeycomb (0.323), triangular (0.857), and Kagomé (0.462) lattices.
This fits with the observation that all bulk dimers, other than those on
$M_{0}^{+}$, are completely constrained.

_Results (quantum)_ — The QDM can be defined on $S_{\mathcal{N}}$ by replacing
the square tiles (plaquettes) of Ref. 10 with $S_{0}$ tiles. Explicitly, on
any spectre $S_{0,i}$ we can define $|r_{i}\rangle$ and $|b_{i}\rangle$ to be
the quantum states with the red and blue dimer placements in Fig. 1
respectively. The Hamiltonian then reads

$\displaystyle\\!\\!\\!\\!\\!\\!\hat{H}\\!\\!=\\!\\!\\!\\!\\!\sum_{S_{0,i}\in
S_{\mathcal{N}}}\\!\\!\\!\\!\\!-t\left(|r_{i}\rangle\langle
b_{i}|+|b_{i}\rangle\langle r_{i}|\right)+V\left(|r_{i}\rangle\langle
r_{i}|+|b_{i}\rangle\langle b_{i}|\right)$ (5)

where $t$ and $V$ are real and $t$ is positive. The terms weighted by $-t$ can
be thought of as defining a kinetic energy operator which enacts ‘flips’
$|r\rangle\leftrightarrow|b\rangle$. The terms weighted by $V$ define a
potential energy operator which counts ‘flippable’ plaquettes of the form
$|r\rangle$ or $|b\rangle$.

Eq. (5) is well studied in the square lattice, where $|r_{i}\rangle$ denotes
two vertical dimers, and $|b_{i}\rangle$ two horizontal dimers, on square $i$.
The so-called Rokhsar Kivelson (RK) point $t=V$ separates ordered phases with
different symmetries. The order is set by the sign of $V/t$ which either
attempts to maximise or minimise the number of flippable plaquettes [10, 11,
27]. In contrast, on the spectre tiling $S_{\mathcal{N}}$ the only flippable
plaquettes are $M_{0}^{+}$ tiles. Their number is entirely fixed. This heavy
constraint decouples the problem into one of matching independent $M_{0}^{+}$
tiles with quantum dimers. Each tile $M_{0,i}^{+}$ admits two energy
eigenstates which we denote:

$\displaystyle|\pm_{i}\rangle=\left(|r_{i}\rangle\pm|b_{i}\rangle\right)/\sqrt{2}$
(6)

with corresponding energies $V\mp t$. The ground state of Eq. (5) is therefore

$\displaystyle\hat{H}\prod_{M_{0,i}^{+}}|+_{i}\rangle=(V-t)N_{\textrm{Mystic}}\prod_{M_{0,i}^{+}}|+_{i}\rangle.$
(7)

All excited states can be formed by swapping individual $|+_{i}\rangle$ for
$|-_{i}\rangle$ at a cost of $2t$ energy per swap.

_Discussion_ — Removing a dimer from a perfect matching results in two
unmatched vertices. These can be thought of as particle-like ‘monomer’ defects
which can move independently of one another by dimer re-arrangements [11].
Specifically, each monomer lies at the end of an ‘alternating path’, a set of
edges alternately uncovered and covered by dimers. Switching which edges are
covered and uncovered moves the monomer along the path. The spectre again has
an interesting structure in this regard. Note for example that each green
alternating path in Fig. 2 terminates only on the boundary and the gold dimer.
The same structure holds for all $S_{\mathcal{N}}$. Deleting the gold dimer to
create a pair of monomers, one of the pair can move to the boundary along any
green path; in the thermodynamic limit it can escape to infinity. In fact any
test pair of monomers has this same feature: one will esape to the boundary,
and the other localises on an Upper Mystic $M_{0}^{+}$.

In QDMs on previously studied planar bipartite graphs, such as the square
lattice, the RK point $t=V$ constitutes a deconfined quantum critical point
between ordered phases [28]. Deconfinement means that test-monomers can be
separated to infinite distance at finite energy cost [29, 30, 11]. In general,
since QDMs on bipartite graphs map to compact (matter-free) quantum
electrodynamics [31, 29, 13], and since deconfined phases cannot exist in
compact 2+1D $U(1)$ gauge theories [32], the RK point cannot be part of a
deconfined phase existing over a range of $V/t$.

Remarkably, the behaviour in the (bipartite) spectre tiling appears to be at
odds with this statement. By the argument just given, any pair of test
monomers can be infinitely separated. Doing so preserves the number of
flippable plaquettes, so costs no energy according to Eq. (5) at any $V/t$.
Test monomers are therefore deconfined over all $V/t$. We suggest that the
result of Ref. 32 may survive because there seems to be no obvious mapping to
a compact $U(1)$ gauge theory, since the vertices in the spectre tiling
connect to variable numbers of edges. Another difference to previous studies
is that all previously known bipartite RK points were characterised by
algebraic dimer correlations [11]; spectre dimer correlations, being
completely uncorrelated between different $M_{0}^{+}$ tiles, are not algebraic
at any $V/t$.

QDMs on non-bipartite graphs behave qualitatively differently. Here, they _do_
admit deconfined phases spanning a continuous range of $V/t$ [33, 11, 13, 34],
and their emergent gauge field descriptions are $\mathbb{Z}_{2}$ rather than
$U(1)$ [35, 36, 13]. The spectre tiling can be made non-bipartite by omitting
the gold vertex in Fig. 1 whenever it is not forced by the tiling. Fig. 1
shows that gold vertices appear only on the boundaries connecting $S_{1}$ and
$M_{1}$ tiles, so some of the intuition developed here may hold in non-
bipartite spectre tilings. Nevertheless, preliminary checks suggest a more
complicated behaviour.

Figure 4: Removing the boundary-touching Mystic $M_{0}$ from region $S_{2}$
allows the freedom formerly localised to the boundary to move into the bulk.
Purple edges always receive a dimer, and black edges never receive a dimer, in
any perfect matching. Orange edges are free to either host a dimer or not.

Returning to the classical model, different weights $w$ can be assigned in Eq.
(1). For example, in the square lattice different weights might be assigned to
horizontal edges compared to vertical edges [1]. However, since aperiodic
tilings lack a unit cell, there is no obvious choice for assigning weights.
One option for $S_{\mathcal{N}}$ tilings is to assign different weights to
edges within regions $S_{1}$ and $M_{1}$ while assigning weights consistently
between different $S_{1}$ and $M_{1}$. Since dimer placements are fixed for
all internal edges other than $M_{0}^{+}$, the corresponding weights factor
out of the partition function. Those edges which never receive a dimer make no
contribution regardless of weight. The partition function can therefore also
be calculated in this more general case, with the sum being over weights of
edges appearing on $M_{0}^{+}$ or the boundary of the tiling.

It is interesting to consider what happens when tiles are added or removed
from the $S_{\mathcal{N}}$ regions while still obeying the spectre tiling
rules. The total number of vertices can become odd, as in region $M_{1}$, or
even but with an imbalance between the numbers of vertices in the bipartite
subgraphs. In both cases there are zero perfect matchings, since dimers
connect vertices on distinct bipartite subgraphs. Another possibility is a
monomer-free region as shown in Fig. 4. Precisely one $M_{0}$ touches the
boundary of any $S_{\mathcal{N}}$ region. Removing this $M_{0}$ from $S_{2}$,
as shown, causes boundaries of some internal $S_{1}$ and $M_{1}$ regions to
gain a degree of freedom (orange edges host 0 or 1 dimers). This region hosts
six perfect matchings excluding those localised around $M_{0}^{+}$. In general
there is a twofold degree of freedom around any graph cycle (closed loop of
edges) which connects to the rest of the graph only via edges not hosting a
dimer. This accounts for the freedom around $M_{0}^{+}$, the boundaries of
$S_{\mathcal{N}}$, and also these more complicated branching structures in
other tile patches.

The complexity of the spectre tiling leads to a number of surprising
simplifications in physical models, permitting exact results where periodic
(and other aperiodic) tilings do not. It remains to be seen if there is
anything deeper about the structure of the tiling which leads to this
simplicity.

Acknowledgments—The authors thank S. Franca and J. Schirmann for helpful
discussions, and A. G. Grushin, R. Moessner, P. d’Ornellas, M. A. Sánchez
Martínez, Z. Ringel, and J. van Wezel for helpful comments on the manuscript.
F.F. was supported by EPSRC grant EP/X012239/1.

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##  Appendix A Appendix A: Free energy per dimer – analytics

In this Appendix we calculate the free energy $f_{N}$ per dimer for the
classical dimer model in an $N$-vertex region generated by $\mathcal{N}$
compositions of a spectre tile. The partition function is

$\mathcal{Z}=2^{N_{\text{Mystic}}+1}$ (8)

and

$\displaystyle f_{N}$ $\displaystyle=\frac{\ln\left(\mathcal{Z}\right)}{N/2}$
(9) $\displaystyle=\frac{N_{\text{Mystic}}+1}{N/2}\cdot\ln\left(2\right).$
(10)

In the bulk of any spectre tiling there are 6 vertices per spectre, found by
summing the 14 vertices weighted by the internal angle the enclosed at each
vertex. The number of spectres in any region is
$N_{\text{Spectre}}+2N_{\text{Mystic}}$, since the Mystic contains two
spectres. Therefore

$N=6\left(N_{\text{Spectre}}+2N_{\text{Mystic}}\right)$ (11)

in the bulk. Using this expression for _all_ vertices will undercount the
vertices on the boundary, but this error will tend to zero in the
thermodynamic limit. The result is

$f_{N}=\frac{N_{\text{Mystic}}+1}{N_{\text{Spectre}}+2N_{\text{Mystic}}}\cdot\frac{\ln\left(2\right)}{3}.$
(12)

Fig. 1 shows the composition rules for the tiles. Counting the tiles gives the
following rule for the number of each tile type after $\mathcal{N}$
compositions starting from a single spectre:

$\left(\begin{array}[]{c}N_{\text{Spectre}}\left(\mathcal{N}\right)\\\
N_{\text{Mystic}}\left(\mathcal{N}\right)\end{array}\right)=\left(\begin{array}[]{cc}7&6\\\
1&1\end{array}\right)^{\mathcal{N}}\left(\begin{array}[]{c}1\\\
0\end{array}\right).$ (13)

The eigenvalues of this matrix are

$\lambda_{\pm}=4\pm\sqrt{15}.$ (14)

To approach the thermodynamic limit $\mathcal{N}\rightarrow\infty$
systematically, note that the eigenvalues obey

$\lambda^{2}=8\lambda-1$ (15)

from which it follows that

$\displaystyle\lambda^{3}$ $\displaystyle=63\lambda-8$ (16)
$\displaystyle\lambda^{4}$ $\displaystyle=496\lambda-63$ (17)
$\displaystyle\lambda^{5}$ $\displaystyle=3905\lambda-496$ (18)

and so on. In general

$\lambda^{n}=L_{n}\lambda-L_{n-1}$ (19)

where $L_{0}=1$, $L_{1}=8$. This generates OEIS A001090:

$L_{n}=1,\text{\,{\hbox{\,}}}8,\,\,63,\,\,496,\,\,3905,\,\,30744\ldots$ (20)

The first few terms in the series of Eq. 13 are

$\left(\begin{array}[]{c}N_{\text{Spectre}}\\\
N_{\text{Mystic}}\end{array}\right)=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\left(\begin{array}[]{c}7\\\
1\end{array}\right),\left(\begin{array}[]{c}55\\\
8\end{array}\right),\left(\begin{array}[]{c}433\\\
63\end{array}\right),\ldots$ (21)

from which we see that

$\displaystyle
N_{\text{Spectre}}\left(\mathcal{N}\right)+N_{\text{Mystic}}\left(\mathcal{N}\right)$
$\displaystyle=L_{\mathcal{N}}$ (22) $\displaystyle
N_{\text{Mystic}}\left(\mathcal{N}\right)$ $\displaystyle=L_{\mathcal{N}-1}.$
(23)

Substituting into Eq. 12 gives

$\displaystyle f_{N\left(\mathcal{N}\right)}$
$\displaystyle=\frac{L_{\mathcal{N}-1}+1}{L_{\mathcal{N}}+L_{\mathcal{N}-1}}\cdot\frac{\ln\left(2\right)}{3}.$
(24)

Since $\lambda_{+}>1$ and $\left|\lambda_{-}\right|<1$ it follows that
$\lambda_{+}$ is a Pisot Vijayaraghavan number. The significance is that in
the thermodynamic limit $\mathcal{N}\rightarrow\infty$ only $\lambda_{+}$ will
survive:

$\lim_{\mathcal{N}\rightarrow\infty}L_{\mathcal{N}}/L_{\mathcal{N}-1}=\lambda_{+}$
(25)

and so

$\lim_{N\rightarrow\infty}f_{N}=\frac{1}{5+\sqrt{15}}\cdot\frac{\ln\left(2\right)}{3}.$
(26)

##  Appendix B Appendix B: Free energy per dimer – numerics

Our raw data are given in Table 1. Note that the final row in the column
$f_{N}$ is partly analytical: it assumes Eq. (1) for $\mathcal{Z}$ in the main
text, but uses the numerical value of $N$. Obtaining this value numerically
requires finding the determinant of an $1665406\times 1665406$ matrix.

$N$ | $N/N_{T}$ | $f_{N}$ | Aitken $N/N_{T}$ | Aitken $f_{N}$
---|---|---|---|---
78 | 8.666667 | 0.035546 |  |
518 | 7.295775 | 0.024086 | 6.177147 | 0.028448
3734 | 6.679785 | 0.023761 | 6.029943 | 0.026321
28006 | 6.363554 | 0.024601 | 6.004131 | 0.026074
214662 | 6.195330 | 0.025225 | 6.000536 | 0.026044
1665406 | 6.105062 | 0.025592* |  |
Table 1: Raw numerical data. $N$ is the number of vertices, $N_{T}$ the number
of $S_{0}$ spectre tiles, and $f_{N}$ the free energy per dimer. Aitken
$N/N_{T}$ has had the Aitken delta squared process applied to $N/N_{T}$ for
faster convergence, and the final column recalculates $f_{N}$ using the new
$N/N_{T}$. Result * is partly analytical (see text).

It is evident from here and Fig. 3 in the main text that the free energy per
dimer $f_{N}$

$f_{N}=\frac{2\ln\left(\mathcal{Z}\right)}{N}$ (27)

is converging incredibly slowly. The issue is that the boundary of the
$S_{\mathcal{N}}$ region becomes highly fractal for large $\mathcal{N}$ (with
a fractal dimension of around 1.44), so that a larger than expected number of
vertices lie on the boundary. One way to see the effect is to find $N/N_{T}$,
the number of vertices $N$ divided by the number of $S_{0}$ spectre tiles
$N_{T}$. As argued above, this value should be 6 in the bulk, and should
therefore tend to 6 as $\mathcal{N}\rightarrow\infty$. Inspecting Table 1
shows a poor convergence. To help with this we apply a nonlinear series
acceleration, Aitken’s delta squared process [26], in which a series $A_{n}$
is transformed to a new series $A_{n}^{\prime}$ with faster convergence:

$A_{n}^{\prime}=\frac{A_{n+1}A_{n-1}-A_{n}^{2}}{A_{n+1}+A_{n-1}-2A_{n}}.$ (28)

Table 1 shows the result of this transformation, which is highly effective. We
then re-calculate $f_{N}$ using

$f_{N}=\frac{2\ln\left(\mathcal{Z}\right)}{(N/N_{T})N_{T}}$ (29)

and the accelerated $N/N_{T}$ terms. The result is a quick convergence to the
analytical value.