# Quantum orders in the frustrated Ising model on the bathroom tile lattice

Sumner N. Hearth  Department of Physics, Boston University, Boston,
Massachusetts 02215, USA  Siddhardh C. Morampudi  Center for Theoretical
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chris R. Laumann  Department of Physics, Boston University, Boston,
Massachusetts 02215, USA

###### Abstract

We determine the zero and finite temperature phase diagram of the fully
frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase
diagram exhibits a wealth of 2+1d physics, including 1\. classical Coulomb
dimer liquids of both square and triangular lattice types; 2\. quantum order-
by-disorder induced phases breaking $\mathbb{Z}_{4}$, $\mathbb{Z}_{6}$, and
$\mathbb{Z}_{8}$ symmetries; 3\. finite temperature Kosterlitz-Thouless (KT)
phases floating over the $\mathbb{Z}_{6}$ and $\mathbb{Z}_{8}$ orders; and, 4.
staircases of (in)-commensurate symmetry breaking phases at intermediate
coupling. We establish this elaborate phase diagram using a combination of
dimer model mapping, perturbation theory, Landau analysis and Stochastic
Series Expansion Quantum Monte Carlo (QMC-SSE). Our results provide a baseline
for studying frustrated magnetism with D-Wave architecture annealers, where
the 4-8 lattice can be embedded naturally without ‘cloning’, reducing the
number of competing energy scales. Simulations with the D-Wave 2000Q
demonstrate qualitative agreement with the high temperature portion of the
phase diagram, but are unable to access the low temperature phases.

###### Contents

  1. I Introduction
  2. II Model and Symmetries
  3. III Classical Limit ($\Gamma=0$)
  4. IV Quantum Model ($\Gamma>0$)
    1. IV.1 Isotropic Model
    2. IV.2 Strongly Anisotropic Model
    3. IV.3 Small Anisotropy Phase Competition
    4. IV.4 Incommensurate Phases
  5. V Quantum Monte Carlo
    1. V.1 Semiclassical RVB Update
  6. VI Realization on a Quantum Annealer
  7. VII Conclusion
  8. VIII Acknowledgements
  9. A Single Mode Approximation
  10. B RVB Autocorrelation Times

##  I Introduction

Programmable quantum annealers have developed significantly in the last decade
and now host thousands of interacting qubits realizing an effective transverse
field Hamiltonian[1, 2, 3]. While these developments are primarily motivated
by the hope that such quantum annealers will efficiently solve classical
optimization problems, there is accumulating evidence that they may be used to
probe the statistical physics of frustrated magnets [4, 5, 6, 7]. Decades of
work in magnetism has revealed the richness and complexity that frustration
imparts on the resulting magnetic phases, and also the limitations of
classical methods for studying them. This raises the question of whether
current or near term quantum annealers can provide new insights into
frustrated magnetic models.

Figure 1: (color online) The direct 4-8 lattice (solid) and the dual Union
Jack lattice (dashed). Spins live on circles. In our ‘gauge’ choice, red/thin
edges are antiferromagnetic and black/thick are ferromagnetic. In the
anisotropic deformation of the model, the thick edges carry coupling
$J^{\prime}$ and the thin have coupling $J$.

This is a delicate undertaking. Typically, the large degeneracy of the low
energy manifold in frustrated models leads to an extreme sensitivity to
perturbations. The complexity of embedding such a model into the available
architecture thus introduces additional opportunities for systematic errors to
creep into the analysis. Motivated by the underlying architecture of the
D-Wave devices – both the original Chimera and the more recent Pegasus
architectures – we identify the geometrically frustrated transverse field
Ising model on the square-octogonal (‘4-8’) lattice as a natural choice for
study as it embeds directly without additional energy scales being introduced
by ‘cloning’, where multiple physical qubits are tied together with strong
couplings in order to represent a single lattice site (see Fig. 1). Prior work
on frustrated magnets with the D-Wave device can in fact be interpreted as
studying particular regimes of the 4-8 model where certain bonds have become
especially strong[4, 5, 6, 7]. We show, using a variety of theoretical and
numerical techniques, that the larger phase diagram exhibits a cornucopia of
classical and quantum phases, capturing a large swathe of statistical physics
over the last few decades. These include two varieties of zero temperature
classical Coulomb spin liquid[8, 9], adjacent quantum ordered-by-disorder
symmetry breaking phases[10, 11, 12], intermediate coupling incommensurate
symmetry breaking phases[13] and several associated Kosterlitz-Thouless (KT)
phases at finite temperature [14, 15, 16].

The results of our case study of the model using the D-Wave 2000Q are
qualitatively consistent with our classical analysis in the high temperature
regime. However, the region of coupling space accessible in the device lies
outside the interesting finite transverse field phases: certainly the low
temperature ordered phases are entirely inaccessible, while the finite
temperature KT phase is at the margin of accessibility and we see at best
qualitative agreement with our computed phase diagram. While this is a
disappointing result for the current device, the parametric improvements
necessary to get into the low temperature phase seem entirely plausible for
next generation devices. We note that the Stochastic Series Expansion Quantum
Monte Carlo (QMC-SSE) techniques which we used in our classical study were
also inefficient in the low temperature regime. This suggests that the D-Wave
devices could reliably surpass existing classical techniques in a few
generations if these parameters can indeed be improved.

In the following, we introduce the model more precisely and analyze its
symmetries (Sec. II) and then turn to a tour of its phase diagram using a
variety of analytic and numerical techniques, beginning with the classical
model at zero transverse field (Sec. III), then the finite transverse field
model in the isotropic (Sec. IV.1) model and the large (Sec. IV.2) and small
(Sec. IV.3) anisotropy limits. We connect these with a study of the
incommensurate phases at intermediate anisotropy (Sec. IV.4). We briefly
sketch the details of our QMC-SSE implementation and semi-classical updates we
introduced to better equilibrate the model in Sec. V. We turn to the results
from the D-Wave annealer in Sec. VI and finally briefly conclude.

##  II Model and Symmetries

The _isotropic_ Frustrated Quantum Ising Model (FQIM) is governed by the
transverse field Ising Hamiltonian

$\displaystyle H$ $\displaystyle=-J\sum_{\langle
ij\rangle}s_{ij}\sigma^{z}_{i}\sigma^{z}_{j}-\Gamma\sum_{i}\sigma^{x}_{i}$ (1)

where $\langle ij\rangle$ runs over the nearest neighbors of the 4-8 lattice,
$\sigma^{x/z}_{i}$ are the Pauli $x$ and $z$ matrices at site $i$, and
$s_{ij}=\pm 1$ determines whether the bond $ij$ is ferro- ($s_{ij}=+1$) or
antiferromagnetic ($s_{ij}=-1$). The couplings $J$ and $\Gamma$ are positive.
We use indices $i,j$ to indicate general sites on the 4-8 lattice. Where
needed, we also label sites $i=(\vec{R},\mu)$ where $\vec{R}$ runs over the
sites of a square Bravais lattice (with lattice constant 1) and $\mu$ runs
over the 4 sites in the unit cell at the north, east, south and west corners
of the elementary square plaquette.

An elementary plaquette $p$ (which can be either a square or an octagon) is
classically frustrated if there are an odd number of antiferromagnet bonds
around its boundary; mathematically, $p$ is frustrated if
$F_{p}=\prod_{\langle ij\rangle\in\partial p}s_{ij}=-1$. The fully frustrated
model has $F_{p}=-1$ for every plaquette. Without loss of generality, we
choose a ‘gauge’ $s_{ij}$ where the antiferromagnet bonds are arranged on the
thick bonds of Fig. 1 – ie. the horizontal and vertical bonds between the
squares and the northeast edge of each square. It is useful to define a
‘gauge’ transformation by $\chi_{i}\in\\{\pm 1\\}$ to flip the $z$-component
of each spin $i$ for which $\chi_{i}=-1$; that is,

$\displaystyle
G(\\{\chi_{i}\\})=\prod_{i}(\sigma^{x}_{i})^{\frac{1-\chi_{i}}{2}}$ (2)

Except for the global Ising flip,

$\displaystyle G_{2}=G(\\{\chi_{i}=-1\\})=\prod_{i}\sigma^{x}_{i}$ (3)

such transformations are manifestly not symmetries of the Ising Hamiltonian
$H$ as they send $H(\\{s_{ij}\\})$ to the distinct Hamiltonian,
$H(\\{\chi_{i}s_{ij}\chi_{j}\\})$. On the other hand, the frustrations $F_{p}$
are ‘gauge’-invariant in that the ‘gauge’-related Hamiltonians have the same
frustrated plaquettes. We use quotes around the term ‘gauge’ to remind the
reader that the FQIM Hamiltonian does not have true local symmetries.

The ‘gauge’ transforms play a useful role in understanding the global symmetry
group of the FQIM Hamiltonian. The space group of the 4-8 lattice is that of
the underlying square Bravais lattice with $\pi/2$ rotation centers at the
center of each of the elementary (diagonal) squares. This group is generated
by lattice translations $T_{x}$, $T_{y}$ and the reflections
$M_{xy}:(x,y)\to(y,x)$ and $M_{x}:(x,y)\to(-x,y)$. In our ‘gauge’, the
Hamiltonian $H$ is clearly invariant under the lattice translations and
$M_{xy}$ (see Fig. 1), but, the antiferromagnetic bond on the northeast edge
of the elementary square is mapped to the northwest edge by $M_{x}$. This can
be corrected by an appropriate ‘gauge’-transformation
$G_{x}=G(\\{e^{i(0,\pi)\cdot\vec{R}}(-1)^{\delta_{\mu,1}}\\})$ which maps the
antiferromagnetic bonds back to the northeast edges (Fig 2). Putting this
together with the global Ising flip, the global symmetry group $\mathcal{G}$
of $H$ is generated by the operators

$\displaystyle\mathcal{G}=\langle G_{2},T_{x},T_{y},M_{xy},G_{x}M_{x}\rangle.$
(4)

Physically, the isotropic FQIM has the full symmetry of the 4-8 lattice, but
the action of the symmetry transformations mixes space and spin degrees of
freedom. This will play an important role in the Landau analysis of the
symmetry breaking orders induced by the transverse field.

Figure 2: The ‘gauge’ transform $G_{x}$ flips the spins marked in solid blue,
mapping the Hamiltonian to another with the same frustrated plaquettes but the
choice of frustrated bonds mirrored across the $y$-axis.

We further consider an _anisotropic_ deformation of the FQIM in which the
ferromagnetic bonds (thick bonds in Fig. 1) come with magnitude $J^{\prime}$
while the antiferromagnetic bonds comes with $J$.

$\displaystyle H=-J\sum_{\begin{subarray}{c}\langle ij\rangle\\\
s_{ij}=-1\end{subarray}}s_{ij}\sigma^{z}_{i}\sigma^{z}_{j}-J^{\prime}\sum_{\begin{subarray}{c}\langle
ij\rangle\\\
s_{ij}=+1\end{subarray}}s_{ij}\sigma^{z}_{i}\sigma^{z}_{j}-\Gamma\sum_{i}\sigma^{x}_{i}$
(5)

This model has manifestly reduced symmetry – if a space group transformation
moves the northeast bond, it can no longer be fixed by a
‘gauge’-transformation which only move the signs $s_{ij}$ around. Thus, the
global symmetry group is reduced to

$\displaystyle\mathcal{G}^{\prime}=\langle G_{2},T_{x},T_{y},M_{xy}\rangle.$
(6)

Classically, the dual representation of the Ising model[17] is expressed in
terms of the ‘broken bond’ variables,

$\displaystyle\tau^{z}_{ij}=s_{ij}\sigma^{z}_{i}\sigma^{z}_{j}$ (7)

which are subject to the constraint

$\displaystyle\prod_{\langle ij\rangle\in\partial
p}\tau_{ij}^{z}=\left(\prod_{\langle ij\rangle\in\partial
p}s_{ij}\right)\left(\prod_{i\in\partial p}(\sigma_{i}^{z})^{2}\right)=F_{p}.$
(8)

If $\tau^{z}=-1$, the corresponding bond in $H$ is broken – i.e., it carries
positive energy. The full quantum Hamiltonian $H$ can be recast as an Ising
gauge theory in the $\tau$ variables,

$\displaystyle H$ $\displaystyle=-J\sum_{\langle
ij\rangle}\tau^{z}_{ij}-\Gamma\sum_{i}\prod_{j\in\partial i}\tau^{x}_{ij}$ (9)

where $j\in\partial i$ runs over the sites neighboring $i$ on the lattice.
Here, $\tau^{x}_{ij}$ is the Pauli $x$ matrix conjugate to $\tau^{z}_{ij}$
associated to bond $\langle ij\rangle$. For simplicity, we have written $H$
for the isotropic case; the anisotropic extension is straightforward. As
usual, the gauge representation, Eq. (9) with constraint Eq. (8), is an exact
rewriting of the FQIM, Eq. (1), up to the global Ising symmetry: every valid
broken bond configuration $\tau^{z}$ fixes a $\sigma^{z}$ configuration up to
a global choice of $\uparrow$ or $\downarrow$.

In the fully frustrated model $F_{p}=-1$, so physical configurations must have
an odd number of broken bonds around any plaquette. The classical ground
states thus have one broken bond per plaquette and can be profitably
reinterpreted as hard-core dimer configurations on the dual union-jack lattice
(dashed lines, Fig. 1) by drawing each broken bond as a dimer bisecting the
direct lattice bond. See Fig 6 for an example such dimer covering
corresponding to a ‘columnar state’ of the dimers. Unlike the essentially
exact gauge representation, the hard-core dimer representation is only useful
at low temperature and transverse field, where the relevant states are close
to the classical ground state manifold. In this case, however, as much is
known about both classical[18] and quantum dimer models[19, 20], it provides a
wealth of guidance.

Figure 3: The zero temperature phase diagram of the 4-8 fully frustrated Ising
model. The blue dot at $J^{\prime}=J,\Gamma=0$ is a square lattice classical
Coulomb liquid. Between the $\mathbb{Z}_{4}$ and $\mathbb{Z}_{2}$ symmetry-
breaking phases there is a _staircase_ of ordered phases with incommensurate
spatial periods. A similar region may exist between $\mathbb{Z}_{4}$ and
$\mathbb{Z}_{6}$.

##  III Classical Limit ($\Gamma=0$)

We start by considering the low temperature phases of the $\Gamma=0$ classical
model, see Fig 5. These can best be understood in the dimer representation on
the dual Union-Jack lattice. This lattice may be viewed as a square lattice
oriented at $45^{\circ}$ to the horizontal with additional horizontal and
vertical bonds. All valid dimer coverings of the square lattice are coverings
of the Union-Jack lattice; any dimer coverings using the additional horizontal
and vertical bonds require additional dimers, and are therefore not ground
states[21]. Since the ground state dimer coverings are in 1-to-1
correspondence with the square lattice coverings we recover the well-known
residual entropy[10] of the well studied square lattice Coulomb dimer liquid,

$\displaystyle\mathcal{S}(T=0)$ $\displaystyle=0.583.$ (10)

The anisotropic deformation strengthening the ferromagnetic bonds by a factor
of $J^{\prime}/J$ bind the spins together and prohibit broken bonds between
them at zero temperature (see bold bonds in Fig 1). Removing the appropriate
edges from the Union-Jack dimer model produces a brick lattice (Fig 4) — a
simple deformation of the Honeycomb lattice. In the classical zero temperature
limit we therefore find a honeycomb lattice dimer liquid with the residual
entropy[22, 11, 23],

$\displaystyle\mathcal{S}^{\prime}(T=0)$ $\displaystyle=0.323.$ (11)

Figure 4: At $T=\Gamma=0$ and $J^{\prime}>J$, the classical ground states are
in 2-1 correspondence with a dimer liquid on the dual brick lattice (shown
dashed). This is a deformed honeycomb lattice, dual to the triangular lattice.

Weakening the same bonds ($J^{\prime}/J<1$) prefers a particular dimer
position on each unit cell, leading to a unique staggered dimer ground state.
In the spin language this state breaks the global $\mathbb{Z}_{2}$ symmetry,
since dimer states are in a 2-to-1 correspondence with spin configurations.

The $T,J^{\prime}\ll J$ model, after a gauge transformation, locks spins
together within each unit cell and can be mapped to the 2D square lattice
ferromagnetic Ising model with bond strength $J^{\prime}$. With $J^{\prime}$
the only accessible energy scale we expect a thermal phase transition to
disorder with critical temperature following $T_{c}\propto J^{\prime}$.

For $J^{\prime}/J>0$ the model has identical zero temperature ground states to
those of the square lattice ZZD (_zig-zag domino_) model studied by André _et
al_[11]. They predict no direct finite temperature transition and no classical
(thermal) order-by-disorder for $J^{\prime}\geq J$. We expect the same for the
analogous 4-8 model.

Figure 5: The classical $\Gamma=0$ phase diagram. At $T=0$ there are two
regimes: for $J^{\prime}>0$ the ground states are identical to those of the
ZZD Square Lattice[11] model including a square lattice dimer liquid at
$J^{\prime}=J$ and a honeycomb lattice dimer liquid at $J^{\prime}>J$; for
$J^{\prime}=0$ the system decouples into disconnected length 4 Ising chains.

##  IV Quantum Model ($\Gamma>0$)

###  IV.1 Isotropic Model

To study the ordering effects of a transverse field on the lattice we return
to the isotropic case. A small transverse field lifts the ground state
degeneracy of the square lattice dimer liquid, breaking lattice symmetries
through _quantum order-by-disorder_[24, 20]. Motivated by the $T=0$ classical
mapping to the fully frustrated square lattice we consider symmetry breaking
phases which correspond to columnar (Fig 6) and plaquette orders of the
quantum dimer model. It is still controversial whether the columnar,
plaquette, and a mixed phase is realized on the square lattice.

Figure 6: An example of a columnar state found at $J^{\prime}\geq J$. Dimers,
which live on the dual lattice, are shown as red rectangles. A _flippable
plaquette_ , such as the one marked with a blue square, lives between each
pair of parallel adjacent dimers.

As in that model, the relevant orders can be organized into a complex order
parameter $\phi$ with $\mathbb{Z}_{8}$ clock symmetry. For the 4-8 lattice, we
construct $\phi$ explicitly by considering the a _single mode
approximation_[23, 25, 26] (SMA) for magnon excitation in the large $\Gamma$
paramagnetic phase.

In the infinite $\Gamma$ limit, the ground state is
$\ket{\rightarrow\rightarrow...}$. Delocalized magnon excitations are created
by the operator $\sum a^{\pm}_{\mu}e^{\pm ikR}\sigma^{z}_{R,\mu}$. For the
isotropic model they have minimum energy at $k=(\frac{\pi}{2},-\frac{\pi}{2})$
with

$\displaystyle E$ $\displaystyle=2\Gamma-(1+\sqrt{2})J$ (12) $\displaystyle
a^{\pm}$
$\displaystyle=\frac{1}{2}(e^{\pm\frac{3i\pi}{4}},e^{\pm\frac{i\pi}{2}},e^{\mp\frac{3i\pi}{4}},1).$
(13)

Eq (12) predicts a phase transition at
$\Gamma_{c}~{}=~{}\frac{1+\sqrt{2}}{2}J$ as the gap closes and magnons
condense. The condensation occurs at finite momentum, breaking lattice
symmetries. Motivated by this pattern, we consider the complex order parameter
$\phi$ with momentum $k=(\frac{\pi}{2},-\frac{\pi}{2})$,

$\displaystyle\phi$
$\displaystyle=\sum_{R,\mu}a^{+}_{\mu}e^{ikR}\sigma^{z}_{R,\mu}$ (14)

The lattice symmetry group acts on the 2D operator space spanned by $\phi$ and
$\phi^{\dagger}$. The most salient feature is that $G_{x}M_{x}M_{xy}$ acts as
a generator of $\mathbb{Z}_{8}$ symmetry, $G_{x}M_{x}M_{xy}:\phi\rightarrow
e^{-\frac{3i\pi}{4}}\phi$. Furthermore
$T_{x}G_{x}M_{x}:\phi\rightarrow\phi^{\dagger}$ acts as complex conjugation on
the $\phi$ plane.

With this symmetry action in hand, we construct a Landau free energy
functional for the condensation transition.

$\displaystyle\phi$ $\displaystyle\equiv|\phi|e^{i\theta}$ (15) $\displaystyle
f$
$\displaystyle\simeq\sum_{n=1}^{4}c_{n}|\phi|^{2n}+g_{8}|\phi|^{8}\mbox{cos}(8\theta)$
(16)

The first allowed anisotropy for $\phi$ is at eighth order and describes an
eight-fold degenerate symmetry breaking ground state. A value of $g_{8}>0$
favors four columnar states and their global spin flips, whereas $g_{8}<0$
favors plaquette states. We expect to find columnar states based on the
surrounding phases in parameter space, however the analogous region of the
phase diagram on the square lattice is controversial[27, 28].

Figure 7: Temperature-transverse field phase diagram of isotropic model
($J^{\prime}=J$). An example quantum annealing path is shown (see Fig. 19).

As a discrete symmetry breaking order, the $\mathbb{Z}_{8}$ phase should
persist to finite T. The eight-fold anisotropy is irrelevant, however, at the
quantum critical point ($T=0$, $\Gamma=\Gamma_{c}$) where we expect a $3D$ XY
phase transition to the quantum paramagnet. Around this point the phase
boundaries are controlled by the universal critical exponents of the $3D$ XY
transition, producing a quantum critical fan. The thermal transition is not
direct, however, and the thermal critical point splits into a KT critical
phase[14, 15, 16] with an upper ($T_{c}^{+}$) and lower ($T_{c}^{-}$) finite
temperature transition. We expect that the KT phase terminates at $\Gamma=0$
and $T=0$ in the dimer liquid. This can be seen by André _et al_ ’s
analysis[11] of the _ZZD_ model in which thermal fluctuations simply disorder
the square lattice dimer liquid at $T=0$. The upper critical temperature for
the transition from KT to a thermal paramagnet scales with the energy
contribution from flippable plaquettes (Fig 6), predicting a phase boundary
which initially scales as $T^{+}_{c}\propto\Gamma^{2}$. The phase diagram for
the isotropic model is shown in Fig 7.

We confirm the qualitative $\Gamma>0$ phase diagram using QMC-SSE (see Sec.
V). First we confirm the presence of a KT phase below $T_{c}^{+}$ by analyzing
sampled spin configurations at $\Gamma=J/2<\Gamma_{c}$ and varying $\beta=1/T$
and lattice sizes. The susceptibility of $\phi$ collapses with a universal
scaling function[4]. This data collapse indicates $T^{+}_{c}\approx 0.166J$
for $\Gamma=\frac{J}{2}$, see Fig 8.

The lower temperature phases and boundaries are more challenging to simulate.
By taking line cuts varying $\Gamma$ at $T=J/10<T^{+}_{c}$ we map the
paramagnetic transition out of the KT phase, Fig 9. This finite temperature
transition is expected to occur below the $\Gamma_{c}$ predicted by the single
mode approximation, and indeed we find a transition around $\Gamma\approx
1.1J$. Detailed scaling is inaccessible due to the slowing of the simulation
at low temperatures.

The transition to the ordered $\mathbb{Z}_{8}$ phase lies at a temperature
well below the upper critical temperature. Actually observing the
$\mathbb{Z}_{8}$ order phase at low temperature is numerically challenging as
we expect[15] $T_{c}^{-}\sim T_{c}^{+}/8^{2}$. Furthermore competition between
columnar and plaquette ordering patterns make this phase transition
particularly difficult to study, as has been observed with analogous states on
the square lattice[28].

Figure 8: Data collapse for susceptibility of $\phi$ near the thermal phase
boundary between the KT phase and the paramagnet. $2\Gamma=J=J^{\prime}$, with
$t=\frac{T-T_{c}^{+}}{T_{c}^{+}}$. Note $T$ decreases from left to right
towards $T_{c}^{+}$ which lies at $+\infty$ on these axes. See Sec. V for
simulation details. Figure 9: Average magnitude $|\phi|$ at $\beta J=10$ for
various $L\times L$ isotropic lattice sizes showing crossover into a
paramagnet near $\Gamma_{c}\sim 1.1J$. Error bars obscured by trendlines.

###  IV.2 Strongly Anisotropic Model

There are two anisotropic limits to consider: $J^{\prime}\ll J$ and
$J^{\prime}\gg J$. For both we start our analysis from the classical ground
state. The $J^{\prime}\ll J$ region is comparatively simple because the
ordered _staggered_ phase persists until a critical $\Gamma_{c}$ or $T_{c}$,
after which the model transitions to a paramagnet. In the $J^{\prime}\gg J$
limit we find a story similar to the isotropic case; a classical $T=0$ dimer
liquid, now on a honeycomb lattice, has its ground state degeneracy lifted by
a transverse field. Prior work on the triangular fully frustrated Ising model,
and SMA analysis on the effective brick lattice (Fig 4) both suggest a
$\mathbb{Z}_{6}$ symmetry breaking ordering pattern which can be captured by
the complex order parameter,

$\displaystyle\psi$ $\displaystyle=\sum_{R,\mu}b_{\mu}e^{ikR}\sigma^{z}_{\mu}$
(17) $\displaystyle b$
$\displaystyle=\frac{1}{2}(e^{\frac{2i\pi}{3}},e^{\frac{2i\pi}{3}},e^{-\frac{2i\pi}{3}},1)$
(18)

By analysing the action of $\mathcal{G}^{\prime}$ on $(\psi,\psi^{\dagger})$;
we find a 6-fold clock symmetry is generated by $G_{2}T_{x}:\psi\rightarrow
e^{-\frac{i\pi}{3}}\psi$. $T_{x}^{2}M_{xy}:\psi\rightarrow\psi^{\dagger}$ acts
as conjugation. The corresponding Landau free energy admits anisotropy at
$6^{th}$ order,

$\displaystyle f^{\prime}$
$\displaystyle\simeq\sum_{n=1}^{3}c^{\prime}_{n}|\psi|^{2n}+g_{6}|\psi|^{6}\mbox{cos}(6\theta).$
(19)

Again, the anisotropy is irrelevant at the 3D XY quantum critical point. At
finite temperature we expect a KT phase between upper and lower critical
temperatures. The phase diagram is very similar to that of the isotropic model
(Fig 7), albeit with a critical temperature which initially scales as
$\Gamma^{4}$. This regime has been studied before[4] for a specific value of
$J^{\prime}=1.8J$.

###  IV.3 Small Anisotropy Phase Competition

For small anisotropy, where $|\frac{J^{\prime}}{J}-1|\ll 1$, energy
corrections explicitly favor a subset of the $\mathbb{Z}_{8}$ ground states:
columnar phases aligned with one diagonal or the other. This results in two
distinct $\mathbb{Z}_{4}$ symmetry breaking columnar phases on either side of
the isotropy line. At larger anisotropy, where $J^{\prime}/J$ is far from $1$,
the columnar phases should give way to $\mathbb{Z}_{2}$ or $\mathbb{Z}_{6}$
phases as discussed previously.

Let us estimate the boundary separating these phases by comparing the
variational energy calculated for each. First consider the phase boundaries in
the small $\Gamma$ limit, near the classical line. Small $J^{\prime}$ and
$\Gamma$ perturbations about the isotropic classical point, written
$\epsilon=\frac{J^{\prime}-J}{J}$ and $\gamma=\frac{\Gamma}{J}$, modify $H$ by
a small term $V$:

$\displaystyle H$ $\displaystyle=J\sum_{\langle
ij\rangle}s_{ij}\sigma_{i}^{z}\sigma_{j}^{z}+V$ (20) $\displaystyle V/J$
$\displaystyle=\epsilon\sum_{\begin{subarray}{c}\langle ij\rangle\\\
s_{ij}=-1\end{subarray}}\sigma_{i}^{z}\sigma_{j}^{z}+\gamma\sum_{i}\sigma_{i}^{x}$
(21)

We find in the $\epsilon>0$ region, the competition is between
$\mathbb{Z}_{4}$ states and $\mathbb{Z}_{6}$ states. First, we consider the
variational energy of the columnar state like the one in Fig 6, labeled as
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{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.45831pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. Nearby low energy
states can be reached by flipping two spins in any of the flippable
plaquettes. We label the state made by flipping the plaquette at unit cell $R$
as $\ket{\leavevmode\hbox to8.46pt{\vbox
to8.46pt{\pgfpicture\makeatletter\hbox{\hskip 1.0pt\lower-1.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{R}$. There are as
many of these states as there are unit cells, so the Hamiltonian in the dimer
language[19] restricted to these states is written

$H/J=\sum_{R}\epsilon\ket{\leavevmode\hbox to8.46pt{\vbox
to8.46pt{\pgfpicture\makeatletter\hbox{\hskip 1.0pt\lower-1.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{R}\bra{\leavevmode\hbox
to8.46pt{\vbox to8.46pt{\pgfpicture\makeatletter\hbox{\hskip
1.0pt\lower-1.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{R}-\gamma^{2}\left(\ket{\leavevmode\hbox
to8.46pt{\vbox to8.46pt{\pgfpicture\makeatletter\hbox{\hskip
1.0pt\lower-1.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{R}\bra{\leavevmode\hbox
to8.46pt{\vbox to8.46pt{\pgfpicture\makeatletter\hbox{\hskip
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.45831pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+h.c.\right).$ (22)

For the general Hamiltonian there are also terms linear in $\gamma$ from
flipping single spins, such as $\gamma\ket{\leavevmode\hbox to8.46pt{\vbox
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{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{R}\bra{\leavevmode\hbox
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{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@moveto{1.0pt}{6.45831pt}\pgfsys@curveto{1.0pt}{7.0106pt}{0.55229pt}{7.45831pt}{0.0pt}{7.45831pt}\pgfsys@curveto{-0.55229pt}{7.45831pt}{-1.0pt}{7.0106pt}{-1.0pt}{6.45831pt}\pgfsys@curveto{-1.0pt}{5.90602pt}{-0.55229pt}{5.45831pt}{0.0pt}{5.45831pt}\pgfsys@curveto{0.55229pt}{5.45831pt}{1.0pt}{5.90602pt}{1.0pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
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{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@moveto{7.45831pt}{0.0pt}\pgfsys@curveto{7.45831pt}{0.55229pt}{7.0106pt}{1.0pt}{6.45831pt}{1.0pt}\pgfsys@curveto{5.90602pt}{1.0pt}{5.45831pt}{0.55229pt}{5.45831pt}{0.0pt}\pgfsys@curveto{5.45831pt}{-0.55229pt}{5.90602pt}{-1.0pt}{6.45831pt}{-1.0pt}\pgfsys@curveto{7.0106pt}{-1.0pt}{7.45831pt}{-0.55229pt}{7.45831pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@moveto{7.45831pt}{6.45831pt}\pgfsys@curveto{7.45831pt}{7.0106pt}{7.0106pt}{7.45831pt}{6.45831pt}{7.45831pt}\pgfsys@curveto{5.90602pt}{7.45831pt}{5.45831pt}{7.0106pt}{5.45831pt}{6.45831pt}\pgfsys@curveto{5.45831pt}{5.90602pt}{5.90602pt}{5.45831pt}{6.45831pt}{5.45831pt}\pgfsys@curveto{7.0106pt}{5.45831pt}{7.45831pt}{5.90602pt}{7.45831pt}{6.45831pt}\pgfsys@closepath\pgfsys@moveto{6.45831pt}{6.45831pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.45831pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{0.0pt}{6.45831pt}\pgfsys@lineto{6.45831pt}{6.45831pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$, however they make
identical contributions to all classical ground states and can be omitted when
comparing the states. At $\epsilon=0$ and $\gamma>0$ the state is independent
of $\gamma$, the energy per unit cell is simply
$\langle\mathcal{H}\rangle\propto-\gamma^{2}J$. For $\epsilon>0$, with finite
$\gamma$, we find $\langle\mathcal{H}\rangle\propto\epsilon J$ as resonant
plaquettes break the strengthened bonds. To leading order in each expansion
parameter we find the variational energy per unit cell to be
$\langle\mathcal{H}\rangle=(\epsilon-\gamma^{2})J$.

Repeating a similar procedure for the $\mathbb{Z}_{6}$ phase we need to use
flippable plaquettes on the brick lattice, made by rotating 3 dimers along the
alternating walls of the bricks.

$H/J=-\sum_{R}\gamma^{4}\left(\ket{\leavevmode\hbox to10.61pt{\vbox
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{2.15277pt}{-3.7286pt}\pgfsys@moveto{3.15277pt}{-3.7286pt}\pgfsys@curveto{3.15277pt}{-3.17632pt}{2.70506pt}{-2.7286pt}{2.15277pt}{-2.7286pt}\pgfsys@curveto{1.60048pt}{-2.7286pt}{1.15277pt}{-3.17632pt}{1.15277pt}{-3.7286pt}\pgfsys@curveto{1.15277pt}{-4.2809pt}{1.60048pt}{-4.7286pt}{2.15277pt}{-4.7286pt}\pgfsys@curveto{2.70506pt}{-4.7286pt}{3.15277pt}{-4.2809pt}{3.15277pt}{-3.7286pt}\pgfsys@closepath\pgfsys@moveto{2.15277pt}{-3.7286pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{-2.15277pt}{-3.7286pt}\pgfsys@moveto{-1.15277pt}{-3.7286pt}\pgfsys@curveto{-1.15277pt}{-3.17632pt}{-1.60048pt}{-2.7286pt}{-2.15277pt}{-2.7286pt}\pgfsys@curveto{-2.70506pt}{-2.7286pt}{-3.15277pt}{-3.17632pt}{-3.15277pt}{-3.7286pt}\pgfsys@curveto{-3.15277pt}{-4.2809pt}{-2.70506pt}{-4.7286pt}{-2.15277pt}{-4.7286pt}\pgfsys@curveto{-1.60048pt}{-4.7286pt}{-1.15277pt}{-4.2809pt}{-1.15277pt}{-3.7286pt}\pgfsys@closepath\pgfsys@moveto{-2.15277pt}{-3.7286pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{-4.30554pt}{0.0pt}\pgfsys@moveto{-3.30554pt}{0.0pt}\pgfsys@curveto{-3.30554pt}{0.55229pt}{-3.75325pt}{1.0pt}{-4.30554pt}{1.0pt}\pgfsys@curveto{-4.85783pt}{1.0pt}{-5.30554pt}{0.55229pt}{-5.30554pt}{0.0pt}\pgfsys@curveto{-5.30554pt}{-0.55229pt}{-4.85783pt}{-1.0pt}{-4.30554pt}{-1.0pt}\pgfsys@curveto{-3.75325pt}{-1.0pt}{-3.30554pt}{-0.55229pt}{-3.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-4.30554pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{-2.15277pt}{3.7286pt}\pgfsys@moveto{-1.15277pt}{3.7286pt}\pgfsys@curveto{-1.15277pt}{4.2809pt}{-1.60048pt}{4.7286pt}{-2.15277pt}{4.7286pt}\pgfsys@curveto{-2.70506pt}{4.7286pt}{-3.15277pt}{4.2809pt}{-3.15277pt}{3.7286pt}\pgfsys@curveto{-3.15277pt}{3.17632pt}{-2.70506pt}{2.7286pt}{-2.15277pt}{2.7286pt}\pgfsys@curveto{-1.60048pt}{2.7286pt}{-1.15277pt}{3.17632pt}{-1.15277pt}{3.7286pt}\pgfsys@closepath\pgfsys@moveto{-2.15277pt}{3.7286pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@moveto{2.15277pt}{-3.7286pt}\pgfsys@moveto{3.15277pt}{-3.7286pt}\pgfsys@curveto{3.15277pt}{-3.17632pt}{2.70506pt}{-2.7286pt}{2.15277pt}{-2.7286pt}\pgfsys@curveto{1.60048pt}{-2.7286pt}{1.15277pt}{-3.17632pt}{1.15277pt}{-3.7286pt}\pgfsys@curveto{1.15277pt}{-4.2809pt}{1.60048pt}{-4.7286pt}{2.15277pt}{-4.7286pt}\pgfsys@curveto{2.70506pt}{-4.7286pt}{3.15277pt}{-4.2809pt}{3.15277pt}{-3.7286pt}\pgfsys@closepath\pgfsys@moveto{2.15277pt}{-3.7286pt}\pgfsys@fill\pgfsys@invoke{
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Since the bricks do not include dimers which break $J^{\prime}$ bonds we end
up with an energy per unit cell simply modified by $-\gamma^{4}J$. Competition
between these two energies determines the shape of the boundary between the
$\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$ phases at low $\epsilon$ and low
$\gamma$. To leading order this is given by $\epsilon_{c}\propto\gamma^{2}$.
In the $J^{\prime}<J$ region the staggered phase competes with a columnar
phase (rotated 90 degrees from that shown in Fig 6 so as to break a single
$J^{\prime}$ bond per unit cell). The same treatment predicts a similar phase
boundary below the $J^{\prime}=J$ line: $\epsilon_{c}\propto-\gamma^{2}$.
These estimates also motivate the $T_{c}$ boundaries shown in Fig 7 and Fig
10.

Notably KT phases are not supported by Clock Models with $q\leq 4$ states[15].
Despite the isotropic and strongly anisotropic models having KT phases, some
of the phases of the weakly anisotropic models, such as the $\mathbb{Z}_{4}$
phase, do not support KT phases and instead transition directly into the
thermal paramagnet above a critical temperature.

Figure 10: Temperature-transverse field phase diagram for weak anisotropy
$\frac{J^{\prime}-J}{J}\ll 1$. The $\psi$ order parameter with
$\mathbb{Z}_{6}$ anisotropy permits an intermediate temperature KT phase. No
such phase exists for $\phi$ with $\mathbb{Z}_{4}$. Double lines indicate
first order transitions between these symmetry incommensurate phases;
alternatively, it is possible that the phase transition splits into a region
of incommensurate phases.

###  IV.4 Incommensurate Phases

Between the weakly anisotropic and strongly anisotropic limits previously
described, we find a collection of additional ground state ordering patterns
with distinct momenta. SMA in the large $\Gamma$ limit predicts a smoothly
varying magnon condensation momentum $\vec{k}_{min}=\pm(k_{min},-k_{min})$ as
a function of $J^{\prime}$, see Fig 11. We can trace $k_{min}$ as we increase
$J^{\prime}$. As seen previously, $J^{\prime}=J$ predicts
$k_{min}=\frac{2\pi}{4}$. Increasing to $J^{\prime}\approx 2J$ moves $k_{min}$
towards $\frac{2\pi}{3}$. Between these values, however, SMA predicts ordering
patterns which are incommensurate with the underlying lattice[13]. We expect
the the model to lock into ground states with smaller unit cells, in other
words smaller denominators $q$ in $k_{min}=\frac{p\pi}{q}$. In the
$J^{\prime}\gg J$ limit SMA predicts $k_{min}\sim\frac{4\pi}{5}$. This
disagrees with the SMA prediction on the large $J^{\prime}$ effective brick
lattice (Fig 4), as well as the perturbative expansion around the classical
$\Gamma=0$ line, so we expect no $k_{min}=\frac{4\pi}{5}$ phase to appear in
the phase diagram.

Figure 11: Evolution of energy of lowest magnon with momentum $\vec{k}=(k,-k)$
with anisotropy $J^{\prime}/J$. Minimum momentum configurations for each value
of $J^{\prime}/J$ are traced.

We can predict the approximate phase boundaries by numerically comparing the
eigenenergies of magnon excitations. Expanding $H_{m}$ (34) around $k_{min}=0$
(which corresponds to the _staggered_ phase) predicts a critical value of
$J_{c}^{\prime}=\left(\frac{1+\sqrt{5}}{3+\sqrt{5}}\right)J$ at which the zero
mode is no longer the first to condense. Magnons with momentum
$k=\frac{2\pi}{5}$ and those with $k=\frac{2\pi}{4}$ have equal energy at
$J^{\prime}\approx 0.89J$. For large enough $\Gamma$ we therefore expect a
first-order phase transition near $J^{\prime}\approx 0.89J$ out of a phase
with $k=\frac{2\pi}{5}$, and another into a $\mathbb{Z}_{4}$ symmetry breaking
phase with $k=\frac{2\pi}{4}$ (Fig 12). There may, however, be additional
symmetry breaking phases between these. Similarly we predict a transition
between the upper $\mathbb{Z}_{4}$ breaking phase and the $\mathbb{Z}_{6}$
breaking phase (with $k=\frac{2\pi}{3}$) to occur near $J^{\prime}\approx
1.27J$ (Fig 13). Note that these predictions are from SMA and performed in the
infinite $\Gamma$ limit, far above the transition into the paramagnet. Despite
this, in both Fig 12 and Fig 13 the predicted phase boundaries line up very
well with observed transitions from QMC samples.

Figure 12: Structure factor at momentum $(k,-k)$ at varying $J^{\prime}$.
Samples at $\beta J=10$ and $\Gamma=0.9J$. Results of QMC study on 24x24
lattice. Vertical dashed lines indicate illustrative transitions predicted by
the SMA (left, bifurcation of minimum $k$ in magnon dispersion, cf. Fig. 11;
right, crossing of energy at $k=2\pi/4$ and $k=2\pi/5$). Figure 13: Transition
from $\mathbb{Z}_{4}$ phase to a quasi-long range ordered $\mathbb{Z}_{6}$ KT
phase. QMC data from a 12x12 lattice at $\beta J=10$ and $\Gamma=J$. Vertical
dashed line indicates energy crossing of $k=2\pi/3$ and $k=2\pi/4$ in SMA
dispersion. Error bars obscured by trendlines.

##  V Quantum Monte Carlo

We confirm many features of this complicated phase diagram using QMC-SSE[29,
30, 31, 32] to sample $z$-aligned spin configurations $\ket{\psi}$ from the
lattice according to their Boltzmann weights, $\bra{\psi}e^{-\beta
H}\ket{\psi}$. These samples can then be used to compute observables. SSE
requires all terms in the Hamiltonian to be positive; we use the transverse
field Ising model Hamiltonian plus a constant offset restricted to $\Gamma\geq
0$:

$H=\sum_{\langle
ij\rangle}J_{ij}(\sigma_{i}^{z}\sigma_{j}^{z}+I)-\sum_{i}\Gamma(\sigma_{i}^{x}+I)$
(24)

We use typical diagonal and cluster updates, as well as a custom semiclassical
update for improving performance near the $\Gamma=0$ limit (Sec. V.1). This
update exploits flippable plaquettes more readily than cluster updates by
moving diagonal terms between bonds. We also use Replica swapping / parallel
tempering[33] for sweeps across parameters $\beta$, $\Gamma$, and
$J^{\prime}$.

To study phase transitions we take the sampled $z$-basis spin configurations
and act on them with the relevant complex order parameter operators (e.g. Eq
(14), (18)). Within the appropriate phase the order parameters acquire a non-
zero vacuum expectation value; the Monte-Carlo sampling density supports this
Fig 14(a)-14(c).

To find phase transitions we look at averages of the magnetization
$m=\sqrt{\phi^{*}\phi}$, or susceptibility $\chi=m^{2}$. Each of these
averages is across 4 independent simulations and 1000 samples for each,
samples are taken after 100 or 1000 QMC sweeps (depending on parameters),
beginning after a 100,000 sweep thermalization period. These sampling rates
were determined by measuring the autocorrelation time of the bond variables in
the $KT_{8}$ phase. A sweep consists of diagonal update, a cluster update, and
a thermal spin-flip update on any variables unconstrained by imaginary time
operators. At low $\Gamma$ we additionally perform $\frac{N}{2}$ semiclassical
updates with $N$ the total number of spins. These do not make a significant
difference to autocorrelation times for $\Gamma>T$ and can be omitted for such
simulations.

(a) $\Gamma=J$ and $\beta J=10$

(b) $\Gamma=0.8J$ and $\beta J=256$

(c) $J^{\prime}=1.1J$, $\Gamma=J$, $\beta J=10$

Figure 14:  Probability density of $\phi$ sampled in complex plane from QMC-
SSE. Note the bulls eye structure indicates that the amplitude has developed
an expectation value even if the phase continues to fluctuate as in the (a) KT
phase. At lower $T$, (b) shows subtle evidence of $\mathbb{Z}_{8}$ symmetry
breaking corresponding to a columnar phase. With anisotropy, (c) shows 4-fold
symmetry breaking corresponding to columnar order.

###  V.1 Semiclassical RVB Update

Multibranch cluster updates[30, 34] perform poorly as the model approaches the
classical low $\Gamma$ regime and the entire SSE graph forms a single cluster
connected by two-body interactions. To improve autocorrelation times for the
simulation near the classical limit we have developed an update which flips
smaller clusters of spins, motivated by the classical ring exchange updates
for dimer models.

Consider an Ising-symmetric model with diagonal 2-body operators
($H^{d}_{ij}=J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}$) and offdiagonal 1-body
operators ($H^{t}_{i}=\Gamma\sigma^{x}_{i}$). We can make $H^{t}$ constant
across all matrix elements, $[H^{t}_{i}]_{ss^{\prime}}=-\Gamma$, and ensure no
diagonal matrix elements are negative by adding a constant to the Hamiltonian

$H=(J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}+JI)-\Gamma(\sigma_{x}+I).$ (25)

The RVB update consists of flipping clusters spanning real space and imaginary
time, similarly to the multibranch cluster update. This update differs in that
we allow 2-body diagonal operators to be moved around on the surface of the
cluster. The multibranch update can then be viewed as the special case where
there are no operators on the surface, and therefore no operators need to be
moved.

Figure 15: An example cluster for the semi-classical RVB update. The four
relevant segments are separated by dashed lines. Spin world lines are
represented by the white space between vertical black lines. The red shaded
cluster will be flipped and the blue two-body operator moved by the update if
accepted.

To visualize this update, first consider world lines for each spin
(Represented in Fig 15 as the white space between vertical black lines), and
boundaries (horizontal lines) defined by the constant 1-body operators. Since
the 1-body operators have constant matrix elements across all indices, when
flipping a world line we may stop at any of the 1-body operators without
changing the weight of the graph. Define a cluster by selecting subsections of
the world lines (highlighted in red). Each subsection must begin and end at
one of the 1-body operators, like the multibranch cluster update. A convenient
definition of the cluster is the set of spins within it at each point in
imaginary time: $\mathcal{C}(\tau)$.

To flip the spins in the cluster we will move the 2-body operators with one
leg inside the cluster, and one leg outside. These are the operators whose
weights may be changed by the new spin configuration. To simplify the update,
for each operator to be moved at imaginary time $\tau$, we will only consider
new positions at the same $\tau$. At each moment in imaginary time we will
calculate the acceptance probability of moving a 2-body operator on the
border, $p(\tau)$. We can group contiguous regions of imaginary time with
identical classical spin configurations in or around the cluster into
_segments_ with the same $p(\tau)$. The total flip probability for segment
$\alpha$ is then $p_{\alpha}=p(\tau)^{n_{\alpha}}$ where $n_{\alpha}$ the
number of operators needing to be moved in that segment. Segment delimiters
are marked as dashed lines in Fig 15.

Let $\ket{\psi_{\alpha}}$ be the classical spin state in and around the
cluster in segment $\alpha$. Let $\ket{\psi^{\prime}_{\alpha}}$ be the
similarly defined spin state after the cluster has been flipped. Define
$W_{\alpha}=\sum_{\langle
i/j\rangle}\bra{\psi_{\alpha}}H^{d}_{ij}\ket{\psi_{\alpha}}$ the energy
contribution from diagonal terms along the border of the cluster, and
similarly $W^{\prime}_{\alpha}$ for $\psi^{\prime}_{\alpha}$.

$\displaystyle\langle i/j\rangle\equiv\\{i,j\in\langle
ij\rangle~{}|~{}\left(i\in\mathcal{C}(\tau)\right)\oplus\left(j\in\mathcal{C}(\tau)\right)\\}$
(26)

The notation $\langle i/j\rangle$ indicates nearest neighbors $i$ and $j$
where exactly one of $i$ or $j$ is inside the cluster at a given imaginary
time slice.

For an update rule from state $a$ to $b$, detailed balance requires
$\mathbf{P}_{accept}\frac{\mathbf{P}_{a\rightarrow
b}}{\mathbf{P}_{b\rightarrow a}}=\frac{W^{\prime}}{W}$ with $W$ the total
Boltzmann weight of the graph. For each segment there are $n_{\alpha}$
operators to move, each one represents a term in $H^{d}$, let

$\displaystyle W_{\alpha}$ $\displaystyle=\sum_{\langle
i/j\rangle}\bra{\psi_{\alpha}}H^{d}_{ij}\ket{\psi_{\alpha}}=\sum_{l}w_{\alpha,l}$
(27) $\displaystyle W_{\alpha}^{\prime}$
$\displaystyle=\sum_{l}w^{\prime}_{\alpha,l}$ (28)

where each $w_{l}$ represents one of the sites along the border of the
cluster, and an entry in $\langle i/j\rangle$. The proposed state $b$ is made
by flipping the spins in the cluster then moving all the $n_{\alpha}$ terms on
the border to new positions $l^{\prime}$ with probability
$\frac{w^{\prime}_{l^{\prime}}}{W^{\prime}_{\alpha}}$. Let
$\mathbf{P}_{a\rightarrow b}$ be the probability of selecting a specific state
$b$ after the cluster has flipped from state $a$,

$\displaystyle\mathbf{P}_{a\rightarrow b}$
$\displaystyle=\prod_{\alpha}\prod_{i=1}^{n_{\alpha}}\frac{w^{\prime}_{\alpha,b_{i}}}{W^{\prime}_{\alpha}}$
(29) $\displaystyle\mathbf{P}_{b\rightarrow a}$
$\displaystyle=\prod_{\alpha}\prod_{i=1}^{n_{\alpha}}\frac{w_{\alpha,a_{i}}}{W_{\alpha}}$
(30)

where $a_{i}$ and $b_{i}$ are used to indicate the new positions for each of
the $n_{\alpha}$ operators to make state $a$ and $b$.

We can now define the acceptance weight for segment $\alpha$ as

$\displaystyle
p_{\alpha}=\left(\frac{W^{\prime}_{\alpha}}{W_{\alpha}}\right)^{n_{\alpha}}$
(31)

and show that it satisfies detailed balance with a metropolis style update
rule. An operator in segment $\alpha$ which moves from position $a_{i}$ to
$b_{i}$ has a relative weight change
$\frac{w^{\prime}_{\alpha,b_{i}}}{w_{\alpha,a_{i}}}$, implying the net graph
weight change is
$\frac{W^{\prime}}{W}=\prod_{\alpha}\prod_{i}^{n_{\alpha}}\left(\frac{w^{\prime}_{\alpha,b^{\prime}_{i}}}{w_{\alpha,a_{i}}}\right)$.
We can now solve for the probability of accepting the update:

$\displaystyle\mathbf{P}_{accept}$
$\displaystyle=\frac{W^{\prime}}{W}\frac{\mathbf{P}_{b\rightarrow
a}}{\mathbf{P}_{a\rightarrow b}}$
$\displaystyle=\left(\prod_{\alpha,i}\frac{w^{\prime}_{\alpha,b_{i}}}{w_{\alpha,a_{i}}}\right)\left(\frac{\prod_{\alpha,i}\frac{w_{\alpha,b_{i}}}{W_{\alpha}}}{\prod_{\alpha,i}\frac{w^{\prime}_{\alpha,b_{i}}}{W^{\prime}_{\alpha}}}\right)$
$\displaystyle=\prod_{\alpha,i}\left(\frac{W^{\prime}_{\alpha}}{W_{\alpha}}\right)=\prod_{\alpha}p_{\alpha}$
(32)

As is standard in metropolis updates, $\mathbf{P}_{accept}$ is taken as the
minimum of this calculated value and $1$. By choosing clusters to update based
on the flippable plaquettes in the expected ground state, we increase the
likelihood of having segments with classically flippable clusters, and thus
$p_{\alpha}=1$.

By randomly selecting small candidate clusters we were able to improve the
autocorrelation times in the small $\Gamma$ regime (See Appendix. B). This
update move is limited by the choice of cluster selection, and we have yet to
investigate applying other cluster based update moves to the cluster selection
stage, though some candidates exist such as the sweeping cluster update[35].
Furthermore, although the RVB update was motivated by classical ring exchange
updates, there may be room for improvement by adapting other algorithms
developed for classical clock models[36] in a similar way.

##  VI Realization on a Quantum Annealer

To experimentally test some of the predicted phases we use a programmable
quantum device. Initially built to study classical optimization problems, the
D-Wave 2000Q is a superconducting qubit based quantum annealer which simulates
a TFIM Hamiltonian.

$H=A(s)\sum_{i}\sigma^{x}_{i}+B(s)\left(\sum_{\langle
ij\rangle}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}+h_{i}\sigma_{i}^{z}\right)$ (33)

Here, $J_{ij}$ and $h_{i}$ are programmable parameters, and $s$ is the
annealing variable which tunes the relative strengths of $A(s)$ and $B(s)$.
The default annealing schedule takes $s$ from $0$ to $1$, starting at $B(0)=0$
and ending at $A(1)=0$, each following a curve specific to the machine[37].
The set of available couplings is given by the architecture; the Chimera
architecture of the D-Wave 2000Q provides up to 2048 qubits and 4196 tunable
couplings arranged in a 16x16 grid with unit cells of 8 qubits each, a
subsection is shown in Fig 16.

Figure 16: Subsection of D-Wave’s Chimera Architecture. Spin variables live on
vertices with tunable Ising interactions shown as edges. Highlighted edges
(identical color scheme to Fig 1) correspond the embedding of the 4-8 lattice
with open boundary conditions. All other couplings disabled. Figure 17:
Samples from the D-Wave 2000Q can be profitably visualized through the dimer
mapping.

To corroborate the evidence provided by Quantum Monte Carlo for the predicted
phase diagram, we embedded the 4-8 lattice isotropic Ising model in the
Chimera architecture. Some defects are introduced by missing spin variables or
bonds. These are machine-dependent defects and can not be completely avoided,
see Fig 17 for an example embedding and sampled dimer state. Each unit cell in
the Chimera graph contains 8 spin variables, allowing two separate sheets of
the 4-8 lattice to be embedded together then tied at their borders. Both
cylindrical[4] and open boundary conditions can be embedded.

We can alter the default annealing schedule to allow the system to equilibrate
at nonzero transverse field. Rather than scaling directly from $s=0$ to $1$,
we scale $s$ to chosen value $s_{\circ}$ such that
$\Gamma/|J|=A(s_{\circ})/B(s_{\circ})$ and allow the system to thermalize. The
machine has a low physical temperature, $T\sim 14mK$, and relatively large
coupling strengths $\beta B(s=1)\sim 41$ (and $\beta A(s=0)\sim 34$). However
the couplings vary as a function of the annealing schedule $s$, therefore a
limitation of this approach is that since both $A$ and $B$ change as a
function of $s$, we find a lower bound for $T/|J|$ for a given choice of
$\Gamma/|J|$, producing an annealing path which varies in both $\Gamma/|J|$
and $T/|J|$ (Fig 19). Furthermore the D-Wave 2000Q must bring the transverse
field to $\Gamma=0$ before taking spin measurements. The timescale of this
quench is not negligible compared to the inverse energy scales, and may be a
source of noise. An additional machine limitation is the precision of the
$J_{ij}$ and $h_{i}$ couplings, which may be shifted from their expected
values. These shifts may be addressed by shimming[4] – performing random
measurements and using average magnetization and correlations to modify
couplings. Previous work[4, 5] has additionally included more advanced noise
reduction techniques, such as adjusting bonds based on symmetries of the
model, altering annealing schedules, and performing reversed or repeated
annealings from seeded ordered states.

Figure 18: Evidence of $\mathbb{Z}_{8}$ KT quasi-long range ordering in
D-Wave. Data from 2048 samples for each point. Cylindrical b.c. used 1866
qubits on two sheets, Open b.c. used 996 qubits on a single sheet. Figure 19:
Trajectory through the $T/\Gamma$ plane. $|\phi|$ shown for Cylindrical b.c.
The phase boundary is drawn consistent with theory (at small $\Gamma$) and QMC
line cuts in Fig 7; we have not systematically mapped out the entire boundary.

By sampling at the lowest allowed $T/|J|$ for each choice of $\Gamma$, or in
other words using the largest allowed coupling strengths, we find a
significant increase in $|\phi|$ around the expected transverse fields. Since
larger $\Gamma$ values must be quenched and pass through phase boundaries we
expect the response to be smeared toward larger $\Gamma$ values. The result
are summarized in Fig 18-19. Qualitatively, we find $\phi$ to be consistent
with the predictions and numerics in preceding sections, although the data
quality falls far below that which QMC is able to offer.

##  VII Conclusion

The fully frustrated 4-8 lattice contains a deceptively large variety of 2D
phases of matter. The phase diagram is developed in low and large $\Gamma$
limits for both the isotropic and anisotropic models, and numerical methods
support our theoretical arguments. The D-Wave 2000Q quantum annealer is
nominally able to realize the frustrated lattice, and by altering the
annealing schedule we could measure the ordering effects of a transverse
field. However, the support offered by the annealer was of lower quality than
that provided by QMC especially as the D-Wave device was limited to relatively
high temperature regions of the phase diagram at finite transverse field.
Accessible parameters are within an order of magnitude of the couplings needed
to observe the quantum ordered or KT phases and thus may well be realizable in
the next generation devices. Our study suggests that future generations of the
D-Wave annealer may actually contribute to physical understanding of
frustrated models beyond those accessible with current classical techniques.

##  VIII Acknowledgements

The authors are grateful to C. Chamon, A. Chandran, R. Moessner, A. Sandvik,
K-H. Wu, S. Zhou for useful discussions. This research used Ising, Los Alamos
National Laboratory’s D-Wave quantum annealer. Ising is supported by NNSA’s
Advanced Simulation and Computing program. C.R.L. acknowledges support from
the NSF through grant PHY-1752727 and the generous hospitality of the Aspen
Center for Physics, which is supported by NSF grant PHY-1607611.

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##  Appendix A Single Mode Approximation

We can find the relevant order parameters by estimating which delocalized
magnon excitations in the paramagnetic region are the first to condense. The
Bloch Hamiltonian restricted to the single magnon states with momentum $k$ is
given by

$\displaystyle H_{m}$
$\displaystyle=\left(\begin{matrix}0&-J&-Je^{ik_{y}}&J^{\prime}\\\
-J&0&J^{\prime}&-Je^{ik_{x}}\\\ -Je^{-ik_{y}}&J^{\prime}&0&J^{\prime}\\\
J^{\prime}&-Je^{-ik_{x}}&J^{\prime}&0\end{matrix}\right)+2\Gamma.$ (34)

The minimum eigenenergy is given by

$\displaystyle E_{k}$
$\displaystyle=2\Gamma-J\sqrt{3+2\sqrt{1-\mbox{sin}(k_{x})\mbox{sin}(k_{y})}}$
(35) $\displaystyle E_{(\frac{\pi}{2},-\frac{\pi}{2})}$
$\displaystyle=2\Gamma-(1+\sqrt{2})J$ (36)

##  Appendix B RVB Autocorrelation Times

To illustrate the benefit of the RVB update in the small $\Gamma$ regime we
measured the exponential fit to $e^{-t/\tau}$ of the autocorrelation graphs
for bond variables, sampled over a variety of transverse field values (Fig
20). We see convergence around $\Gamma=J$ where the RVB does not significantly
contribute compared to the cluster update.

Figure 20: Autocorrelation time for bond variables with and without the RVB
update step. Data taken from 32 independent runs on a $4\times 4$ isotropic
4-8 lattice at $\beta J=5$.