Classical Analog of Quantum Models in Synthetic Dimensions

The Potts model Wu (1982) is a generalization of the Ising model in which the degrees of freedom $S_{j}=1,2,3,\cdots q$ on each lattice site take on $q$ possible values, with the Ising model corresponding to $q=2$. The energy is $-J_{0}$ when adjacent $S_{j}$ share a common value, and zero otherwise. A considerable literature exists concerning the Potts model, with many generalizations, including multi-spin interactions, dilution, external fields, etc. The physics of these is well explored, both analytically and numerically in Ref. Wu (1982). Relevant to the work presented here, the two-dimensional Potts model exhibits a first order phase transition when $q>4$.

We add a term to the Potts model to describe a situation in which the energy is also decreased by $J_{1}$ if neighboring spins $S_{j}$ differ by $\pm 1$. That is,

$\displaystyle E=-J_{0}\sum_{\langle ij\rangle}\delta_{S_{i},S_{j}}-J_{1}\sum_{\langle ij\rangle}\delta_{S_{i},S_{j}+1}-J_{1}\sum_{\langle ij\rangle}\delta_{S_{i},S_{j}-1}$

(2)

We denote by $N=L^{2}$ the number of spatial sites of a lattice with linear dimension $L$. It’s reasonable to employ a Periodic Boundary Condition (PBC) in both the real dimensions as well as the synthetic dimension. In this language, the additional $J_{1}$ term corresponds to a coupling favoring molecules on neighboring sites having adjacent positions in the synthetic dimension, such as is present in the $V$ term of Eq. (1). In the results which follow we set $J_{0}=1$ as our energy scale.

For $q=3$, Eq. 2 is the same as the $q=3$ Potts model with $J_{0}\rightarrow J_{\rm eff}\equiv J_{0}-J_{1}$ (for $J_{0}>J_{1})$. Thus our model would share all Pott’s properties, including a second order transition at $T_{c}=J_{\rm eff}/{\rm ln}(1+\sqrt{3})$. This exact mapping is true only for $q=3$, but is nevertheless suggestive. We do not study the small $q$ limit further here, since, in extending the interactions to include also $\delta_{S_{i},S_{j}+1}$ and $\delta_{S_{i},S_{j}-1}$, for small $q$ the “range” (three) becomes close to the “lattice size” $q$ in the synthetic direction.

The clock model Ortiz et al. (2012); Li et al. (2020) is another well known description of classical phase transitions admitting a controllable discrete set of values $n_{i}=1,2,3,\cdots p$ on each site. It should be noted that, as is the standard notation, the number of Potts degrees of freedom is denoted by $q$, with $p$ being used for the Clock model. The clock model energy is,

$\displaystyle E=-J_{0}\sum_{\langle ij\rangle}{\rm cos}\bigg{(}\frac{2\pi}{p}(n_{i}-n_{j})\bigg{)}\,\,.$

(3)

The clock model shows qualitatively different behavior at different $p$ values, as we will demonstrate happens in our model as well. Fig. 1 shows the phase diagram of the clock model. Unlike the Ising and Potts models, which have single critical points, the clock model, for $p>4$ exhibits two distinct transition temperatures, between which there is a Berezinskii-Kosterlitz-Thouless (BKT) phase. The higher-temperature transition occurs at a critical temperature $T_{c,1}\sim J_{0}$, while the lower transition has $T_{c,2}\sim{\cal O}(1/p^{2})$, vanishing in the XY limit $p\rightarrow\infty$. These two distinct transitions occur because the $\mathbb{Z}_{p}$ breaking of the full continuous symmetry of the XY model is irrelevant for temperatures above $T_{c,2}$ when $p>4$ José et al. (1977); Cardy (1980). As we will show in the sections below, Eq. 2 exhibits similarity to the clock model in the sense of exhibiting three phases, with an intermediate ‘sheet phase’ separating the high temperature disordered and low temperature $\mathbb{Z}_{q}$ breaking phases.

To explore the statistical mechanics of Eq. 2 quantitatively, we will primarily use the simple classical single site update Metropolis Monte Carlo method. We measure observables including the internal energy $E$, the specific heat $C=dE/dT$, and the average synthetic distance $D$ between neighboring real-space sites, where

$\displaystyle D\equiv\frac{1}{2N}\sum_{\langle i,j\rangle}|S_{i}-S_{j}|\,\,.$

(4)

The summation runs over all neighboring sites $i$ and $j$, and $|S_{i}-S_{j}|$ measures the distance of site $i$ and $j$ in the ‘synthetic dimension’. The normalization is to the number of bonds $2N$ on the square lattice. $D$ is defined to loosely capture a transition into a “sheet” phase between a low temperature phase where all $S_{i}$ are identical and the high temperature disordered phase. We will define this sheet phase more precisely in Section 4.

To characterize the phases further, we define

$\displaystyle P_{nn}(d)\equiv\frac{1}{2N}\sum_{\langle ij\rangle}\delta\big{(}\,d-|S_{i}-S_{j}|\,\big{)}$

(5)

which measures the fractional likelihood that the synthetic distance $|S_{i}-S_{j}|$ between near neighbor sites $\langle ij\rangle$ is exactly $d$. $D$ and $P_{nn}(d)$ are related by $D=\sum_{d}d\,P_{nn}(d)$.

The mean field theory of the standard, many-state Potts model can be found, for example, in Mittag and Stephen (1974). The addition of a second energy scale for next nearest neighbors presents some new features which we now consider. For ease of computation, we assign the Potts spin at one site $S$ an accompanying spin variable $\lambda(S)=\omega^{S}$ ($S=0,1,...,q-1$). Here, $\omega=e^{2i\pi/q}$ is the complex increment between spin variables. We then use the identity,

$\displaystyle\delta_{S,S^{\prime}\pm 1}=\frac{1}{q}\sum^{q}_{k=1}\omega^{k(S-S^{\prime}\mp 1)}=\frac{1}{q}\sum^{q}_{k=1}\lambda(S)^{k}\lambda(S^{\prime})^{q-k}\omega^{\mp k}$

(6)

which allows us to convert Kronecker delta functions in the Hamiltonian to products of spin variables and $\omega$. Applying this definition to Eq. 2, the energy between two sites is

$\displaystyle E_{S,S^{\prime}}=-\frac{1}{q}\sum_{k=1}^{q}\big{[}J_{0}+J_{1}\big{(}\omega^{k}+\omega^{-k}\big{)}\big{]}\lambda(S)^{k}\lambda(S^{\prime})^{q-k}$

(7)

We introduce the mean field by defining a set of $q-1$ order parameters, $\langle\lambda^{k}\rangle=R\,e^{ik\theta}$ Mittag and Stephen (1974). $\theta$ serves as the ‘alignment’ of the mean field and can take on values of $\theta=0,\frac{2\pi(1)}{q},...,\frac{2\pi(q-1)}{q}$; if $\theta$ is $\frac{2\pi(4)}{q}$, then the field is aligned relative to $S=5$ (akin to an origin). As the Potts model views neighboring sites as being the same spin, off by one, or of any other value, we are free to choose our field alignment and hence set $\theta=0$ for convenience. Replacing the spin variable $\lambda(S^{\prime})^{q-k}$ with order parameter $\langle\lambda^{q-k}\rangle=R$ leaves us with

	$\displaystyle H_{S}=-\frac{nR}{q}\sum_{k=1}^{q}\big{[}J_{0}\omega^{Sk}+J_{1}\big{(}\omega^{(S+1)k}+\omega^{(S-1)k}\big{)}\big{]}$		(8)
	$\displaystyle E_{S}=-nR\big{[}J_{0}\delta_{S,0}+J_{1}\big{(}\delta_{S-1,0}+\delta_{S+1,0}\big{)}\big{]}$		(9)

where $n$ is the number of nearest neighbor sites ($n=4$ for a square lattice). Taking the expectation value $\langle\lambda\rangle$ and setting it equal to $R$, we arrive at the self-consistent equation

$\displaystyle R=\frac{\text{Tr}\big{(}\lambda e^{-\beta H_{S}}\big{)}}{\text{Tr}\big{(}e^{-\beta H_{S}}\big{)}}=\frac{\big{(}e^{\beta nJ_{0}R}-1\big{)}+2\big{(}e^{\beta nJ_{1}R}-1\big{)}\cos\frac{2\pi}{q}}{e^{\beta nJ_{0}R}+2e^{\beta nJ_{1}R}+q-3}$

(10)

which at $J_{1}=0$ reduces to the self-consistent equation of the standard Potts mean field theory Mittag and Stephen (1974).

The resulting phase diagram with $q=8$ is given in the heat map of Fig. 2. The blue region shows the high temperature, disordered phase where the order parameter $R=0$. For $J_{1}/J_{0}<1$ there is the ‘conventional’ low $T$ ferromagnetic phase of the Potts model (bright green) where Potts variables on different sites are identical. However a second ordered phase appears (cool green) for $J_{1}/J_{0}>1$ in which $R={\rm Re}(e^{2\pi i/q})=\sqrt{2}/2=0.707$ for $q=8$. In this region, the Potts variables on adjacent sites differ by $\pm 1$. We refer to this as an ‘antiferromagnetic’ regime since adjacent spins are forced to take distinct values. We will argue, based on the approximation-free Monte Carlo solution in the next section, in which we can take more sensitive ‘snapshots’ of the spin configuration, that an additional intermediate regime arises in which the Potts variables on adjacent sites are either identical, or differ by $\pm 1$. In the heat map of Fig. 2, this behavior is seen in the dark blue region that emerges for $J_{1}/J_{0}=1$, where $0.707<R<1$ as if the system was allowing for the coexistence of nearest- and next nearest-neighboring interactions between ferromagnetic and antiferromagnetic ordering. The critical point at $J_{1}/J_{0}=0$ is consistent with the analytic value $T_{c}=12/(7\,{\rm ln}7)=0.8810$ for $q=8$.

We turn now present results obtained with Monte Carlo.

It is known for the pure clock model that $q=5$ is a critical value for the appearance of a BKT phase at intermediate temperature, and that the phase diagram is qualitatively the same for any $q>5$, i.e. differing only in the range of temperature over which the intermediate phase is stable, as seen in Fig. 1. Similarly, we observe a intermediate regime that differs from the ground state behavior for $q>6$ in our model as well. To explore the novel intermediate region, we focus our study on just two values: $q=8$ and $q=16$. This choice also allows us to gain some insight into quantitative changes with $q$ without excessive redundancy. We will demonstrate later that instead of creating pairs of vortexes as in the clock model, a ‘sheet’ is formed in the intermediate temperature region when the $J_{1}$ interaction sufficiently strong.

We begin by showing the energy per site $E(T)/N$ for $J_{1}=0.0,0.2,0.8,1.15$ with $q=16$ (Fig. 3a) and $q=8$ (Fig. 3b). For $J_{1}<J_{0}=1$ the ground state has all the sites taking the same spin value, and an energy per site $E_{0}/N=-2J_{0}$. We will call this ground state the Ferromagnetic (FM) phase. For $J_{1}>J_{0}=1$, it is energetically favored to have adjacent sites with $S_{j}=S_{i}\pm 1$ so that $E_{0}/N=-2J_{1}$ at $T=0$. We call this the Antiferromagnetic (AFM) phase. In both panels, the known first order transition at $T_{c}=J_{0}/{\rm ln}(1+\sqrt{q})$ of the conventional Potts model ($J_{1}=0.0$) is shown for comparison. This first order character remains evident at small $J_{1}$, however, occurring at a (slightly) reduced $T_{c}$. We interpret the lowering of $T_{c}$ to arise from the fact that $J_{1}$ initially competes with the tendency of $J_{0}$ to make all values of $S_{j}$ identical.

As $J_{1}$ increase further, however, $T_{c}$ begins to grow. Here our picture is that the two interactions behave in a more cooperative way, raising the critical temperature. Notably, it can be seen that when $J_{1}$ is small, $T_{c}$ predicted by mean field theory is slightly higher than that predicted by Monte Carlo data, which is to be expected, as mean field theory overestimates $T_{c}$ by neglecting fluctuations. Simultaneously, the evolution of $E(T)$ becomes more gradual, suggesting a continuous phase transition, or even a crossover. Studies on a finer mesh of $J_{1}$ indicate the change of order of transition appears to coincide with the value of $J_{1}$ for which $T_{c}$ is minimized. As can be seen from the $J_{1}=0.8$ curve, aside from the now-continuous transition at $T\sim 0.7$, a second feature emerges at $T\sim 0.2$. This is a first indication of the existence of an intermediate regime.

The change from first-order character is further revealed in the specific heat. Figure 4 gives $C(T)$ for the same sequence of $J_{1}$, again for $q=16$ and $q=8$. For $J_{1}=0.0,0.2$, the first-order discontinuity of $E(T)$ in the thermodynamic limit is signalled by the very large value of the finite lattice derivative, exceeding by more than an order of magnitude the peaks at $J_{1}=0.8,1.15$ where the transition is second order. Framed more precisely, at a first order transition the specific heat on a finite lattice scales as the volume of the system, $C(T)\sim N=L^{2}$, whereas for a second order transition $C(T)\sim L^{\alpha/\nu}$. Here $\alpha$ and $\nu$ are the critical exponents for the specific heat and correlation length respectively. As a consequence, the specific heat peaks are much smaller in the second order region.

As noted already in the data for $E(T)$, Fig. 3, for $J_{1}=0.8,1.15$ an additional, lower temperature, specific heat peak emerges. This suggests the existence of an intermediate regime between the ordered phase at low temperature and the disordered phase at high temperature. We shall focus our discussion to the intermediate regime in the next section. Given the qualitative similarity of the results for $q=8$ and $q=16$, in the next section we will restrict our analysis to the larger value of $q=16$.

We now turn to examining the nature of the intermediate region, in which we believe a ‘sheet’ forms, analogous to that of the quantum model Feng et al. (2022). Fig. 5 shows the average distance $D$ in synthetic dimension between nearest neighbors on the lattice, as a function of $T$, for $J_{1}=0.0,0.2,0.8$ and $1.15$, that is, horizontal sweeps of the phase diagram of Fig. 2. For all $J_{1}$ values, we see $D$ asymptote at high temperature to $D=q/4$, which indicates a disordered phase where the spins randomly (probability $1/q$) and independently occupy different synthetic position.. (Note our use of PBC in the synthetic direction imposes a maximal value $D=q/2$.)

For $J_{1}=0.0$ and $0.2$, $D$ increases from $0$ to the high temperature limit of $q/4$ through a direct jump, suggesting a single first order phase transition from the ordered phase to the disordered phase, like the Potts model. However, for $J_{1}=0.8$, $D$ shows a plateau which coincides with the temperature range between the two specific heat peaks in Fig. 4. In this region, $J_{1}$ is sufficiently strong to allow neighboring spins $S_{i}$ and $S_{j}$ to take on values $S_{j}=S_{i}\pm 1$ in addition to $S_{j}=S_{i}$ favored by $J_{0}$. The precise value of $D$ depends on $J_{1}$, which will control the relative likelihood of $S_{j}=S_{i}\pm 1$ and $S_{j}=S_{i}$. If those cases are all equally probable, for example, then $D\sim 2/3$. We will refer to this as the sheet phase, because spatially neighboring spins exist in a range of three adjacent values, and, as we shall see, this persists to larger separations, so that the configuration looks like a sheet in $2+1$ dimensions. This is to be distinguished from the Ferromagnetic phase, where all spins are identical.

For $J_{1}=1.15$, the ground state is different. $D=1$ at low temperature is expected now since all the neighboring spins favor taking values that differ by one. This change of ground state can be checked in the $T=0$ limit of $E$ in Fig. 3, which takes on the value of $E=-2J_{1}$ as expected. With increasing $T$, neighboring spins in the system start to take on values that are the same, thus the system enters the sheet regime, before going into the disordered phase at high temperature, as was the case with $J_{1}=0.8$. That the intermediate $T$ region of $J_{1}=0.8$ and $J_{1}=1.15$ are of the same nature, and no transition is between them, will be corroborated below.

The existence of the intermediate sheet region can be qualitatively understood by looking at the system’s free energy. For $J_{1}<1$, at very low but finite temperature, excited states become accessible. The lowest-energy excitations are the ones where one site takes a spin value that differs by one from its neighbors’. This costs an energy of $\Delta E=(J_{0}-J_{1})$. Comparing to a Potts model with $J=(J_{0}-J_{1})$, it’s evident that our model should retain its FM order for a range of finite low temperatures, since all of its excitations cost more energy than that of the Potts model’s, which is know to have a FM ground state at low temperature. However, upon increasing $T$, a manifold of excited states that we call ’sheet state’ will proliferate. The sheet state refers to a state where all the neighboring spins take on spin values that are either the same or differ by one. Such states have extensive energy $\Delta E\propto(J_{0}-J_{1})$, and entropy per site $\Delta S=\ln 3$. The latter can be understood by a counting similar to that of degenerate ground states for $J_{1}>1$, which is outlined in the Appendix Classical Analog of Quantum Models in Synthetic Dimensions. This means sheet states will have a free energy contribution $\Delta F\propto(J_{0}-J_{1})-T\ln 3$, which goes negative at $T\gtrsim(J_{0}-J_{1})/\ln 3$. We will show in the next paragraph that the system makes a transition into a sheet-state regime at a finite temperature for $J_{1}\gtrsim J_{1c}\sim 0.4$ for $q=16$. As we continue to increase $T$, the system will go into a disordered state as expected. We recall that it was also observed in $E(T)$ and $C(T)$ plots that at the same $J_{1c}$ where sheet phase appears, the transition to the high temperature disordered phase becomes continuous. We argue that this is plausible as the system in the sheet regime could have different symmetry from its ground state. This also raises the question of whether the transition from the ground state to the sheet regime is a true phase transition. We will come back to this question when discussing the phase diagram.

To support the existence of the sheet regime further, we measure $P_{nn}(|S_{i}-S_{j}|)$. In the upper panel, we see $P_{nn}(0)\sim 1$ while both $P_{nn}(1)$ and $P_{nn}(2)\sim 0$ at low $T$ for $J_{1}=0.8$. This tells us all the spins are taking the same value as they should in the FM phase. As we increase $T$ past the temperature of the lower heat capacity peak, $P_{nn}(0)$ starts to drop significantly, while $P_{nn}(1)$ increases. Meanwhile $P_{nn}(2)$ is still very close to $0$. This is to say, in this region, neighboring spins are equally likely to take values that are identical and off by one, but they dislike taking values that are further separated, the defining characteristic of the sheet phase, and consistent with Fig. 5. As we increase $T$ further, all the curves go to the same limit, resembling the disordered phase where all spin values are equally likely.

Similarly, in the upper panel, the low $T$ limit of $J_{1}=1.15$ is showing a clear feature of the AFM phase. In the intermediate $T$ region between the two heat capacity peaks, the decrease of $P_{nn}(1)$ and the increase of $P_{nn}(0)$ are both significant, while $P_{nn}(2)$ is kept at a low level. This is similar to the behavior at $J_{1}=0.8$. The difference is that here we’re coming to the sheet regime from an AFM phase instead of a FM phase. This picture will be validated by the machine learning method explained in the next section.

Both supervised and unsupervised machine learning algorithms have been used recently to find critical points of classical and quantum models of magnetism Wang (2016); Hu et al. (2017); Carrasquilla and Melko (2017); Zhang et al. (2019); Canabarro et al. (2019); van Nieuwenburg et al. (2017); Lee and Kim (2019). Many of the most impressive applications of machine learning are those using unsupervised algorithms, since no prior knowledge of the phase diagram is needed. Recently, for example, phase transitions of the $q$-state Potts model were analyzed with a series of unsupervised techniques including Principal Component Analysis and k-means clustering, among others Tirelli et al. (2022). While the results of these methods were certainly impressive, we now explore the use of an unsupervised method known as the learning by confusion (LBC) approach, which uses neural networks instead of dimensional reduction or clustering algorithms. This method is becoming very widespread, and was even used to examine the $q$-state Potts model in a recent study Richter-Laskowska et al. (2022). In this section, we outline the LBC method and the details of our implementation, and subsequently present our results.

The LBC algorithm, while being completely unsupervised, can be thought of as a series of attempts at supervised machine learning. Raw spin configurations generated from Monte Carlo are used as input data, which are then purposely given incorrect labels. In particular, ‘fake’ critical points $T_{c}^{\prime}$ are postulated: for each, all configurations corresponding to temperatures below $T_{c}^{\prime}$ are given a label of 0, and those above are given a label of 1. The data are then split into training and testing data sets. A neural network is trained with the labeled data, and the accuracy is computed against the test data split. As each new $T_{c}^{\prime}$ is chosen, the neural network is re-trained, and a new accuracy is then calculated.

The key observation is the following: When $T_{c}^{\prime}$ is very large (i.e. higher than all the temperatures sampled in the Monte Carlo), all data, training and test, are labeled 0, and the neural net can learn that trivial labeling with 100% accuracy. The same is true for $T_{c}^{\prime}$ very small. Likewise, when $T_{c}^{\prime}$ arrives at a value close to the true transition temperature, the neural network will be able to classify most of the configurations correctly, yielding a very high accuracy. For other values of $T_{c}^{\prime}$, the neural network will fail to distinguish some configurations from each other, resulting in comparatively low accuracies. It follows, then, that the accuracy vs temperature plots will have a ‘W’ shape, with the middle peak corresponding to the correct $T_{c}$. In cases with two phase transitions, accuracy vs temperature plots result in a double ‘W’ shape, with the two peaks corresponding to the two critical temperatures. This process can also be done holding temperature constant, and varying other degrees of freedom such as $J_{1}$.

Although the LBC method has had major success in recent years, it is important to note that, because we are using finite sized lattices, it is possible that the accuracy peak corresponds to a crossover rather than a true phase transition. While there is work indicating that the shape of the peak corresponds to the order of the phase transition Richter-Laskowska et al. (2022), it is still certainly possible that a peak corresponds to a crossover. As such, we turn to more traditional methods to more closely characterize the nature of the LBC accuracy peaks.

Traditionally, the LBC method is used with a fully connected neural network, which is expressive enough to distinguish phase transitions in many cases van Nieuwenburg et al. (2017); Lee and Kim (2019). In our case, however, such neural networks performed quite poorly, so convolutional neural networks (CNNs) were used instead, which are much better at picking out subtle spatial information from the data. It may be possible to get successful results with fully connected neural networks, but more data or a more complex training protocol may need to be used, slowing down the algorithm dramatically. Fig. 7 shows an example of the LBC method using CNNs, illustrating congruent results with traditional methods.

We tested the LBC approach on data generated from Monte Carlo simulations using $q=8$, as it is sufficiently high for the existence of the sheet regime, but moving to larger $q$ values would increase the time for computation, as explained in the Appendix. We also tested the LBC scheme at many different values of $J_{1}$ to get more comprehensive results. As expected, LBC peaks and specific heat peaks were in strong agreement, both in cases with and without the sheet regime. Fig. 8 shows the resulting phase diagram generated from many runs similar to that of Fig. 7. In addition to sweeps across temperature, we also present multiple runs holding temperature constant, and sweeping across $J_{1}$ to further validate our results.

The use of the LBC approach applied to this model is not to claim superiority over traditional methods such as C(T) peaks, but rather to highlight its application to a specific problem in which the phase diagram was unknown. By benchmarking the LBC method in this way, we aim to show its effectiveness and reliability. The ultimate goal is to help expand the toolkit available for dealing with novel or poorly understood systems. In this way, we hope to underscore the potential of the LBC approach to provide new insights, which will become useful in areas where more traditional methods might face challenges.

Based on the analysis in the above sections, we present in Fig. 8 the phase diagrams of our model for $q=8$ and $q=16$, as well as snapshots of states in each of the phases. As we found in Sec. IV, $q=8$ and $q=16$ have qualitatively the same behavior, which is confirmed in panels (a) and (d). While there are slight differences in the specific transition temperatures, both phase diagrams have very similar structures. Lower $q$ values such as $q=4$, however, are likely to yield much different phase diagrams, as is the case in the $q$ state clock model Ortiz et al. (2012).

At sufficiently low $T$, the system stays at its FM or AFM ground states for $J_{1}<1$ and $J_{1}>1$ respectively, snapshots of which are also shown in Fig. 8. In the FM phase, all spins line up, whereas in the AFM phase, spins deviates by exactly one spin value from that of its neighbor’s, rendering a fluctuating-looking snapshot.

Below a critical value of $J_{1c}$ ($J_{1c}\sim 0.4$ for $q=16$ and $J_{1c}\sim 0.5$ for $q=8$), the system acts like a Potts model, which transitions into the disordered phase through a first order phase transition, as indicated by the blue line in the phase diagram. Increasing $J_{1}$ only slightly lowers $T_{c}$, which can be understood since $J_{1}$ makes it harder for the system to develop ferromagnetic order.

Above the critical value of $J_{1c}$, as we increase $T$, the system develops a sheet regime, before a phase transition into the disordered phase. A snapshot of the sheet regime is shown, where neighboring spins are able to take the same value as well as off-by-one values. For this reason, if we treat the spin values as a synthetic dimension, we’ll see a sheet in the 2 spatial + 1 synthetic dimension. For $J_{1}<1$, the smooth transition in $E(T)$ in Fig. 3 and the size independence of the peak in $C(T)$ in Fig. 4 indicates that the boundary between the FM and sheet region is either a crossover or a KT transition. Similar size independence of heat capacity is observed in the $XY$ model as well Nguyen and Boninsegni (2021). This is marked as a dashed magenta boundary in the phase diagrams. For $J_{1}>1$, the transition from the AFM to the sheet region looks continuous as well if we look at the $E(T)$ plot in Fig. 3. However, we believe it’s likely a second order phase transition as Fig. 4 shows clear size dependence of the heat capacity peak for $J_{1}=1.15$, as opposed to $J_{1}=0.8$. Furthermore, the FM and AFM phases are distinct, in that there should be a phase transition dividing the two regions. This provides further evidence that the boundary from the AFM phase to the sheet regime is a phase transition as opposed to a crossover. We have marked the phase boundary pink to differentiate from the first order transition happening between the FM and sheet phase.

Finally for all $J_{1}>J_{1c}$, the system will eventually transition to a disordered phase through a second order phase transition, indicated by the green boundary in the phase diagram. We are be able to corroborate this claim while locating the transition temperature more accurately by using the Binder ratio $\frac{\langle M^{4}\rangle}{\langle M^{2}\rangle^{2}}$. Here $M$ is defined as $M=\frac{1}{N}\sum_{j}e^{\frac{2\pi S_{j}i}{q}}$. It can be seen from Fig. 9 that the positions of the corresponding heat capacity peaks are very close to the critical temperatures indicated by the Binder ratio crossing points. However we should mention that the Binder ratio fails to capture other phase boundaries, thus we’re not showing the Binder ratio in a broader temperature range here.

In this paper we have explored the phase diagram of a variant of the Potts model in which the energy is lowered not only when the degrees of freedom $S_{i}=1,2,\cdots q$ on adjacent spatial sites are identical, but where there is an additional energy lowering when they differ by $\pm 1$, that is when $S_{j}=S_{i}\pm 1$. Such a situation mimics that of experiments with ultracold dipolar particles in synthetic dimensions, for example ultracold polar molecules in an optical trap in which each molecule has a ladder of possible rotational states with $z$ component of the angular momentum $m_{i}$ of molecule $i$, and inter-molecular dipole interactions favor $m_{j}=m_{i}\pm 1$.

Although a precise connection between the two models is limited by the absence of quantum fluctuations in our classical analog, our results reveal several interesting qualitative links. In particular, we observe the emergence of a low temperature sheet regime which is very similar to one of the defining characteristics of the molecular system.

Prior studies Glumac and Uzelac (1998) of the three state Potts model have considered the effect of the range of the interaction $V(r)\sim 1/r^{1+\sigma}$ on the order of the transition, finding first order character for $\sigma\lesssim 0.7$. The context was that of a lattice with one (physical) dimension and long range (power law decaying) interactions. Our specific perspective here has been to formulate a classical model which serves as an analog of quantum models of sheet formation in synthetic dimension. Our work, which incorporates an energy which is lowered when adjacent sites (in two ‘physical’ dimensions) have variables which are identical (the usual Potts model) or differ by $\pm 1$ (our extension) can be interpreted as a similar extension of the ‘range’ of the interaction, but only in the third ‘synthetic’ direction.

Finally, we emphasize that determination of the phase diagrams of such models poses an interesting new challenge to machine learning and related information-theoretic Negrete et al. (2018, 2021) approaches being developed for statistical mechanics. We have shown here that learning by confusion van Nieuwenburg et al. (2017); Lee and Kim (2019) provides a powerful and accurate means to gain insight, and, significantly, that it is able to capture a ‘double-$W$’ structure of the accuracy which might be expected in the presence of two successive finite temperature transitions.

M.C., Y.Z., M.C, and R.T.S acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0022311. K.H. acknowledges support from the Welch Foundation (C-1872) and the National Science Foundation (PHY-1848304). K.H. thanks the Aspen Center for Physics, supported by the National Science Foundation grant PHY-1066293, and the KITP, supported in part by the National Science Foundation under Grant PHY-1748958, for their hospitality while part of this work was performed.