Symmetry Breaking in an Extended O(2) Model

Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model in two dimensions that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term $-h_q\cos(q\varphi)$. Its coupling $h_q$ allows to smoothly interpolate between the O(2) model ($h_q=0$) and a $q$-state clock model ($h_q\rightarrow\infty$). In the latter case, a $q$-state clock model can also be defined for noninteger values of $q$. Thus, such a limit can also be considered as an analytic continuation of an ordinary $q$-state clock model to noninteger $q$. In previous work, we established the phase diagram for noninteger $q$ in the infinite coupling limit ($h_q\rightarrow\infty$). We showed that there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we seek to establish the phase diagram at finite values of the coupling using Monte Carlo and tensor methods. We show that for noninteger $q$, the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer $q=2,3,4$, we know there is a second-order phase transition at infinite coupling (i.e. the well-known clock models). At finite coupling, we find that the critical exponents for $q=3,4$ vary with the coupling, and for $q=4$ the transition may turn into a BKT transition at small coupling. We comment on the similarities and differences of the phase diagrams with those of quantum simulators of the Abelian-Higgs model based on ladder-shaped arrays of Rydberg atoms.

.The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer , there is a second-order phase transition at low temperature and a crossover at high temperature.In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods.The results may be relevant for configurable Rydberg-atom arrays.

Introduction
For the quantum simulation of quantum field theories, one needs to discretize space and truncate the fields.For example, one could make a Z  approximation of a continuous  (1) symmetry.Given the limited qubits available, it is important to optimize the discretization procedure, and it may be be useful to have a family of models that allows one to continously interpolate among various possible discretizations.
We consider a class of extended-O(2) models by adding a term  cos() to the classical  (2) model.The continuously tunable parameters  and  allows us to interpolate among various models.When  = 0, the symmetry-breaking term is turned off, and this is just the  (2) model.When  = ∞, the  (2) symmetry is completely broken, and the spins take only discrete angles  = 2/ for integer .For finite values of , we can study the effect of the symmetry-breaking.
The parameter  allows us to tune from the Ising model ( = 2), through the -state clock models, and to the  model ( → ∞).We allow also noninteger , which allows us to interpolate among all of these models.Previously, we studied the  → ∞ limit of this model [1,2], and we found that for noninteger , there is a crossover at small  and an Ising phase transition at large .We present here some preliminary results for finite  by looking at the specific heat and entanglement entropy.These early results suggest that the  = ∞ phase diagram for noninteger  seems to persist to all  > 0. However, a rigorous determination of the phase diagram awaits the completion of our ongoing finite size scaling study of this model.

The Extended-O(2) Model
We study a two-dimensional classical spin model obtained by adding a symmetry-breaking term to the action of the classical O(2) model When  = 0, this is the classic XY model, which is known to have a BKT transition.When  > 0, the second term breaks periodicity and we must choose  ∈ [ 0 ,  0 + 2) for some choice  0 .In this work, we use  0 = 0.When  → ∞, the continuous angle  is forced into the discrete "clock" angles as illustrated in Figure 1.For integer , this is the ordinary -state clock model with Z  symmetry.
For noninteger , this defines an interpolation of the clock model to noninteger values of .For noninteger , divergence from ordinary clock model behavior is driven by the "leftover" angle The choice  ∈ [0, 2) results in a hard cutoff at  = 0.In practice, for noninteger , we found that this cutoff skews the distribution of angles severely enough that the model at finite  does not smoothly connect to the model at infinite  when  → ∞.To fix this, we shifted the angle domain such that  ∈ [−, 2 − ) to move the hard cutoff away from the clock angle at  = 0. We chose  = (1 −  /).The  model has a disordered phase at small , a BKT transition near   = 1.12, and a critical phase at large .We previously established the phase diagram of the  = ∞ plane [1].For integer , it is the well-studied clock model.For noninteger , there is a crossover at small  and a second-order phase transition of the Ising universality class at large .Establishing the phase diagram at finite- is the goal of the current and ongoing work.
Previously, we studied the  → ∞ limit of this model [1,2].In this limit, we were able to replace the action with but then directly restrict the previously continuous angles to the discrete values given by Eq. ( 2).Using Markov Chain Monte Carlo (MCMC) and tensor renormalization group (TRG) methods, we were able to map out the  = ∞ phase diagram of the model.For noninteger , we found a crossover at small  and a second-order phase transition of the Ising universality class at large .We are currently working to understand the phase diagram at finite-.See Figure 2.

Leon Hostetler
We define the internal energy as and the specific heat as where . . .denotes the ensemble average.The entanglement entropy is a quantum quantity, however, one can introduce an analog for classical lattice systems via a reduced density matrix where the degrees of freedom of the system are partially integrated out.On a tensor network this can be realized as a partial trace of indices.We define the entanglement entropy as where    are the eigenvalues of a reduced density matrix ρ .A detailed discussion can be found in [3].

Preliminary Results
When we studied the model in the  → ∞ limit, we were able to treat the spin degrees of freedom as discrete.This allowed us to use an MCMC heatbath algorithm to explore the model at small volumes and a TRG method to study it at large volumes.The model is more difficult to study when  is finite.The spin degrees of freedom are now continuous.MCMC heatbath is no longer an option, so we're left with the Metropolis algorithm, which suffers from low acceptance rates and leads to large autocorrelations for noninteger .Furthermore, our TRG method was designed for the  → ∞ limit, and extending it to finite- proved difficult.We needed to make some algorithmic improvements.On the Monte Carlo side, we implemented a biased Metropolis heatbath algorithm [4], which is designed to approach heatbath acceptance rates.To explore large volumes, we implemented a tensorial approach that used Gaussian quadrature.
Recent previous work on a very similar model includes [6].There, the authors studied integer  ≥ 5 in the  → 0 limit as a symmetry-breaking perturbation to the  (2) model.They found that a second transition (in addition to the BKT transition that occurs at  = 0) occurs for any finite  > 0.
In the present work, we consider also intermediate and large values of  as well as noninteger values of .Our initial exploration of the model at finite- was performed using MCMC on small lattices, but then we used tensor methods to perform a large volume survey of the parameter space.
In general, a plot of the specific heat versus temperature or  shows a peak whenever there is phase transition or a crossover.Thus, a "heatmap" of the specific heat can serve as a proxy for the phase diagram.In Figure 3, we show three such heatmaps of the specific heat in the - In our model,  is treated as a coupling attached to the interaction term of the energy function, and then the Boltzmann factor is  − .In their model,  is factored out, such that the Boltzmann factor is  − , and  therefore functions as a true thermodynamic inverse temperature.We believe the phase diagrams of these two models are qualitatively the same since the two models differ only by a rescaled .plane for three different values of .These heatmaps were computed by TRG with  = 1024 and Δ = Δ = 0.08.In the left panel,  = 16.This already looks like the  = ∞ phase diagram, where for noninteger  there is a crossover at small , and a second-order phase transition of the Ising universality class at large .We see that there are discontinuities as  crosses integer values.At large values of , the spin angles strongly prefer the "clock angles"  = 2/ as illustrated in Figure 1.As  crosses an integer, an additional clock angle is introduced into the model, hence the discontinuites at integer  when  is large.In the middle panel,  = 1.At this intermediate value of , the discontinuities have disappeared, and what remains are smooth curves.In the right panel,  = 0.1.At this small value of , the  (2) symmetry is barely broken by the addition of the  cos() term, and the specific heat "phase diagram" seems to be smoothly connecting to the  phase diagram at  = 0.
In Figure 3, we see that at intermediate values of , a heatmap of the specific heat develops smooth curves.This suggests the possibility of a continuous line of phase transitions even across integer values of , or perhaps even of more exotic phases.For example, the heatmap of the specific heat at  = 1 shows similarities to the phase diagram of certain Rydberg atom systems [5], which also involve energy functions with continuously tunable parameters.
In Figure 4, we look more closely at  = 1 in the neighborhood of  = 3.In the left panel, we have the specific heat, which shows fairly smooth behavior even as one crosses an integer value of .In the right panel, we look at the entanglement entropy.From the heatmap of the entanglement entropy, we see that the discontinuity at  = 3, which disappears for the specific heat as one dials  toward zero, persists however for the entanglement entropy.In fact, the entanglement entropy takes a value of ln 3 along the line  = 3 for sufficiently large -implying a line of Z 3 order along this

Summary and Ongoing Work
We are studying the O(2) model extended with a symmetry-breaking term  cos().The model has two continuously tunable parameters, which allows us to tune from the Ising model, through the -state clock models, and to the  model by varying  from  = 2 to  → ∞, and it allows us to tune from the XY model to a given -state clock model by varying  from  = 0 to  = ∞.
Previously, we established the  = ∞ plane of the 3 parameter (, , ) phase diagram.When is an integer, we recover the classic -state clock model which has a single second-order phase transition for  = 2, 3, 4 and two BKT transitions for  ≥ 5.When  is noninteger, we get a crossover and a second-order phase transition.Between integer values of , the phase transition lines are smooth and continuous, however, across integer values of  they are not.For example, between 3 <  ≤ 4, there is a continuous second-order transition line and similarly for 4 <  ≤ 5.However, there is a discontinuity at  = 4. On the other hand, at  = 0, there is a single continuous line of BKT transitions, corresponding to the  model for all values of .The question we are faced with is how does the  = ∞ phase diagram connect to the  = 0 phase diagram as  is varied?Does the  = ∞ phase diagram persist to all finite values of , or is there some value of  > 0 at which the phase transition lines are smooth and continuous?
In this work, we perform a preliminary exploration of the finite- region of the phase diagram.The specific heat, which develops a peak near values of  corresponding to crossovers or phase transitions, can serve as a proxy for the phase diagram.As we tune to smaller values of , the discontinuities that exist at infinite  disappear and the lines become smooth and continuousgiving the possibility that smooth and continuous phase transition lines occur across integer values of .However, if we look at the entanglement entropy, we see that the original discontinuities seem to persist-suggesting that the  = ∞ phase diagram may persist to all  > 0.
To rigorously characterize the phase diagram at finite , we are currently performing a finite size scaling study of this model to extract the critical exponents , , , , and  using MCMC and TRG.We expect to have those results very soon and to give a detailed picture of the phase diagram at finite  in a forthcoming paper.

Figure 3 :
Figure 3: Heatmaps of the specific heat from TRG with  = 1024.(Left panel) Here  = 16.This already looks like the  = ∞ phase diagram, where for noninteger  there is a crossover at small , and a second-order phase transition of the Ising universality class at large .We see that there are discontinuities as  crosses integer values.At large values of , the spin angles strongly prefer the "clock angles"  = 2/ as illustrated in Figure 1.As  crosses an integer, an additional clock angle is introduced into the model, hence the discontinuites at integer  when  is large.(Middle panel) Here  = 1.At this intermediate value of , the discontinuities have disappeared, and what remains are smooth curves.(Right panel) Here  = 0.1.At this small value of , the  (2) symmetry is barely broken by the addition of the  cos() term, and the specific heat "phase diagram" seems to be smoothly connecting to the  phase diagram at  = 0.

Figure 4 :
Figure 4: Here, we look more closely at  = 1 in the neighborhood of  = 3.In the left panel, we have the specific heat, which shows fairly smooth behavior even as one crosses an integer value of .In the right panel, we look at the entanglement entropy, which shows that a discontinuity at integer values of  persists to small values of .The entanglement entropy takes a value of ln 3 along the line  = 3 for sufficiently large -implying a line of Z 3 order along this line.Within the lobe region, the entanglement entropy takes a value of ln 2-implying a region of Z 2 order.