Emergent O() Symmetry in a Series of 3D Potts Models
We study the -state Potts model on the simple cubic lattice with ferromagnetic interactions in one lattice direction, and antiferromagnetic interactions in the two other directions. As the temperature decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O() model with . This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller we find, for and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at , and the system effectively reduces to a disordered two-dimensional Potts antiferromagnet. These results are obtained by means of Monte Carlo simulations and finite-size scaling.
According to the hypothesis of universality, critical phase transitions fall in classes determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry may emerge in the order parameter, and the associated critical behavior may become very rich.
Examples of this phenomenon are known, in particular in two dimensions (2D). In the -state clock model , the spins are confined to a plane and take discrete directions . The ferromagnetic ground state is -fold degenerate, reflecting the symmetry. For , however, the 2D clock model exhibits a Berezinskii-Kosterlitz-Thouless (BKT) transition as the temperature is lowered, and quasi-long-range order with continuous O(2) symmetry emerges at , just as in the rotationally invariant XY model. Emergent symmetries are also found in many other physical systems, including a spin ice system , deconfined quantum critical points , high- superconductors  and so forth , and are often accompanied by very interesting phenomena. For example, in the spin ice , emergent SU(2) symmetry leads to an unusual phase transition with a jump in the order parameter, which is a feature of discontinuous transitions, whereas the the domain wall tension vanishes, which is a feature of continuous transitions. In a class of models with and U(1) symmetry , emergent supersymmetry at the Ising-BKT multicritical point leads to new critical behavior, with unusual scaling of the correlation length.
The Potts model  has spins with values that interact as , reflecting permutation symmetry. The antiferromagnetic model on the simple cubic lattice breaks an effective symmetry at low temperatures, and O(2) symmetry emerges at the critical point . In line with these findings, a BKT transitions with emergent O(2) symmetry can arise on simple cubic lattices with a finite thickness . For similar models with one might thus expect emergent O() symmetry, i.e., isotropy in dimensions. While this scenario is consistent with numerical results , the situation is not entirely clear .
It has been hypothesized  that in high- superconductors, the magnetic and superconducting degrees of freedom can merge into a critical state with effective SO(5) symmetry. However, as argued by Fradkin et al. , the symmetry of the corresponding O(5) fixed point is easily broken since the components of the order parameter are inequivalent on the microscale. This applies as well to other systems  for which higher emergent symmetries had been proposed.
A different critical behavior occurs in two-dimensional systems with mixed interactions—i.e., ferromagnetic (FM) in one direction and antiferromagnetic (AF) in the other. The mixed Potts model on the square lattice undergoes a BKT-like transition, and O(2) symmetry emerges in the low-temperature range . Rich phenomena also occur in the the mixed Ising () model on the multi-layered triangular lattice .
In this paper, we study -state mixed Potts models on the simple cubic lattice, with FM couplings in the direction and AF couplings in the plane. Using cluster-type Monte Carlo algorithms, we find continuous phase transitions for as the temperature is lowered. The critical behavior of these systems is consistent with O() universality in 3D, with . This result may hold more generally, i.e., for at least some . To our knowledge, such an emergent O() symmetry in the 3D Potts model has, apart from the antiferromagnet, not been reported in literature.
2Model, Algorithm, critical points, and critical exponents
The reduced Hamiltonian of the mixed Potts model is
where the spins take values . With , the sum in the first term, taken over all nearest-neighbor sites in layer , defines AF couplings. The second term defines FM couplings in the direction. We refer to the temperature as .
Cluster Monte Carlo methods are very effective for simulation of FM lattice models , while the efficiency of the Wang-Swendsen-Kotecký (WSK) algorithm  for AF Potts models depends on the lattice type and temperature. For mixed interactions, we apply a single-cluster algorithm merging elements of the Wolff method for FM models ) and the WSK algorithm. A combination with the geometric cluster algorithm , which employs lattice symmetries, is still needed for effective simulations of systems up to at sufficiently low temperatures. In addition we applied Metropolis sweeps.
The sampled observables include the staggered susceptibility , the uniform susceptibility , their Binder ratios and , and the specific heat :
where is the system volume, and the energy density. and are defined as
with (with ) the density of state- spins on sublattice , namely
with spin coordinates . The sublattice is defined by the parity of . The and are the order parameters of the model, exposing a possible symmetry breaking of the model. It should be noted that, in the AF Potts model, the type of order in the low-temperature phases depends essentially on entropy effects, apart from the energy effect. For example, the low temperature phase of the 3-state AF Potts model on the simple cubic lattice displays long-range order with one sublattice frozen in one of the Potts states, while the spins on the other sublattices are free to randomly take one of the other Potts states . The maximal entropy of the latter sublattice explains the existence of this type of state. The phase transition to this state is thus, at least in part, entropy-driven, similar to behavior found for certain two-dimensional Potts antiferromagnets .
We have investigated the model (Equation 1) for and 6, with periodic boundary conditions. The procedure involved three steps, specified here for .
First, we simulate for several at a number of temperatures taken in a wide range. Each data point is based on Monte Carlo steps (MCSs). Each MCS consists of 5 Wolff-cluster updates of the WSK type, 5 geometric-cluster updates, 5 Metropolis sweeps, and data sampling. Each different simulation uses a different random seed, and starts from a random initial configuration, after which about data samples are discarded to allow for equilibration of the system. These simulations are distributed over different CPU cores. After their completion, the resulting data are collected, and the averages of the physical variables and their error bars are calculated. Plots of and in Figure 1 yield an approximate critical point . This is also seen in the scaling of the specific heat, in the left panel of Figure 2.
Next, in order to determine the critical exponents of this phase transition we simulate near the estimated , with MCS taken at each data point. The data scale as
where is the thermal exponent, is the correction-to-scaling exponent, and , and are unknowns. A least-squares fit of this formula to the data yields , and .
In the last step, we simulate at and fit the data of by
with spatial dimensionality . The magnetic exponent follows as . The uniform susceptibility was also fitted by Eq. (Equation 4), with replaced by another magnetic exponent . This fit gives .
The results for , and 6 used the same procedure. The critical points and exponents are listed in Table ?. A comparison with the exponents and of the O() model  shows that the phase transition of the -state mixed Potts model fits the universality class of the O() model with .
3Histogram of the O(n) symmetry
The remarkable emergence of 3D O() universality in these models invites the construction of order parameter histograms, by representing the spins as vectors. These vectors are symmetrically distributed in -dimensional space, such that their scalar product matches the pair potential of Eq. (Equation 1), which can be written as
For example, in the case of these vectors span a regular triangle, i.e., with . For , the vectors are three-dimensional ones, and span a regular tetrahedron:
Based on this vector representation, the magnetization is sampled separately for sublattices and . This yields the components of and in Eqs. (Equation 2) and ( ?)
Figures Figure 3(a)-(c) display the histograms of the staggered magnetization for systems, projected on two Cartesian axes, for , 4, and 5 respectively. The histograms are the same for any choice of the axes. The apparent isotropy shows the emergent O() symmetry. For , the symmetry persists in a finite system for a range below , as shown in Figure 3(d).
Figure 4 shows that Potts ferromagnets behave differently. It compares the histogram of the orientation of the staggered magnetization of the mixed Potts model, projected on the plane, to similar plots for the magnetization of ferromagnets with and 3.
4First order transition of the model
In the range below , the 4-state mixed Potts model displays jumps in and , near in Figs. Figure 1 and Figure 2. In the middle range , vanishes for , while converges to a nonzero value. In contrast, both and converge to nonzero values in the low range . Thus different symmetries are broken on the two sides of . The histograms of and show a broken symmetry for , with for the permutation of the two sublattices, and for the symmetric group for the four-state Potts model. In a typical configuration at , one Potts state dominates one sublattice, and the remaining states randomly occur on the other sublattice. This implies the breaking of the corner-cubic symmetry described by the four vectors for , preceded by a sublattice sign . In a typical configuration at , one sublattice is dominated by two random spin states, and the other sublattice by the other states. The staggered magnetization vector then points at one face of a cube, signaling a broken face-cubic symmetry. The histograms of are shown in Figure 5(a) for and Figure 5(b) for , clearly displaying the broken corner-cubic and face-cubic symmetries. Similarly, Figs. Figure 5(d) and Figure 5(e) show the histogram of for , and , respectively. The transition at is not well visible in the ordinary energy density , but it is exposed by the energy-like quantity based on the next-nearest neighbor correlations in the - planes, expressed as
where denote the next-nearest neighboring sites in the planes. Figure 6 shows the curves of versus and the curves of versus , the curves of are featureless for this transition, but the curves of at for large system sizes show an obvious energy gap. This result reflects the stronger next-nearest neighbor correlations in the corner-cubic phase.
The 3-state mixed Potts model breaks the symmetry in the whole range , as shown in Figs. Figure 5(c) and (f). But for the , another similar discontinuous transition appears at , which breaks a symmetry in the low- range . The degeneracy in the intermediate range is , where denotes the binomial coefficient. In a typical configuration at , the spins on one sublattice randomly take two states, and the other spins randomly take the remaining three states.
A similar discontinuous transition may occur in the 6-state mixed Potts model. We observed that the symmetry is broken at low , and that another ordered phase exists at intermediate . But we did not find a jump in or for systems up to . This may still be due to a strong finite-size effect.
At zero temperature the model reduces to a square-lattice AF Potts model , which is Néel ordered for , critical for , and disordered for . Figure 7 summarizes the phase behavior of the -state mixed Potts models.
5Conclusion and discussion
In summary, our results indicate that -state Potts models on the simple cubic lattice with mixed FM and AF interactions display continuous phase transitions, with critical exponents in the O() universality class with . The order parameter displays this emergent symmetry at criticality. In the low temperature ranges of the and 5 models, perhaps also for , a discontinuous transition occurs between two ordered phases. For , the model crosses over to the square-lattice AF Potts model, which is disordered for .
Although the temperature is the only variable, the -state mixed Potts model displays diverse and enigmatic phenomena. The O() symmetry is not at all obvious in the Hamiltonian, but it nevertheless emerges, and controls the critical properties of the continuous phase transition at . In the sense of universality, the 3-state mixed Potts model is similar to the O(2) model with a perturbation, which also displays an emergent O(2) symmetry at criticality. For , the mixed Potts model has a low-temperature ordered phase that spontaneously breaks the -dimensional face-cubic symmetry. The histogram of the order parameter at criticality shows an emergent O() symmetry, and the estimated thermal exponent is a decreasing function of , consistent with the O(1) universality. In the analogous case of the pure antiferromagnetic Potts model, the low-temperature ordered phase also breaks the 3D face-cubic symmetry and that the Monte Carlo simulation up to also yields critical exponents consistent with the O(3) universality class . Since the cubic perturbation is expected to be relevant for the O(3) model , one cannot fully exclude that for , these phase transitions are of weak first order or belong to another universality class. However, in either case the question still remains why the effects of this perturbation are invisible in our analysis of finite systems. On the basis of our systematic study of the -state mixed Potts models, we conclude that any symmetry-lowering perturbations of the emergent symmetries are strongly suppressed, allowing the possibility that the ordering transitions fit exactly in the O() universality classes.
Finally, we mention that the mixed Potts model resembles a square-lattice quantum Potts antiferromagnet in a transverse field . The dimension in the classical model corresponds with imaginary time in the Suzuki-Trotter formulation of the quantum model. The present series of mixed Potts models may provide a simple example where quantum fluctuations give rise to rich behavior.
We thank Cristian D. Batista for valuable discussions. This work is supported by the National Science Foundation of China (NSFC) under Grant Nos. 11205005 and 11275185, and by the Anhui Provincial Natural Science Foundation under Grant No. 1508085QA05.
- J. Tobochnik, Phys. Rev. B 26, 6201 (1982). Erratum of this reference: J. Tobochnik, Phys. Rev. B 27, 6972 (1983).
- L. D. C. Jaubert, J. T. Chalker, P. C. W. Holdsworth, and R. Moessner, Phys. Rev. Lett. 105, 087201(2010).
- T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004).
- S.-C. Zhang, Science 275, 1089 (1997); E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 77, 909 (2004).
- L. Huijse, B. Bauer, and E. Berg, Phys. Rev. Lett. 114, 090404(2015).
- F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
- J. S. Wang, R. H. Swendsen, and R. Kotecký, Phys. Rev. Lett. 63, 109 (1989).
- S. Miyashita, J. Phys. Soc. Jpn. 66, 3411(1997).
- C.-X. Ding, W.-A. Guo, and Y. Deng, Phys. Rev. B 90, 134420 (2014).
- M. Itakura, Phys. Rev. B 60, 6558 (1999).
- A. I. Mudrov and K. B. Varnashev, Phys. Rev. E 58, 5371 (1998).
- E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015).
- M. Quartin and S. L. A. de Queiroz, J. Phys. A: Math. Gen. 36, 951 (2003).
- S.-Z Lin, Y. Kamiya, G.-W. Chern, and C. D. Batista, Phys. Rev. Lett. 112, 155702 (2014).
- R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987).
- U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
- C. Dress and W. Krauth, J. Phys. A 28, L597 (1995).
- J. R. Heringa and H. W. J. Blöte, Physica A 232, 369 (1996).
- J. R. Heringa and H. W. J. Blöte, Phys. Rev. E 57, 4976 (1998).
- Y. Deng, Y. Huang, J. L. Jacobsen, J. Salas, and A. D. Sokal, Phys. Rev. Lett. 107, 150601 (2011); Q. N. Chen, M. P. Qin, J. Chen, Z. C. Wei, H. H. Zhao, B. Normand, and T. Xiang, Phys. Rev. Lett. 107, 165701 (2011).
- M. Hasenbusch, Phys. Rev. B 82, 174433 (2010).
- S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, J. Stat. Phys. 157, 869 (2014).
- Q. Liu, Y. Deng, T. M. Garoni, and H. W. J. Blöte, Nucl. Phys. B, 859, 107 (2012).
- J. Salas and A. D. Sokal, J. Stat. Phys. 92, 729 (1998).
- J. Lou, A. W. Sandvik, and L. Balents, Phys. Rev. Lett. 99, 207203 (2007).
- J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61, 15136 (2000); R. Folk and Yu. Holovatch, and T. Yavor’skii, Phys. Rev. B. B 61, 15114 (2000); K. B. Varnashev, Phys. Rev. B 61, 14660 (2000).
- J. Sólyom and P. Pfeuty, Phys. Rev. B 24, 218 (1981).
- M. Campostrini, J. Nespolo, A. Pelissetto, and E. Vicari, Phys. Rev. E 91, 052103 (2015).
- Y.-W. Dai, S. Y. Cho, M. T. Batchelor, and H.-Q. Zhou, Phys. Rev. E 89, 062142 (2014).