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arXiv:1508.04538 · Physical review. B./Physical review. B

Emergent O( n ) symmetry in a series of three-dimensional Potts models

Chengxiang Ding, Henk W. J. Blöte, Youjin Deng

Published Sep 1, 2016• 13 citations
arXivHTMLPDF

Abstract

Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly 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 emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-${T}_{c}$ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of $q$-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O($n$) symmetry with $n$=$q$-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O($n$=$q$-1) universality classes.

DOI
10.1103/physrevb.94.104402

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Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly 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 emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc{T}_{c}Tc​ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of qqq-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(nnn) symmetry with nnn=qqq-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(nnn=qqq-1) universality classes.
We study the q-state Potts model on the simple cubic lattice with ferromagnetic interactions along one lattice direction, and antiferromagnetic interactions along the two other directions. As the temperature T decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O(n) model with n=q-1. This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller T we find, for q=4 and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at T=0, 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.
Using Monte Carlo simulations in the frame of stochastic series expansion (SSE), we study the three-state quantum Potts model. The cluster algorithm we used is a direct generalization of that for the quantum Ising model. The simulations include the one dimensional and two dimensional ferromagnetic three-state quantum Potts model and the two dimensional antiferromagnetic three-state quantum Potts model. Our results show that the phase transition of the one dimensional ferromagnetic quantum Potts model belongs to the same universality class of the two dimensional classical Potts model, the two dimensional ferromagnetic quantum Potts model undergoes a first order transition, which is also in analogy to its classical correspondence. The phase transition of the antiferromagnetic quantum Potts model is continuous, whose universality class belongs to the three-dimensional classical XY model, owing to an `emergent' O(2) symmetry at the critical point, although its ordered phase breaks the Z_6 symmetry.
In certain Mott-insulating dimerized antiferromagnets, triplet excitations of the paramagnetic phase display both three-particle and four-particle interactions. When such a magnet undergoes a quantum phase transition into a magnetically ordered state, the three-particle interaction becomes part of the critical theory provided that the lattice ordering wave vector is zero. One microscopic example is the staggered-dimer antiferromagnet on the square lattice, for which deviations from O(3) universality have been reported in numerical studies. Using both symmetry arguments and microscopic calculations, we show that a nontrivial cubic term arises in the relevant order-parameter quantum field theory, and we assess its consequences using a combination of analytical and numerical methods. We also present finite-temperature quantum Monte Carlo data for the staggered-dimer antiferromagnet which complement recently published results. The data can be consistently interpreted in terms of critical exponents identical to that of the standard O(3) universality class, but with anomalously large corrections to scaling. We argue that the cubic interaction of critical triplons, although irrelevant in two spatial dimensions, is responsible for the leading corrections to scaling due to its small scaling dimension.
Universality is a fundamental concept in modern physics. For the q-state Potts model, the critical exponents are merely determined by the order-parameter symmetry S_{q}, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment, i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of 0<q<2. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyperuniversality that the critical exponents are the same for 0<q<2 and, moreover, the Ising system (q=2) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be q dependent. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.
We report Monte Carlo simulations of a three-layer coupled q=3 Potts model on the hexagonal lattice (z=6) using Metropolis single-spin-flip dynamics. The model features intra-layer coupling β, inter-layer coupling α, and a quenched memory field w that explicitly breaks Z₃ symmetry. Through finite-size scaling analysis of system sizes L = 9, 18, 27, 36, 54, 81, 96, and 128, we extract a set of critical exponents centered on the three-body order parameter C₃. The anomalous dimension η ≈ 1.41 and effective correlation dimension D_eff = 2 − η ≈ 0.59 fall outside all known two-dimensional universality classes, suggesting a new universality class driven by the irreducible three-body coupling. Five independent scaling laws (Fisher, Josephson, Rushbrooke, Widom, hyperscaling) are satisfied to residuals < 10⁻³. L = 96 and L = 128 data are preliminary; high-statistics runs are ongoing.
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent q=0.60(2)q=0.60(2)q=0.60(2), however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the Z6{Z}_{6}Z6​ symmetry-breaking phase, with critical exponent ys=0.41(2){y}_{\mathrm{s}}=0.41(2)ys​=0.41(2). The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent q=0.37(1)q=0.37(1)q=0.37(1); however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.