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Physical Review E

Effects of discreteness on gauge glass models in two and three dimensions

Satoshi Yotsuyanagi, Yusuke Suemitsu, Yukiyasu Ozeki

Published Apr 23, 2009• 2 citations
Publisher

Abstract

The q-state clock gauge glass model is studied to see the effect of discreteness on the Kosterlitz-Thouless (KT) transition and the ferromagnetic (FM) critical phenomenon in random systems. The nonequilibrium relaxation analysis is applied. In two dimensions, the successive transitions of paramagnetic (PM), KT, and FM phases are investigated along the Nishimori line for q=6, 8, 10, 12, 14, 16, and 1024 (recognized as infinity) cases. For the upper critical temperature, it is found that the transition temperature is almost the same as in the continuous case for all q values. The lower transition temperature is found to be proportional to 1/q2. In three dimensions, the critical behavior of the PM-FM transition is studied along the Nishimori line for q=6, 8, 16, and 1024 cases. It is found that the spin discreteness is irrelevant, and the transition belongs to the same universality class as in the (continuous) XY case.

DOI
10.1103/physreve.79.041138

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The q-state clock gauge glass model is studied to see the effect of discreteness on the Kosterlitz-Thouless (KT) transition and the ferromagnetic (FM) critical phenomenon in random systems. The nonequilibrium relaxation analysis is applied. In two dimensions, the successive transitions of paramagnetic (PM), KT, and FM phases are investigated along the Nishimori line for q=6, 8, 10, 12, 14, 16, and 1024 (recognized as infinity) cases. For the upper critical temperature, it is found that the transition temperature is almost the same as in the continuous case for all q values. The lower transition temperature is found to be proportional to 1/q2. In three dimensions, the critical behavior of the PM-FM transition is studied along the Nishimori line for q=6, 8, 16, and 1024 cases. It is found that the spin discreteness is irrelevant, and the transition belongs to the same universality class as in the (continuous) XY case.
The critical exponents are estimated for the gauge glass model in two dimensions, in which only the Kosterlitz-Thouless (KT) phase appears in the low-temperature regime. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents: the static exponent η and the dynamical exponent z. Since the system exhibits criticality in the whole KT phase, we estimate the exponents on the boundary as well as inside the KT phase. The static exponent η depends on both the temperature and the strength of randomness, while the dynamical one z is almost constant throughout the KT phase, including the boundary.
The ferromagnetic critical exponents are estimated along the phase boundary for the gauge glass model in three dimensions. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents. Together with the previously obtained results for the ± J Ising model in two and three dimensions, universality classes are examined for these disordered models. The result reveals a direct numerical confirmation of the Harris criterion for typical models with positive, zero and negative values of α 0 , the exponent for the specific heat in the regular case. In addition, we show a nonuniversal behavior of the dynamical exponent z with respect to the randomness for these models irrespective of α 0 .
The Kosterlitz-Thouless (KT) transition is investigated for gauge glass models in two dimensions by means of the nonequilibrium relaxation (NER) method. Two kinds of models, which have the same symmetry, are analyzed. Using the scaling analysis of the NER function on a large lattice with L=1000L=1000L=1000, we confirm the KT transition numerically for both models. This indicates the stability of the KT phase against a small disorder, which was previously claimed by perturbation expansion and renormalization group arguments.
We investigate the temperature-disorder (TTT-$\ensuremath{\sigma}$) phase diagram of a three-dimensional gauge glass model, which is a cubic-lattice nearest-neighbor XY\mathit{XY}XY model with quenched random phase shifts Axy{A}_{\mathit{xy}}Axy​ at the bonds. We consider the uncorrelated phase-shift distribution $P({A}_{\mathit{xy}})~\mathrm{exp}[(\mathrm{cos}{A}_{\mathit{xy}})/\ensuremath{\sigma}]$, which has the pure XY\mathit{XY}XY model, and the uniform distribution of random phase shifts as extreme cases at $\ensuremath{\sigma}=0$ and $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$, respectively, and which gives rise to equal magnetic and overlap correlation functions when $T=\ensuremath{\sigma}$. Our study is mostly based on numerical Monte Carlo simulations. While the high-temperature phase is always paramagnetic, at low temperatures there is a ferromagnetic phase for weak disorder (small $\ensuremath{\sigma}$) and a glassy phase at large disorder (large $\ensuremath{\sigma}$). These three phases are separated by transition lines with different magnetic and glassy critical behaviors. The disorder induced by the random phase shifts turns out to be irrelevant at the paramagnetic-ferromagnetic transition line, where the critical behavior belongs to the 3D XY\mathit{XY}XY universality class of pure systems; disorder gives rise only to very slowly decaying scaling corrections. The glassy critical behavior along the paramagnetic-glassy transition line belongs to the gauge-glass universality class, with a quite large exponent $\ensuremath{\nu}=3.2(4)$. These transition lines meet at a multicritical point M, located at ${T}_{M}={\ensuremath{\sigma}}_{M}=0.7840(2)$. The low-temperature ferromagnetic and glassy phases are separated by a third transition line, from M down to the T=0T=0T=0 axis, which is slightly reentrant.
We investigate the magnetic and glassy transitions of the square-lattice XY model in the presence of random phase shifts. We consider two different random-shift distributions: the Gaussian distribution and a slightly different distribution (cosine distribution) which allows the exact determination of the Nishimori line where magnetic and overlap correlation functions are equal. We perform Monte Carlo simulations for several values of the temperature and of the variance of the disorder distribution, in the paramagnetic phase close to the magnetic and glassy transition lines. We find that, along the transition line separating the paramagnetic and the quasi-long-range order phases, magnetic correlation functions show a universal Kosterlitz-Thouless behavior as in the pure XY model, while overlap correlations show a disorder-dependent critical behavior. This behavior is observed up to a multicritical point which, in the cosine model, lies on the Nishimori line. Finally, for large values of the disorder variance, we observe a universal zero-temperature glassy critical transition, which is in the same universality class as that occurring in the gauge-glass model.
The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.