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arXiv:cond-mat/9805300 · Physical review. B, Condensed matter

Glassy Potts model: A disordered Potts model without a ferromagnetic phase

Enzo Marinari, Stefano Mossa, Giorgio Parisi A

Published Apr 1, 1999• 20 citations
arXivHTMLPDF

Abstract

We introduce a Potts model with quenched, frustrated disorder, which enjoys a gauge symmetry that forbids spontaneous magnetization and allows the glassy phase to extend from ${T}_{c}$ down to $T=0.$ We study numerically the four-dimensional model with $q=4$ states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder cumulant, and the divergence of the overlap susceptibility. We show that the dynamical behavior of the system is characterized by aging.

DOI
10.1103/physrevb.59.8401

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We introduce a Potts model with quenched, frustrated disorder, which enjoys a gauge symmetry that forbids spontaneous magnetization and allows the glassy phase to extend from Tc{T}_{c}Tc​ down to T=0.T=0.T=0. We study numerically the four-dimensional model with q=4q=4q=4 states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder cumulant, and the divergence of the overlap susceptibility. We show that the dynamical behavior of the system is characterized by aging.
Spin systems exposed to the influence of random magnetic fields are paradigmatic examples for studying the effect of quenched disorder on condensed-matter systems. In this context, previous studies have almost exclusively focused on systems with Ising or continuous symmetries, while the Potts symmetry, albeit being of fundamental importance also for the description of realistic physical systems, has received very little attention. In the present study, we use a recently developed quasiexact method for determining ground states in the random-field Potts model to study the problem with four states. Extending the protocol applied for the three-state model, we use extensive finite-size scaling analyses of the magnetization, Binder parameter, energy cumulant, specific heat, and the connected as well as disconnected susceptibilities to study the magnetic ordering transition of the model. In contrast to the system in the absence of disorder, we find compelling evidence for a continuous transition, and we precisely determine the critical point as well as the critical exponents, which are found to differ from the exponents of the three-state system as well as from those of the random-field Ising model.
We perform numerical simulations, including parallel tempering, a four-state Potts glass model with binary random quenched couplings using the JANUS application-oriented computer. We find and characterize a glassy transition, estimating the critical temperature and the value of the critical exponents. Nevertheless, the extrapolation to infinite volume is hampered by strong scaling corrections. We show that there is no ferromagnetic transition in a large temperature range around the glassy critical temperature. We also compare our results with those obtained recently on the ``random permutation'' Potts glass.
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 ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
We study the aging of the two-dimensional fully-frustrated Ising and threestate Potts models by means of Monte Carlo simulations. The system is initially prepared at infinite temperature and then quenched either at its critical temperature or in the low-temperature phase for the three-state Potts model. Autocorrelation and response functions are shown to satisfy the scaling behavior conjectured for aging ferromagnets if one assumes the existence of logarithmic corrections. The dynamical and autocorrelation exponents are compatible with � = d and z = 2 for the Ising model and probably the threestate Potts model both at and below Tc. The fluctuation-dissipation ratio goes asymptotically to different non-trivial values at Tc and does not vanish below Tc.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.