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arXiv:1002.4288 · Journal of Statistical Mechanics Theory and Experiment

Critical behavior of three-dimensional disordered Potts models with many states

R. A. Baños, Andrés De la Cruz, L. A. Fernández, A. Gordillo-Guerrero, Jose Miguel Gil-Narvion, Marco Guidetti, A. Maiorano, Filippo Mantovani, Enzo Marinari, Víctor Martín-Mayor, Jorge Monforte-Garcia, A. Muñoz Sudupe, Denis Navarro, Giorgio Parisi A, Sergio Perez-Gaviro, Juan Jesus Ruiz-Lorenzo, Beatriz Seoane, Sebastiano Fabio Schifano, Alfonso Tarancón, R. Tripiccione, David Yllanes

Published May 5, 2010• 13 citations
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Abstract

We study the 3D disordered Potts model with p = 5 and 6. Our numerical simulations (that severely slow down for increasing p ) detect a very clear spin glass phase transition. We evaluate the critical exponents and the critical value of the temperature, and we use known results at lower p values to discuss how they evolve for increasing p . We do not find any sign of the presence of a transition to a ferromagnetic regime.

DOI
10.1088/1742-5468/2010/05/p05002

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We study the 3D disordered Potts model with p = 5 and 6. Our numerical simulations (that severely slow down for increasing p ) detect a very clear spin glass phase transition. We evaluate the critical exponents and the critical value of the temperature, and we use known results at lower p values to discuss how they evolve for increasing p . We do not find any sign of the presence of a transition to a ferromagnetic regime.
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 study the critical behavior of the short-range ppp-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for p=3p=3p=3, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.273$ for a Gaussian distribution of interactions and ${T}_{\mathrm{c}}\ensuremath{\simeq}0.377$ for a $\ifmmode\pm\else\textpm\fi{}J$ (bimodal) distribution. These results indicate that the lower critical dimension of the three-state Potts glass is below 3. By contrast, the correlation length of the ten-state (p=10)(p=10)(p=10) Potts glass in three dimensions remains small even at very low temperatures and thus shows no sign of a transition. In four dimensions we find that the p=3p=3p=3 Potts glass with Gaussian interactions has a spin-glass transition at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.
The results of computer simulations of the critical behavior of three-dimensional structures described by disordered Potts models with q = 3 and q = 4 spin states are discussed. Systems with linear dimensions L = 20–60 with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 are examined. It is found that in a three-dimensional Potts model with q = 3 spin states, introducing nonmagnetic impurities induces a second order phase transition. In a Potts model with q = 4, on the other hand, a shift from a first order phase transition to a second order phase transition is observed at high dilution (p = 0.65). Numerical values of the critical parameters are found using finite-size scaling theory and it is shown that they form a universality class corresponding to disordered systems.
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 have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.
Phase transitions induced by varying the strength of disorder in the large- q state Potts model in 3d are studied by analytical and numerical methods. By switching on the disorder, the transition stays of first order, but different thermodynamical quantities display essential singularities. Only for strong enough disorder will the transition be soften into a second-order one, in which case the ordered phase becomes non-homogeneous at large scales, while the non-correlated sites percolate the sample. In the critical regime the critical exponents are found universal: β/ν = 0.60(2) and ν = 0.73(1).