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arXiv:1001.3737 · Journal of Statistical Mechanics Theory and Experiment

Magnetic and glassy transitions in the square-latticeXYmodel with random phase shifts

Vincenzo Alba, Andrea Pelissetto, Ettore Vicari

Published Mar 11, 2010• 17 citations
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Abstract

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.

DOI
10.1088/1742-5468/2010/03/p03006

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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.
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 study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
We investigate the zero-temperature glassy transitions in the square-lattice ±J Ising model, with bond distribution P(J{xy})=pδ(J{xy}-J)+(1-p)δ(J{xy}+J) ; p=1 and p=1/2 correspond to the pure Ising model and to the Ising spin glass with symmetric bimodal distribution, respectively. We present finite-temperature Monte Carlo simulations at p=4/5 , which is close to the low-temperature paramagnetic-ferromagnetic transition line located at p≈0.89 , and at p=1/2 . Their comparison provides a strong evidence that the glassy critical behavior that occurs for 1-p{0}<p<p{0} , p{0}≈0.897 , is universal, i.e., independent of p . Moreover, we show that glassy and magnetic modes are not coupled at the multicritical zero-temperature point where the paramagnetic-ferromagnetic transition line and the T=0 glassy transition line meet. On the theoretical side we discuss the validity of finite-size scaling in glassy systems with a zero-temperature transition and a discrete Hamiltonian spectrum. Because of a freezing phenomenon which occurs in a finite volume at sufficiently low temperatures, the standard finite-size scaling limit in terms of TL(1/ν) does not exist; the renormalization-group invariant quantity ξ/L should be used instead as basic variable.
We study the square-lattice XY model in the presence of random phase shifts. We consider two different disorder distributions with zero average shift and investigate the low-temperature quasi-long-range order phase which occurs for sufficiently low disorder. By means of Monte Carlo simulations we determine several universal quantities which are then compared with the analytic predictions of the random spin-wave theory. We observe a very good agreement which indicates that the universal long-distance behavior in the whole low-disorder low-temperature phase is fully described by the random spin-wave theory.
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 consider the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ model defined on a simple cubic lattice and study its behavior close to the multicritical Nishimori point, where the paramagnetic-ferromagnetic, the paramagnetic-glassy, and the ferromagnetic-glassy transition lines meet in the $T\text{\ensuremath{-}}p$ phase diagram (ppp characterizes the disorder distribution and gives the fraction of ferromagnetic bonds). For this purpose, we perform Monte Carlo simulations on cubic lattices of size $L\ensuremath{\leqslant}32$ and a finite-size-scaling analysis of the numerical results. The magnetic-glassy multicritical point is found at p∗=0.76820(4){p}^{*}=0.768\phantom{\rule{0.2em}{0ex}}20(4)p∗=0.76820(4), along the Nishimori line given by $2p\ensuremath{-}1=\mathrm{tanh}(J∕T)$. We determine the renormalization-group dimensions of the operators that control the renormalization-group flow close to the multicritical point, y1=1.02(5){y}_{1}=1.02(5)y1​=1.02(5), y2=0.61(2){y}_{2}=0.61(2)y2​=0.61(2), and the susceptibility exponent $\ensuremath{\eta}=\ensuremath{-}0.114(3)$. The temperature and crossover exponents are $\ensuremath{\nu}=1∕{y}_{2}=1.64(5)$ and $\ensuremath{\phi}={y}_{1}∕{y}_{2}=1.67(10)$, respectively. We also investigate the model-AAA dynamics, obtaining the dynamic critical exponent z=5.0(5)z=5.0(5)z=5.0(5).