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Journal of Physics A Mathematical and General

Glassy behaviour and semi-local invariance in Ising model with four-spin interaction

Adam Lipowski

Published Nov 7, 1997• 59 citations
Publisher

Abstract

We show that the rapidly quenched high-temperature phase of the 3D Ising model, with four-spin interaction, evolves into the glassy phase with higher energy than in the low-temperature phase which can be reached upon slow cooling. The mechanism which traps the dynamics in the glassy phase is a consequence of a shape dependence of the energy of excitations. The semi-local invariance in this model is lifted by (sufficiently strong) thermal fluctuations to the local gauge invariance.

DOI
10.1088/0305-4470/30/21/012

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We study a class of homogeneous finite-dimensional Ising models which were recently shown to exhibit glassy properties. Monte Carlo simulations of a particular three-dimensional model in this class show that the glassy phase obtained under slow cooling is dominated by large-scale excitations whose energy E(l) scales with their size l as E(l) approximately l(Theta) with Theta approximately 1.33(20). Simulations suggest that in another model of this class, namely the four-spin model, energy is concentrated mainly in linear defects, making the domain walls tensionless in this case also. Two-dimensional variants of these models are trivial and most likely the energy of excitations scales with the exponent Theta=1.
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.
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Slow cooling dynamics of the Ising ppp-spin interaction spin-glass model
We have studied dynamical behaviour of the infinite-range Ising spin glass model with ppp-spin interaction above and below the transition into the non-ergodic phase. The transition is continuous at sufficiently high external magnetic field. The dynamic critical exponent of the power-law decay of the autocorrelation function at the transition point is shown to decrease smoothly to zero as the field approaches the ``tricritical'' point from above; at lower fields the transition is discontinuous. The {\it slow cooling} approach is used to study the nonergodic behavior below the transition at zero external field. It is shown that the anomalous response function Δ(t,t′)Δ(t,t')Δ(t,t′) contains δδδ-function as well as regular contributions at {\it any} temperature below the phase transition. No evidence of the second phase transition (known to exist within the static replica solution of the same model) is found. At lower enough temperatures the {\it slow cooling} solution approaches the one known for the standard SK model.
Experimentally derived physical properties of 3d Ising spin glasses are discussed and compared to what is anticipated from theory and simulations of model systems. It is found that the spin glass experiences a zero-field phase transition to a low-temperature spin-glass phase, that there is no phase transition in a magnetic field and that nonequilibrium and chaos effects often dominate the dynamics of the low-temperature glassy phase. These results are in agreement with findings from Monte Carlo simulations and with properties derived from the droplet model.
We provide strong evidence that the effective spin-spin interaction in a multimodal confocal optical cavity gives rise to a self-induced glassy phase, which emerges exclusively from the peculiar Euclidean correlations and is not related to the presence of disorder as in standard spin glasses. As recently shown, this spin-spin effective interaction is both nonlocal and nontranslational invariant, and randomness in the atoms' positions produces a spin glass phase. Here we consider the simplest feasible disorder-free setting, where atoms form a one-dimensional regular chain and we study the thermodynamics of the resulting effective Ising model. We present extensive results showing that the system has a low-temperature glassy phase. The model depends on the adimensional parameter α=(a/w_{0})^{2}, a being a lattice spacing and w_{0} an interaction length scale. Notably, for rational values of α=p/q, the number of metastable states at low temperature grows exponentially with q and the problem of finding the ground state rapidly becomes computationally intractable, suggesting that the system develops high-energy barriers and ergodicity breaking occurs.