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Physical review. B, Condensed matter

Static and dynamic magnetic response of spin-glass models with short-range interactions

Wolfgang Kinzel, Kurt Binder

Published Feb 1, 1984• 52 citations
Publisher

Abstract

The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility $\frac{{M}_{\mathrm{eq}}}{H}$ is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau ${H}_{c}^{\mathrm{eq}}(T)$ tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at $T=0$, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures ${T}_{f}$, in striking similarity to experimental data, but ${T}_{f}=0$ is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from $\frac{{M}_{\mathrm{eq}}}{H}$ below a certain critical field ${H}_{c}(t)$, $t$ being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane ${H}_{c}(t)$ extrapolates to a time-dependent freezing temperature ${T}_{c}(t)$, and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.

DOI
10.1103/physrevb.29.1300

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The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility MeqH\frac{{M}_{\mathrm{eq}}}{H}HMeq​​ is found, which diverges for small fields with a power law MeqH\ensuremath∝H1\ensuremathΔ\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}HMeq​​\ensuremath∝H\ensuremathΔ1​, \ensuremathΔ=3.5\ifmmode±\else\textpm\fi0.5\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5\ensuremathΔ=3.5\ifmmode±\else\textpm\fi0.5. The boundary of the plateau Hceq(T){H}_{c}^{\mathrm{eq}}(T)Hceq​(T) tends to zero temperature as Hceq(T)\ensuremath∝T\ensuremathΔ{H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}Hceq​(T)\ensuremath∝T\ensuremathΔ, consistent with a scaling description appropriate for a static phase transition at T=0T=0T=0, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures Tf{T}_{f}Tf​, in striking similarity to experimental data, but Tf=0{T}_{f}=0Tf​=0 is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from MeqH\frac{{M}_{\mathrm{eq}}}{H}HMeq​​ below a certain critical field Hc(t){H}_{c}(t)Hc​(t), ttt being the time scale over which the field is applied. In the H\ensuremath−TH\ensuremath{-}TH\ensuremath−T plane Hc(t){H}_{c}(t)Hc​(t) extrapolates to a time-dependent freezing temperature Tc(t){T}_{c}(t)Tc​(t), and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below Tc{T}_{c}Tc​. Approaching Tc{T}_{c}Tc​ from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at Tc{T}_{c}Tc​ the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below Tc{T}_{c}Tc​. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below Tc{T}_{c}Tc​. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below Tc{T}_{c}Tc​, with a temperature-dependent exponent $\ensuremath{\nu}$. Near Tc{T}_{c}Tc​, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at T=0T=0T=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below Tc{T}_{c}Tc​. Approaching Tc{T}_{c}Tc​ from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at Tc{T}_{c}Tc​ the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below Tc{T}_{c}Tc​. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below Tc{T}_{c}Tc​. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below Tc{T}_{c}Tc​, with a temperature-dependent exponent $\ensuremath{\nu}$. Near Tc{T}_{c}Tc​, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at T=0T=0T=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.
By a recursive method numerically exact free energies are calculated for square $L\ifmmode\times\else\texttimes\fi{}L$ Ising lattices, with 6< L< 186<~L<~186< L< 18, for several kinds of frozen-in bond disorder: (i) bonds $\ifmmode\pm\else\textpm\fi{}J$ with various concentrations of negative bonds; (ii) bonds distributed according to a Gaussian distribution. Ground states of these systems are identified, the response to "ordering fields" is studied, and the correlation function 〈S0SR〉T2{〈{S}_{0}{S}_{R}〉}_{T}^{2}〈S0​SR​〉T2​ is calculated as a function of temperature for various distances RRR in the lattice. This correlation is found to decay strongly (presumably exponentially) with increasing RRR even at temperatures distinctly below the apparent freezing temperature Tf{T}_{f}Tf​ of previous Monte Carlo simulations; this "freezing transition" is hence unambiguously identified as a nonequilibrium effect. However, the correlation length is found to become long ranged at low temperatures, and it is suggested that a phase transition still occurs at T=0T=0T=0; while in the Gaussian model the spin-glass order parameter q(T=0)=1q(T=0)=1q(T=0)=1, it is found that $q\ensuremath{\equiv}0$ in the $\ifmmode\pm\else\textpm\fi{}J$ model where rather a power-law decay of correlations 〈S0SR〉T=02{〈{S}_{0}{S}_{R}〉}_{T=0}^{2}〈S0​SR​〉T=02​ occurs. Performing Monte Carlo simulations for precisely the same systems, the cooling times necessary to reach the true ground states of the system are identified, as well as the simulation times necessary to reach thermal equilibrium for the correlation functions. These times are found to increase so strongly with LLL that for systems of macroscopic size the correct thermal equilibrium is probably irrelevant for experimental purposes. Rather a statistical mechanics based on the many long-lived metastable states would be required.
By a recursive method numerically exact free energies and correlation functions are calculated for finite two- and three-dimensional Ising lattices with bonds chosen randomly either ±J or from a gaussian distribution. While (dynamic) Monte Carlo simulations of precisely the same systems show the onset of Edwards-Anderson ordering at rather sharply defined nonzero freezing temperature Tf, the exact static calculations imply that there is no static order parameter qEA = [<Si≳2T]av at nonzero temperature. The correlation length, which would become infinite when qEA would start to become nonzero, is found at Tf to be of the order of a few lattice spacings only. As a result, the freezing-in of these models is identified as a dynamic phenomenon, and not a phase transition associated with static Edwards-Anderson order. The specific heat of the gaussian spin glass is found to vary nearly linear with temperature. On the other hand, in two-dimensional Ising systems with random fields a nonzero (Mattis-like) spin glass order is found.