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Journal of Physics C Solid State Physics

Real-space renormalization group calculations for spin glasses and dilute magnets

A. P. Young, Robin B. Stinchcombe

Published Dec 28, 1976• 118 citations
Publisher

Abstract

The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.

DOI
10.1088/0022-3719/9/24/012

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Journal of the Physical Society of Japan•Jun 29, 2006•1 citation
The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.
The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.
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.
We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards–Anderson model to discuss the existence of the spin glass (SG) phase. We derive the RG equations under the replica symmetric (RS) ansatz and the resulting flow diagram indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that this failure at the MCP is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.