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Physical review. B, Condensed matter

Gauge-invariant method for the ±Jspin-glass model

John A. Blackman, J. Poulter

Published Sep 1, 1991• 39 citations
Publisher

Abstract

This paper deals with the \ifmmode\pm\else\textpm\fi{}J Ising model in two dimensions in the general case where there is a concentration p of negative bonds. We are particularly interested in exploring the Pfaffian (or combinatorial) method for treating this problem because a Green's-function formalism can be developed in which the frustration manifests itself in a gauge-invariant form through temperature-dependent singularities. We describe the formalism in detail and show how the disappearance of ferromagnetic order at a critical value of p can be related to a transition from localized to extended states analogous to the Anderson transition. Calculations are performed for the ground-state energy and entropy, which are obtained exactly for large samples. We introduce an exponent \ensuremath{\rho} that describes the underlying singularity structure of the Green's-function formalism and we show how this is related to the exponent of the spin-spin correlation function of the Ising problem. A value for this exponent of \ensuremath{\eta}=0.4 is obtained.

DOI
10.1103/physrevb.44.4374

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Finite-size scaling analysis of exact ground states for ± J spin glass models in two dimensions

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With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.

arXiv:cond-mat/9612116•Europhysics Letters (EPL)•Jul 1, 1997•81 citations
This paper deals with the \ifmmode\pm\else\textpm\fi{}J Ising model in two dimensions in the general case where there is a concentration p of negative bonds. We are particularly interested in exploring the Pfaffian (or combinatorial) method for treating this problem because a Green's-function formalism can be developed in which the frustration manifests itself in a gauge-invariant form through temperature-dependent singularities. We describe the formalism in detail and show how the disappearance of ferromagnetic order at a critical value of p can be related to a transition from localized to extended states analogous to the Anderson transition. Calculations are performed for the ground-state energy and entropy, which are obtained exactly for large samples. We introduce an exponent \ensuremath{\rho} that describes the underlying singularity structure of the Green's-function formalism and we show how this is related to the exponent of the spin-spin correlation function of the Ising problem. A value for this exponent of \ensuremath{\eta}=0.4 is obtained.
We develop an exact gauge-invariant method for studying the two-dimensional $\ifmmode\pm\else\textpm\fi{}J$ spin glass. It is applied to the case of an arbitrary concentration of $(1\ensuremath{-}p)$ positive and p negative bonds and is thus a generalization of the more commonly studied p=50p=50%p=50 model. The ground-state properties are examined and in particular it is shown that the spin correlation exponent \ensuremath{\eta} remains constant over the range pc<p<50{p}_{c}<p<50%.pc​<p<50 The value obtained is $\ensuremath{\eta}=0.34\ifmmode\pm\else\textpm\fi{}0.02.$ A wide range of values for \ensuremath{\eta} is quoted in the literature. We indicate possible reasons for the discrepancies and indicate that there are potential advantages in doing calculations at concentrations markedly lower than 50%.
Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to 143{14}^{3}143 using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions of the concentration p of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps P(q)P(q)P(q) is used to investigate how the spin-glass phase evolves with increasing concentration p.p.p. The spin-glass order is characterized by a broad distribution of overlaps which extends down to q=0.q=0.q=0.
We investigate the ground state properties of the ± J Ising model with asymmetric probability weight in two dimensions. We propose a numerical transfer matrix method which enables us to calculate directly the ground-state properties such as correlation functions, defect energy and entropy at T =0. We find that the spin glass correlation function decays in a power law like g~(r)∼r−η~\tilde{g}(r)\sim{r}^{-\tilde{\eta}}g~​(r)∼r−η~​ with η~≃0.21\tilde{\eta}\simeq0.21η~​≃0.21 in the range 0.5≤p≲0.850.5\leq{p}\lesssim0.850.5≤p≲0.85, suggesting the weak universality of spin glass-the concentration of + J bond is denoted by p . At the ferromagnetic phase boundary locating around p =0.89, the ferromagnetic correlation function decays as g(r)∼r−η~g(r)\sim{r}^{-\tilde{\eta}}g(r)∼r−η~​ with η≃0.240±0.016, which suggests the weak universality of ferromagnetic critical exponent from the non-random critical point ( p =1, T =2.269···) to the ground sate counterpart. The behavior of correlation functions, and also that of defect energies indicate the existence of an extra-phase, called the random antiphase state, in 0.85≲p≲0.890.85\lesssim{p}\lesssim0.890.85≲p≲0.89.
The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$
At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-driven ferromagnet-to-paramagnet transition controlled by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix -- the Hamiltonian of a noninteracting quantum problem -- this RG procedure corresponds to an iterative diagonalization of the quantum Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size: log⁡εmin−1∼Lψ\log \varepsilon_{\it min}^{-1} \sim L^ψlogεmin−1​∼Lψ. The tunneling exponent ψψψ is equal to the spin stiffness exponent θcθ_cθc​ characterizing the zero-temperature fixed point.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.