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Journal of the Physical Society of Japan

Ground State Properties of the ±JIsing Model in Two Dimensions

Yukiyasu Ozeki

Published Oct 15, 1990• 61 citations
Publisher

Abstract

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 \(\tilde{g}(r)\sim{r}^{-\tilde{\eta}}\) with \(\tilde{\eta}\simeq0.21\) in the range \(0.5\leq{p}\lesssim0.85\), 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 \(g(r)\sim{r}^{-\tilde{\eta}}\) 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 \(0.85\lesssim{p}\lesssim0.89\).

DOI
10.1143/jpsj.59.3531

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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 (\tilde{g}(r)\sim{r}^{-\tilde{\eta}}) with (\tilde{\eta}\simeq0.21) in the range (0.5\leq{p}\lesssim0.85), 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)\sim{r}^{-\tilde{\eta}}) 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\lesssim{p}\lesssim0.89).
The ± J Ising model with asymmetric probability weight of ± bonds in two dimensions is studied in the p - T plane by the numerical transfer-matrix method. We calculate finite-width susceptibilities and finite-width correlation lengths of long strips of various sizes up to 14×10 5 . Finite-size scaling analysis suggests that the random antiphase state exists adjacent to the ferromagnetic phase. The boundary separating these phases is vertical (parallel to the temperature axis) in the phase diagram. The weak universality seems to hold along the ferromagnetic critical line.
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.
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%.
We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration p∊[0,1]p∊[0,1]p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension LLL up to 700700700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
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 $\ensuremath{\beta}=0.639(20).$
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.