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Journal of Physics A Mathematical and General

The specific heat of the two-dimensional JIsing model

Hidetsugu Kitatani, Toshimasa Chino, Hideshi Ohya

Published Apr 8, 2003• 0 citations
Publisher

Abstract

The specific heat of the two-dimensional ±J Ising model has been investigated by the numerical transfer matrix method and Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes a negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. The region mentioned above is found to be fairly large in the p – T plane ( p is the concentration of the ferromagnetic bond and T is the temperature). Moreover, it includes the Nishimori line. Namely, it includes a part of the paramagnetic–ferromagnetic phase boundary, on which the specific heat cannot diverge. The present result indicates that the specific heat does not diverge at least on a part of the paramagnetic–ferromagnetic phase boundary above the multicritical point, which is in conflict with previous results.

DOI
10.1088/0305-4470/36/16/302

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The specific heat of the two-dimensional ±J Ising model has been investigated by the numerical transfer matrix method and Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes a negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. The region mentioned above is found to be fairly large in the p – T plane ( p is the concentration of the ferromagnetic bond and T is the temperature). Moreover, it includes the Nishimori line. Namely, it includes a part of the paramagnetic–ferromagnetic phase boundary, on which the specific heat cannot diverge. The present result indicates that the specific heat does not diverge at least on a part of the paramagnetic–ferromagnetic phase boundary above the multicritical point, which is in conflict with previous results.
The specific heat of the two-dimensional ±J\pm J±J Ising model has been investigated by the numerical transfer matrix method and Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes the negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. The region mentioned above is found to be fairly large in the p−Tp-Tp−T plane (ppp is the concentration of the ferromagnetic bond and TTT is the temperature). Moreover, it includes the Nishimori line. Namely, it includes a part of the paramagnetic-ferromagnetic phase boundary, on which the specific heat cannot diverge. The present result indicates that the specific heat does not diverge at least on a part of the paramagnetic-ferromagnetic phase boundary above the multicritical point, which is in conflict with previous results.
The value of the critical exponent ν of the two-dimensional ± J Ising model along the ferromagnetic-nonferromagnetic phase boundary in the p - T plane ( p is the concentration of + J bonds; T is the temperature) has been calculated by the use of the transfer matrix method and the Roomany-Wyld approximation for long strips with finite width ( L ×10 5 : L ≤18), and has been found to take almost constant value, 1.0, at temperature above the Nishimori line. By the same method, a lower bound of the critical concentration between paramagnetic phase and the random antiphase state (RAS phase) has been calculated at some temperature below the Nishimori line in the p - T plane.
The ferromagnetic-nonferromagnetic phase boundary on the two-dimensional ± J Ising model in the p - T plane is determined by the use of the coherent-anomaly method (CAM) proposed by Suzuki, where p is the concentration of the + J bond, and T the temperature. The phase boundary below the Nishimori line has been found to be vertical at p =0.884±0.001 to the p -axis, which is consistent with the previous results obtained under the assumption of Suzuki's weak universality. The critical exponent γ p ( T ) has also been determined along the phase boundary, which shows a crossover tendency.
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.
The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.