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Journal of the Physical Society of Japan

The Critical Exponentνalong the Ferromagnetic-Nonferromagnetic Phase Boundary in the Two-Dimensional ±JIsing Model

Hidetsugu Kitatani, Takehiko Oguchi

Published May 15, 1992• 30 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.61.1598

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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 ± 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.
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 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.
We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T *=0.9527(1), p *=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T *, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T *. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν−η=d−2=0.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.