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Journal of the Physical Society of Japan

Ferromagnetic-Nonferromagnetic Phase Boundary on the Two-Dimensional ±JIsing Model

Hidetsugu Kitatani, Takehiko Oguchi

Published Nov 15, 1990• 41 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.3823

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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 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 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 ± 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.
We explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional ±J\pm J±J random-bond Ising model, where antiferromagnetic bonds occur with probability ppp and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of +J+J+J and −J-J−J bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the (p,T)(p,T)(p,T) plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as pc=0.1000(2)p_c = 0.1000(2)pc​=0.1000(2), and we extract the associated critical exponents ν=1.26(1)ν= 1.26(1)ν=1.26(1), β/ν=0.85(1)β/ν= 0.85(1)β/ν=0.85(1), γ/ν=0.264(5)γ/ν= 0.264(5)γ/ν=0.264(5), and fractal dimension df≈1.11d_f \approx 1.11df​≈1.11. The value of pcp_cpc​ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin systems.
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.
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.