Abstract
We analytically estimate the locations of phase transition points in the ground state for the $\pm J$ random bond Ising model with asymmetric bond distributions on the square lattice. We propose and study the percolation transitions for two types of bond shared by two non-frustrated plaquettes. The present method indirectly treats the sizes of clusters of correlated spins for the ferromagnetic and spin glass orders. We find two transition points. The first transition point is the phase transition point for the ferromagnetic order, and the location is obtained as $p_c^{(1)} \approx 0.895 \, 399 \, 54$ as the solution of $[p^2 + 3 (1-p)^2 ]^2 \, p^3 - \frac{1}{2} = 0$. The second transition point is the phase transition point for the spin glass order, and the location is obtained as $p_c^{(2)} = \frac{1}{4} [2 + \sqrt{2 (\sqrt{5} - 1)}] \approx 0.893 \, 075 \, 69$. Here, $p$ is the ferromagnetic bond concentration, and $1 - p$ is the antiferromagnetic bond concentration. The obtained locations are reasonably close to the previously estimated locations. This study suggests the presence of the intermediate phase between $p_c^{(1)}$ and $p_c^{(2)}$; however, since the present method produces remarkable values but has no mathematical proof for accuracy yet, no conclusions are drawn in this article about the presence of the intermediate phase.
Highlights
The establishment of reliable theories of spin glasses has been one of the most challenging problems in statistical physics for years.1), 2), 3), 4), 5) In this article, our main interests are to make the determination of the structure of the phase diagram for spin glasses, and to clarify the properties of the phases.Figure 1 shows a schematic phase diagram for the ±J random bond Ising model on the square lattice
The present study suggests the presence of the intermediate phase
We analytically estimated the locations of phase transition points in the ground state for the ±J random bond Ising model with asymmetric bond distributions on the square lattice
Summary
The establishment of reliable theories of spin glasses has been one of the most challenging problems in statistical physics for years.1), 2), 3), 4), 5) In this article, our main interests are to make the determination of the structure of the phase diagram for spin glasses, and to clarify the properties of the phases. We do not mention the problem6) of the presence or absence of the spin glass phase at a finite temperature for the square lattice. We mention the locations of the phase transition points p(c1) and p(c2) at zero temperature for the square lattice and mention the intermediate phase between p(c1) and p(c2). We propose and study the percolation transitions for two types of bond shared by two nonfrustrated plaquettes. Miyazaki8) proposed an analytical method that estimates the location of the phase transition point at zero temperature by analyzing the typeset using PTPTEX.cls Ver.0.9
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