Phase Transitions in a Modified Ising Spin Glass Model: A Tensor-Network-based Sampling Approach

Abstract

Phase transitions in a modified Nishimori model, including the model considered by Kitatani, on a two-dimensional square lattice are investigated using a tensor-network-based sampling scheme. In this model, generating bond configurations is computationally demanding because of the correlated random interactions. The employed sampling method enables hierarchical and independent sampling of both bonds and spins. This approach allows high-precision calculations for system sizes up to $L=256$. The results provide clear numerical evidence that the spin-glass and ferromagnetic transitions are separated on the Nishimori line, supporting the existence of an intermediate Mattis-like spin-glass phase. This finding is consistent with the reentrant transition numerically observed in the two-dimensional Edwards-Anderson (EA) model. Furthermore, critical exponents estimated via finite-size-scaling analysis indicate that the universality class of the transitions differs from that of the standard independent and identically distributed EA model.

Figures (6)

• Figure 1: Schematic phase diagram for the Edwards-Anderson model. The vertical axis represents the temperature $T=\beta^{-1}$, and the horizontal axis indicates the disorder strength. The phase diagram shows paramagnetic (PM), ferromagnetic (FM), and spin-glass (SG) phases. The Nishimori line (NL), drawn as a dashed curve, passes through the multicritical point at $\beta_{\text{MCP}}$ on the phase boundary. The curve of the ferromagnetic phase boundary toward $\beta_\text{G}>\beta_{\text{MCP}}$ at low temperatures indicates the occurrence of a reentrant transition.
• Figure 2: Schematic phase diagram for the Kitatani model. The vertical axis represents the temperature $T=\beta^{-1}$, and the horizontal axis indicates the disorder strength. The dashed curve denotes the Nishimori line for the Kitatani model, defined by $\beta=K_p+a$, while the dotted curve represents the Nishimori line for the corresponding EA model, $\beta=K_p$. The multicritical point $\beta_{\mathrm{MCP}}$ and the ferromagnetic transition point $\beta_\mathrm{X}$ are indicated on the dotted and dashed curves, respectively. A possible intermediate Mattis-like spin glass phase is expected to emerge between the paramagnetic and ferromagnetic phases along the dotted curve in the region $\beta_{\mathrm{MCP}}<\beta<\beta_{\mathrm{X}}$.
• Figure 3: Inverse-temperature dependence of the Binder ratios for (a) the spin-glass order parameter $g_q$ and (b) the ferromagnetic order parameter $g_m$ in the two-dimensional Kitatani model with $a=0.45$. The system sizes examined are $L=32$, $64$, $128$, and $256$. Error bars are smaller than the symbol sizes where they are not visible.
• Figure 4: Finite-size-scaling plots for (a) $g_q$ and (b) $g_m$. The horizontal axis represents the scaling variable $(\beta-\beta_c^\phi)L^{1/\nu_\phi}$, with the critical point $\beta_c^\phi$ and $1/\nu_\phi$ for the order parameter $\phi$ via Bayesian scaling analysis. The obtained parameters are $\beta_c^q=1.0506(5)$ and $1/\nu_q=0.644(25)$ for the spin-glass transition, and $\beta_c^m=1.0533(6)$ and $1/\nu_m=0.630(26)$ for the ferromagnetic transition.
• Figure 5: Finite-size-scaling plots of the squared order parameters: (a) spin-glass order $[\langle q^2\rangle]$ and (b) the ferromagnetic order $[\langle m^2\rangle]$. The vertical axis is scaled as $[\langle\phi^2\rangle]L^{2\beta_\phi/\nu_\phi}$ to exhibit the anomalous dimension. The critical inverse temperatures $\beta_c^\phi$ are fixed to the estimates obtained from the Binder ratio analysis. The estimated scaling parameters are $1/\nu_q=0.668(23)$ and $2\beta_q/\nu_q=0.188(4)$, and $1/\nu_m=0.663(25)$ and $2\beta_m/\nu_m=0.145(4)$.