Locations of multicritical points for spin glasses on regular lattices

We present an analysis leading to precise locations of the multicritical points for spin glasses on regular lattices. The conventional technique for determination of the location of the multicritical point was previously derived using a hypothesis emerging from duality and the replica method. In the present study, we propose a systematic technique, by an improved technique, giving more precise locations of the multicritical points on the square, triangular, and hexagonal lattices by carefully examining the relationship between two partition functions related with each other by the duality. We can find that the multicritical points of the $\pm J$ Ising model are located at $p_c=0.890813$ on the square lattice, where $p_c$ means the probability of $J_{ij}=J(>0)$, at $p_c=0.835985$ on the triangular lattice, and at $p_c=0.932593$ on the hexagonal lattice. These results are in excellent agreement with recent numerical estimations.

Figure 1

Phase diagram of the $\pm J$ Ising model on a two-dimensional lattice (left panel) and on higher dimensions (right panel). The vertical axis expresses the temperature $T$, and the horizontal line denotes the concentration $p$ of the antiferromagnetic interactions. The multicritical point is described by the black point (MCP). The Nishimori line is described by the dashed line. For higher dimensions, not only the ferromagnetic (FM) and paramagnetic phases (PM) but also the spin-glass phase (SG) exists.

Figure 2

Schematic picture of the duality and the renormalization flow for the replicated $\pm J$ Ising model.

Figure 3

A self-dual hierarchical lattice. The number $r$ represents the step of the construction of the hierarchical lattice. The renormalized inverse temperature is expressed by ${\beta }^{\left(l\right)}$, where $l$ is the step of the renormalization.

Figure 4

Example of the partial summation for the square lattice and the cluster. The top figures describe one of the types of the summation for the square lattice. The bottom figures express the cluster for the principal Boltzmann factor.

Figure 5

Duality for the triangular lattice. After the duality transformation, the partition function on the triangular lattice is transformed into another partition function on the hexagonal lattice.

Figure 6

Example of the summation for the triangular lattice and the cluster. The top figures express one of the types of the summation for the triangular lattice. The bottom figures express the cluster for the evaluation of the principal Boltzmann factor.

Figure 7

Dual lattice for the cluster in Fig. 4. The black colored sites are free spins as the targets of the summation and the white ones are fixed in the up-pointing directions.

Figure 8

Two patterns of the clusters for the improved technique on the square lattice. The top figures express the clusters for the evaluations of $Z^{\left(s\right)}$ and $Z^{*\left(s\right)}$. The bottom figures represent the dual lattices for the clusters, on which the partition functions are denoted by $Z_{\mathrm{D}}^{\left(s\right)}$. The filled circles are the targets of the summation and white ones are fixed to up directions $\left\{S_i\right\}=1$.

Figure 9

Two patterns of the clusters for the improved technique on the triangular lattice. The left-hand side for each type of the approximations is for the partition function $Z_{\mathrm{TR}}^{\left(s\right)}$, and the right-hand side is for the dual partition function $Z_{\mathrm{TR}}^{*\left(s\right)}$, for which the star-triangle transformation is needed. We use the same symbols as in Fig. 8.

Figure 10

Other approximations by use of two clusters on the square lattice. The small lattice $t=1$ includes 12 bonds and $t=2$ has 24 bonds.

Figure 11

Relationships between $Y$ and $Z_{\mathrm{TR}}^{\left(1\right)}$, and between $Y^{*}$ and $Z_{\mathrm{TR}}^{*\left(1\right)}$. The spin $S^{\prime }$ at the center of the star shape is of the summation for the star-triangle transformation.

Figure 12

(Color online) Phase diagram of the $\pm J$ Ising model on the square lattice by the $s=1$ improved technique. The vertical axis is the temperature, and the horizontal axis is the probability for $J_{ij}=J>0$ of the $\pm J$ Ising model. The thick dashed line (blue) is by the conventional conjecture and the thick solid line (blue) is by the improved technique. The thin dashed line (red) is the Nishimori line.

Figure 13

Several approximations for estimations of the slope at $T_c$. The numbers of bonds in each pattern are 36 $(s=3)$, 40 $(t=3)$, 64 $(s=4)$, and 60 $(t=4)$.

Figure 14

(Color online) Phase diagram of the bond-diluted Ising model on the square lattice by the $s=1$ improved technique. The vertical axis is the temperature, and the horizontal axis is the probability for $J_{ij}=J>0$.