Multicritical points for spin-glass models on hierarchical lattices

The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry, and the replica method. We find that the conjecture does not give the exact answer but leads to locations slightly away from the numerically reliable data. We propose an improved conjecture to give more precise predictions of the multicritical points than the conventional one. This improvement is inspired by a different point of view coming from the renormalization group and succeeds in deriving very consistent answers with many numerical data.

Figure 1

Three mutually dual pairs of hierarchical lattices investigated in Ref. [12]. The numbers 1–3 denote these three mutually dual pairs of hierarchical lattices studied in the presented paper.

Figure 2

Additional mutually dual pairs of hierarchical lattices studied in the presented paper. The solid lines denote bonds replaced by the renormalized bonds at each renormalization. Bonds shown dashed stay unrenormalized.

Figure 3

Self-dual hierarchical lattices. After renormalization of bond moving and decimation, the self-dual hierarchical lattices become the structure like the Wheatstone bridge.

Figure 4

Two renormalization steps. The left-hand side is bond moving. The right-hand side is decimation.

Figure 5

Construction of one of the self-dual hierarchical lattices. The number $s$ denotes the construction step.

Figure 6

Results for the $\pm J$ (upper panel) and Gaussian (lower panel) Ising models along the Nishimori line on the self-dual hierarchical lattice with $b=3$, see Fig. 3. The conjecture states that the multicritical points are located at $p_c=0.889972$ and $J_0/J^2=1.02127$, respectively. The white square denotes the probability of the paramagnetic phase and the black one represents that of the ferromagnetic phase. The error bars represent $1/\sqrt{1000}$ reflecting the number of the sampling pools.

Figure 7

Size effect for the $\pm J$ Ising model on the self-dual hierarchical lattice with $b=3$. The black and white marks denote the probabilities of the paramagnetic and ferromagnetic phases, respectively. The symbols $\square$ and $■$ are for ${10}^3$ sampling pools with ${10}^6$ bonds, $△$ and $▲$ are for those with $5\times {10}^6$ bonds, and $○$ and $●$ are with $9\times {10}^6$ bonds.

Figure 8

A schematic picture to consider the renormalization flow and the duality for the replicated $\pm J$ Ising model.

Figure 9

The phase boundary by the conventional and improved conjectures for a self-dual hierarchical lattice with $b=3$. 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 bold dashed line is by the conventional conjecture and the bold solid line is by the improved conjecture. The thin dashed line is the Nishimori line.