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arXiv:0706.2052 · Journal of the Physical Society of Japan

Multicritical Points of Potts Spin Glasses on the Triangular Lattice

Masayuki Ohzeki

Published Oct 24, 2007• 3 citations
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Abstract

We predict the locations of several multicritical points of the Potts spin glass model on the triangular lattice. In particular, continuous multicritical lines, which consist of multicritical points, are obtained for two types of two-state Potts (i.e., Ising) spin glasses with two- and three-body interactions on the triangular lattice. These results provide us with numerous examples to further verify the validity of the conjecture, which has succeeded in deriving highly precise locations of multicritical points for several spin glass models. The technique, called the direct triangular duality, a variant of the ordinary duality transformation, directly relates the triangular lattice with its dual triangular lattice in conjunction with the replica method.

DOI
10.1143/jpsj.76.114003

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We predict the locations of several multicritical points of the Potts spin glass model on the triangular lattice. In particular, continuous multicritical lines, which consist of multicritical points, are obtained for two types of two-state Potts (i.e., Ising) spin glasses with two- and three-body interactions on the triangular lattice. These results provide us with numerous examples to further verify the validity of the conjecture, which has succeeded in deriving highly precise locations of multicritical points for several spin glass models. The technique, called the direct triangular duality, a variant of the ordinary duality transformation, directly relates the triangular lattice with its dual triangular lattice in conjunction with the replica method.
The Ising spin glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture that relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Toward this end we precisely determine the global phase diagrams for these six hierarchical spin glasses, using up to $2.5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities ddd and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with d=2d=2d=2 or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.
We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent $\ensuremath{\nu}$ governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude $\ensuremath{\eta}$, (iii) the conformal anomaly ccc, (iv) the finite-size susceptibility exponent $\ensuremath{\gamma}/\ensuremath{\nu}$, and (v) the set of multifractal exponents ${{\ensuremath{\eta}}_{k}}$ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.
The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-para-magnetic phase boundary is probed, with the following estimates for the associated critical indices: $\ensuremath{\nu}=1.49(2)$, $\ensuremath{\gamma}=2.71(4)$, ${\ensuremath{\eta}}_{1}=0.183(3)$, which are distinctly different from the percolation values.
We use numerical transfer-matrix methods to investigate properties of the multicritical point of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and nonlinear zero-field uniform susceptibilities, on strip of widths $4\ensuremath{\leqslant}L\ensuremath{\leqslant}16$ sites, from which we estimate the conformal anomaly ccc, the decay-of-correlations exponent $\ensuremath{\eta}$, and the linear and nonlinear susceptibility exponents $\ensuremath{\gamma}∕\ensuremath{\nu}$ and ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}$, with the help of finite-size scaling and conformal invariance concepts. Our results are c=0.46(1)c=0.46(1)c=0.46(1); $0.187\ensuremath{\lesssim}\ensuremath{\eta}\ensuremath{\lesssim}0.196$; $\ensuremath{\gamma}∕\ensuremath{\nu}=1.797(5)$; ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\ensuremath{\eta}=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=2\ensuremath{\gamma}∕\ensuremath{\nu}+d$ (with space dimensionality d=2d=2d=2) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.
We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [$tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1$ for the $\ifmmode\pm\else\textpm\fi{}J$ distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori's manifold is expected to control multicriticality for a wider class of models.