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arXiv:cond-mat/0303128 · Physical review. B, Condensed matter

Correlation-function distributions at the Nishimori point of two-dimensional Ising spin glasses

S. L. A. de Queiroz, Robin B. Stinchcombe

Published Oct 8, 2003• 16 citations
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Abstract

The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions $P(C)$ and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions $P(C)$ provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1\ensuremath{-}C)P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of $P(C)$ are observed near $C=1$ and $C=0,$ in partial accord with a simple scaling scheme which preserves the gauge symmetry.

DOI
10.1103/physrevb.68.144414

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The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions P(C)P(C)P(C) and associated moments of spin-spin correlation-functions C on strips. The angular dependence of the shape of correlation function distributions P(C)P(C)P(C) provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of (1\ensuremath−C)P(C)(1\ensuremath{-}C)P(C)(1\ensuremath−C)P(C) reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power-law divergences of P(C)P(C)P(C) are observed near C=1C=1C=1 and C=0,C=0,C=0, in partial accord with a simple scaling scheme which preserves the gauge symmetry.
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.
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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.
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.
We reexamine the disorder-dominated multicritical point of the two-dimensional $\ifmmode\pm\else\textpm\fi{}J$-Ising model, known as the Nishimori point (NP). At the NP we investigate numerically and analytically the behavior of the disorder correlator, familiar from the self-dual description of the pure critical point of the two-dimensional Ising model. We consider the logarithmic average and the qthq\mathrm{th}qth moments of this correlator in the ensemble average over randomness, for continuous q in the range 0<q<2.5,0<q<2.5,0<q<2.5, and demonstrate their conformal invariance. At the NP we find, in contrast to the self-dual pure critical point, that the disorder correlators exhibit multiscaling in q which is different from that of spin-spin correlators and that their scaling dimension becomes negative for q>1q>1q>1 and q<0.q<0.q<0. Using properties on the Nishimori line we show that the first moment (q=1)(q=1)(q=1) of the disorder correlator is exactly one for all separations. The spectrum of scaling dimensions at the NP is not parabolic in q.
We consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferromagnetic and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\ensuremath{\gamma}/\ensuremath{\nu},$ $\ensuremath{\alpha}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure-Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds nondiverging specific heats. A percolationlike ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.