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Physical review. B, Condensed matter

Hyperscaling, dimensional reduction, and the random-field Ising model

Ho-Fai Cheung

Published May 1, 1986• 19 citations
Publisher

Abstract

We show that the hyperscaling law is modified whenever the relevant fixed point of the renormalization-group transformation is at zero temperature (infinite coupling). Instead of 2-\ensuremath{\alpha}=d\ensuremath{\nu}, the hyperscaling law now reads 2-\ensuremath{\alpha}=d'\ensuremath{\nu} where d' is the anomalous dimension which is equal to d-\ensuremath{\Vert}y\ensuremath{\Vert} and y is the exponent for the leading irrelevant operator that flows to zero temperature. We discuss the random-field Ising model in detail. We also perform a numerical domain-wall renormalization-group analysis for the d=3 model. For a Gaussian random-field distribution, our results are consistent with a second-order transition at low temperatures, and \ensuremath{\Vert}y\ensuremath{\Vert} is found to be 1.3--1.7. For a \ifmmode\pm\else\textpm\fi{}h distribution, the transition appears to be first order at T=0. .AE

DOI
10.1103/physrevb.33.6191

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We show that the hyperscaling law is modified whenever the relevant fixed point of the renormalization-group transformation is at zero temperature (infinite coupling). Instead of 2-\ensuremath{\alpha}=d\ensuremath{\nu}, the hyperscaling law now reads 2-\ensuremath{\alpha}=d'\ensuremath{\nu} where d' is the anomalous dimension which is equal to d-\ensuremath{\Vert}y\ensuremath{\Vert} and y is the exponent for the leading irrelevant operator that flows to zero temperature. We discuss the random-field Ising model in detail. We also perform a numerical domain-wall renormalization-group analysis for the d=3 model. For a Gaussian random-field distribution, our results are consistent with a second-order transition at low temperatures, and \ensuremath{\Vert}y\ensuremath{\Vert} is found to be 1.3--1.7. For a \ifmmode\pm\else\textpm\fi{}h distribution, the transition appears to be first order at T=0. .AE
The scaling laws derived by Grinstein (1976) for the random-field Ising model (RFIM) are rederived on the assumption that the transition is second order and that the critical behaviour is controlled by a zero-temperature fixed point. The scaling laws involve three independent exponents nu , eta and gamma , the last appearing in a modified hyperscaling relation, 2- alpha =(d-y) nu . It is argued that such hyperscaling modifications are a general feature of phase transitions controlled by zero-temperature fixed points. Explicit evaluation of the RFIM exponents in d=2+ epsilon dimensions, yields, to order epsilon , 1/ nu = epsilon , eta =1- epsilon /2 and y=1+ epsilon /2. The exponent nu is different from that of the pure model in (d-y) dimensions implying that no exact 'dimensional reduction' is possible near two dimensions.
A renormalization-group analysis is presented for the internal energies near a second-order phase boundary under random fields, presuming flow to a strong-coupling fixed point. In agreement with the recent result of Bray and Moore, we derive a modified hyperscaling relation, 2-\ensuremath{\alpha}=(d-${y}_{\mathrm{\ensuremath{-}}}$)\ensuremath{\nu}, where ${y}_{\mathrm{\ensuremath{-}}}$ is the strong-coupling runaway eigenvalue exponent. We find the bound ${y}_{\mathrm{\ensuremath{-}}}$\ensuremath{\le}d/2 for Ising or n>1 component spins. For the three-dimensional random-field Ising transition, this result is incompatible with the separately reported exponents \ensuremath{\alpha}\ensuremath{\simeq}0 and \ensuremath{\nu}\ensuremath{\simeq}1.5.
A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
Abstract We report a high-precision numerical estimation of the critical exponent α of the specific heat of the random-field Ising model in four dimensions. Our result α = 0.12 ( 1 ) indicates a diverging specific-heat behavior and is consistent with the estimation coming from the modified hyperscaling relation using our estimate of θ via the anomalous dimensions η and η ¯ . Our analysis benefited from a high-statistics zero-temperature numerical simulation of the model for two distributions of the random fields, namely a Gaussian and Poissonian distribution, as well as recent advances in finite-size scaling and reweighting methods for disordered systems. An original estimate of the critical slowing down exponent z of the maximum-flow algorithm used is also provided.