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Europhysics Letters (EPL)

Breakdown of Hyperscaling in Random Systems—An Inequality

Moshe Schwartz

Published Aug 1, 1991• 29 citations
Publisher

Abstract

It is shown that under certain physical conditions a d -dependent inequality relating the critical exponents of quenched random systems may be obtained. This inequality rules out the usual hyperscaling relation. Application of the inequality to the random field Ising system points out some discrepancies between experimental results.
DOI
10.1209/0295-5075/15/7/014

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In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
The scaling laws relating the critical exponents characterizing the ferromagnetic phase transition are shown to be modified in the presence of quenched random magnetic fields. The hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$, for example, becomes $(d+{\ensuremath{\lambda}}_{u})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$, where the index ${\ensuremath{\lambda}}_{u}$ is negative and is related to the range of the ferromagnetic exchange interactions. This breakdown of hyperscaling results from a singular dependence of the thermodynamic functions on the scaling field associated with an irrelevant operator.
In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension D = 6 . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.
Random-field ferromagnetism systems are discussed, using an equivalent annealed system. It is shown how the critical exponents of the random system are related to the exponents of the pure system. The critical exponent eta is expressed in terms of a dimensionality reduction d'=d-2+ eta (d'). The lower critical dimension for the Ising model is two and for the continuous models it is four. Very strong arguments are given for a hyperscaling relation 2- alpha =d'v and for all the exponents to be given by a dimensionality reduction.
Abstract Upper and lower bounds are estimated of the two‐point correlation function exponent, η, of the Ising model, as a function of the lattice dimensionality, d. The average value of these two bounds together with the correlation length critical exponent v(d) evaluated by Thompson, lead to a set of critical indexes which satisfy the hyperscaling relation. This set of exponents also shows an adequate agreement with the results obtained by Le‐Guillou and Zinn‐Justin for the Ising model in non‐integer dimensions, in a large range of the parameter d.