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Physical Review Letters

Ferromagnetic Phase Transitions in Random Fields: The Breakdown of Scaling Laws

G. Grinstein

Published Oct 4, 1976• 277 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.37.944

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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\ensuremathν=2\ensuremath−\ensuremathαd\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}d\ensuremathν=2\ensuremath−\ensuremathα, for example, becomes (d+\ensuremathλu)\ensuremathν=2\ensuremath−\ensuremathα(d+{\ensuremath{\lambda}}_{u})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}(d+\ensuremathλu​)\ensuremathν=2\ensuremath−\ensuremathα, where the index \ensuremathλu{\ensuremath{\lambda}}_{u}\ensuremathλu​ 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 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 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.
Equilibrium properties of the phase transition of a ferromagnetic spin system under a quenched random magnetic field are briefly summarized. Based upon a path integral formulation for stochastic processes, the n→0 method of replica is extended to treat critical relaxation of quenched random spin systems. As an application, the dynamic scaling hypothesis is examined for the random field problem by means of 1/m-expansion method (m; the number of order parameter components) and the renormalization group method near 6 space dimensions. It is shown that d→d−2 rule does not hold for the dynamic critical exponent of the characteristic frequency.
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
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.