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physica status solidi (b)

The Effect of Higher‐Order Cumulants in the Random‐Field Ising Model

V. Janiš

Published Dec 1, 1986• 4 citations
Publisher

Abstract

Abstract It is shown that below a critical dimension dc = 4 higher‐order random‐field cumulants contribute to the divergent part of the free energy. They renormalize the standard diagrammatic expansion into the most divergent terms in weak random fields in such a way that the perturbative effective dimensionality in the scaling region of the d < 4 model is two. It is argued that any dimensional reduction based on a diagrammatic expansion is doubtful in lower dimensions.

DOI
10.1002/pssb.2221380218

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arXiv:cond-mat/0410118•Physical Review Letters•Dec 23, 2004•87 citations
Abstract It is shown that below a critical dimension dc = 4 higher‐order random‐field cumulants contribute to the divergent part of the free energy. They renormalize the standard diagrammatic expansion into the most divergent terms in weak random fields in such a way that the perturbative effective dimensionality in the scaling region of the d < 4 model is two. It is argued that any dimensional reduction based on a diagrammatic expansion is doubtful in lower dimensions.
Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.
The effects of weak random fields and random higher-rank anisotropies are investigated for n-component magnets with n\ensuremath{\ge}2 for low temperatures near four dimensions. All of the random fields and anisotropies are marginal at the lower critical dimension, dc{d}_{c}dc​=4. By use of supersymmetry, it is shown that at zero temperature the formal perturbation expansion in powers of the strengths, {${\ensuremath{\Delta}}_{\ensuremath{\mu}}$}, of the random fields and all the anisotropies only depend on a simple combination, \ensuremath{\Delta}\ifmmode \tilde{}\else \~{}\fi{}, of the {${\ensuremath{\Delta}}_{\ensuremath{\mu}}$}. Furthermore, this expansion in powers of \ensuremath{\Delta}\ifmmode \tilde{}\else \~{}\fi{} is equivalent to the expansion of the pure n-component system in powers of T in two dimensions less. It would be natural to conclude that this implies a generalized dimensional reduction for exponents; however, we present renormalization-group calculations which indicate that this is not valid. Functional renormalization-group recursion relations are derived which couple together all the random fields and anisotropies. It is demonstrated that, in contrast to earlier claims, there is no perturbative fixed point of the renormalization group in 4+\ensuremath{\epsilon} dimensions. The flows go into regimes where nonperturbative effects are important and it is argued that dimensional reduction is likely to break down. This is the first example of which we are aware of a problem for which an infinite number of marginal operators play an important role. Various suggestions concerning the behavior of random-anisotropy magnets are discussed in the light of the present results.
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.
The Niemeijer-Van Leeuwen cumulant expansion is used in the renormalization-group analysis of the critical-point behavior of the Ising model. Second- and third-order results are obtained for the simple cubic and square lattices, respectively, for a variety of cell sizes, using a parameter-dependent renormalization transformation introduced by Kadanoff and Houghton. Although the results in two dimensions are in general good, there is no evidence of any convergence, with the best values for the critical exponents and critical temperature being given by the second-order theory. In three dimensions the values found for the critical temperature and magnetic scaling index for a cell of 27 spins are reasonable, but the corresponding thermal scaling index is quite poor.
We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.