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Physical review. B, Condensed matter

Random fields, random anisotropies, nonlinearσmodels, and dimensional reduction

Daniel S. Fisher

Published Jun 1, 1985• 147 citations
Publisher

Abstract

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, ${d}_{c}$=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.

DOI
10.1103/physrevb.31.7233

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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, {\ensuremathΔ\ensuremathμ{\ensuremath{\Delta}}_{\ensuremath{\mu}}\ensuremathΔ\ensuremathμ​}, of the random fields and all the anisotropies only depend on a simple combination, \ensuremath{\Delta}\ifmmode \tilde{}\else ~{}\fi{}, of the {\ensuremathΔ\ensuremathμ{\ensuremath{\Delta}}_{\ensuremath{\mu}}\ensuremathΔ\ensuremathμ​}. 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.
We discuss the breakdown of the Parisi-Sourlas supersymmetry (SUSY) and of the dimensional-reduction (DR) property in the random field Ising and O( N N ) models as a function of space dimension d d and/or number of components N N . The functional renormalization group (FRG) predicts that this takes place below a critical line d_\mathrm{DR}(N) d D R ( N ) . We revisit the perturbative FRG results for the RFO( N N )M in d=4+\epsilon d = 4 + ϵ and carry out a more comprehensive investigation of the nonperturbative FRG approximation for the RFIM. In light of this FRG description, we discuss the perturbative results in \epsilon=6-d ϵ = 6 − d recently derived for the RFIM by Kaviraj, Rychkov, and Trevisani. We stress in particular that the disappearance of the SUSY/DR fixed point below d_\mathrm{DR} d D R arises as a consequence of the nonlinearity of the FRG equations and cannot be found via the perturbative expansion in \epsilon=6-d ϵ = 6 − d (nor in 1/N 1 / N ). We also provide an error bar on the value of the critical dimension d_\mathrm{DR} d D R for the RFIM, which we find around 5.11±0.09 5.11 ± 0.09 , by studying several successive orders of the nonperturbative FRG approximation scheme.
We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy O(N)O(N)O(N) models near d=4d=4d=4, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at two-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at the two-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long-range order in d<4d<4d<4, are shown to fit into the picture predicted by our recent nonperturbative FRG approach.
We study the critical behavior and phase diagram of the ddd-dimensional random field O(N)O(N)O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension dDR(N){d}_{\text{DR}}(N)dDR​(N) and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole (N,d)(N,d)(N,d) plane. Below dDR(N){d}_{\text{DR}}(N)dDR​(N), our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.
We study by the perturbative Functional Renormalization Group (FRG) the Random Field and Random Anisotropy O(N)O(N)O(N) models near d=4d=4d=4, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at 2-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at 2-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long range order in d<4d<4d<4, are shown to fit into the picture predicted by our recent non-perturbative FRG approach.
Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.
The large-distance behaviors of the random field and random anisotropy O(N)O(N)O(N) models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg (N=3)(N=3)(N=3) model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field O(N)O(N)O(N) model the correlation length is found to be finite at the arbitrarily weak disorder for any N>3.N>3.N>3. The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.