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arXiv:cond-mat/0510344 · Physical Review Letters

Random-Field Spin Models beyond 1 Loop: A Mechanism for Decreasing the Lower Critical Dimension

Pierre Le Doussal, Kay Jörg Wiese

Published May 15, 2006• 52 citations
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Abstract

The functional renormalization group for the random-field and random-anisotropy O(N) sigma models is studied to 2 loop. The ferromagnetic-disordered (F-D) transition fixed point is found to next order in d = 4 + epsilon for N > N(c) (N(c) = 2.834 740 8 for random field, N(c) = 9.441 21 for random anisotropy). For N < N(c) the lower critical dimension d = d(lc) plunges below d(lc) = 4: we find two fixed points, one describing the quasiordered phase, the other is novel and describes the F-D transition. d(lc) can be obtained in an (N(c)-N) expansion. The theory is also analyzed at large N and a glassy regime is found.

DOI
10.1103/physrevlett.96.197202

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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.
The influence of a local anisotropy of random orientation on a ferromagnetic phase transition is studied for two cases of anisotropy axis distribution. To this end a model of a random anisotropy magnet is analyzed by means of the field theoretical renormalization group approach in two loop approximation refined by a resummation of the asymptotic series. The one-loop result of Aharony indicating the absence of a second-order phase transition for an isotropic distribution of random anisotropy axis at space dimension d<4d<4d<4 is corroborated. For a cubic distribution the accessible stable fixed point leads to disordered Ising-like critical exponents.
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 stability of fixed points in the two-loop renormalization group for the random field O(N)\mathrm{O}(N)O(N) spin model in 4+ϵ4+ϵ4+ϵ dimensions. We solve the fixed-point equation in the 1∕N1∕N1∕N expansion and ϵϵϵ expansion. In the large-NNN limit, we study the stability of all fixed points. We solve the eigenvalue equation for the infinitesimal deviation from the fixed points under physical conditions on the random anisotropy function. We find that the fixed point corresponding to dimensional reduction is singly unstable and others are unstable or unphysical. Therefore, one has no choice other than dimensional reduction in the large-NNN limit. The two-loop $\ensuremath{\beta}$ function enables us to find a compact area in the (d,N)(d,N)(d,N) plane where the dimensional reduction breaks down. We calculate higher-order corrections in the 1∕N1∕N1∕N and ϵϵϵ expansions to the fixed point. Solving the corrected eigenvalue equation nonperturbatively, we find that this fixed point is singly unstable also for sufficiently large NNN and the critical exponents show a dimensional reduction.
We consider the random-field O(N\mathrm{O}(NO(N) spin model with long-range exchange interactions which decay with distance rrr between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance rrr as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as NNN is finite.
The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis.Expressions for the renormalization group functions are calculated up to the two-loop order and investigated both by an ε = 4 -d expansion and directly at space dimension d = 3 by means of the Pad é-Borel resummation.One accessible stable fixed point indicating a 2nd order ferromagnetic phase transition with dilute Ising-like critical exponents is obtained.