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arXiv:cond-mat/0511096 · Physical Review Letters

Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models

Matthieu Tissier, Gilles Tarjus

Published Mar 1, 2006• 81 citations
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Abstract

By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.

DOI
10.1103/physrevlett.96.087202

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By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.
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.
We study the phases and phase transitions of a disordered one-dimensional quantum qqq-state clock Hamiltonian using large-scale Monte Carlo simulations. Making contact with earlier results, we confirm that the clean, translational invariant version of the model, for q=6q=6q=6, hosts an intermediate emergent quasi-long-range ordered (QLRO) phase between the symmetry-broken true long-range ordered (TLRO) phase and the disordered (paramagnetic) phase. With increasing disorder strength, the quasi-long-range ordered phase shrinks and finally vanishes at a multicritical point, beyond which there is a direct transition from the TLRO phase to the paramagnetic phase. After establishing the phase diagram, we characterize the critical behaviors of the various quantum phase transitions in the model. We find that weak disorder is an irrelevant perturbation of the Berezinskii-Kosterlitz-Thouless transitions that separate the QLRO phase from the TLRO and paramagnetic phases. For stronger disorder, some of the critical exponents become disorder-dependent already before the system reaches the multicritical point. We also show that beyond the multicritical point, the direct transition from the TLRO phase to the paramagnetic phase is governed by an infinite-randomness critical point in line with strong-disorder renormalization group predictions. While our numerical results are for q=6q=6q=6, we expect the qualitative features of the behavior to hold for all q>4q>4q>4.
We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.
We investigate three-dimensional O(N) spin models driven with a uniform velocity over a random field. Within a spin-wave approximation, it is shown that in the strong driving regime the model with N=2 exhibits a quasi-long-range order in which the spatial correlation function decays in a power-law form. Furthermore, for the cases that N=2 and 3, we numerically demonstrate a nonequilibrium phase transition between the quasi-long-range order phase and the disordered phase, which turns out to resemble the Kosterlitz-Thouless transition in the two-dimensional pure XY model in equilibrium.