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arXiv:cond-mat/0701256 · Physical Review B

Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study

Andrei A. Fedorenko, Florian Kühnel

Published May 21, 2007• 29 citations
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Abstract

We study the long-distance behavior of the $O(N)$ model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation $x$ using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.

DOI
10.1103/physrevb.75.174206

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We study the long-distance behavior of the O(N)O(N)O(N) model in the presence of random fields and random anisotropies correlated as \ensuremath∼1∕xd\ensuremath−\ensuremathσ\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}\ensuremath∼1∕xd\ensuremath−\ensuremathσ for large separation xxx using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double \ensuremathϵ=d\ensuremath−4\ensuremath{\epsilon}=d\ensuremath{-}4\ensuremathϵ=d\ensuremath−4 and \ensuremathσ\ensuremath{\sigma}\ensuremathσ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in (d,\ensuremathσ,N)(d,\ensuremath{\sigma},N)(d,\ensuremathσ,N) parameter space and compute the critical exponents to first order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ and \ensuremathσ\ensuremath{\sigma}\ensuremathσ. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of 3He\ensuremath-A^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}3He\ensuremath-A in aerogel.
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 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.
In this note, we clarify the stability of the large-N functional RG fixed points of the order/disorder transition in the random-field (RF) and random-anisotropy (RA) O(N) models. We carefully distinguish between infinite N, and large but finite N. For infinite N, the Schwarz-Soffer inequality does not give a useful bound, and all fixed points found in cond-mat/0510344 (Phys. Rev. Lett. 96, 197202 (2006)) correspond to physical disorder. For large but finite N (i.e. to first order in 1/N) the non-analytic RF fixed point becomes unstable, and the disorder flows to an analytic fixed point characterized by dimensional reduction. However, for random anisotropy the fixed point remains non-analytic (i.e. exhibits a cusp) and is stable in the 1/N expansion, while the corresponding dimensional-reduction fixed point is unstable. In this case the Schwarz-Soffer inequality does not constrain the 2-point spin correlation. We compute the critical exponents of this new fixed point in a series in 1/N and to 2-loop order.
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 study the surface scaling behavior of a semi-infinite d-dimensional O(N) spin system in the presence of a quenched random field and random anisotropy disorders. It is known that above the lower critical dimension d(LC) = 4 the infinite models undergo a paramagnetic-ferromagnetic transition for N > N(c) (N(c) = 2.835 for the random field and N(c) =9.441 for random anisotropy). For N < N(c) and d < d(LC) there exists a quasi-long-range-order phase with a zero order parameter and a power-law decay of spin correlations. Using a functional renormalization group, we derive the surface scaling laws that describe the ordinary surface transition for d > d(LC) and the long-range behavior of spin correlations near the surface in the quasi-long-range-order phase for d < d(LC). The corresponding surface exponents are calculated to one-loop order. The obtained results can be applied to the surface scaling of periodic elastic systems in disordered media, amorphous magnets, and (3)He-A in aerogel.
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