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arXiv:cond-mat/0507096 · Physical Review B

Effect of second-rank random anisotropy on critical phenomena of a random-field O(N) spin model in the large N limit

Yoshinori Sakamoto, Hisamitsu Mukaida, Chigak Itoi

Published Oct 5, 2005• 8 citations
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Abstract

We study the critical behavior of a random field $\mathrm{O}(N)$ spin model with a second-rank random anisotropy term in spatial dimensions $4<d<6$, by means of the replica method and the $1∕N$ expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the $1∕N$ expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large $N$ limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large $N$. Thus, we conclude that the dimensional reduction holds for sufficiently large $N$.

DOI
10.1103/physrevb.72.144405

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We study the critical behavior of a random field O(N)\mathrm{O}(N)O(N) spin model with a second-rank random anisotropy term in spatial dimensions 4<d<64<d<64<d<6, by means of the replica method and the 1∕N1∕N1∕N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the 1∕N1∕N1∕N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large NNN limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large NNN. Thus, we conclude that the dimensional reduction holds for sufficiently large NNN.
We study the critical behavior of a random field O(N ) spin model included a second-rank random anisotropy term in spatial dimensions 4 < d < 6, by means of the replica method and the 1/N -expansion.We assume the replica symmetry, and study the stability of replica symmetric saddle point against the fluctuation due to the second-rank random anisotropy.Consequently, the replica symmetric saddle point is shown to be stable in the large N limit.Thus, the dimensional reduction holds in the large N limit.
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.
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 reconsider the problem of the critical behavior of a three-dimensional O(m)O(m)O(m) symmetric magnetic system in the presence of random-anisotropy disorder with a generic trimodal random-axis distribution. By introducing nnn replicas to average over disorder it can be coarse grained to a ${\ensuremath{\phi}}^{4}$ theory with an $m\ifmmode\times\else\texttimes\fi{}n$ component order parameter and five coupling constants taken in the limit of $n\ensuremath{\rightarrow}0$. Using a field theory approach we renormalize the model to two-loop order and calculate the $\ensuremath{\beta}$ functions within the $\ensuremath{\varepsilon}$ expansion and directly in three dimensions. We analyze the corresponding renormalization group flows with the help of the Pad\'e-Borel resummation technique. We show that there is no stable fixed point accessible from physical initial conditions whose existence was argued in previous studies. This may indicate the absence of a long-range ordered phase in the presence of random-anisotropy disorder with a generic random-axis distribution.
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.
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 $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation xxx 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.