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arXiv:cond-mat/0603195 · Physical Review B

Stability of fixed points in the (4+ϵ) -dimensional random field O(N) spin model for sufficiently large N

Yoshinori Sakamoto, Hisamitsu Mukaida, Chigak Itoi

Published Aug 2, 2006• 12 citations
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Abstract

We study the stability of fixed points in the two-loop renormalization group for the random field $\mathrm{O}(N)$ spin model in $4+ϵ$ dimensions. We solve the fixed-point equation in the $1∕N$ expansion and $ϵ$ expansion. In the large-$N$ 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-$N$ limit. The two-loop $\ensuremath{\beta}$ function enables us to find a compact area in the $(d,N)$ plane where the dimensional reduction breaks down. We calculate higher-order corrections in the $1∕N$ and $ϵ$ expansions to the fixed point. Solving the corrected eigenvalue equation nonperturbatively, we find that this fixed point is singly unstable also for sufficiently large $N$ and the critical exponents show a dimensional reduction.

DOI
10.1103/physrevb.74.064402

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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 \ensuremathβ\ensuremath{\beta}\ensuremathβ 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 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.
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 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 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.
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.
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.