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Progress of Theoretical Physics Supplement

Effect of Second-Rank Random Anisotropy on Critical Phenomena of Random Field O(N) Spin Model

Yoshinori Sakamoto, Hisamitsu Mukaida, Chigak Itoi

Published Jan 1, 2005• 1 citation
Publisher

Abstract

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.

DOI
10.1143/ptps.157.139

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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 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 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.
Non-symmetric multicritical points in random magnetic mixtures with competing spin anisotropies are studied by means of the renormalized field theory and the replica method. Nine β functions and two critical indices η are evaluated up to O(ε2)[ε=4-d(d: space dimensionality)]. The investigated model consists of two terms: the anisotropic exchange energy previously studied by Fishman-Aharony and the random off-diagonal anisotropy energy (RAM) recently proposed by Mukamel-Grinstein. Many randomly mixed magnets have the latter term due to the spin lattice coupling or the dipole-dipole interaction. Random off-diagonal anisotropy energy couples longitudinal and transverse spin components in this system. Owing to the existence of this term, the second-order tetracritical point predicted by Fishman-Aharony should be replaced by the unstable coupled tetracritical point, which corresponds to the first-order or smeared phase transition.
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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 study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to εεε approximation the renormalization group equations taking into account possible replica symmetry breaking.