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arXiv:cond-mat/0311576 · Physical Review E

Spin models with random anisotropy and reflection symmetry

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

Published Sep 10, 2004• 12 citations

Abstract

We study the critical behavior of a general class of cubic-symmetric spin systems in which disorder preserves the reflection symmetry s(a) --> -s(a) , s(b) --> s(b) for b not equala . This includes spin models in the presence of random cubic-symmetric anisotropy with probability distribution vanishing outside the lattice axes. Using nonperturbative arguments we show the existence of a stable fixed point corresponding to the random-exchange Ising universality class. The field-theoretical renormalization-group flow is investigated in the framework of a fixed-dimension expansion in powers of appropriate quartic couplings, computing the corresponding beta functions to five loops. This analysis shows that the random Ising fixed point is the only stable fixed point that is accessible from the relevant parameter region. Therefore, if the system undergoes a continuous transition, it belongs to the random-exchange Ising universality class. The approach to the asymptotic critical behavior is controlled by scaling corrections with exponent Delta= -alpha(r) , where alpha(r) approximately -0.05 is the specific-heat exponent of the random-exchange Ising model.

DOI: 10.1103/physreve.70.036104

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We study the critical behavior of a general class of cubic-symmetric spin systems in which disorder preserves the reflection symmetry s(a) --> -s(a) , s(b) --> s(b) for b not equala . This includes spin models in the presence of random cubic-symmetric anisotropy with probability distribution vanishing outside the lattice axes. Using nonperturbative arguments we show the existence of a stable fixed point corresponding to the random-exchange Ising universality class. The field-theoretical renormalization-group flow is investigated in the framework of a fixed-dimension expansion in powers of appropriate quartic couplings, computing the corresponding beta functions to five loops. This analysis shows that the random Ising fixed point is the only stable fixed point that is accessible from the relevant parameter region. Therefore, if the system undergoes a continuous transition, it belongs to the random-exchange Ising universality class. The approach to the asymptotic critical behavior is controlled by scaling corrections with exponent Delta= -alpha(r) , where alpha(r) approximately -0.05 is the specific-heat exponent of the random-exchange Ising model.

We reconsider the problem of the critical behavior of a three-dimensional

O(m)

symmetric magnetic system in the presence of random-anisotropy disorder with a generic trimodal random-axis distribution. By introducing

n

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.

The influence of a local anisotropy of random orientation on a ferromagnetic phase transition is studied for two cases of anisotropy axis distribution. To this end a model of a random anisotropy magnet is analyzed by means of the field theoretical renormalization group approach in two loop approximation refined by a resummation of the asymptotic series. The one-loop result of Aharony indicating the absence of a second-order phase transition for an isotropic distribution of random anisotropy axis at space dimension

d<4

is corroborated. For a cubic distribution the accessible stable fixed point leads to disordered Ising-like critical exponents.

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the β-functions and the critical exponents.

The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis.Expressions for the renormalization group functions are calculated up to the two-loop order and investigated both by an ε = 4 -d expansion and directly at space dimension d = 3 by means of the Pad é-Borel resummation.One accessible stable fixed point indicating a 2nd order ferromagnetic phase transition with dilute Ising-like critical exponents is obtained.

We consider the Ginzburg-Landau

\mathrm{MN}

model that describes

MN

-vector cubic models with

O(M)

-symmetric couplings. We compute the renormalization-group functions to six-loop order in

d=3.

We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model

(M=1).

We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For

M>~2

we show that the

O(M)

fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.

We study the O(2) model with Z_{4}-symmetric perturbations within the framework of the nonperturbative renormalization group (RG) for spatial dimensionality d=2 and 3. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In d=3 the system is controlled by the XY, Ising, and low-T fixed points in the presence of a dangerously irrelevant anisotropy coupling λ. In d=2 the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior, and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.