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arXiv:cond-mat/0305041 · Physical review. B, Condensed matter

Crossover from random-exchange to random-field critical behavior in Ising models

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

Published Sep 24, 2003• 8 citations
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Abstract

We compute the crossover exponent $\ensuremath{\varphi}$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as ${\mathrm{Fe}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2}$ and ${\mathrm{Mn}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2},$ when applying an external magnetic field. Our result $\ensuremath{\varphi}=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.

DOI
10.1103/physrevb.68.092409

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We compute the crossover exponent \ensuremathφ\ensuremath{\varphi}\ensuremathφ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as FexZn1\ensuremath−xF2{\mathrm{Fe}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2}Fex​Zn1\ensuremath−x​F2​ and MnxZn1\ensuremath−xF2,{\mathrm{Mn}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2},Mnx​Zn1\ensuremath−x​F2​, when applying an external magnetic field. Our result \ensuremathφ=1.42(2)\ensuremath{\varphi}=1.42(2)\ensuremathφ=1.42(2) for the crossover exponent is in good agreement with the available experimental estimates.
A field-theoretic treatment of Grinstein and Luther's model for describing critical phenomena in two- and three-dimensional dilute magnetic systems is presented. With the use of the loop expansion and a simple resummation technique, all the relevant features of the $\ensuremath{\epsilon}$-expansion treatment are confirmed. A crossover to random critical behavior is observed for a number of spin components M<McM<{M}_{c}M<Mc​, where ${M}_{c}\ensuremath{\simeq}1.19$ in two dimensions and ${M}_{c}\ensuremath{\simeq}2.01$ in three dimensions (3D). The first reliable evaluation of the critical exponents for the dilute 3D Ising model is also given.
The equality φ = γ between the exponent φ, which describes crossover from zero to finite-random-field critical behaviour, and the order parameter susceptibility exponent, γ, is shown to be replaced by φ > γ when the random exchange dominates. Rough renormalization group estimates yield φ/γ ≃ 1.1 for three-dimensional random Ising models. The implications to experiments on dilute antiferromagnets in uniform fields are discussed. Similarly to spin glasses, the nonlinear susceptibility of the latter is shown to diverge as ( T - T N ) -0.7 .
We study the crossover behaviors that can be observed in the high-temperature phase of three-dimensional dilute spin systems, using a field-theoretical approach. In particular, for randomly dilute Ising systems we consider the Gaussian-to-random and the pure-Ising-to-random crossover, determining the corresponding crossover functions for the magnetic susceptibility and the correlation length. Moreover, for the physically interesting cases of dilute Ising, XY, and Heisenberg systems, we estimate several universal ratios of scaling-correction amplitudes entering the high-temperature Wegner expansion of the magnetic susceptibility, of the correlation length, and of the zero-momentum quartic couplings.
The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.
The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.
The critical behavior of the random-field Ising model has been a puzzle for a long time. Different theoretical methods predict that the critical exponents of the random-field ferromagnet in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components. This result contradicts the experiments and simulations. We calculate the critical exponents of the random-field O(N) model with the (4+ε)-expansion and obtain values different from the critical exponents of the pure ferromagnet in 2+εdimensions. In contrast to the previous approaches we take into account an infinite set of relevant operators emerging in the problem. We demonstrate how these previously missed relevant operators lead to the breakdown of the (6-ε)-expansion for the random-field Ising model.