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Journal of Physics C Solid State Physics

Random magnetic fields in the n-vector model: the large-n limit

Paolo Carra, J. T. Chalker

Published Mar 30, 1985• 4 citations

Abstract

The authors solve exactly the large-n limit of the n-vector model with a quenched random field, allowing for the possibilities of both long-range correlations in the random field and long-range exchange interactions. It should be emphasised that, although the ideal Bose gas is frequently assumed identical to the large-n limit of the n-vector model, their identity has only been demonstrated for models with full translational symmetry (Gunton and Buckingham 1968, Stanley 1968). Indeed, without such symmetry, there are examples where several kinds of chemical potential must be introduced into the Bose gas in order to preserve the relationship with the n-vector model (Knops 1973, Fishman and Ziman 1982). In fact, the authors find that the two models behave equivalently. In addition, they are able to extend earlier results in the various ways necessary to present a comparison of the large-n limit with recent work (Chang and Abrahams 1984, Kardar et al. 1983) using epsilon -expansions about the lower and upper critical dimension some discrepancies emerge.

DOI: 10.1088/0022-3719/18/9/025

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The authors solve exactly the large-n limit of the n-vector model with a quenched random field, allowing for the possibilities of both long-range correlations in the random field and long-range exchange interactions. It should be emphasised that, although the ideal Bose gas is frequently assumed identical to the large-n limit of the n-vector model, their identity has only been demonstrated for models with full translational symmetry (Gunton and Buckingham 1968, Stanley 1968). Indeed, without such symmetry, there are examples where several kinds of chemical potential must be introduced into the Bose gas in order to preserve the relationship with the n-vector model (Knops 1973, Fishman and Ziman 1982). In fact, the authors find that the two models behave equivalently. In addition, they are able to extend earlier results in the various ways necessary to present a comparison of the large-n limit with recent work (Chang and Abrahams 1984, Kardar et al. 1983) using epsilon -expansions about the lower and upper critical dimension some discrepancies emerge.

We have investigated the dynamics of the isotropic N-vector model with long-range exchange couplings in random magnetic fields using a 1/N expansion. The leading order is exactly solved, showing the existence of a ferromagnetic phase separated from the disordered paramagnetic phase by a line. The critical behaviour of the system has been examined in the next-to-leading order of the 1/N expansion, showing that the critical exponents are by no means related to the ones of the pure system in d-2 dimensions. The ordered phase has been also investigated in the next-to-leading order, revealing a typical Goldstone behaviour of the non time-persistent part of the transverse fluctuations. For the longitudinal fluctuations, two different types of coexistence singularities emerge, one from the non time-persistent (as in the pure systems), vanishing with the temperature, and a more divergent one from the time-persistent part of the correlation.

Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.

The effect of the long-range correlated random magnetic fields which behave like

〈h(x)h(y)〉

$\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (

m

is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.

The possibility of long-range order under quenched random magnetic fields can be deduced from strong-coupling rescaling behavior. The lower critical dimensions of

{d}_{l}=2

and 4 are indicated, respectively, for Ising and

n

-component spin models with

n>1

.