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Journal of Physics C Solid State Physics

Scaling theory of the random-field Ising model

Alan J. Bray, M A Moore

Published Oct 10, 1985• 206 citations
Publisher

Abstract

The scaling laws derived by Grinstein (1976) for the random-field Ising model (RFIM) are rederived on the assumption that the transition is second order and that the critical behaviour is controlled by a zero-temperature fixed point. The scaling laws involve three independent exponents nu , eta and gamma , the last appearing in a modified hyperscaling relation, 2- alpha =(d-y) nu . It is argued that such hyperscaling modifications are a general feature of phase transitions controlled by zero-temperature fixed points. Explicit evaluation of the RFIM exponents in d=2+ epsilon dimensions, yields, to order epsilon , 1/ nu = epsilon , eta =1- epsilon /2 and y=1+ epsilon /2. The exponent nu is different from that of the pure model in (d-y) dimensions implying that no exact 'dimensional reduction' is possible near two dimensions.

DOI
10.1088/0022-3719/18/28/006

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Physical Review Letters•Feb 3, 1986•471 citations
The scaling laws derived by Grinstein (1976) for the random-field Ising model (RFIM) are rederived on the assumption that the transition is second order and that the critical behaviour is controlled by a zero-temperature fixed point. The scaling laws involve three independent exponents nu , eta and gamma , the last appearing in a modified hyperscaling relation, 2- alpha =(d-y) nu . It is argued that such hyperscaling modifications are a general feature of phase transitions controlled by zero-temperature fixed points. Explicit evaluation of the RFIM exponents in d=2+ epsilon dimensions, yields, to order epsilon , 1/ nu = epsilon , eta =1- epsilon /2 and y=1+ epsilon /2. The exponent nu is different from that of the pure model in (d-y) dimensions implying that no exact 'dimensional reduction' is possible near two dimensions.
We show that the hyperscaling law is modified whenever the relevant fixed point of the renormalization-group transformation is at zero temperature (infinite coupling). Instead of 2-\ensuremath{\alpha}=d\ensuremath{\nu}, the hyperscaling law now reads 2-\ensuremath{\alpha}=d'\ensuremath{\nu} where d' is the anomalous dimension which is equal to d-\ensuremath{\Vert}y\ensuremath{\Vert} and y is the exponent for the leading irrelevant operator that flows to zero temperature. We discuss the random-field Ising model in detail. We also perform a numerical domain-wall renormalization-group analysis for the d=3 model. For a Gaussian random-field distribution, our results are consistent with a second-order transition at low temperatures, and \ensuremath{\Vert}y\ensuremath{\Vert} is found to be 1.3--1.7. For a \ifmmode\pm\else\textpm\fi{}h distribution, the transition appears to be first order at T=0. .AE
A renormalization-group analysis is presented for the internal energies near a second-order phase boundary under random fields, presuming flow to a strong-coupling fixed point. In agreement with the recent result of Bray and Moore, we derive a modified hyperscaling relation, 2-\ensuremath{\alpha}=(d-${y}_{\mathrm{\ensuremath{-}}}$)\ensuremath{\nu}, where ${y}_{\mathrm{\ensuremath{-}}}$ is the strong-coupling runaway eigenvalue exponent. We find the bound ${y}_{\mathrm{\ensuremath{-}}}$\ensuremath{\le}d/2 for Ising or n>1 component spins. For the three-dimensional random-field Ising transition, this result is incompatible with the separately reported exponents \ensuremath{\alpha}\ensuremath{\simeq}0 and \ensuremath{\nu}\ensuremath{\simeq}1.5.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T→0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α+2β+γ=1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν≈1.0, β≈0.05, γ̄≈2.9, γ≈1.5 and α≈−0.55.
A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.