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Physical review. B, Condensed matter

Modified hyperscaling relation for phase transitions under random fields

A. Nihat Berker, Susan R. McKay

Published Apr 1, 1986• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.33.4712

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Europhysics Letters (EPL)•May 10, 1994•0 citations
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\ensuremath−{y}_{\mathrm{\ensuremath{-}}}y\ensuremath−​)\ensuremath{\nu}, where y\ensuremath−{y}_{\mathrm{\ensuremath{-}}}y\ensuremath−​ is the strong-coupling runaway eigenvalue exponent. We find the bound y\ensuremath−{y}_{\mathrm{\ensuremath{-}}}y\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.
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
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.
A highly detailed renormalization-group study is applied to the equilibrium statistical mechanics of the random-field Ising model in three dimensions, with analysis of the renormalization-group flows of over 8 500 000 quantities that determine the quenched probability distribution. We determine the strong-coupling fixed distribution that controls the finite-temperature phase boundary. This boundary is second order but has a magnetization behavior that is phenomenally close to a discontinuity, with the exponent \ensuremath{\beta}=0.0200\ifmmode\pm\else\textpm\fi{}0.0005. Magnetization curves are obtained, demonstrating that, for example, the calculated M changes from 0 to 0.407, almost half saturation, as (Tc{\mathit{T}}_{\mathit{c}}Tc​-T)/Tc{\mathit{T}}_{\mathit{c}}Tc​ is varied from 0 to 0.2947\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$. However, a rigorous argument is given showing that the magnetization M cannot be truly discontinuous at phase transitions controlled by a strong-coupling random-field fixed distribution, in renormalization-group studies that employ the Migdal-Kadanoff local recursion relations.
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.
To avoid the complicated topology of surviving clusters induced by standard strong disorder RG in dimension d > 1, we introduce a modified procedure called 'boundary strong disorder RG' where the order of decimations is chosen a priori . We apply this modified procedure numerically to the random transverse field Ising model in dimension d = 2. We find that the location of the critical point, the activated exponent ψ ≃ 0.5 of the infinite-disorder scaling, and the finite-size correlation exponent ν FS ≃ 1.3 are compatible with the values obtained previously using standard strong disorder RG. Our conclusion is thus that strong disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze the RG flows within the two phases in more detail, to show explicitly the presence of various correlation length exponents: we measure the typical correlation exponent ν typ ≃ 0.64 for the disordered phase (this value is very close to the correlation exponent of the pure two-dimensional quantum Ising model), and the typical exponent ν h ≃ 1 for the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at infinite-disorder critical points. We also measure, within the disordered phase, the fluctuation exponent ω ≃ 0.35 which is compatible with the directed polymer exponent in (1 + 1) dimensions.