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Europhysics Letters (EPL)

Critical exponents of the U ( n ) vector spin glasses

J Stein

Published Jun 20, 1996• 2 citations
Publisher

Abstract

The critical exponents of the classical spin glasses with O ( m − 2 n )⊗ U ( n ) symmetry are obtained to order 2 of the -expansion around six dimensions, where = 6 − d . This new unitary symmetry is realized in the m -vector ( m ⩾ 2) spin glasses with anisotropic quenched random Dzyaloshinskii-Moriya (DM) interaction. Such a random DM interaction with n correlated couplings leads to critical exponents which are different from those of any isotropic m -vector spin glass and thus belong to new universality classes.

DOI
10.1209/epl/i1996-00520-y

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The critical exponents of the classical spin glasses with O ( m − 2 n )⊗ U ( n ) symmetry are obtained to order 2 of the -expansion around six dimensions, where = 6 − d . This new unitary symmetry is realized in the m -vector ( m ⩾ 2) spin glasses with anisotropic quenched random Dzyaloshinskii-Moriya (DM) interaction. Such a random DM interaction with n correlated couplings leads to critical exponents which are different from those of any isotropic m -vector spin glass and thus belong to new universality classes.
We study the classical n-vector spin glass with anisotropic quenched random Dzyaloshinskii - Moriya (DM) interaction. A random DM interaction with m independent and separated couplings defines a generalized gauge-glass model with a rotational symmetry and a broken global reflection invariance. It is shown that this model is in the same universality class as the random-gauge XY (`gauge glass') model. With an additional uniaxial anisotropy, a crossover from Ising-like to gauge-glass critical behaviour is found for a sufficiently large variance of the DM interaction. A new situation arises when there is correlation between the separated random DM couplings. We show that the critical behaviour of a spin glass with two correlated couplings of the anisotropic DM interaction is in a new universality class. The critical exponents and of this model are calculated at two-loop order near six dimensions. We also present a simplified and more rigorous field-theoretic analysis of the gauge-glass model.
Vector spin glasses with random short-range anisotropic couplings are studied using the defect-wall renormalization-group method. By calculating the correlation length exponent \ensuremath{\nu} and the fractal dimension of the defect wall, ds{\mathit{d}}_{\mathit{s}}ds​, we obtain the first direct evidence that in two dimensions these systems have a zero-temperature transition in the same universality class as the Ising spin glass. We also present results which strongly suggest that these systems have a transition at nonzero temperature in three dimensions.
Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.
Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures βcβ_cβc​, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for βcβ_cβc​ and the exponent ννν from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable τ=1−β2/βc2τ= 1-β^2/β_c^2τ=1−β2/βc2​ and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents γγγ and ννν, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.
Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.