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Physical Review Letters

Critical Dynamics of a Heisenberg Spin-Glass

John Hertz, Richard A. Klemm

Published May 22, 1978• 28 citations
Publisher

Abstract

The spin-glass transition of a magnetic material with quenched impurities is studied with use of perturbative mode-mode coupling theory, without the use of the replica method. In mean-field theory we find that the transport coefficient $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Gamma}}$ diverges just above the spin-glass freezing temperature ${T}_{f}$ like ${(T\ensuremath{-}{T}_{f})}^{\ensuremath{-}\frac{1}{2}}$ for $d>4$ and ${(T\ensuremath{-}{T}_{f})}^{\frac{2}{d\ensuremath{-}1}}$ for $2<d<4$.

DOI
10.1103/physrevlett.40.1397

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The spin-glass transition of a magnetic material with quenched impurities is studied with use of perturbative mode-mode coupling theory, without the use of the replica method. In mean-field theory we find that the transport coefficient \ensuremathΓ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Gamma}}\ensuremathΓ\ifmmode~\else˜\fi​ diverges just above the spin-glass freezing temperature Tf{T}_{f}Tf​ like (T\ensuremath−Tf)\ensuremath−12{(T\ensuremath{-}{T}_{f})}^{\ensuremath{-}\frac{1}{2}}(T\ensuremath−Tf​)\ensuremath−21​ for d>4d>4d>4 and (T\ensuremath−Tf)2d\ensuremath−1{(T\ensuremath{-}{T}_{f})}^{\frac{2}{d\ensuremath{-}1}}(T\ensuremath−Tf​)d\ensuremath−12​ for 2<d<42<d<42<d<4.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
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A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large- N limit. A spin-glass transition is found for all values of the spin S. The quantum critical regime associated with the quantum transition at S = 0 and the various regimes in the spin-glass phase at high spin are analyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intriguing connections between the equilibrium properties of the quantum problem and the out-of-equilibrium dynamics of classical models are pointed out.
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