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Journal of Physics C Solid State Physics

Renormalisation group approach to the critical dynamics of a Heisenberg spin glass

John Hertz, Richard A. Klemm

Published Jul 14, 1979• 1 citation
Publisher

Abstract

The dynamical transport coefficient Gamma for a Heisenberg spin glass with mode-mode coupling and a conserved spin is calculated using the expansion applied to the time dimensionality. The upper critical time dimension d t c equals 2 for the spatial dimensionality d>4 and equals d/2 for 2<d<4. To leading order in epsilon =d t c -d t , Gamma is found to exhibit critical 'speeding up'.

DOI
10.1088/0022-3719/12/13/008

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The dynamical transport coefficient Gamma for a Heisenberg spin glass with mode-mode coupling and a conserved spin is calculated using the expansion applied to the time dimensionality. The upper critical time dimension d t c equals 2 for the spatial dimensionality d>4 and equals d/2 for 2<d<4. To leading order in epsilon =d t c -d t , Gamma is found to exhibit critical 'speeding up'.
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 Tf{T}_{f}Tf​ like ${(T\ensuremath{-}{T}_{f})}^{\ensuremath{-}\frac{1}{2}}$ for d>4d>4d>4 and ${(T\ensuremath{-}{T}_{f})}^{\frac{2}{d\ensuremath{-}1}}$ for 2<d<42<d<42<d<4.
Closed non-linear equations are derived through mode-coupling approximations for the correlation functions and susceptibilities for the Edwards-Anderson model of a classical Heisenberg system with nearest-neighbour random exchange couplings. These equations, which exhibit full dynamical stability, lead to a self-consistent treatment of spin-density propagation and spin-current relaxation. They are found to describe a transition from a paramagnetic to a spin glass phase, which is viewed as one where the dynamics changes from ergodic to a non-ergodic behaviour. Near the critical temperature T c the low-frequency correlation functions are obtained as dynamical scaling laws governed by a critical frequency omega c , which slows down proportionally to epsilon 2 , with epsilon =(T-T c )/T c . Approaching the critical point from the paramagnetic side the diffusivity vanishes linearly with epsilon . The susceptibility exhibits a symmetric cusp. The drastic changes of the spin dynamics at the transition point show up most clearly for the spin-current spectra. The numerical solution of the equations is discussed.
The phenomenological renormalisation group transformation, introduced by Nightingale (1976), is extended to a form suitable for the study of a quantum mechanical system at T=0. The spin- 1 / 2 Ising model in a transverse field Gamma , in one and two dimensions, is considered. The new scaling form, called phenomenological time scaling, is derived by exploiting the connection between a d dimensional quantum mechanical system at T=0 and a (d+1) dimensional classical system in the anisotropic limit of weakly coupled chains. Estimates are made of the critical field Gamma c and the exponent nu associated with the vanishing of the first excitation energy at T=0. These results also provide some information about the thermal critical behaviour of the anisotropic three-dimensional Ising model.
We extend a previ\'ous study of quantum critical dynamics in ferromagnets to the isotropic antiferromagnet. The model we consider here also includes so-called "mode-mode coupling" terms. The quantum limit is found to have the characteristic dimensionality dQ=6{d}_{Q}=6dQ​=6 versus the classical value dC=4{d}_{C}=4dC​=4. A renormalization-group expansion to order $\ensuremath{\epsilon}=6\ensuremath{-}d$ is used to evaluate spin and staggered spin transport quantities ${\ensuremath{\Gamma}}_{M}$ and ${\ensuremath{\Gamma}}_{N}$ in the limit of low critical temperature TN{T}_{N}TN​. Dynamic scaling is found to break down in the quantum case and scaling regions of quantum, classical, and "mixed" statistics are discussed in light of this breakdown.