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Journal of Applied Physics

Universality classes of dynamical behavior (abstract)

Jian-Min Liu, Gerhard H. Muller

Published Nov 15, 1991• 0 citations
Publisher

Abstract

In studies of dynamic correlation functions the focus is, in general, on their long-time asymptotic behavior and the singularity structure of the associated spectral densities. It turns out that the analysis of the same dynamic correlation functions from a quite different perspective is equally useful and revealing. The focus of our study is on the properties of spectral densities at high frequencies, specifically their decay law, expressible asΦ0■ω■ ∼exp■ − ω2■λ■, in terms of a characteristic exponent λ. That decay law governs the growth rate of the sequence of recurrents which determine the relaxation function in the continued-fraction representation. The value of λ contains valuable information on the underlying dynamical processes taking place in the system, which, in some sense, is complementary to that inferred from the singularity structure of spectral densities. A detailed study of the dynamics of various quantum and classical spin models has prompted us to introduce the concept of ‘‘universality class’’ for a classification of dynamical behavior on the basis of the characteristic exponent λ. For the equivalent-neighbor XYZ model1 we are able to demonstrate four different prototype universality classes: λ=0 (compact support), λ=1 (Gaussian decay), λ=2 (exponential decay), λ=3 (stretched exponential decay), which can be interpreted in terms of basic notions of classical dynamics. Finally, we present the exact T=∞ dynamic correlation functions for a two-sublattice Heisenberg antiferromagnet with uniform intersublattice interaction and zero intrasublattice interaction.

DOI
10.1063/1.350013

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Recently an elegant and quite powerful finite-system approach to determine the exponents ηx and ηz from simple spectral properties has been proposed. For critical systems, the two exponents can be expressed in terms of finite-size spectral gaps as follows: η(N)x=2ΔE01(N)/ΔE(N), η(N)z=2ΔE00(N)/ΔE(N). Here ΔE(N) is the finite-size gap between the ground state (SzT=0,k=0) and the lowest excitation at k=2π/N; ΔE01(N) is the gap to the lowest ‖SzT‖=1 excitations (at k=π), and ΔE00(N) is the gap to the next lowest SzT=0 excited state. The η(N) sequence is then extrapolated to N→∞. For XY models, differences between s=1/2 and s≥1 appear. For s=1/2, the excitations which determine ΔE00(N) and ΔE(N) are degenerate, which implies that ηz=1/2, in agreement with the exact analytic result. For spin-1, however, the next lowest SzT=0 state is located at k=2π/N instead of k=π, and is therefore identical to the state which determines the gap ΔE. The resulting equality ΔE=ΔE00 implies ηz=2, as in the spin-1/2 case. In fact, our result corresponds to power-law decay for all s, and hence we differ from Schulz and Ziman, who claim the out-of-plane correlation function decays exponentially for s>1/2. For the in-plane correlation function, the spectral gap method again agrees with the exact result ηx=0.5 for s=1/2. The consensus of this and other numerical methods for s=1 gives a value ηx≂0.20, considerably different from the case of s=1/2. Hence it is tempting to conjecture that ηx is s dependent, implying that XY models belong to different universality classes for different s. However, a finite-size study of the conformal anomaly produces the result that c=1, independent of s. This situation is further discussed.
The long-time behavior of certain fast-decaying infinite temperature correlation functions on one-, two-, and three-dimensional lattices of classical spins with various kinds of nearest-neighbor interactions is studied numerically, and evidence is presented that the functional form of this behavior is either simple exponential or exponential multiplied by cosine. Due to the fast characteristic timescale of the long-time decay, such a universality cannot be explained on the basis of conventional Markovian assumptions. It is suggested that this behavior is related to the chaotic properties of the spin dynamics.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.
The dynamics of the classical two-sublattice XYZ model with uniform intersublattice interaction and zero intrasublattice interaction is completely integrable for arbitrary system sizes. This makes the system amenable to an exact analysis of dynamic correlation functions. Here we present some exact results for the case with isotropic interaction (XXX model). The dynamical properties of the two-sublattice XYZ model are compared with those of the equivalent-neighbor XYZ model and categorized into universality classes of dynamical behavior.