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Journal of Applied Physics

The recursion method applied to the T=0 dynamics of the 1D s=1/2 Heisenberg and XY models

V. S. Viswanath, Gerhard H. Muller

Published Nov 15, 1991• 16 citations
Publisher

Abstract

The frequency-dependent spin autocorrelation functions for the 1D s=1/2 Heisenberg and XY models at zero temperature are determined by the recursion method. These applications further demonstrate the efficacy of a new calculational scheme developed for the termination of continued fractions. A special feature of the recursion method highlighted here is its capability to predict the exponent of the infrared singularities in spectral densities.

DOI
10.1063/1.350036

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The frequency-dependent spin autocorrelation functions for the 1D s=1/2 Heisenberg and XY models at zero temperature are determined by the recursion method. These applications further demonstrate the efficacy of a new calculational scheme developed for the termination of continued fractions. A special feature of the recursion method highlighted here is its capability to predict the exponent of the infrared singularities in spectral densities.
We use the recursion method to study the spectral and dynamical properties of the one-dimensional (1D) s=1/2 XXZ method with planar anisotropy at T=0. Distinct methods of continued-fraction analysis have been developed for the weak-coupling and strong-coupling regimes of the corresponding lattice fermion system. The weak-coupling analysis presented here yields detailed information on the spectral-weight distribution in dynamic structure factors and spin autocorrelation functions, notably on the infrared singularities produced by critical fluctuations, and on the bound states for the case of attractive fermion interaction. The same method is then applied to the charge dynamics of the 1D t-J model for strongly correlated electrons. There it yields similar yet distinct results in the regime of weak exchange coupling. The results for renormalized bandwidths of particle-hole excitations are consistent with available results for charge velocities, and the results for the infrared exponent in the charge dynamic structure factor agree with existing results for the exponent of the equal-time charge correlation function.
The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ tsumq{sum}_{\mathbf{q}}sumq​${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.
We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.
This dissertation reports the results of three loosely connected research projects. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis for the investigation of the dynamical properties at zero temperature of a particular quantum many-body system, including models of localized electron spins and models of strongly correlated itinerant electrons. We have studied these four models: two-dimensional (2D) s = 1/2 Heisenberg antiferromagnet (square lattice), the 2D s = 1/2 Heisenberg model with competing nearest-neighbor (Jsb1J\\sb1Jsb1) and next-nearest neighbor (Jsb2J\\sb2Jsb2) interactions, the 1D t-J model, and the 1D t-JsbzJ\\sb{z}Jsbz model.
Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensional 1 XY and XXZ models at T= oo are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the X X X model bear the unmistakable signature of spin diffusion in the form of a squareroot infrared divergence in the spectral density.