Journal of Applied Physics

Recursion method in quantum spin dynamics: The art of terminating a continued fraction

V. S. Viswanath, Gerhard H. Muller

Published May 1, 1990• 46 citations

Abstract

The results obtained from applications of the recursion method to quantum many-body dynamics can be greatly improved if an appropriate termination function is employed in the continued-fraction representation of the corresponding relaxation function. We present a general recipe for the construction and use of such termination functions along with two applications in spin dynamics. The method can be adapted to any other problem in quantum many-body dynamics.

DOI: 10.1063/1.345859

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Journal of Applied Physics

Recursion method in quantum spin dynamics: The art of terminating a continued fraction

V. S. Viswanath, Gerhard H. Muller

Published May 1, 1990• 46 citations

Publisher

Abstract

DOI: 10.1063/1.345859

Related papers

Lanczos recursion on a quantum computer for the Green's function and ground state

Thomas E. Baker

A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground-state wave function on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wave function does not need to be reprepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground-state wave function on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wave function does not need to be reprepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.

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The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈 ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ (t) ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ 〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈 ${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$ (t) ${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$ 〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.

Physical Review ANov 1, 199037 citations

1/2

chains and ladders as well as energy transport in the mixed-field Ising model in one dimension.

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

(t)

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈

{\mathit{S}}_{\mathit{i}}^{\mathit{x}}

(t)

{\mathit{S}}_{\mathit{i}}^{\mathit{x}}

〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.