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Physical Review Letters

Exact dynamical behavior near the critical point in the transverse Ising model

Chahn Lee, Satoru Kobayashi

Published Feb 27, 1989• 18 citations
Publisher

Abstract

In the one-dimensional transverse Ising model, we calculate exactly the time evolution and the admittance of the magnetization and also of the energy near the critical point. The results for the above two quantities are shown to coincide with each other, and are explicitly represented in terms of functions of the inverse static susceptibility. The analyses are made by using the recurrence-relations method developed on the basis of Mori's continued-fraction representation.

DOI
10.1103/physrevlett.62.1061

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In the one-dimensional transverse Ising model, we calculate exactly the time evolution and the admittance of the magnetization and also of the energy near the critical point. The results for the above two quantities are shown to coincide with each other, and are explicitly represented in terms of functions of the inverse static susceptibility. The analyses are made by using the recurrence-relations method developed on the basis of Mori's continued-fraction representation.
General feature of dynamical properties of systems near the second order phase transition are studied by using the continued fraction representation. It is shown that time-evolution of an observable relevant to the second order phase transition exhibits no relaxation at the critical point, It is also shown that under a certain condition the relation of the dynamic critical exponent to the exponent of the static susceptibility can be derived. As an explicit example, the relaxation function, admittance and the random forces of the transverse Ising model are exactly calculated near the critical point to be given as functions of the inverse static susceptibility. These results confirm the correctness of the general argument given above.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
The dynamics of the three-dimensional anisotropic spin-\textonehalf{} Ising model in a transverse field is studied above Tc{T}_{c}Tc​ by a method in which the coupled-chain problem is reduced to that of a single chain in an effective field. The problem of a single chain is studied by means of Mori's continued-fraction expansion in a four-pole approximation. The moments are obtained by using the time-dependent correlation function calculated in the ccc-cyclic and in the improved ccc-cyclic approximations [J. Phys. A 13, 223 (1980)]. A comparison between these two approximations is done by calculating the critical temperature as function of the tunneling field. The one-dimensional correlation length is obtained, and it agrees well with the experimental results for the quasi-one-dimensional ferroelectric CsD2{\mathrm{D}}_{2}D2​PO4{\mathrm{O}}_{4}O4​.