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Journal of Physics A Mathematical and General

Critical dynamics of quantum chains by position space renormalisation group

R B Stinchombe, Raimundo R. dos Santos

Published Jul 11, 1985• 2 citations
Publisher

Abstract

A decimation transformation is applied to the homogeneous equations of motion for creation and annihilation operators of kink-like excitations of the transverse Ising and XY chains at zero temperature. The authors obtain the exact dispersion relations and their dynamic scaling form. The static behaviour is derived as the zero frequency limit of the recursion relations.

DOI
10.1088/0305-4470/18/10/008

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Journal of Physics A Mathematical and General•Dec 1, 1981•10 citations
A decimation transformation is applied to the homogeneous equations of motion for creation and annihilation operators of kink-like excitations of the transverse Ising and XY chains at zero temperature. The authors obtain the exact dispersion relations and their dynamic scaling form. The static behaviour is derived as the zero frequency limit of the recursion relations.
The critical dynamics of the isotropic XY-model on the one-dimensional superlattice is considered in the framework of the position space renormalization group theory. The decimation transformation is introduced by considering the equations of motion of the operators associated to the excitations of the system, and it corresponds to an extension of the procedure introduced by Stinchcombe and dos Santos (J. Phys. A18, L597 (1985)) for the homogeneous lattice. The dispersion relation is obtained exactly and the static and dynamic scaling forms are explicitly determined. The dynamic critical exponent is also obtained and it is shown that it is identical to the one of the XY-model on the homogeneous chain.
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.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.