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Journal of Physics C Solid State Physics

Exact scalings of pure and dilute quantum transverse Ising chains

Robin B. Stinchcombe

Published Jun 10, 1981• 10 citations
Publisher

Abstract

A phenomenological scaling method is used to obtain exact renormalisation-group transformations for pure and bond-diluted transverse Ising chains at zero temperature. An exact discussion is given of basic critical properties of the two systems, and the exponents characterising the quantum and percolative critical behaviour of the correlation length of the dilute chain are found to be nu lambda = nu rho =1.

DOI
10.1088/0022-3719/14/16/003

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A phenomenological scaling method is used to obtain exact renormalisation-group transformations for pure and bond-diluted transverse Ising chains at zero temperature. An exact discussion is given of basic critical properties of the two systems, and the exponents characterising the quantum and percolative critical behaviour of the correlation length of the dilute chain are found to be nu lambda = nu rho =1.
Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.
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.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.