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Journal of Physics C Solid State Physics

On the zero temperature critical behaviour of the xy model in a transverse field

J Goldhirsch

Published Dec 28, 1979• 11 citations
Publisher

Abstract

A previously developed bosonisation scheme is applied to the investigation of the zero-temperature critical behaviour of the XY model in a transverse field. It is shown how this method yields, in a simple and straightforward way, the results previously obtained using cumulants. The meaning of these results is briefly analysed.
DOI
10.1088/0022-3719/12/24/006

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The scaling theory of finite size effects in the limiting bulk behaviour is extended to treat crossover phenomena. The method is used to study quantum critical phenomena in the XY chain in a transverse field at zero temperature, through the scaling of the longitudinal susceptibilities and of the energy gaps between the ground and two first excited states. As expected, no abrupt change in critical exponents is observed for small anisotropy. Because of the finiteness of the system, but the limiting isotropic and anisotropic regions display quite distinct critical behaviour, in good agreement with known results. Another interesting result obtained is that the two energy gaps examined vanish at the critical line with the same critical exponent.
The critical behaviour of the susceptibility of the Ising model with a transverse field is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T are given as functions of Δ/ k T for the quadratic, triangular, simple cubic, b.c.c. and f.c.c. lattices. The universality of the critical index is concluded from the analysis of the coefficients and the critical temperature is obtained as function of the transverse field for the five kinds of lattices.