Effect of disorder on quantum phase transitions in anisotropic XY spin chains in a transverse field

TL;DR

This paper investigates how disorder affects quantum phase transitions in anisotropic XY spin chains in a transverse field, revealing that certain transitions fall into the same universality class as the random transverse field Ising model, with specific effects depending on the type of disorder.

Contribution

It provides exact analytical results on the critical properties of disordered XY spin chains, connecting them to known universality classes and analyzing the impact of different types of disorder.

Findings

Disorder preserves the universality class of the Ising transition.

Random anisotropy transition remains in the same universality class.

Disorder in isotropic exchange or transverse field destroys the anisotropy transition.

Abstract

We present some exact results for the effect of disorder on the critical properties of an anisotropic XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real space renormalization group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the $x$ direction to a ferromagnet in the $y$ direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a non-zero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground state energy has an essential singularity. The results obtained for the dynamical critical exponent, the typical correlation length, and the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.