The isotropic XY model on the inhomogeneous periodic chain

The static and dynamic properties of the isotropic XY-model (s = 1/2) on the inhomogeneous periodic chain, composed of N segments with n different exchange interactions and magnetic moments, in a transverse field h are obtained exactly at arbitrary temperatures. The properties are determined by introducing the generalized Jordan-Wigner transformation and by reducing the problem to a diagonalization of a finite matrix of n-th order. The diagonalization procedure is discussed in detail and the critical behaviour induced by the transverse field, at T = 0, is presented. The quantum transitions are determined by analyzing the behaviour of the induced magnetization, defined as (1/n) ∑n m=1 μm < S z j,m > where μm is the magnetic moment at site m within the segment j, as a function of the field, and the critical fields determined exactly. The dynamic correlations, < Sz j,m(t)S z j ,m (0) >, and the dynamic susceptibility χzz q (ω) are also obtained at arbitrary temperatures. Explicit results are also presented in the limit T = 0, where the critical behaviour occurs, for the static susceptibility χzz q (0) as a function of the transverse field h, and for the frequency dependency of dynamic susceptibility χzz q (ω). Also in this limit, the transverse time-correlation < Sx j,m(t)S x j ′ ,m ′ (0) >, the dynamic and isothermal susceptibilities, χxx q (ω) and χ xx T , are obtained for the transverse field greater or equal than the saturation field.