The excitation spectrum of bosons and fermions on the one-dimensional superlattice

Abstract

A Hamiltonian which is expressed as quadratic form of fermion or boson operators on the one-dimensional superlattice (closed chain) is considered. Each unit cell of the superlattice is composed of two subcells, which alternate along with the lattice. The quadratic terms of the Hamiltonian are restricted to operators associated with nearest-neighbour sites, so that the solution on each subcell can be obtained in closed form for any arbitrary number of sites. The elementary excitations of the system are obtained by using an extension of the site transfer matrix technique introduced by Barnas and Hillebrands in 1993. In view of the Hamiltonian considered, the most general quadratic form of creation and annihilation, the solution obtained can be applied to a large class of problems in superlattices, provided that they can be described by boson or fermion operators. This unified solution represents a generalization of the known solutions for particular systems obtained using special operator representations. An explicit application is presented for the spin-1/2 XY model in a transverse field, and for the isotropic model it is shown that the results are identical with the exact results recently obtained by the present authors using the Green function method.