Ground states of theXY-model

Abstract

Ground states of theX Y-model on infinite one-dimensional lattice, specified by the Hamiltonian $$---J\left[ {\sum {\left\{ {(1 + \gamma )\sigma _x^{(j)} \sigma _x^{(j)} + (1 - \gamma )\sigma _y^{(j)} \sigma _y^{(j + 1)} } \right\} + 2\lambda \sum {\sigma _z^{(j)} } } } \right]$$ with real parametersJ≠0,γ andλ, are all determined. The model has a unique ground state for |λ|≧1, as well as forγ=0, |λ|<1; it has two pure ground states (with a broken symmetry relative to the 180° rotation of all spins around thez-axis) for |λ|<1,γ≠0, except for the known Ising case ofλ=0, |λ|=1, for which there are two additional irreducible representations (soliton sectors) with infinitely many vectors giving rise to ground states.