Ground states of theXY-model

Ground states of theX Y-model on infinite one-dimensional lattice, specified by the Hamiltonian $$---Jleft[ {sum {left{ {(1 + gamma )sigma _x^{(j)} sigma _x^{(j)} + (1 - gamma )sigma _y^{(j)} sigma _y^{(j + 1)} } right} + 2lambda sum {sigma _z^{(j)} } } } right]$$ 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. The ergodic property of ground states under the time evolution is proved for the uniqueness region of parameters, while it is shown to fail (even if the pure ground states are considered) in the case of non-uniqueness region of parameters.