Spin-1 Kitaev model in one dimension

Abstract

We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S>½ on each site and the Hamiltonian is JΣ_nS_n^xS_n+1^y. The cases where S is integer and half-odd integer are qualitatively different. We show that there is a Z₂-valued conserved quantity W_n for each bond (n,n+1) of the system. For integer S, the Hilbert space can be decomposed into 2^N sectors, of unequal sizes. The number of states in most of the sectors grows as d^N, where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d=(√5+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-½ model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term λΣ_nW_n, and show that this has gapless excitations in the range λ₁^c≤λ≤λ₂^c. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points λ₁^c and λ₂^c.