Quasifree States in Some One-Dimensional Quantum Spin Models

We use numerical methods to investigate the $SU_q(N)$ Perk–Schultz spin chain at the special quantum parameter value $q=-e^{i\pi/N}$. We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter $q$ and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers $n_i$ of particles of different types ($i=0,1,\dots,N-1$) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of $q$. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous $SU_q(N-1)$ vertex model that is involved in constructing the system of Bethe-ansatz equations of a matrioshka structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.