Rigorous studies on boundary effects of a dimerized S=1 Ising chain with single-ion anisotropy

We investigate the influences of the boundary conditions on the ground state properties of a dimerized S=1 Ising chain with single-ion anisotropy, which are solved exactly by means of a mapping to the spin-1/2 Ising chain with the alternating transverse fields and the Jordan-Wigner transformation. We obtain the exact results of the minimal energy gap Δ0 for exciting a fermion quasi-particle, the minimal energy gap Δh for exciting a hole and the ground state phase diagram under various boundary conditions. The results show that the boundary conditions do not change the quantum phase transition points of the system, but the minimal energy gaps in the cases of periodic and open boundary conditions are quantitatively different. When the dimerized transverse single-ion anisotropies parameter D1>D2, the ground state is lied in the non-hole systems with arbitrary lattice sites. But if D1