One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥ 0. The ground states of this model contain the Haldane state, large-D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.KEYWORDS: Heisenberg chain, exact diagonalization, conformal field theory, dimer phase, Haldane phase, large-D phase §1. IntroductionOne dimensional antiferromagnetic Heisenberg chains have been the subject of recent investigations by numerous groups. The various phenomena of current interest are shown to be due to quantum effects. A uniform antiferromagnetic Heisenberg chain is known to have a gapless ground state for half-integer-spin. Especially, the exact solution is available for S = 1/2 chain.1) In contrast, for the integer-spin,2) there is a gap between the first excited state and ground state.One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation undergo a transition from the Haldane state to the dimer state as the strength of bond alternation increases. This transition has been studied by many methods.3-5) On the other hand, the phase boundaries and critical properties of S = 1 Heisenberg chains with uniaxial single-ion anisotropy 6) were determined by Glaus and Schneider using a finite size scaling method. The physical picture of both states was clarified by Nijs and Rommelse in terms of string order 7) and by Tasaki using a stochastic geometric representation.8) Using an exact diagonalization method and a phenomenological renormalization group approach, Tonegawa et al. calculated the phase diagram of the S = 1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy. They found the Haldane phase, Néel phase, dimer phase and large-D phase in the ground state. No phase transition was found, however, between the dimer phase and large-D phase.9) In this paper, we calculate the ground state phase diagram of a one demensional S = 1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy using the twisted boundary condition method.5, 10) This improves the accuracy of the transition points. Using conformal field theory and the level spectroscopy method, we obtain the * E-mail: chenwei@riron.ged.saitama-u.ac.jpLuttinger liquid parameter K and the critical exponent ν of the energy gap.This paper is organized as follows. In the next section, the model Hamiltonian is defined and the numerical results are presented. From the exact diagonalization data with the twisted boundary condition, the ground state phase diagram is obtained. The Luttinger liquid parameter K is calculated by conformal field theory and the level spectroscopy method at the transition p...