Numerical Study of the $S=1$ Antiferrromagnetic Spin Chain with Bond Alternation

We study the $S=1$ quantum spin chain with bond alternation ${\cal H}=\sum _i (1-(-1)^iδ)\vect{S}_i\cdot \vect{S}_{i+1}$ by the density matrix renormalization group method recently proposed by Steven R. White (\PRL{69}{3844}{1993}). We find a massless point at $δ_c =0.25 \pm 0.01$. We also find the edge states in the region $δ<δ_c$ under the open boundary condition, which disappear in the region $δ>δ_{c}$. At the massless point, the spin wave velocity $v_s$ is $3.66 \pm 0.10$ and the central charge $c$ is $1.0\pm 0.15$. Our results indicate that a continuous phase transition occurs at the massless point $δ=δ_c$ accompanying breaking of the hidden $Z_2\times Z_2$ symmetry.

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