Spin-Peierls transition of the first order in $S=1\mathrm{}$ antiferromagnetic Heisenberg chains

We investigate a one-dimensional $S=1\mathrm{}$ antiferromagnetic Heisenberg model coupled to a lattice distortion by a quantum Monte Carlo method. Investigating the ground-state energy of the static bond-alternating chain, we find that the instability to a dimerized chain depends on the value of the spin-phonon coupling, unlike the case of $S=\frac{1}{2}.$ The spin state is the dimer state or the uniform Haldane state depending on whether the lattice distorts or not, respectively. At an intermediate value of the spin-phonon coupling, we find the first-order transition between the two states. We also find the coexistence of the two states.