A quantum Monte Carlo algorithm is constructed starting from the standard perturbation expansion in the interaction representation. The resulting configuration space is strongly related to that of the stochastic series expansion (SSE) method, which is based on a direct power-series expansion of $\exp \left(-\beta H\right).\mathrm{}$ Sampling procedures previously developed for the SSE method can therefore be used also in the interaction representation formulation. The method is tested on the $S=1/2\mathrm{}$ Heisenberg chain. Then, as an application to a model of great current interest, a Heisenberg chain including phonon degrees of freedom is studied. Einstein phonons are coupled to the spins via a linear modulation of the nearest-neighbor exchange. The simulation algorithm is implemented in the phonon occupation-number basis, without Hilbert space truncations, and is exact. Results are presented for the magnetic properties of the system in a wide temperature regime, including the $T\to 0$ limit where the chain undergoes a spin-Peierls transition. Some aspects of the phonon dynamics are also discussed. The results suggest that the effects of dynamic phonons in spin-Peierls compounds such as ${\mathrm{GeCuO}}_3$ and ${\alpha }^{\prime }-{\mathrm{NaV}}_2{\mathrm{O}}_5$ must be included in order to obtain a correct quantitative description of their magnetic properties, both above and below the dimerization temperature.

• Received 6 June 1997

DOI:https://doi.org/10.1103/PhysRevB.56.14510