Abstract

Heisenberg model is a common approach for treating the interaction of spins on a lattice, and Bonner and Fisher used it to investigate the temperature dependence of the magnetic susceptibility and the convergence to the thermodynamic limit on antiferromagnetic spin chains. Numerous approaches like the Stochastic Series Expansion (SSE) using Quantum Monte Carlo (QMC) have been developed to investigate the model further. However, one of the assumptions the Heisenberg model makes is that the lattice is stationary and the couplings constant, which would otherwise depend on the positions of the coupled spins. Imaginary time path integrals can be used to include the nuclear motion and sample both the phase space and the spin space simultaneously for any potential and coupling constant functional forms. Here, several methods for treating the Heisenberg model including nuclear motion have been developed and tested, with the main ones, based on the Exact Diagonalisation and Stochastic Series Expansion, named Ring Polymer Exact Diagonalisation (RPED) and Ring Polymer Stochastic Series Expansion (RPSSE). Using these methods, the effects of spin-phonon coupling on antiferromagnetic spin chains are investigated, both classical and quantum in origin.