Towards a quantum Monte Carlo for lattice systems

Abstract

In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.