Stochastic phase-space methods for lattice models

Grand-canonical inverse-temperature calculations of a single mode Bose-Hubbard model are presented, using the Gaussian phase space representation. Simulation of 100 particles is achieved in the ground state, having started with a low-particle-number thermal state. A preliminary foray into a three-mode lattice is made, but the sampling error appears to be too large for the simple approach taken here to be successful in larger systems. The quantum (real-time) dynamics of a one-dimensional Bose gas with two-particle losses are investigated. The Positive-P equations for this system are unstable, and this causes Positive-P simulations to `die' after a certain amount of time. Gauges are used to (sometimes partially) stabilise the equations. The effects on simulation times of various gauges, branching methods, and non-square diffusion matrix factorisations on simulation times are investigated. Despite the absence of repulsive inter-particle interactions, it is observed that $g^{(2)}$ rises above 1 at a finite particle separation. A phase space method for spin systems is introduced, based on SU(2) coherent states. This is essentially a spin analogue of the Positive-P method. The system of stochastic differential equations arising out of this method require weighted averages to be taken, and the weights can vary exponentially, leading to inefficient sampling. For the case of the Ising model, a transform is made to a set of equations which relaxes (in a dummy time variable) to the partition function at a given temperature, and allows unweighted ensemble averages to be taken. This allows accurate simulations to be achieved at a range of temperatures, with nearest-neighbour correlation functions agreeing with theory. This represents a proof of principle for the use of stochastic phase space methods in spin systems, and furthermore the method should be suited to open spin systems, at least for a small number of qubits