Optimisation of complex integration contours at higher order

Abstract

We continue our study of contour deformation as a practical tool for dealing with the sign problem using the $d$-dimensional Bose gas with non-zero chemical potential as a toy model. We derive explicit expressions for contours up to the second order with respect to a natural small parameter and generalise these contours to an ansatz for which the evaluation of the Jacobian is fast ($O(1)$). We examine the behaviour of the various proposed contours as a function of space-time dimensionality, the chemical potential, and lattice size and geometry and use the mean phase factor as a measure of the severity of the sign problem. In turns out that this method leads to a substantial reduction of the sign problem and that it becomes more efficient as space-time dimensionality is increased. Correlations among contributions to $\mathrm{Im}⟨S⟩$ play a key role in determining the mean phase factor and we examine these correlations in detail.

30 pages, 17 figures

References in corpus (2)

• Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential
• Deep Learning Beyond Lefschetz Thimbles