Optimisation of complex integration contours at higher order

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