A simple approach towards the sign problem using path optimisation
We suggest an approach for simulating theories with a sign problem that relies on optimisation of complex integration contours that are not restricted to lie along Lefschetz thimbles. To that end we consider the toy model of a one-dimensional Bose gas with chemical potential. We identify the main contribution to the sign problem in this case as coming from a nearest neighbour interaction and approximately cancel it by an explicit deformation of the integration contour. We extend the obtained expressions to more general ones, depending on a small set of parameters. We find the optimal values of these parameters on a small lattice and study their range of validity. We also identify precursors for the onset of the sign problem. A fast method of evaluating the Jacobian related to the contour deformation is proposed and its numerical stability is examined. For a particular choice of lattice parameters, we find that our approach increases the lattice size at which the sign problem becomes serious from L ≈ 32 to L ≈ 700. The efficient evaluation of the Jacobian (O(L) for a sweep) results in running times that are of the order of a few minutes on a standard laptop.
KeywordsLattice field theory simulation
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- S. Lawrence, Beyond thimbles: sign-optimized manifolds for finite density, in 36th International Symposium on Lattice Field Theory (Lattice 2018), East Lansing, MI, U.S.A., July 22–28, 2018 (2018) [arXiv:1810.06529] [INSPIRE].