Abstract
We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action (“Lefschetz thimble”). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.
Highlights
JHEP05(2016)053 need to determine the combination of thimbles equivalent to the original real integral, a task of great complexity to say the least.1 The need for a multi-thimble integration, at least in some models, was evidenced in [14] where we considered a simple soluble 0 + 1 dimensional fermionic model previously used as a toy model for the sign problem [15, 16]
We describe a family of such manifolds that interpolate between the tangent space at one critical point and the union of relevant thimbles
We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive
Summary
Where the integration is over real fields φ. The quartic interaction can be eliminated by introducing an auxiliary field φ and, upon integrating over χ, χ the partition function is This model has a sign problem for non-zero values of μ and it has been used as a toy model for testing ideas such as complex Langevin dynamics [15, 16, 20] and hybrid Monte Carlo on Lefschetz thimbles [11, 12]. A semiclassical estimate of the contribution of the other thimbles, which is suppressed by their higher action, suggests that they have the right order of magnitude to explain the discrepancies with the exact result This is, only a plausibility argument as the semiclassical expansion is not reliable for the parameters considered and the possibility that the discrepancy arises from a problem in the algorithm computing the contribution of the main thimble should not be excluded
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