Abstract
Lefschetz thimbles and complex Langevin dynamics both provide a means to tackle the numerical sign problem prevalent in theories with a complex weight in the partition function, e.g. due to nonzero chemical potential. Here we collect some findings for the quartic model, and for U(1) and SU(2) models in the presence of a determinant, which have some features not discussed before, due to a singular drift. We find evidence for a relation between classical runaways and stable thimbles, and give an example of a degenerate fixed point. We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics.
Highlights
JHEP10(2014)159 and current advances have focussed on better control over the residual sign factor in the case of a single thimble [31, 32]
We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics
We have studied and contrasted the distributions sampled by complex Langevin dynamics and the Lefschetz thimbles in a number of models with a complex action, focussing on examples with a determinant in the Boltzmann weight
Summary
In ref. [35] a first comparison between Lefschetz thimbles and complex Langevin dynamics was made for the quartic model in zero dimensions. The eigenvectors in the complex plane correspond to the directions of the stable and the unstable thimble This is illustrated in figure 1 for the quartic model (2.2), where the fixed points are indicated with blue circles and the classical Langevin drift with (normalised) arrows, for a specific choice of parameters, σ = 1, h = 1 + i. For x = 0 the corresponding action is real for all y, the x = 0 axis corresponds to the unstable thimble associated with the fixed point on the x = 0 axis (the instability follows from the Hessian, and from the fact that z → ±i∞ is not in the region of convergence of the original integral). It is interesting to compare both approaches with the Lefschetz thimbles
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