Abstract
QCD in 0+1 dimensions is numerically solved via thimble regularization. In the context of this toy model, a general formalism is presented for SU(N) theories. The sign problem that the theory displays is a genuine one, stemming from a (quark) chemical potential. Three stationary points are present in the original (real) domain of integration, so that contributions from all the thimbles associated to them are to be taken into account: we show how semiclassical computations can provide hints on the regions of parameter space where this is absolutely crucial. Known analytical results for the chiral condensate and the Polyakov loop are correctly reproduced: this is in particular trivial at high values of the number of flavors N_f. In this regime we notice that the single thimble dominance scenario takes place (the dominant thimble is the one associated to the identity). At low values of N_f computations can be more difficult. It is important to stress that this is not at all a consequence of the original sign problem (not even via the residual phase). The latter is always under control, while accidental, delicate cancelations of contributions coming from different thimbles can be in place in (restricted) regions of the parameter space.
Highlights
The very first proposal of thimble regularization was intended to extend our capabilities to properly define quantum field theories [1,2,3]
We have already stated that a contribution from each critical point is expected: this is a direct consequence of the fact that they are all sitting on the original domain of integration.1Collecting contributions of more than one thimble to solve the theory is in general a delicate issue
We showed that QCD in 0 þ 1 dimensions can be effectively computed in thimble regularization
Summary
The very first proposal of thimble regularization was intended to extend our capabilities to properly define quantum field theories [1,2,3]. The basic idea underlying thimble regularization amounts to deforming the original domain of integration of a given field theory into a new one, which is made of one or more manifolds. Solving the sign problem via a deformation of the integration domain is conceptually satisfying and the thimble approach is potentially very powerful. The paper is organized as follows: in Sec. II we discuss 0 þ 1 QCD and how to treat it in thimble regularization, in particular enlightening the role of symmetry and discussing the semiclassical approximation (to put this work in the right perspective we end Sec. II putting forward a few caveats); in Sec. III we present our results, both by flat, crude Monte Carlo and by importance sampling in the steepest ascents space; in Sec. IV we present our conclusions.
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