Towards sampling complex actions

Abstract

Path integrals with complex actions are encountered for many physical systems ranging from spin-or mass-imbalanced atomic gases and graphene to quantum chromo-dynamics at finite density to the non-equilibrium evolution of quantum systems. Many computational approaches have been developed for tackling the sign problem emerging for complex actions. Among these, complex Langevin dynamics has the appeal of general applicability. One of its key challenges is the potential convergence of the dynamics to unphysical fixed points. The statistical sampling process at such a fixed point is not based on the physical action and hence leads to wrong predictions. Moreover, its unphysical nature is hard to detect due to the implicit nature of the process. In the present work we set up a general approach based on a Markov chain Monte Carlo scheme in an extended state space. In this approach we derive an explicit real sampling process for generalized complex Langevin dynamics. Subject to a set of constraints, this sampling process is the physical one. These constraints originate from the detailed-balance equations satisfied by the Monte Carlo scheme. This allows us to re-derive complex Langevin dynamics from a new perspective and establishes a framework for the explicit construction of new sampling schemes for complex actions.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part III: Shock and rarefaction waves in RG flows reveal limitations of the $N \rightarrow \infty$ limit in $O(N)$-type models

“…Since the equations in x and y are related by the coordinate transformation y = 1 2 x 2 , the solutions and also characteristics curves are directly related. For simplicity we solve the characteristic equations for the flow equation (47) in the rescaled invariant y and then compute the corresponding curves in x using the coordinate transformation. A direct solution of the characteristic equations for the flow equation (45) in x is also possible and shares a lot of computations with the slightly simpler computation in y.…”

“…In Fig. 10 we present numerical results for the RG flow in the rescaled invariant y using the flow equation (47) with the piecewise constant initial condition of Eq. ( 16) obtained with the KNP O(∆y 1 ) scheme discussed in the previous subsection of this appendix.…”

“…( 16) obtained with the KNP O(∆y 1 ) scheme discussed in the previous subsection of this appendix. The flow equation (47) with the piecewise constant initial condition of Eq. ( 16) constitutes two Riemann problems as outlined in the beginning of this paper in Sec.…”

“…Appendix E: The FRG flow equation formulated in the O(N )-invariant in the large-N limit Before we discuss our numerical results for the flow equation (47) formulated in the 1 N -rescaled invariant y ≡ 1 2 x 2 in Sub.Sec. E 2 of this App., we briefly introduce the employed numerical KNP scheme in the next Sub.Sec.…”

“…Refs. [3,24,33,[35][36][37][38][39][40][41][42][43][44][45][46][47]. Even a lot of aspects of the large-/infinite-N limit have been discussed already within this setup, cf.…”