New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble

140

A three-dimensional effective lattice theory of Polyakov loops is derived from QCD by expansions in the fundamental character of the gauge action, u, and the hopping parameter, κ, whose action is correct to κ n u m with n + m = 4. At finite baryon density, the effective theory has a sign problem which meets all criteria to be simulated by complex Langevin as well as by Monte Carlo on small volumes. The theory is valid for the thermodynamics of heavy quarks, where its predictions agree with simulations of full QCD at zero and imaginary chemical potential. In its region of convergence, it is moreover amenable to perturbative calculations in the small effective couplings. In this work we study the challenging cold and dense regime. We find unambiguous evidence for the nuclear liquid gas transition once the baryon chemical potential approaches the baryon mass, and calculate the nuclear equation of state in the limit of heavy baryons. In particular, we find a negative binding energy per nucleon causing the condensation, whose absolute value decreases exponentially as mesons get heavier. For decreasing meson mass, we observe a first order liquid gas transition with an endpoint at some finite temperature, as well as a gap between the onset of isospin and baryon condensation.

Section: Acknowledgmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Heavy dense QCD and nuclear matter from an effective lattice theory

Langelage

Neuman

Philipsen

2014

140

“…The main approaches which have been pursued are complex Langevin dynamics and integration along Lefschetz thimbles [26][27][28][29][30][31][32][33][34][35]. In the latter, the original path of integration is deformed in order to pass through the fixed (or critical) points of the complex action, which typically reside in the complexified space.…”

Section: Introductionmentioning

confidence: 99%

“…This is typically referred to as a global sign problem. The numerical implementation of this approach [28] has been tested in a number of models, including the interacting Bose gas at nonzero chemical potential in four dimensions [29,31], which has a severe sign problem [10],…”

Section: Introductionmentioning

confidence: 99%

Some remarks on Lefschetz thimbles and complex Langevin dynamics

Aarts

Bongiovanni

Seiler

et al. 2014

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.

“…When the flow time τ becomes infinitely large, M τ contracts to a set of Lefschetz thimbles, and the GLTM reduces to the so-called Lefschetz-thimble method [3]. Since…”

Section: Generalized Lefschetz-thimble Methods (Gltm)mentioning

confidence: 99%

Combining the complex Langevin method and the generalized Lefschetz-thimble method

Nishimura

Shimasaki

2017

Abstract:The complex Langevin method and the generalized Lefschetz-thimble method are two closely related approaches to the sign problem, which are both based on complexification of the original dynamical variables. The former can be viewed as a generalization of the stochastic quantization using the Langevin equation, whereas the latter is a deformation of the integration contour using the so-called holomorphic gradient flow. In order to clarify their relationship, we propose a formulation which combines the two methods by applying the former method to the real variables that parametrize the deformed integration contour in the latter method. Three versions, which differ in the treatment of the residual sign problem in the latter method, are considered. By applying them to a single-variable model, we find, in particular, that one of the versions interpolates the complex Langevin method and the original Lefschetz-thimble method.