Abstract
In this paper we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.
Highlights
It is well known that the culprit behind this lack of results is the infamous sign problem which is prohibiting numerical simulations when μ/T > 1 or μ > mπ/2
This seems to point to the absence of an overlap problem in the reweighted complex Langevin (RCL) method, even though the sign problem is clearly present in the phase transition region
Using the data for N = 6, 12, 24, 48 we find a naive volume scaling for the RCL method that is proportional to exp(0.3 × N ) for the number of configurations necessary to get the same accuracy for all values of N
Summary
We discuss a random matrix theory inspired model [19, 32] for QCD at finite baryon density originally proposed by Stephanov [18]. The unquenched partition function of the Stephanov model was cast analytically in a form that allows for either an easy numerical evaluation at finite N , or that allows for a complete analytical solution via a saddle point approximation in the thermodynamic limit where N → ∞ [18, 22]. In the large N limit, it can be evaluated by a saddle-point approximation, see appendix A It has a pion-condensation phase for μ > mπ/2 corresponding to the parameter domain when the quark mass is inside the support of the eigenvalues. In order to study the properties of the Langevin algorithm we will perform numerical simulations of the Stephanov model employing the CL algorithm and test its convergence properties by comparing the obtained numerical data for the chiral condensate and the baryon density with analytical results computed using the partition function (2.11). In several cases we will simulate the Osborn model in order to display potential issues that might arise for the CL method when switching from a model without a phase transition to one where the sign problem triggers a phase transition
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