Complex Langevin Algorithm Research Articles

Complex Langevin simulation of a random matrix model at nonzero chemical potential

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.

Progress on Complex Langevin simulations of a finite density matrix model for QCD

We study the Stephanov model, which is an RMT model for QCD at finite density, using the Complex Langevin algorithm. Naive implementation of the algorithm shows convergence towards the phase quenched or quenched theory rather than to intended theory with dynamical quarks. A detailed analysis of this issue and a potential resolution of the failure of this algorithm are discussed. We study the effect of gauge cooling on the Dirac eigenvalue distribution and time evolution of the norm for various cooling norms, which were specifically designed to remove the pathologies of the complex Langevin evolution. The cooling is further supplemented with a shifted representation for the random matrices. Unfortunately, none of these modifications generate a substantial improvement on the complex Langevin evolution and the final results still do not agree with the analytical predictions.

Surmounting the sign problem in nonrelativistic calculations: A case study with mass-imbalanced fermions

The calculation of the ground state and thermodynamics of mass-imbalanced Fermi systems is a challenging many-body problem. Even in one spatial dimension, analytic solutions are limited to special configurations and numerical progress with standard Monte Carlo approaches is hindered by the sign problem. The focus of the present work is on the further development of methods to study imbalanced systems in a fully non-perturbative fashion. We report our calculations of the ground-state energy of mass-imbalanced fermions using two different approaches which are also very popular in the context of the theory of the strong interaction (Quantum Chromodynamics, QCD): (a) the hybrid Monte Carlo algorithm with imaginary mass imbalance, followed by an analytic continuation to the real axis; and (b) the Complex Langevin algorithm. We cover a range of on-site interaction strengths that includes strongly attractive as well as strongly repulsive cases which we verify with non-perturbative renormalization group methods and perturbation theory. Our findings indicate that, for strong repulsive couplings, the energy starts to flatten out, implying interesting consequences for short-range and high-frequency correlation functions. Overall, our results clearly indicate that the Complex Langevin approach is very versatile and works very well for imbalanced Fermi gases with both attractive and repulsive interactions.

Does the complex Langevin method give unbiased results?

We investigate whether the stationary solution of the Fokker-Planck equation of the complex Langevin algorithm reproduces the correct expectation values. When the complex Langevin algorithm for an action $S(x)$ is convergent, it produces an equivalent complex probability distribution $P(x)$ which ideally would coincide with $e^{-S(x)}$. We show that the projected Fokker-Planck equation fulfilled by $P(x)$ may contain an anomalous term whose form is made explicit. Such term spoils the relation $P(x)=e^{-S(x)}$, introducing a bias in the expectation values. Through the analysis of several periodic and non-periodic one-dimensional problems, using either exact or numerical solutions of the Fokker-Planck equation on the complex plane, it is shown that the anomaly is present quite generally. In fact, an anomaly is expected whenever the Langevin walker needs only a finite time to go to infinity and come back, and this is the case for typical actions. We conjecture that the anomaly is the rule rather than the exception in the one-dimensional case, however, this could change as the number of variables involved increases.

Onset transition to cold nuclear matter from lattice QCD with heavy quarks.

Lattice QCD at finite density suffers from a severe sign problem, which has so far prohibited simulations of the cold and dense regime. Here we study the onset of nuclear matter employing a three-dimensional effective theory derived by combined strong coupling and hopping expansions, which is valid for heavy but dynamical quarks and has a mild sign problem only. Its numerical evaluations agree between a standard Metropolis and complex Langevin algorithm, where the latter is free of the sign problem. Our continuum extrapolated data approach a first order phase transition at μ(B) ≈ m(B) as the temperature approaches zero. An excellent description of the data is achieved by an analytic solution in the strong coupling limit.

A numerical attempt on the chiral Schwinger model

The chiral Schwinger model is formulated in the wilson-fermion formulation on the lattice and then simulated by the complex langevin algorithm. The simulation is done both without and with gauge fixing to the Lorentz gauge for the compact gauge links. Some preliminary results are presented which indicate that the complex langevin is well behaving with the complex chiral fermion determinant.

Sketching the phase diagram for a Yukawa model with dynamical chiral fermions

Smith-Swift-type lattice models with dynamical chiral fermions lead to complex effective actions. This makes their numerical simulation problematic. We report on the application of a Complex Langevin algorithm to this problem. In exploratory simulations on 4 4 and 6 4 lattices, we do not find any runaway solutions. The phase diagram which emerges from these simulations seems to agree with analytical expectations, and with the phase diagram seen in unquenched studies of vectorlike models.

Complex Langevin solution to an effective theory of lattice QCD.

By means of a complex Langevin algorithm, we simulate on the full group space an effective three-dimensional theory of QCD at finite temperature and finite chemical potential. We study the effect of the chemical potential on the first-order character of the deconfinement phase transition.

Complex Langevin solution to an effective theory of compact lattice QED

By means of a complex Langevin algorithm, we simulate an effective three-dimensional lattice theory which arises from compact lattice QED at finite temperature and finite chemical potential. The complex process coverges remarkably well.

Complex Langevin simulation of the SU(3) spin model with nonzero chemical potential.

We study the effective three-dimensional SU(3) spin model with nonzero chemical potential. This model describes the strong-coupling, large-fermion-mass limit of QCD at finite temperature and baryon density. The results obtained with a complex Langevin algorithm are encouraging for a future simulation of QCD with nonzero chemical potential. The Langevin simulations converge even for large values of the chemical potential, and excellent agreement with exact solutions in the extreme strong-coupling ($\ensuremath{\beta}=0$) limit has been found.