Complex Langevin Dynamics Research Articles

Stability of complex Langevin dynamics in effective models

The sign problem at nonzero chemical potential prohibits the use of importance sampling in lattice simulations. Since complex Langevin dynamics does not rely on importance sampling, it provides a potential solution. Recently it was shown that complex Langevin dynamics fails in the disordered phase in the case of the three-dimensional XY model, while it appears to work in the entire phase diagram in the case of the three-dimensional SU(3) spin model. Here we analyse this difference and argue that it is due to the presence of the nontrivial Haar measure in the SU(3) case, which has a stabilizing effect on the complexified dynamics. The freedom to modify and stabilize the complex Langevin process is discussed in some detail.

Complex Langevin dynamics in the SU(3) spin model at nonzero chemical potential revisited

The three-dimensional SU(3) spin model is an effective Polyakov loop model for QCD at nonzero temperature and density. It suffers from a sign problem at nonzero chemical potential. We revisit this model using complex Langevin dynamics and assess in particular the justification of this approach, using analyticity at small mu^2 and the criteria for correctness developed recently. Finite-stepsize effects are discussed in some detail and a higher-order algorithm is employed to eliminate leading stepsize corrections. Our results strongly indicate that complex Langevin dynamics is reliable in this theory in both phases, including the critical region. This is in sharp contrast to the case of the XY model, where correct results were obtained in only part of the phase diagram.

Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finite chemical potential

We demonstrate analytically that complex Langevin dynamics can solve the sign problem in one-dimensional QCD in the thermodynamic limit. In particular, it is shown that the contributions from the complex and highly oscillating spectral density of the Dirac operator to the chiral condensate are taken into account correctly. We find an infinite number of classical fixed points of the Langevin flow in the thermodynamic limit. The correct solution originates from a continuum of degenerate distributions in the complexified space.

On the convergence of complex Langevin dynamics: the three-dimensional XY model at finite chemical potential

The three-dimensional XY model is studied at finite chemical potential using complex Langevin dynamics. The validity of the approach is probed at small chemical potential using imaginary chemical potential and continuity arguments, and at larger chemical potential by comparison with the world line method. While complex Langevin works for larger beta, we find that it fails for smaller beta, in the region of the phase diagram corresponding to the disordered phase. Diagnostic tests are developed to identify symptoms correlated with incorrect convergence. We argue that the erroneous behaviour at smaller beta is not due to the sign problem, but rather resembles dynamics observed in complex Langevin simulations of simple models with complex noise.

Adaptive stepsize and instabilities in complex Langevin dynamics

Stochastic quantization offers the opportunity to simulate field theories with a complex action. In some theories unstable trajectories are prevalent when a constant stepsize is employed. We construct algorithms for generating an adaptive stepsize in complex Langevin simulations and find that unstable trajectories are completely eliminated. To illustrate the generality of the approach, we apply it to the three-dimensional XY model at nonzero chemical potential and the heavy dense limit of QCD.

Complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas

Stochastic quantization can potentially be used to simulate theories with a complex action due to a nonzero chemical potential. We study complex Langevin dynamics in the relativistic Bose gas analytically, using a mean field approximation. We concentrate on the region with a Silver Blaze problem and discuss convergence, stability, fixed points, and the severeness of the sign problem. The real distribution satisfying the extended Fokker-Planck equation is constructed and its nonlocal form is explained. Finally, we compare the mean field results in finite volume with the numerical data presented in Ref. [1].

Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential

A nonperturbative study of field theories with a complex action, such as QCD at finite baryon density, is difficult due to the sign problem. We show that the relativistic Bose gas at finite chemical potential has a sign and "silver blaze" problem, similar to QCD. We then apply stochastic quantization and complex Langevin dynamics to study this theory with nonperturbative lattice simulations. Independence of chemical potential at small and a transition to a condensed phase at large chemical potential are found. Lattices of size N4, with N=4, 6, 8, 10, are used. We show that the sign problem is severe, however, we find that it has no negative effect using this approach. This improves the prospects of applying stochastic quantization to QCD at nonzero density.

Two complex problems on the lattice: transport coefficients and finite chemical potential

After a few remarks about the problem of extracting transport coefficients from lattice QCD calculations, I report on recent developments in applying stochastic quantization and complex Langevin dynamics to field theories with a complex action due to a nonzero chemical potential. First results demonstrate that the sign problem poses no obstacle for this approach, even in the thermodynamic limit. I conclude with a comparison of two simple one-link models, describing a euclidean system at finite chemical potential and a Minkowski system in real time.

Stochastic quantization at finite chemical potential

A nonperturbative lattice study of QCD at finite chemical potential is complicated due to the complex fermion determinant and the sign problem. Here we apply the method of stochastic quantization and complex Langevin dynamics to this problem. We present results for U(1) and SU(3) one link models and QCD at finite chemical potential using the hopping expansion. The phase of the determinant is studied in detail. Even in the region where the sign problem is severe, we find excellent agreement between the Langevin results and exact expressions, if available. We give a partial understanding of this in terms of classical flow diagrams and eigenvalues of the Fokker-Planck equation.