Erhard Seiler

Complex Langevin method: When can it be trusted?

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

Complex Langevin: etiology and diagnostics of its main problem

The complex Langevin method is a leading candidate for solving the so-called sign problem occurring in various physical situations. Its most vexing problem is that sometimes it produces ‘convergence to the wrong limit’. In this paper we carefully revisit the formal justification of the method, identifying points at which it may fail and derive a necessary and sufficient criterion for correctness. This criterion is, however, not practical, since its application requires checking an infinite tower of identities. We propose instead a practical test involving only a check of the first few of those identities; this raises the question of the ‘sensitivity’ of the test. This sensitivity as well as the general insights into the possible reasons of failure (the etiology) are then tested in two toy models where the correct answer is known. At least in those models the test works perfectly.

Gauge cooling in complex Langevin for lattice QCD with heavy quarks

Abstract We employ a new method, “gauge cooling”, to stabilize complex Langevin simulations of QCD with heavy quarks. The results are checked against results obtained with reweighting; we find agreement within the estimated errors, except for strong gauge coupling in the confinement region. The method allows us to go to previously unaccessible high densities.

Controlling complex Langevin dynamics at finite density

At nonzero chemical potential the numerical sign problem in lattice field theory limits the use of standard algorithms based on importance sampling. Complex Langevin dynamics provides a possible solution, but it has to be applied with care. In this review, we first summarise our current understanding of the approach, combining analytical and numerical insight. In the second part we study SL(C, ℂ) gauge cooling, which was introduced recently as a tool to control complex Langevin dynamics in nonabelian gauge theories. We present new results in Polyakov chain models and in QCD with heavy quarks and compare various adaptive cooling implementations.

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.

Some remarks on Lefschetz thimbles and complex Langevin dynamics

A bstractLefschetz 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.

Monte Carlo simulation of non-compact QCD with stochastic gauge fixing

Abstract A non-compact lattice model of quantum chromodynamics is studied numerically. Whereas in Wilsons lattice theory the basic variables are the elements of a compact Lie group, the present lattice model resembles the continuum theory in that the basic variables A are elements of the corresponding Lie algebra, a non-compact space. The lattice gauge invariance of Wilsons theory is lost. As in the continuum, the action is a quartic polynomial in A , and a stochastic gauge fixing mechanism - which is covariant in the continuum and avoids Faddeev-Popov ghosts and the Gribov ambiguity — is also transcribed to the lattice. It is shown that the model is self-compactifying, in the sense that the probability distribution is concentrated around a compact region of the hyperplane div A = 0 which is bounded by the Gribov horizon. The model is simulated numerically by a Monte Carlo method based on the random walk process. Measurements of Wilson loops, Polyakov loops and correlations of Polyakov loops are reported and analyzed. No evidence of confinement is found for the values of the parameters studied, even in the strong coupling regime.

Some more remarks on the Witten–Veneziano formula for the η′ mass

Abstract We discuss some subtleties in connection with the new attempts to provide a firm basis for ths Witten–Veneziano formula.

On the construction of quantized gauge fields

In this paper the construction of the two-dimensional abelian Higgs model begun in two earlier articles is completed. First we show how to remove the remaining ultraviolet cutoff on the gauge field, then we construct the infinite volume limit and verify the axioms of Osterwalder and Schrader for the expectation values of gauge invariant local fields. Finally it is shown that an auxiliary gauge field mass that was introduced to avoid infrared problems can be safely removed.

Monte Carlo study of noncompact lattice QCD

Abstract We study a noncompact version of lattice QCD4 by Monte Carlo simulation. The results indicate that the model possesses asymptotic freedom but does not confine static quarks. It looks rather like a self-Higgsed system.