arXiv subjects

Erhard Seiler

Publications and source records attributed to Erhard Seiler.

At least 19 recordsLinked to original sources

arXiv2025

Complex Langevin simulations with a kernel

We discuss recent developments regarding the use of kernels in complex Langevin simulations. In particular, we outline how a kernel can be used to solve the problem of wrong convergence in a simple toy model. Since conventional correctness criteria for complex Langevin results are only necessary but not sufficient, the correct convergence of complex Langevin simulations is not always straightforward to assess. Hence, we furthermore discuss a condition for correctness that we have recently derived, which is both necessary and sufficient. Finally, we outline a machine-learning approach for finding suitable kernels in lattice gauge theories and present preliminary results of its application to the heavy-dense limit of QCD.

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arXiv2025

Necessary and sufficient conditions for correctness of complex Langevin

We derive a family of correctness conditions for complex Langevin simulations. In particular, we show that if in a given theory the expectation values of all observables within a particular space satisfy the theory's Schwinger-Dyson equations as well as certain bounds, then these expectation values are necessarily correct. In fact, these findings are not only valid in the context of complex Langevin simulations, but they also hold for general probability densities on complex manifolds, given an initial complex density on a real manifold. We stress that, while the proposed conditions are necessary and sufficient in a mathematical sense, their practical use is not to prove the correctness of obtained simulation results. Rather, they are mainly useful for detecting incorrect convergence. In particular, we test these criteria in a few simple one- and two-dimensional toy models and find that they are indeed capable of ruling out incorrect results without the need of exact solutions.

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arXiv2024

The Role of Integration Cycles in Complex Langevin Simulations

Complex Langevin simulations are an attempt to solve the sign (or complex-action) problem encountered in various physical systems of interest. The method is based on a complexification of the underlying degrees of freedom and an evolution in an auxiliary time dimension. The complexification, however, does not come without drawbacks, the most severe of which is the infamous 'wrong convergence' problem, stating that complex Langevin simulations sometimes fail to produce correct answers despite their apparent convergence. It has long been realized that wrong convergence may - in principle - be fixed by the introduction of a suitable kernel into the complex Langevin equation, such that the conventional correctness criteria are met. However, as we discuss in this work, complex Langevin results may - especially in the presence of a kernel - still be affected by unwanted so-called integration cycles of the theory spoiling them. Indeed, we confirm numerically that in the absence of boundary terms the complex Langevin results are given by a linear combination of such integration cycles, as put forward by Salcedo & Seiler. In particular, we shed light on the way different choices of kernel affect which integration cycles are being sampled in a simulation and how this knowledge can be used to ensure correct convergence in simple toy models.

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arXiv2024

Kernels and integration cycles in complex Langevin simulations

The method of complex Langevin simulations is a tool that can be used to tackle the complex-action problem encountered, for instance, in finite-density lattice quantum chromodynamics or real-time lattice field theories. The method is based on a stochastic evolution of the dynamical degrees of freedom via (complex) Langevin equations, which, however, sometimes converge to the wrong equilibrium distributions. While the convergence properties of the evolution can to some extent be assessed by studying so-called boundary terms, we demonstrate in this contribution that boundary terms on their own are not sufficient as a correctness criterion. Indeed, in their absence complex Langevin simulation results might still be spoiled by unwanted so-called integration cycles. In particular, we elaborate on how the introduction of a kernel into the complex Langevin equation can - in principle - be used to control which integration cycles are sampled in a simulation such that correct convergence is restored.

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arXiv2023

Complex Langevin: Correctness criteria, boundary terms and spectrum

The Complex Langevin (CL) method to simulate `complex probabilities', ideally produces expectation values for the observables that converge to a limit equal to the expectation values obtained with the original complex `probability' measure. The situation may be spoiled in two ways: failure to converge and convergence to the wrong limit. It was found long ago that `wrong convergence' is caused by boundary terms; non-convergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here we propose a class of criteria which allow to rule out boundary terms and at the same time bad spectrum. Ruling out boundary terms in the equilibrium distribution arising from a CL simulation implies that the so-called convergence conditions are fulfilled. This in turn has been shown to guarantee that the expectation values of holomorphic observables are given by complex linear combinations of $\exp(-S)$ over various integration cycles. If the spectrum is pathological, however, the CL simulation in general does not reproduce the integral over the desired real cycle.

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arXiv2021

Complex Langevin boundary terms in lattice models

In complex Langevin simulations, the insufficient decay of the probability density near infinity leads to boundary terms that spoil the formal argument for correctness. We present a formulation of this term that is cheaply measurable in lattice models, and in principle allows also the direct estimation of the systematic error of the CL method. Results for a toy model, 3d XY model and HDQCD are presented.

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arXiv2021

Complex Langevin: Boundary terms at poles of the drift

The complex Langevin method is a general method to treat systems with complex action, such as QCD at nonzero density. The formal justification relies on the absence of certain boundary terms, both at infinity and at the unavoidable poles of the drift force. Here I focus on the boundary terms at these poles for simple models, which so far have not been discussed in detail. The main result is that those boundary terms (for the "un-evolved" observables) arise after running the Langevin process for a finite time and vanish again as the Langevin time goes to infinity. This is in contrast to the boundary terms at infinity, which can be found to occur in the long time limit (cf. the contribution by Dénes Sexty).

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arXiv2020

Complex Langevin: Boundary terms at poles

We discuss the problem of possible boundary terms at poles of the drift in the complex Langevin method, which spoil correctness of the method. For the simplest, however paradigmatic cases we can find complete answers. Lessons for more generic cases as well as open mathematical problems are discussed.

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arXiv2019

Complex Langevin and boundary terms

As is well known the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. We also show how some simple modification stabilizes the CL process in such a way that it can produce results agreeing with direct integration. Besides explicitly demonstrating the connection between boundary terms and correct convergence our analysis also suggests a correctness criterion which could be applied in realistic lattice simulations.

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arXiv2018

Complex Langevin: Boundary terms and application to QCD

We employ the Complex Langevin method for simulation of complex-valued actions. First, we show how to test for convergence of the method by explicitely computing boundary terms and demonstrate this in a model. Then we investigate the deconfinement phase transition of QCD with $N_f=2$ Wilson-fermions using the Complex Langevin Method and. We give preliminary results for the transition temperatures up to $μ/T_c(μ=0)\approx 5$ and compute the curvature coefficient $κ_2$.

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arXiv2018

Schwinger-Dyson equations and line integrals

The Complex Langevin (CL) method sometimes shows convergence to the wrong limit, even though the Schwinger-Dyson Equations (SDE) are fulfilled. We analyze this problem in a more general context for the case of one complex variable. We prove a theorem that shows that under rather general conditions not only the equilibrium measure of CL but any linear functional satisfying the SDE on a space of test functions is given by a linear combination of integrals along paths connecting the zeroes of the underlying measure and noncontractible closed paths. This proves rigorously a conjecture stated long ago by one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL. one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL.

math-ph↗

arXiv2017

Positive Representations of a Class of Complex Measures

We study the problem of constructing positive representations of complex measures. In this paper we consider complex densities on a direct product of $U(1)$ groups and look for representations by probability distributions on the complexification of those groups. After identifying general necessary and sufficient conditions we propose several concrete realizations. Finally we study some of those realizations in examples representing problems in abelian lattice gauge theories.

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arXiv2017

Status of Complex Langevin

I review the status of the Complex Langevin method, which was invented to make simulations of models with complex action feasible. I discuss the mathematical justification of the procedure, as well as its limitations and open questions. Various pragmatic measures for dealing with the existing problems are described. Finally I report on the progress in the application of the method to QCD, with the goal of determining the phase diagram of QCD as a function of temperature and baryonic chemical potential.

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arXiv2017

Getting even with CLE

In the landscape of approaches toward the simulation of Lattice Models with complex action the Complex Langevin (CL) appears as a straightforward method with a simple, well defined setup. Its applicability, however, is controlled by certain specific conditions which are not always satisfied. We here discuss the procedures to meet these conditions and the estimation of systematic errors and present some actual achievements.

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arXiv2017

Beyond Complex Langevin Equations: positive representation of a class of complex measures

A positive representation for a set of complex densities is constructed. In particular, complex measures on a direct product of U(1) groups are studied. After identifying general conditions which such representations should satisfy, several explicit realizations are proposed. Their utility is illustrated in few concrete examples representing problems in abelian lattice gauge theories.

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arXiv2017

Complex Langevin dynamics and zeroes of the fermion determinant

QCD at nonzero baryon chemical potential suffers from the sign problem, due to the complex quark determinant. Complex Langevin dynamics can provide a solution, provided certain conditions are met. One of these conditions, holomorphicity of the Langevin drift, is absent in QCD since zeroes of the determinant result in a meromorphic drift. We first derive how poles in the drift affect the formal justification of the approach and then explore the various possibilities in simple models. The lessons from these are subsequently applied to both heavy dense QCD and full QCD, and we find that the results obtained show a consistent picture. We conclude that with careful monitoring, the method can be justified a posteriori, even in the presence of meromorphicity.

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arXiv2016

On complex Langevin dynamics and zeroes of the measure I: Formal proof and simple models

In the complex Langevin approach to lattice simulations at nonzero density, zeroes of the fermion determinant lead to a meromorphic drift and hence a need to revisit the theoretical derivation. We discuss how poles in the drift affect the formal justification of the approach and then explore the various potential issues in simple models, in a manner that is applicable to heavy dense and full QCD.

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arXiv2016

Discussion of the loop formula for the fermionic determinant

A formula expressing the fermionic determinant (a large order polynomial) as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and are indexed by loops of increasing length.

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