Complex Langevin simulations and the QCD phase diagram: recent developments
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Review
Complex Langevin simulations and the QCD phase diagram: recent developments Felipe Attanasio1, Benjamin Jäger2,a , Felix P. G. Ziegler2 1 2
Department of Physics, University of Washington, Box 351560, Seattle, WA 98195-1560, USA CP3-Origins and Danish IAS, Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
Received: 31 May 2020 / Accepted: 19 September 2020 / Published online: 6 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Communicated by Laura Tolos
Abstract In this review we present the current state-of-theart on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
1 Introduction Strongly coupled quantum matter encompasses some of the most interesting problems in modern physics. Monte Carlo methods are commonly used to perform numerical simulations of theories not amenable to perturbative expansions. These methods typically rely on the path integral formulation of the theory in Euclidean space-time to have Boltzmannlike weights, which can be interpreted as probability distributions. This allows the generation of field configurations distributed according to these weights, whence observables can be sampled. Real-time theories, QCD at finite baryon density, and nonrelativistic bosons, amongst others, however, have complex actions and, therefore, complex weights in their path integrals. This forbids a probabilistic interpretation of the path integral measure. Moreover, this poses a big numerical challenge as oscillatory contributions from the generated configurations must cancel precisely in order to give accurate answers. This is known as the sign problem.
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In this work, we focus on the complex Langevin (CL) method. It is an extension of stochastic quantisation [1], where real dynamical variables are allowed to take complex values. After some early works [2,3], it was realised that the method was plagued by runaway solutions and convergence to wrong limits [4–7]. More recently, complex Langevin simulations experienced a revival [8–14], with studies focusing on understanding its properties and why it sometimes failed [15–17]. This progress led to the use of adaptive step size for the numerical integration [18], which has improved the numerical stability and reduced the problem of runaway solutions. In addition, criteria for correctness [19,20] that allow a posteriori checks were formulated. Another byproduct of the resurgence of the complex Langevin method was the invention of the gauge cooling technique [21,22], inspired by gauge fixing [10], to limit excursions on the complex mani
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