Complex Langevin Research Articles

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; nonconvergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here, we propose a class of criteria that allow one 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. Published by the American Physical Society 2024

Assessment of the partial saddle point approximation in field-theoretic polymer simulations.

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "signproblem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the signproblem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the signproblem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.

Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel

The complex Langevin (CL) method is a promising approach to overcome the sign problem that occurs in real-time formulations of quantum field theories. Using the Schwinger-Keldysh formalism, we study SU(Nc) gauge theories with CL. We observe that current stabilization techniques are insufficient to obtain correct results. Therefore, we revise the discretization of the CL equations on complex time contours, find a time reflection symmetric formulation and introduce a novel anisotropic kernel that enables CL simulations on discretized complex time paths. Applying it to SU(2) Yang-Mills theory in 3+1 dimensions, we obtain unprecedentedly stable results that we validate using additional observables and that can be systematically improved. For the first time, we are able to simulate non-Abelian gauge theory on time contours whose real-time extent exceeds its inverse temperature. Thus, our approach may pave the way towards an ab-initio real-time framework of QCD in and out of equilibrium with a potentially large impact on the phenomenology of heavy-ion collisions.

Complex Langevin approach to interacting Bose gases

Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to standard Monte Carlo techniques, simulations beyond their scope almost exclusively rely on quantum-mechanical techniques, which are inherently limited to small particle numbers. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional non-relativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent/-species models.

Field-Theoretic Simulation Method to Study the Liquid-Liquid Phase Separation of Polymers.

Liquid-liquid phase separation (LLPS) is a process that results in the formation of a polymer-rich liquid phase coexisting with a polymer-depleted liquid phase. LLPS plays a critical role in the cell through the formation of membrane-less organelles, but it also has a number of biotechnical and biomedical applications such as drug confinement and its targeted delivery. In this chapter, we present a computational efficient methodology that uses field-theoretic simulations (FTS) with complex Langevin (CL) sampling to characterize polymer phase behavior and delineate the LLPS phase boundaries. This approach is a powerful complement to analytical and explicit-particle simulations, and it can serve to inform experimental LLPS studies. The strength of the method lies in its ability to properly sample a large ensemble of polymers in a saturated solution while including the effect of composition fluctuations on LLPS. We describe the approaches that can be used to accurately construct phase diagrams of a variety of molecularly designed polymers and illustrate the method by generating an approximation-free phase diagram for a classical symmetric diblock polyampholyte.

Controlling complex Langevin simulations of lattice models by boundary term analysis

One reason for the well known fact that the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit has been identified long ago: it is 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, in a previous paper we have studied the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. Here we continue this type of analysis for more physically interesting models, focusing on the boundaries at infinity. We start with abelian and non-abelian one-plaquette models, then we proceed to a Polyakov chain model and finally to high density QCD (HDQCD) and the 3D XY model. We show that the direct estimation of the systematic error of the CL method using boundary terms is in principle possible.

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.

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 of us (L. L. S.) and explains a fact observed in nonergodic cases of CL.

Interacting bosons at finite angular momentum via complex langevin

Interacting bosons at finite angular momentum via complex langevin

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.

Field-Theoretic Simulations of Fluctuation-Stabilized Aperiodic “Bricks-and-Mortar” Mesophase in Miktoarm Star Block Copolymer/Homopolymer Blends

A new class of thermoplastic elastomers possessing unusual mechanical properties has recently been discovered in binary blends of A-b-(B-b-A′)n miktoarm star block copolymers and A homopolymers that spontaneously form an unusual, thermodynamically stable, aperiodic “bricks-and-mortar” (B&M) mesophase morphology. The B&M mesophase is believed to be stabilized by thermal fluctuations as in the well-known case of the bicontinuous microemulsion phase. Here, two-dimensional field-theoretic simulations are used to study the equilibrium self-assembly of such miktoarm polymer binary blends. As expected, the B&M mesophase is not present in the mean-field phase diagram obtained with self-consistent field theory, but complex Langevin (CL) simulations, which fully incorporate thermal fluctuation effects, reveal dramatic changes to the phase diagram. A region of strong fluctuations results in the emergent stabilization of the B&M mesophase in a broad composition channel positioned between microphase separation and mac...

Test for a universal behavior of Dirac eigenvalues in the complex Langevin method

We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at non-zero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for correct convergence case, while it follows the Ginibre ensemble for wrong convergence case. The Wigner surmise is physically reasonable from the chemical potential independence of the ChRMT. The Ginibre ensemble is known to be favored in a phase quenched QCD at finite chemical potential. Our result suggests a possibility that the originally universal behavior of the NNS distribution is preserved even in the CL method for correct convergence case.

Effects of thermal fluctuations on directed self-assembly in cylindrical confinement

We investigate the directed self-assembly (DSA) of cylinder-forming block copolymers inside cylindrical guiding templates. To complement and corroborate our experimental investigations, we use field-theoretic simulations to examine the fluctuation-induced variations in the size and position of the cylindrical microdomain that forms in the middle of the guiding hole. Our study goes beyond the usual mean-field approximation and self-consistent field theory simulations (SCFT) and incorporates the effects of thermal fluctuations in the description of the self-assembly process using complex Langevin (CL) dynamics. In addition to CL simulations, we present an efficient SCFT-based approach that can inform about the positional error of the formed cylinders. In this new scheme, an external chemical-potential field is applied to displace the inner cylinder away from its centered, lowest energy configuration. In both our experimental and modeling efforts, we focus on two wall-wetting conditions: (1) minor-block-attractive sidewalls and bottom substrates and neutral top surfaces and (2) neutral sidewalls, substrates, and top surfaces. For both cases, we explore the properties of the formed cylinders, including fluctuations in the center position and the size of the domain, for various prepattern conditions. Our results indicate robust critical dimensions (CDs) of the DSA cylinders relative to the prepattern CD, with a standard deviation <0.9 nm. Likewise, we find that the DSA cylinders are accurately registered in the center of the guiding hole, with deviations in the hole-in-hole distance on the order of ∼0.7 to 1.4 nm, translating to errors in the hole-to-hole distance of ∼1 to 2 nm.

A multi-species exchange model for fully fluctuating polymer field theory simulations.

Field-theoretic models have been used extensively to study the phase behavior of inhomogeneous polymer melts and solutions, both in self-consistent mean-field calculations and in numerical simulations of the full theory capturing composition fluctuations. The models commonly used can be grouped into two categories, namely, species models and exchange models. Species models involve integrations of functionals that explicitly depend on fields originating both from species density operators and their conjugate chemical potential fields. In contrast, exchange models retain only linear combinations of the chemical potential fields. In the two-component case, development of exchange models has been instrumental in enabling stable complex Langevin (CL) simulations of the full complex-valued theory. No comparable stable CL approach has yet been established for field theories of the species type. Here, we introduce an extension of the exchange model to an arbitrary number of components, namely, the multi-species exchange (MSE) model, which greatly expands the classes of soft material systems that can be accessed by the complex Langevin simulation technique. We demonstrate the stability and accuracy of the MSE-CL sampling approach using numerical simulations of triblock and tetrablock terpolymer melts, and tetrablock quaterpolymer melts. This method should enable studies of a wide range of fluctuation phenomena in multiblock/multi-species polymer blends and composites.

Coherent states formulation of polymer field theory

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first direct numerical simulations of the coherent-states (CS) formulation of polymer field theory. In contrast with Edwards' well-known auxiliary-field (AF) framework, the CS formulation does not contain an embedded nonlinear, non-local, implicit functional of the auxiliary fields, and the action of the field theory has a fully explicit, semi-local, and finite-order polynomial character. In the context of a polymer solution model, we demonstrate that the new CS-CL dynamical scheme for sampling fluctuations in the space of coherent states yields results in good agreement with now-standard AF-CL simulations. The formalism is potentially applicable to a broad range of polymer architectures and may facilitate systematic generation of trial actions for use in coarse-graining and numerical renormalization-group studies.

Comparison of Pseudospectral Algorithms for Field-Theoretic Simulations of Polymers

We compare three pseudospectral algorithms for mean-field polymer self-consistent field theory (SCFT) simulations and beyond mean-field field-theoretic simulations (FTS) using the complex Langevin (CL) sampling technique. In agreement with a study by Stasiak and Matsen, we find that for SCFT the fourth-order algorithm developed by Ranjan, Qin, and Morse usually outperforms the other pseudospectral algorithms. In contrast, for CL simulations we find that the second-order algorithm adapted to SCFT by Rasmussen and co-workers often outperforms the fourth-order methods not only in computational speed but also, surprisingly, in accuracy at a fixed contour resolution. This suggests an intricate coupling between pseudospectral schemes for chain contour integration and pseudotime stepping methods in CL simulations. Finally, the nominal stability of the algorithms is examined for the case of homogeneous fields.

Numerical Solutions of the Complex Langevin Equations in Polymer Field Theory

Using a diblock copolymer melt as a model system, we show that complex Langevin (CL) simulations constitute a practical method for sampling the complex weights in field theory models of polymeric f...

Field‐theoretic simulations of polyelectrolyte complexation

Field‐theoretic simulations of polyelectrolyte complexation

Complex Langevin equation and the many-fermion problem

We study the utility of a complex Langevin (CL) equation as an alternative for the Monte Carlo (MC) procedure in the evaluation of expectation values occurring in fermionic many-body problems. We find that a CL approach is natural in cases where non-positive definite probability measures occur, and remains accurate even when the corresponding MC calculation develops a severe ``sign problem''. While the convergence of CL averages cannot be guaranteed in principle, we show how convergent results can be obtained in three examples ranging from simple one-dimensional integrals over quantum mechanical models to a schematic shell model path integral.

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.