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Lefschetz thimble-inspired weight regularizations for complex Langevin simulations

by Kirill Boguslavski, Paul Hotzy, David I. Müller

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Paul Hotzy
Submission information
Preprint Link: https://arxiv.org/abs/2412.02396v1  (pdf)
Date submitted: 2024-12-08 10:34
Submitted by: Hotzy, Paul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
  • Nuclear Physics - Theory
Approach: Computational

Abstract

Complex Langevin (CL) is a computational method to circumvent the numerical sign problem with applications in finite-density quantum chromodynamics and the real-time dynamics of quantum field theories. It has long been known that, depending on the simulated system, CL does not always converge correctly. In this work, we provide numerical evidence that the success or failure of the complex Langevin method is deeply tied to the Lefschetz thimble structure of the simulated system. This is demonstrated by constructing weight function regularizations that deform the thimbles of systems with compact domains. Our results indicate that CL converges correctly when the regularized system exhibits a single relevant compact thimble. We introduce a bias correction to retrieve the values of the original theory for parameter sets where a direct complex Langevin approach fails. The effectiveness of this method is illustrated using several toy models, including the cosine model and the SU(2) and SU(3) Polyakov chains. Finally, we discuss the opportunities and limitations of this regularization approach for lattice field theories.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted

Reports on this Submission

Report #2 by Rasmus Larsen (Referee 1) on 2025-1-27 (Invited Report)

Strengths

1) Paper shows connection between correct convergence of Complex Langevin and the amount of Lefschetz thimbles contributing.

2) Provides a clear description of the regularization methods used and shows how to recover the correct results for the explored models.

3) Well written paper, with good descriptions of Complex Langevin and Lefschetz thimbles and a good summary of other techniques.

Weaknesses

1) It is uncertain how well the explored regularization will work in more complicated models with possibly hundreds of thimbles contributing.

Report

The paper deals with the sign problem, which is observed in many finite density and real time physical systems. The main part of the paper is to explore the problem with wrong convergence of Complex Langevin, which is a technique often used to solve the sign problem. The paper focuses on relatively simple models to show how fast enough convergence of the distribution and correct results are related to the amount of thimbles that contribute to the solution. The paper gives a range of physically motivated models, which are simple enough to be solved, but complex enough to give a good insight into the problems with wrong converge. The paper shows how regularization methods by introducing extra weights can be used to manipulate the convergence and thimble behavior of the models. The paper shows how the introduction of a strong enough regularization that puts the zero of the distribution on the real axis can be used to solve the explored models and how it affects the thimble structure.

The paper is well written and gives a good quick summary of the different techniques explored in the literature. The regularization methods and explored models are well described. The regulation method is solid, while the authors are honest in their explorations of the limitations of the method, as a too strong regularization can lead to too large statistical errors. I think the topics discussed in the paper are important and while the solved models are in many ways toy models, they represent well the level at which we currently are able to understand the convergence problems with Complex Langevin. I therefore consider the general acceptance criteria for the paper fulfilled.

I think the paper provides a good link between the convergence of Complex Langevin and the behavior of Lefschetz thimbles (Expectations 1). The paper shows how the 2 explored regularization methods recover the correct results and how this is related to the number of Lefschetz thimbles contributing (expectation 3). The paper builds on similar methods from a series of papers. The level of breakthrough of the paper can therefore be discussed. I do however consider the level of breakthrough in terms of showing the connection between correct convergence and Lefschetz thimbles to be an important one, while also recovering the correct solution to be a strong enough result for the paper to be published (expectation 4).

I have a few comments below, but would be happy to recommend the paper for publication if adequate response/corrections to the comments are made.

Regards Rasmus Larsen

Requested changes

Comments:

1)
below eq. 27
<O>_R = r <O>_G should be <O>_R = <O>_G as
<O>_R = int dx O*r*G / int dx r*G
so the r’s should cancel.

eq. 28 seems to agree with this assessment.

2)
Figure 4, 10 and 14 left. What are the different colors? Would guess Re and Im, but was not shown anywhere (besides the right figures).

3)
Is there a good reason for why r=-5 in section 5 and r=-25 in section 6? How were these values found? Why negative?

4)
There should be a minus for derivative of Tr[P^-1] in eq. 55 just as there correctly is for K_j in eq 52
as dTr(P^a)/dtau_j = i*Tr(a*tau_j*P^a) (ignoring the specific insertion position in the chain.)


Suggestions (feel free to ignore):
1s)
eq. 30 is correct but does come a bit out of nowhere. A simple comment about it arising from the invariance of of the integral under a shift in a variable x_0 and it is the derivative with respect to x0 of <O> which is zero, that generates these equations, would make it easier to follow.

Recommendation

Ask for minor revision

  • validity: top
  • significance: high
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: excellent

Report #1 by Alexander Rothkopf (Referee 2) on 2025-1-17 (Invited Report)

Strengths

1) proposes a new idea exploiting Dyson-Schwinger equations to improve the viability of an approach by Doi & Tsutsui to achieve correct convergence in complex Langevin simulations.

2) carefully examines the proposed idea in a range of community-standard benchmark scenarios beyond those in literature.

3) well written for practitioners in the field

Weaknesses

1) only a specific form of regularization (affine form) is used, which limits applicability.

2) while showcasing success of the idea of modified actions in relevant example cases, strategy for construction of modification is not easily generalizable.

Report

Simulating quantum theories with a sign problem remains one of the central open goals of modern theoretical physics, as our inability to do so precludes urgently needed progress in various fields, ranging from condensed matter physics to high energy nuclear physics. Complex Langevin (CL), the complexification of the successful stochastic quantisation paradigm, has demonstrated its ability to overcome the sign problem in various systems and parameter ranges but suffers from the well known problem to converge to incorrect solutions without easily to spot pathologies in the simulation result. Prior work over the past decade has unearthed a connection between CL and the analytic structure of the action of the underlying theory, its so-called Lefshetz thimbles. It was uncovered that naively formulated complex Langevin samples in preferred directions that do not necessarily agree with those of the thimbles. In addition the presence of more than a single dominating thimble may lead naive CL to fail to correctly capture the relevant physics by missing some of the thimble structure.

After the community has achieved a first milestone of making possible the aposteriori determination of correct convergence via e.g. the boundary term criteria, it is now paramount to go beyond the state of the art and use the insight gained to develop strategies to actively remedy the incorrect convergence problem.

The authors improve on prior work by Doi and Tsutsui, who proposed a method reminiscent (but qualitatively different) from reweighing. The original idea is to modify the system action such that instead of multiple Thimbles only a single dominant (and compact) Thimble remains that is sampled by a naively formulated CL simulation. Obviously one is not simulating the actual target system and the modification needs to be undone at some step. Doi and Tsutsui attempt to do so at the level of the observables, combining observables sampled with different modifications arranged such that the observable in the original system is recovered. Their method however encounters a step where they have to evaluate the inverse of the so-called reweighing factor, thus suffering from the same numerical difficulty as reweighing itself.

The current paper uses insight from an adjacent field, the study of field theory via Dyson-Schwinger equations (DSE) to make headway. The authors propose to exploit the fact that the DSE in the original theory, are violated in the modified system. I.e. they design new observables, which vanish in the original theory but become finite in the modified theory. They are thus used to extract the information necessary to undo the modification introduced by the modification of the action on the observable level. Their numerical experiments indicate that this approach is viable, as they demonstrate that they can recover to good precision the correct expectation values in several relevant benchmark model systems after simulation with a modified action.

I appreciate the authors' candid discussion of the prospects of the approach to more realistic systems, highlighting potential roadblocks. In a day and age of hyperbole it is good to see a careful assessment of potential difficulties, as well as opportunities.

By improving the regularization method proposed by Doi and Tsutsui, connecting it to the study of field theories in terms of DSE and exploring its use in community standard benchmark systems the paper fulfils the expectation for a publication in SciPost Physics. It is well written, provides sufficient detail for reproduction and features an extensive citation list, so that it also fulfils the general acceptance criteria.

Thus I recommend publication of the paper after the authors have implemented the requested changes given below.

Requested changes

1) p2 i.e. Euclidean field theories -> that can be formulated solely using Euclidean time.

2) p.3 "there is an absence of general, constructive methods to achieve reliable convergence". Since the authors remark in the introduction paragraph that the sign problem is NP hard, it follows that such a general constructive method cannot exist, otherwise P=NP.

3) p.4 perhaps it would be good for the reader unfamiliar with the work of Doi and Tsutsui to mention that one key limitation to their approach is that in their original formulation they need to compute the conventional reweighing factor and thus face the same difficulty as reweighing itself.

4) p.8 The quantity in (19) to investigate the spread of the distribution of CL (boundary terms) is described as observable-independent. I do not agree with this terminology, as (19) chooses one particular observable and different observables can exhibit different tail structures in histograms. The difficulty with the boundary criterion is that no observable may exhibit a tail structure for CL to converge correctly. One may find some, whose histogram decays quickly but there may be others which still exhibit tails.

5) p.10 After a back of the envelope calculation, I am unsure about eq. (28). How do the authors manage to cancel the <O>_\rho contribution from (24)? When subtracting <O>_r2 -<O>_r1 , the denominator comes with two different values of Q. After getting to a common denominator and moving that denominator to the LHS, the terms with <O>_\rho seem to not trivially cancel. What have I overlooked there?

6) p.12 Could the authors give a bit more details in how they arrive at choosing the particular regularization term. For the complex cosine model a similar regularization term was used by Doi and Tsutsui. Does this mean that this is the only viable regularization term or could one modify it?

7) p.13 (comment) While numerical evidence and intuitive arguments are given for the success of the method, it is worrying that the modification term proposed by the method leads to a non-holomorphic effective action. It puts the approach on a similar footing as the dynamical stabilisation approach, which too, while successful in practice does not allow a formal application of the proof of convergence.

8) p.15 "can arise without any numerical signatures in the observables" -> "can arise without any numerical pathologies in the observables".

9) p.19 reduced Polykov -> reduced Polyakov

10) A legend for the colour coding in the scatter plots is missing

11) In the conclusion, could the authors rephrase the first paragraph describing the content of the study. I do not see that the authors have investigated the criterion for correct convergence of CL but have used it to confirm that a proposed modification strategy reproduces the correct results in several model systems.

Recommendation

Ask for minor revision

  • validity: top
  • significance: good
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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