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Lefschetz thimble-inspired weight regularizations for complex Langevin simulations

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Pith's one-line read The paper claims that complex Langevin simulations converge correctly when the system has a single relevant compact Lefschetz thimble, and that a weight regularization plus Dyson-Schwinger bias correction can create that structure and…

desk verdict A genuinely useful bias-correction idea and the cleanest numerical evidence yet for the single-compact-thimble criterion, with a real but non-fatal formal gap around the non-holomorphic periodic regularization. read the letter →

arxiv 2412.02396 v2 pith:Y2YT2GSZ submitted 2024-12-03 hep-lat hep-ph

Kirill Boguslavski , Paul Hotzy , David I. Müller This is my paper · ORCID
classification hep-lathep-ph
keywords complexLangevinsignproblemLefschetzthimblesweightregularizationDyson-SchwingerequationsbiascorrectionPolyakovchainstochasticquantization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish a working rule for when Complex Langevin (CL), a stochastic cure for the numerical sign problem, can be trusted: in every model they test, CL converges correctly when the theory's Lefschetz thimble structure reduces to one relevant compact thimble, and fails when it does not. The authors turn this rule into a method by adding a weight regularization that reshapes the thimbles into that single compact form, then remove the regularization bias with Dyson-Schwinger equations. This restores correct expectation values in the complex cosine model and in SU(2) and SU(3) Polyakov chains at couplings where direct CL fails. The paper also warns that additive regularizations are unlikely to scale to full lattice field theories, so the lasting value is the thimble-based design principle rather than the specific regularization.

What carries the argument

The load-bearing object is the Lefschetz thimble: for a critical point $z_\sigma$ of the action, the thimble is the contour of steepest ascent along which the imaginary part of the action stays constant, and it is called compact when it does not escape to imaginary infinity but terminates at zeros or singularities of the weight. The regularization is the additive modification $\tilde{\rho}(z)=\rho(z)+r\,G(z)+R_0$, with $G$ positive on the real integration domain and chosen so that the regularized weight has exactly one relevant compact thimble for sufficiently large $|r|$. The bias it introduces is removed through the affine identity $\langle O\rangle_\rho = \langle O\rangle_{\tilde{\rho}} + Q\,(\langle O\rangle_{\tilde{\rho}} - \langle O\rangle_R)$, where the ratio $Q=Z_R/Z_\rho$ is extracted without computing the problematic partition function by using Dyson-Schwinger equations, which provide observables whose expectation values vanish in the original theory. Correctness of the regularized simulations is checked with the criterion that the distribution $p(u)$ of the drift magnitude decays faster than any power law.

What would settle it

Take the regularized cosine model of Eq. (34), change the periodic continuation so that the singularities of the effective action lie inside the integration box rather than on its boundary, and measure both the drift-magnitude distribution $p(u)$ and the expectation values against exact integration: a power-law tail in $p(u)$ together with correct expectation values, or exponential decay with wrong values, would show that the single-compact-thimble condition is neither necessary nor sufficient.

Watch

Extended reading notes

Core claim

The central claim, stated in Section 7, is that complex Langevin tends to converge correctly when the system has a single relevant, attractive critical point and the relevant thimble extending from it is compact; the authors offer this as a conjecture supported by all their simulations. They test it by manually regularizing the weight of systems that violate the condition, producing modified models whose thimble structure is single and compact, and then correcting for the modification. The correction works because the regularized expectation value is an affine function of the original one, and the only unknown, the ratio of the two partition functions, is fixed by the Dyson-Schwinger equations of the original theory. The numerical results show that the bias-corrected expectation values agree with exact integration for the complex cosine model and for SU(2) and SU(3) Polyakov chains in parameter regions where unregularized complex Langevin diverges or gives wrong values.

Load-bearing premise

The argument depends on the criterion that complex Langevin converges correctly whenever the distribution of drift magnitudes $p(u)$ decays faster than any power law, and on that criterion remaining valid when the regularized effective action is not holomorphic because the regularization is periodically continued; if that criterion is not sufficient in these non-holomorphic settings, the conclusion of correct convergence is not established.

Editorial extensions

If this is right

  • CL can be made to converge correctly in models where it previously failed by constructing a regularization whose thimble structure is a single compact relevant thimble and then bias-correcting with Dyson-Schwinger equations.
  • The bias-correction formula removes the old obstacle of computing the partition function of the original theory; only regularized expectation values and the ratio $Q$ from Dyson-Schwinger equations are needed.
  • A single compact relevant thimble can be used as a design principle: positive-definite regularization terms that vanish on the boundary of the integration domain, often tied to zeros of the Haar measure, produce such structures in gauge models.
  • Direct additive regularization for full lattice field theories is impractical because it becomes non-local or the bias correction requires exponentially many subset terms, so the practical route lies in kernels or multiplicative regularizations.
  • Multiple compact relevant thimbles are still not enough: the paper's Appendix A shows that CL fails even when all contributing thimbles are compact, strengthening the single-thimble condition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the single-compact-thimble condition holds beyond the toy models, it provides an a priori diagnostic: one could scan the thimble structure of a proposed kernel or regularization before running expensive CL simulations.
  • The DSE-based bias correction is not limited to the additive scheme; any modification whose expectation values relate affinely to the original theory, including multiplicative reweightings when overlap permits, could use the same $Q$-extraction strategy.
  • A testable next step is to apply the same construction to a genuinely higher-dimensional model, such as a complex $\phi^4$ theory in one spatial dimension, where the thimble structure can still be computed numerically, to see whether the compact-single-thimble criterion remains sufficient outside zero-dimensional integrals.
  • Because the paper's correctness checks rely on the Nagata criterion, which is trusted in the non-holomorphic periodic setting, the strongest version of the claim is only as solid as that criterion in that setting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes an additive weight-regularization scheme for complex Langevin (CL) simulations: an extra term R(z) is added to the original weight so that the regularized theory develops a single relevant compact Lefschetz thimble, which the authors argue restores correct CL convergence. A bias-correction formula, Eq. (26), relates expectation values of the regularized theory to those of the original theory through an observable-independent ratio Q, which is extracted using Dyson-Schwinger equations of the original theory. The method is applied to the complex cosine model and to SU(2) and SU(3) Polyakov chain models, with exact integration as a benchmark. The authors find that direct CL fails in the unregularized cases, that regularized CL reproduces the exact regularized expectation values, and that the bias-corrected values agree with the original exact values, typically at the sub-percent level. They also examine the Nagata et al. drift-magnitude criterion p(u) and report exponential decay for the regularized one-dimensional models.

Significance. If the results hold, the paper offers a constructive remedy for a known failure mode of CL and provides some of the clearest numerical evidence to date connecting thimble structure to CL convergence. The main technical novelty is the Dyson-Schwinger-based bias correction, which addresses a limitation of the earlier weight-regularization proposals in Refs. [68-70]. Strengths of the manuscript include exact benchmarks for every model, Fokker-Planck checks for the one-dimensional models, explicit p(u) diagnostics, and a candid discussion of limitations, including the argument in Appendix B that additive regularization is impractical for lattice field theories. The central conjecture, that a single relevant compact thimble is sufficient for correct convergence, is appropriately presented as numerical evidence rather than as a theorem. The main weaknesses are the uncritical use of a holomorphic correctness criterion for the non-holomorphic periodic regularization and an erroneous table in the SU(2) link-model results.

major comments (2)
  1. [Sec. 4, Eq. (35); Sec. 4.2, Fig. 5] The periodic continuation in Eq. (35) makes the regularized weight and hence the drift discontinuous across the lines Re z = ±π for Im z ≠ 0, not merely non-differentiable. The correctness criterion of Ref. [57] is stated in Sec. 2.2 for holomorphic actions, where boundary terms can be controlled through the decay of p(u). The paper applies this criterion to the non-holomorphic regularized cosine model without an argument that the criterion remains sufficient when the drift has such discontinuities. The observation that the boundary segments are repulsive is empirical and parameter-dependent. This matters because Section 7 and the abstract use the p(u) criterion as a principal justification that the regularized CL is correct. I recommend either supplying a boundary-flux or Fokker-Planck check for the discontinuous drift, or explicitly demoting the p(u) check to supportive evidence and relying on the exact-value comparisons of Fig. 4 and Table 2 as the primary correctness test.
  2. [Appendix E, Table 11] Table 11, titled "Bias-corrected results for the regularized SU(2) Polyakov chain link model," lists the exact values of the original theory as 0.275920, 0.268989, 0.142204, etc., which are identical to the regularized exact values in Table 10. The original-theory exact values from Table 9 are 0.339351, 0.212100, 0.147098, etc. As printed, the table contradicts the claim in Fig. 14 that the bias correction recovers the original theory, and it undermines one of the central numerical benchmarks. The exact column must be corrected, and the bias-corrected CL column should be checked to ensure it was not accidentally copied from Table 10.
minor comments (4)
  1. [Sec. 4, after Eq. (35)] The sentence "It appears that the solution seems to be unaffected by these results" should refer to "these discontinuities" rather than "these results," and the text should state explicitly that R is discontinuous, not merely non-differentiable, at the periodicity boundaries for nonzero imaginary part.
  2. [Sec. 4.2, Fig. 5] The dashed line ∝ u^{-2} is presented as a power-law reference, but no fitting range is given for the unregularized tail, and the regularized case is described only visually as exponential; please specify the range and, if possible, a fitted decay rate.
  3. [Sec. 6] For the SU(3) model the p(u) criterion is not shown; the conclusion of correct convergence there rests entirely on the exact-value comparisons in Tables 14-15. This is acceptable, but it should be stated explicitly so the reader does not infer that the criterion was checked in the non-Abelian case.
  4. [Sec. 7 and Abstract] The statement that CL converges correctly when a single relevant compact thimble contributes should be qualified as holding for the models and parameter sets studied, since the paper itself presents this as an empirical conjecture rather than a proven criterion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bias correction is an exact DSE-based identity with external exact-value checks; the single-thimble claim is an explicitly conjectural numerical correlation, not a derived input.

full rationale

The paper's central chain is not circular. The regularized expectation values are defined by Eq. (21), and the bias-correction formula Eq. (26) is an exact algebraic identity expressing the original expectation value in terms of regularized expectation values and an observable-independent ratio Q = Z_R/Z_rho. Q is not fitted to the predicted observables: Section 3.1 constructs O* from the Dyson-Schwinger equations of the original action (Eqs. (30)-(31)) and solves Eq. (29); the same Q is then applied to all powers of the observable, and the results are checked against independently computed exact values (Figs. 4, 10, 14, 16, 17 and the corresponding tables). The claim that CL converges correctly under a single compact relevant thimble is presented as a conjecture ('Our numerical studies suggest that the complex Langevin process tends to correctly converge if there exists a single relevant, attractive critical point and if the relevant thimble extending from the relevant critical point is compact'), not as a theorem derived from the definition of the thimble. The regularization parameter r is manually chosen to realize the desired thimble structure and is not fitted to target observables. The p(u) criterion of Nagata et al. is an external diagnostic; the paper's extension to the non-holomorphic periodic regularization of Eq. (35) is acknowledged as a limitation ('Imposing periodicity in Eq. (34) leads to a non-holomorphic effective action... It appears that the solution seems to be unaffected by these results'), which is a correctness risk rather than a circular reduction. Self-citations [52,53] appear only as background references on kernels and are not load-bearing for the regularization or bias-correction argument.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper's constructive contribution rests on hand-chosen regularizations (r, R0) and on a set of domain assumptions about the correctness criterion and DSE validity. No new physical entities are postulated; the regularization functions G(z) and R[U] are mathematical constructs.

free parameters (5)
  • Regularization strength r = 0.5 (cosine), -5 (SU(2)), -25 (SU(3))
    Chosen by hand to make the regularized weight have a single compact relevant thimble while keeping the bias-correction denominator in Eq. (29) away from zero. Not fitted to target observables.
  • Regularization offset R0 = -exp(i beta) in Eq. (34); implicit in Eq. (43)
    Chosen to place zeros of the regularized weight at the integration boundaries.
  • Adaptive step-size parameters = epsilon=1e-5 (cosine), 1e-4 (gauge); alpha=1e-2; max drift 1e3
    Numerical parameters used to stabilize the CL integration near singularities.
  • Cutoffs for unregularized divergent simulations = |z|<6pi, |Tr[P]|<8 (SU(2)), |Tr[P]|<9 (SU(3))
    Imposed to obtain finite results in cases where CL diverges; the resulting numbers are explicitly shown to be wrong.
  • Number of trajectories and thermalization = 2^16 processes (cosine), 10^4 (SU(2)/SU(3)), theta_therm=100, samples at Delta theta=1 or 0.1
    Statistical settings, not physical parameters.
assumptions (4)
  • domain assumption Exponential decay of the drift-magnitude distribution p(u) is sufficient for correct complex Langevin convergence [57].
    Used in Sec. 4.2 and Sec. 5.1 to diagnose correct vs. wrong convergence without relying on the observables.
  • standard math The action and observables can be extended holomorphically (or meromorphically) so that Picard-Lefschetz thimble decomposition applies.
    Assumed throughout Sec. 2.1; the cosine model regularization is non-holomorphic at the periodic boundaries (Sec. 4), a stated caveat.
  • domain assumption Dyson-Schwinger equations hold for the original theory with vanishing boundary terms.
    Used in Eq. (30) and for the observables O* in Eqs. (36), (45), (47), (58), (59). Requires the boundary term to vanish on the real integration domain.
  • domain assumption Gauge cooling and adaptive time stepping do not bias the CL process.
    Used in Secs. 5.2 and 6, citing previous work [73]; the paper notes gauge cooling cannot fully fix complexified gauge freedom.

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Cite this review

Pith. "Pith review of Lefschetz thimble-inspired weight regularizations for complex Langevin simulations." pith.science (2026). https://pith.science/paper/Y2YT2GSZ

@misc{pith2026241202396,
  author       = {Pith},
  title        = {Pith review of: Lefschetz thimble-inspired weight regularizations for complex Langevin simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2YT2GSZ}},
  note         = {Machine review of arXiv:2412.02396}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2412.02396 by the authors.

Figure 1
Figure 1. Left: Thimble structure of the cosine model with S(z) = iβ cos(z) for real￾valued couplings β. Critical points at zσ = 0,±π,±2π, . . . are shown as black tri￾angles. Extending from these critical points, the (anti-)thimbles are shown as black solid (dashed) curves. The arrows represent the direction of the Langevin drift −∂ S, and the background visualizes the absolute value of the drift. The cosine model ex￾hibits … view at source ↗
Figure 2
Figure 2. Histogram on a logarithmic scale of a complex Langevin process for the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Left panel: Thimble structure of the regularized cosine model with β = 0.5 and r = 0.5. As in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Relative deviation of expectation values of [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The density p(u) of the drift magnitude u from Eq. (19) is shown for both the original (blue) and regularized (red) cosine models at β = 0.5. The original model exhibits a power law decay (indicated by the orange dashed line), which vio￾lates the convergence criterion,…
Figure 6
Figure 6. Figure 6: Thimble structure of the reduced SU(2) Polyakov chain model for the cou [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Relative deviation of expectation values of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Thimble structures (left) and the corresponding CL histograms (right) of the reduced SU(2) Polyakov chain model for the coupling β = (1+i p 3)/2. The same conventions as in Figs. 1 and 3 are used. The relevant thimbles of the unregularized formulation are non-compact l…
Figure 9
Figure 9. Figure 9: Thimble structure of the regularization term [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Relative deviation of expectation values of [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: The density p(u) of the drift magnitude u from Eq. (19) for the reduced SU(2) Polyakov chain model for the couplings β1 = (1+ p 3i)/4 and β2 = (1+ p 3i)/2. For β1 (gray) the density decays exponentially fast, which confirms the criterion for correct convergence. In co…
Figure 12
Figure 12. Figure 12: CL histograms with respect to the real and imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Relative deviation of expectation values of [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Relative deviation of expectation values of [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Normalized histograms of the trace of the Polyakov loop ( [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Relative deviation of expectation values for Tr [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Relative deviation of corrected expectation values for the real ( [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: Thimble structures (left) and the density of the drift magnitude p(u;θ → ∞) (right) of the regularized, reduced SU(2) Polyakov chain model for the coupling β = (1 + i p 3)/2 and regularization force r = 0.5. The considered model is regularized using R(φ) = r sin2 (φ).…

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Forward citations

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Reference graph

Works this paper leans on

76 extracted references · 21 canonical work pages · cited by 4 Pith papers

  1. [57]

    Nagata, J

    K. Nagata, J. Nishimura and S. Shimasaki, Argument for justification of the complex Langevin method and the condition for correct convergence, Phys. Rev. D 94(11), 114515 (2016), doi:10.1103 /PhysRevD.94.114515, 1606.07627. 40 REFERENCES REFERENCES

    arXiv 2016

  2. [1]

    de Forcrand, Simulating QCD at finite density , PoS LAT2009, 010 (2009), doi:10.22323/1.091.0010, 1005.0539

    P . de Forcrand, Simulating QCD at finite density , PoS LAT2009, 010 (2009), doi:10.22323/1.091.0010, 1005.0539

    arXiv 2009

  3. [2]

    M. C. Bañuls and K. Cichy , Review on Novel Methods for Lattice Gauge Theories , Rept. Prog. Phys. 83(2), 024401 (2020), doi:10.1088 /1361-6633/ab6311, 1910.00257

    arXiv 2020

  4. [3]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations , Phys. Rev. Lett. 94, 170201 (2005), doi:10.1103/PhysRevLett.94.170201, cond-mat/0408370

    arXiv 2005

  5. [4]

    Fodor and S

    Z. Fodor and S. D. Katz,Critical point of QCD at finite T and mu, lattice results for physical quark masses, JHEP 04, 050 (2004), doi:10.1088 /1126-6708/2004/04/050, hep-lat/ 0402006

    work page 2004

  6. [5]

    C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F . Karsch, E. Laermann, C. Schmidt and L. Scorzato, The QCD thermal phase transition in the presence of a small chemical potential, Phys. Rev. D66, 074507 (2002), doi:10.1103 /PhysRevD.66.074507, hep-lat/0204010

    arXiv 2002

  7. [6]

    R. V . Gavai and S. Gupta,Pressure and nonlinear susceptibilities in QCD at finite chemical potentials, Phys. Rev. D68, 034506 (2003), doi:10.1103/PhysRevD.68.034506, hep-lat/ 0303013

    work page doi:10.1103/physrevd.68.034506 2003

  8. [7]

    M. G. Alford, A. Kapustin and F . Wilczek, Imaginary chemical potential and finite fermion density on the lattice , Phys. Rev. D 59, 054502 (1999), doi:10.1103/PhysRevD.59.054502, hep-lat/9807039

    arXiv 1999

Show all 76 references
  1. [8]

    D’Elia and M.-P

    M. D’Elia and M.-P . Lombardo,Finite density QCD via imaginary chemical potential, Phys. Rev. D67, 014505 (2003), doi:10.1103 /PhysRevD.67.014505, hep-lat/0209146

    2003 arXiv

  2. [9]

    Bellwied, S

    R. Bellwied, S. Borsanyi, Z. Fodor, J. Günther, S. D. Katz, C. Ratti and K. K. Szabo, The QCD phase diagram from analytic continuation , Phys. Lett. B 751, 559 (2015), doi:10.1016/j.physletb.2015.11.011, 1507.07510

    2015 arXiv

  3. [10]

    Hasenfratz and D

    A. Hasenfratz and D. Toussaint, Canonical ensembles and nonzero density quantum chro- modynamics, Nucl. Phys. B 371, 539 (1992), doi:10.1016 /0550-3213(92)90247-9. 37 REFERENCES REFERENCES

    1992

  4. [11]

    Alexandru, M

    A. Alexandru, M. Faber, I. Horvath and K.-F . Liu, Lattice QCD at finite density via a new canonical approach, Phys. Rev. D72, 114513 (2005), doi:10.1103/PhysRevD.72.114513, hep-lat/0507020

    2005 arXiv

  5. [12]

    H. B. Meyer, A Calculation of the bulk viscosity in SU(3) gluodynamics , Phys. Rev. Lett. 100, 162001 (2008), doi:10.1103 /PhysRevLett.100.162001, 0710.3717

    2008 arXiv

  6. [13]

    H. B. Meyer, A Calculation of the shear viscosity in SU(3) gluodynamics, Phys. Rev. D 76, 101701 (2007), doi:10.1103 /PhysRevD.76.101701, 0704.1801

    2007 arXiv

  7. [14]

    Lawrence, Model-free spectral reconstruction via Lagrange duality(2024), 2408.11766

    S. Lawrence, Model-free spectral reconstruction via Lagrange duality(2024), 2408.11766

    2024 arXiv

  8. [15]

    Chandrasekharan and U.-J

    S. Chandrasekharan and U.-J. Wiese, Meron cluster solution of a fermion sign prob- lem, Phys. Rev. Lett. 83, 3116 (1999), doi:10.1103 /PhysRevLett.83.3116, cond-mat/ 9902128

    1999

  9. [16]

    Pichler, M

    T . Pichler, M. Dalmonte, E. Rico, P . Zoller and S. Montangero, Real-time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks , Phys. Rev. X 6(1), 011023 (2016), doi:10.1103/PhysRevX.6.011023, 1505.04440

    2016 arXiv

  10. [17]

    M. C. Bañuls et al., Simulating Lattice Gauge Theories within Quantum Technologies, Eur. Phys. J. D 74(8), 165 (2020), doi:10.1140 /epjd/e2020-100571-8, 1911.00003

    2020 arXiv

  11. [18]

    N. Klco, J. R. Stryker and M. J. Savage, SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers , Phys. Rev. D 101(7), 074512 (2020), doi:10.1103/PhysRevD.101.074512, 1908.06935

    2020 arXiv

  12. [19]

    C. W . Baueret al., Quantum Simulation for High-Energy Physics , PRX Quantum 4(2), 027001 (2023), doi:10.1103 /PRXQuantum.4.027001, 2204.03381

    2023 arXiv

  13. [20]

    Fodor, S

    Z. Fodor, S. D. Katz and C. Schmidt, The Density of states method at non-zero chemi- cal potential, JHEP 03, 121 (2007), doi:10.1088 /1126-6708/2007/03/121, hep-lat/ 0701022

    2007

  14. [21]

    Langfeld, B

    K. Langfeld, B. Lucini and A. Rago,The density of states in gauge theories, Phys. Rev. Lett. 109, 111601 (2012), doi:10.1103 /PhysRevLett.109.111601, 1204.3243

    2012 arXiv

  15. [22]

    Witten, Analytic Continuation Of Chern-Simons Theory, AMS /IP Stud

    E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS /IP Stud. Adv. Math. 50, 347 (2011), 1001.2933

    2011 arXiv

  16. [23]

    Cristoforetti, F

    M. Cristoforetti, F . Di Renzo and L. Scorzato,New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble, Phys. Rev. D 86, 074506 (2012), doi:10.1103/PhysRevD.86.074506, 1205.3996

    2012 arXiv

  17. [24]

    Di Renzo, S

    F . Di Renzo, S. Singh and K. Zambello,Taylor expansions on Lefschetz thimbles, Phys. Rev. D 103(3), 034513 (2021), doi:10.1103 /PhysRevD.103.034513, 2008.01622

    2021 arXiv

  18. [25]

    Alexandru, G

    A. Alexandru, G. Basar, P . F . Bedaque and N. C. Warrington,Complex paths around the sign problem, Rev. Mod. Phys.94(1), 015006 (2022), doi:10.1103/RevModPhys.94.015006, 2007.05436

    2022 arXiv

  19. [26]

    J. M. Pawlowski, M. Scherzer, C. Schmidt, F . P . G. Ziegler and F . Ziesché, Simulat- ing Yang-Mills theories with a complex coupling , Phys. Rev. D 103(9), 094505 (2021), doi:10.1103/PhysRevD.103.094505, 2101.03938

    2021 arXiv

  20. [27]

    Parisi, ON COMPLEX PROBABILITIES , Phys

    G. Parisi, ON COMPLEX PROBABILITIES , Phys. Lett. B 131, 393 (1983), doi:10.1016/0370-2693(83)90525-7. 38 REFERENCES REFERENCES

    1983 doi

  21. [28]

    Aarts, F

    G. Aarts, F . A. James, E. Seiler and I.-O. Stamatescu,Complex Langevin: Etiology and Di- agnostics of its Main Problem, Eur. Phys. J. C71, 1756 (2011), doi:10.1140/epjc/s10052- 011-1756-5, 1101.3270

    2011 arXiv

  22. [29]

    Abe and K

    Y. Abe and K. Fukushima, Analytic studies of the complex Langevin equation with a Gaus- sian ansatz and multiple solutions in the unstable region , Phys. Rev. D 94(9), 094506 (2016), doi:10.1103 /PhysRevD.94.094506, 1607.05436

    2016 arXiv

  23. [30]

    Nishimura and S

    J. Nishimura and S. Shimasaki, Combining the complex Langevin method and the gener- alized Lefschetz-thimble method, JHEP 06, 023 (2017), doi:10.1007/JHEP06(2017)023, 1703.09409

    2017 arXiv

  24. [31]

    Bluecher, J

    S. Bluecher, J. M. Pawlowski, M. Scherzer, M. Schlosser, I.-O. Stamatescu, S. Syrkowski and F . P . G. Ziegler, Reweighting Lefschetz Thimbles , SciPost Phys. 5(5), 044 (2018), doi:10.21468/SciPostPhys.5.5.044, 1803.08418

    2018 arXiv

  25. [32]

    Alexandru, G

    A. Alexandru, G. Basar and P . Bedaque, Monte Carlo algorithm for simu- lating fermions on Lefschetz thimbles , Phys. Rev. D 93(1), 014504 (2016), doi:10.1103/PhysRevD.93.014504, 1510.03258

    2016 arXiv

  26. [33]

    Alexandru, G

    A. Alexandru, G. Basar, P . F . Bedaque, G. W . Ridgway and N. C. Warrington,Monte Carlo calculations of the finite density Thirring model , Phys. Rev. D 95(1), 014502 (2017), doi:10.1103/PhysRevD.95.014502, 1609.01730

    2017 arXiv

  27. [34]

    Alexandru, G

    A. Alexandru, G. Basar, P . F . Bedaque, S. Vartak and N. C. Warrington, Monte Carlo Study of Real Time Dynamics on the Lattice , Phys. Rev. Lett. 117(8), 081602 (2016), doi:10.1103/PhysRevLett.117.081602, 1605.08040

    2016 arXiv

  28. [35]

    Zambello and F

    K. Zambello and F . Di Renzo, Towards Lefschetz thimbles regularization of heavy-dense QCD, PoS LATTICE2018, 148 (2018), doi:10.22323 /1.334.0148, 1811.03605

    2018 arXiv

  29. [36]

    Di Renzo, S

    F . Di Renzo, S. Singh and K. Zambello, Thimble regularisation of YM fields: crunching a hard problem, PoS LATTICE2021, 233 (2022), doi:10.22323 /1.396.0233, 2112.00062

    2022

  30. [37]

    Parisi and Y.-s

    G. Parisi and Y.-s. Wu, Perturbation Theory Without Gauge Fixing , Sci. Sin. 24, 483 (1981)

    1981

  31. [38]

    Berges and I

    J. Berges and I. O. Stamatescu, Simulating nonequilibrium quantum fields with stochastic quantization techniques , Phys. Rev. Lett. 95, 202003 (2005), doi:10.1103/PhysRevLett.95.202003, hep-lat/0508030

    2005 arXiv

  32. [39]

    Alvestad, R

    D. Alvestad, R. Larsen and A. Rothkopf, Stable solvers for real-time Complex Langevin , JHEP 08, 138 (2021), doi:10.1007 /JHEP08(2021)138, 2105.02735

    2021 arXiv

  33. [40]

    Alvestad, R

    D. Alvestad, R. Larsen and A. Rothkopf, Towards learning optimized kernels for complex Langevin, JHEP 04, 057 (2023), doi:10.1007 /JHEP04(2023)057, 2211.15625

    2023 arXiv

  34. [41]

    N. M. Lampl and D. Sexty , Real time evolution of scalar fields with kernelled Complex Langevin equation (2023), 2309.06103

    2023 arXiv

  35. [42]

    Alvestad, A

    D. Alvestad, A. Rothkopf and D. Sexty ,Lattice real-time simulations with learned optimal kernels, Phys. Rev. D 109(3), L031502 (2024), doi:10.1103 /PhysRevD.109.L031502, 2310.08053

    2024 arXiv

  36. [43]

    Aarts, Can stochastic quantization evade the sign problem? The relativis- tic Bose gas at finite chemical potential , Phys

    G. Aarts, Can stochastic quantization evade the sign problem? The relativis- tic Bose gas at finite chemical potential , Phys. Rev. Lett. 102, 131601 (2009), doi:10.1103/PhysRevLett.102.131601, 0810.2089. 39 REFERENCES REFERENCES

    2009 arXiv

  37. [44]

    Aarts, Complex Langevin dynamics at finite chemical potential: Mean field analysis in the relativistic Bose gas, JHEP 05, 052 (2009), doi:10.1088 /1126-6708/2009/05/052, 0902.4686

    G. Aarts, Complex Langevin dynamics at finite chemical potential: Mean field analysis in the relativistic Bose gas, JHEP 05, 052 (2009), doi:10.1088 /1126-6708/2009/05/052, 0902.4686

    2009 arXiv

  38. [45]

    Aarts, F

    G. Aarts, F . Attanasio, B. Jäger and D. Sexty , The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics , JHEP 09, 087 (2016), doi:10.1007/JHEP09(2016)087, 1606.05561

    2016 arXiv

  39. [46]

    Sexty , Simulating full QCD at nonzero density using the complex Langevin equation , Phys

    D. Sexty , Simulating full QCD at nonzero density using the complex Langevin equation , Phys. Lett. B 729, 108 (2014), doi:10.1016 /j.physletb.2014.01.019, 1307.7748

    2014 arXiv

  40. [47]

    Fodor, S

    Z. Fodor, S. D. Katz, D. Sexty and C. Török, Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: A comparison with multiparameter reweighting, Phys. Rev. D92(9), 094516 (2015), doi:10.1103 /PhysRevD.92.094516, 1508.05260

    2015 arXiv

  41. [48]

    J. B. Kogut and D. K. Sinclair,Applying Complex Langevin Simulations to Lattice QCD at Fi- nite Density, Phys. Rev. D 100(5), 054512 (2019), doi:10.1103/PhysRevD.100.054512, 1903.02622

    2019 arXiv

  42. [49]

    Attanasio, B

    F . Attanasio, B. Jäger and F . P . G. Ziegler,QCD equation of state via the complex Langevin method (2022), 2203.13144

    2022 arXiv

  43. [50]

    Berges, S

    J. Berges, S. Borsanyi, D. Sexty and I. O. Stamatescu, Lattice simulations of real-time quantum fields, Phys. Rev. D 75, 045007 (2007), doi:10.1103 /PhysRevD.75.045007, hep-lat/0609058

    2007 arXiv

  44. [51]

    Berges and D

    J. Berges and D. Sexty , Real-time gauge theory simulations from stochas- tic quantization with optimized updating , Nucl. Phys. B 799, 306 (2008), doi:10.1016/j.nuclphysb.2008.01.018, 0708.0779

    2008 arXiv

  45. [52]

    Boguslavski, P

    K. Boguslavski, P . Hotzy and D. I. Müller, Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel , JHEP 06, 011 (2023), doi:10.1007/JHEP06(2023)011, 2212.08602

    2023 arXiv

  46. [53]

    Boguslavski, P

    K. Boguslavski, P . Hotzy and D. I. Müller, Real-time correlators in 3 +1D thermal lattice gauge theory, Phys. Rev. D109(9), 094518 (2024), doi:10.1103/PhysRevD.109.094518, 2312.03063

    2024 arXiv

  47. [54]

    Aarts, E

    G. Aarts, E. Seiler and I.-O. Stamatescu, The Complex Langevin method: When can it be trusted?, Phys. Rev. D 81, 054508 (2010), doi:10.1103 /PhysRevD.81.054508, 0912. 3360

    2010

  48. [55]

    Aarts, P

    G. Aarts, P . Giudice and E. Seiler,Localised distributions and criteria for correctness in com- plex Langevin dynamics, Annals Phys. 337, 238 (2013), doi:10.1016/j.aop.2013.06.019, 1306.3075

    2013 arXiv

  49. [56]

    Nagata, J

    K. Nagata, J. Nishimura and S. Shimasaki, Justification of the complex Langevin method with the gauge cooling procedure , PTEP 2016(1), 013B01 (2016), doi:10.1093/ptep/ptv173, 1508.02377

    2016 arXiv

  50. [58]

    Scherzer, E

    M. Scherzer, E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin and boundary terms, Phys. Rev. D 99(1), 014512 (2019), doi:10.1103 /PhysRevD.99.014512, 1808. 05187

    2019

  51. [59]

    Scherzer, E

    M. Scherzer, E. Seiler, D. Sexty and I. O. Stamatescu,Controlling Complex Langevin simu- lations of lattice models by boundary term analysis, Phys. Rev. D 101(1), 014501 (2020), doi:10.1103/PhysRevD.101.014501, 1910.09427

    2020 arXiv

  52. [60]

    Seiler, D

    E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin: Correctness cri- teria, boundary terms, and spectrum , Phys. Rev. D 109(1), 014509 (2024), doi:10.1103/PhysRevD.109.014509, 2304.00563

    2024 arXiv

  53. [61]

    Okamoto, K

    H. Okamoto, K. Okano, L. Schulke and S. Tanaka,The Role of a Kernel in Complex Langevin Systems, Nucl. Phys. B 324, 684 (1989), doi:10.1016 /0550-3213(89)90526-9

    1989

  54. [62]

    Okano, L

    K. Okano, L. Schulke and B. Zheng, Kernel controlled complex Langevin simulation: Field dependent kernel, Phys. Lett. B 258, 421 (1991), doi:10.1016 /0370-2693(91)91111-8

    1991

  55. [63]

    Aarts, F

    G. Aarts, F . A. James, J. M. Pawlowski, E. Seiler, D. Sexty and I.-O. Stamatescu, Stability of complex Langevin dynamics in effective models , JHEP 03, 073 (2013), doi:10.1007/JHEP03(2013)073, 1212.5231

    2013 arXiv

  56. [64]

    Aarts, Lefschetz thimbles and stochastic quantization: Complex actions in the complex plane, Phys

    G. Aarts, Lefschetz thimbles and stochastic quantization: Complex actions in the complex plane, Phys. Rev. D 88(9), 094501 (2013), doi:10.1103 /PhysRevD.88.094501, 1308. 4811

    2013

  57. [65]

    Aarts, L

    G. Aarts, L. Bongiovanni, E. Seiler and D. Sexty , Some remarks on Lefschetz thimbles and complex Langevin dynamics, JHEP 10, 159 (2014), doi:10.1007/JHEP10(2014)159, 1407.2090

    2014 arXiv

  58. [66]

    Hayata, Y

    T . Hayata, Y. Hidaka and Y. Tanizaki, Complex saddle points and the sign problem in complex Langevin simulation , Nucl. Phys. B 911, 94 (2016), doi:10.1016/j.nuclphysb.2016.07.031, 1511.02437

    2016 arXiv

  59. [67]

    L. L. Salcedo, Does the complex Langevin method give unbiased results? , Phys. Rev. D 94(11), 114505 (2016), doi:10.1103 /PhysRevD.94.114505, 1611.06390

    2016 arXiv

  60. [68]

    Tsutsui and T

    S. Tsutsui and T . M. Doi, Improvement in complex Langevin dynamics from a view point of Lefschetz thimbles , Phys. Rev. D 94(7), 074009 (2016), doi:10.1103/PhysRevD.94.074009, 1508.04231

    2016 arXiv

  61. [69]

    Tsutsui and T

    S. Tsutsui and T . M. Doi,On a modification method of Lefschetz thimbles, EPJ Web Conf. 175, 11016 (2018), doi:10.1051 /epjconf/201817511016, 1710.06553

    2018

  62. [70]

    T . M. Doi and S. Tsutsui, Modifying partition functions: a way to solve the sign problem , Phys. Rev. D96(9), 094511 (2017), doi:10.1103 /PhysRevD.96.094511, 1709.05806

    2017 arXiv

  63. [71]

    Risken and T

    H. Risken and T . Frank, The Fokker-Planck Equation: Methods of Solution and Applica- tions, Springer Series in Synergetics. Springer Berlin Heidelberg, ISBN 9783540615309 (1996)

    1996

  64. [72]

    Chang and G

    J. Chang and G. Cooper, A practical difference scheme for fokker-planck equations , Journal of Computational Physics 6(1), 1 (1970), doi:https: //doi.org/10.1016/0021- 9991(70)90001-X. 41 REFERENCES REFERENCES

    1970 doi

  65. [73]

    Aarts, L

    G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty and I.-O. Stamatescu, Control- ling complex Langevin dynamics at finite density , Eur. Phys. J. A 49, 89 (2013), doi:10.1140/epja/i2013-13089-4, 1303.6425

    2013 arXiv

  66. [74]

    Montvay and G

    I. Montvay and G. Munster, Quantum fields on a lattice , Cambridge Monographs on Mathematical Physics. Cambridge University Press, ISBN 978-0-521-59917-7, 978-0- 511-87919-7, doi:10.1017 /CBO9780511470783 (1997)

    1997

  67. [75]

    Aarts and I.-O

    G. Aarts and I.-O. Stamatescu, Stochastic quantization at finite chemical potential, JHEP 09, 018 (2008), doi:10.1088 /1126-6708/2008/09/018, 0807.1597

    2008 arXiv

  68. [76]

    Mandl, M

    M. Mandl, M. W . Hansen, E. Seiler and D. Sexty ,Kernels and integration cycles in complex Langevin simulations, In 41st International Symposium on Lattice Field Theory (2024), 2410.13423. 42

    2024 arXiv

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