Complex Langevin dynamics and zeroes of the fermion determinant

Aarts, Gert; Seiler, Erhard; Sexty, Dénes; Stamatescu, Ion-Olimpiu

doi:10.1007/jhep05(2017)044

Cited by 84 publications

(66 citation statements)

References 58 publications

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“…This is a very challenging task owing to the resulting sign problem of doped systems which would lead to a complex Langevin. Such equations have been studied quite extensively in the context LGT 38 and recently applied to the one dimensional Hubbard model 39 and ultra-cold fermionic atoms with unequal masses 40 . The complex Langevin equation was also recently applied to the Holstein-Hubbard model 41 and shown to be very efficient in the parameter range U > g 2 /ω 0 .…”

Section: Discussionmentioning

confidence: 99%

Langevin simulations of a long-range electron-phonon model

Batrouni

Scalettar

2019

Phys. Rev. B

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We present a Quantum Monte Carlo (QMC) study, based on the Langevin equation, of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The bosonic part of the action helps control the variation of the field in imaginary time. As a consequence, the iterative conjugate gradient solution of the fermionic action, which depends on the boson coordinates, converges more rapidly than in the case of electron-electron interactions, such as the Hubbard Hamiltonian. Fourier Acceleration is shown to be a crucial ingredient in reducing the equilibration and autocorrelation times. After describing and benchmarking the method, we present results for the phase diagram focusing on the range of the electron-phonon interaction. We delineate the regions of charge density wave formation from those in which the fermion density is inhomogeneous, caused by phase separation. We show that the Langevin approach is more efficient than the Determinant QMC method for lattice sizes N 8 × 8 and that it therefore opens a potential path to problems including, for example, charge order in the 3D Holstein model.

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Section: Discussionmentioning

confidence: 99%

Langevin simulations of a long-range electron-phonon model

Batrouni

Scalettar

2019

Phys. Rev. B

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“…Simulations using the complex Langevin equation (CLE) [1][2][3][4] can accommodate such complex actions. However, the CLE can only be shown to yield correct values for observables if the space over which the fields evolve is compact, the drift (force) term is holomorphic in the fields, and the solutions are ergodic [5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Applying complex Langevin simulations to lattice QCD at finite density

Kogut

Sinclair

2019

Phys. Rev. D

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We study the use of the complex-Langevin equation (CLE) to simulate lattice QCD at a finite chemical potential (µ) for quark-number, which has a complex fermion determinant that prevents the use of standard simulation methods based on importance sampling. Recent enhancements to the CLE specific to lattice QCD inhibit runaway solutions which had foiled earlier attempts to use it for such simulations. However, it is not guaranteed to produce correct results. Our goal is to determine under what conditions the CLE yields correct values for the observables of interest. Zero temperature simulations indicate that for moderate couplings, good agreement with expected results is obtained for small µ and for µ large enough to reach saturation, and that this agreement improves as we go to weaker coupling.For intermediate µ values these simulations do not produce the correct physics. We compare our results with those of the phase-quenched approximation. Since there are indications that correct results might be obtained if the CLE trajectories remain close to the SU (3) manifold, we study how the distance from this manifold depends on the quark mass and on the coupling. We find that this distance decreases with decreasing quark mass and as the coupling decreases, i.e. as the simulations approach

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“…As in the case of Monte Carlo methods, O(z) CLM can be replaced with a long-time average of the observable calculated for the generated configurations z(t) if ergodicity holds for the Langevin time-evolution. The derivation of the equality (2.4) uses integration by parts, which can be justified if the distribution of z falls off fast enough in the imaginary directions [7,8] as well as near the singularities of the drift term if they exist [9,40]. Recently [10], a subtlety in the use of time-evolved observables in the original argument [7,8] was pointed out, and the derivation of the equality (2.4) has been refined taking account of this subtlety.…”

Section: Complex Langevin Methods (Clm)mentioning

confidence: 99%

“…While the gauge cooling technique [27] (See refs. [10,28] for its justification) has enlarged this range of applicability to the extent that finite density QCD either with heavy quarks [29][30][31] or in the deconfined phase [32,33] can now be investigated, it is not yet clear whether one can investigate it even in the confined phase with light quarks [34][35][36][37][38][39][40][41].…”

Section: Jhep06(2017)023mentioning

confidence: 99%

Combining the complex Langevin method and the generalized Lefschetz-thimble method

Nishimura

Shimasaki

2017

J. High Energ. Phys.

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Abstract:The complex Langevin method and the generalized Lefschetz-thimble method are two closely related approaches to the sign problem, which are both based on complexification of the original dynamical variables. The former can be viewed as a generalization of the stochastic quantization using the Langevin equation, whereas the latter is a deformation of the integration contour using the so-called holomorphic gradient flow. In order to clarify their relationship, we propose a formulation which combines the two methods by applying the former method to the real variables that parametrize the deformed integration contour in the latter method. Three versions, which differ in the treatment of the residual sign problem in the latter method, are considered. By applying them to a single-variable model, we find, in particular, that one of the versions interpolates the complex Langevin method and the original Lefschetz-thimble method.

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Complex Langevin dynamics and zeroes of the fermion determinant

Cited by 84 publications

References 58 publications

Langevin simulations of a long-range electron-phonon model

Langevin simulations of a long-range electron-phonon model

Applying complex Langevin simulations to lattice QCD at finite density

Combining the complex Langevin method and the generalized Lefschetz-thimble method

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