Relativistic Bose gas on a Lefschetz thimble

Cristoforetti, Marco; Renzo, Francesco Di; Mukherjee, Asok K.; Scorzato, L.

doi:10.22323/1.187.0197

Cited by 6 publications

(6 citation statements)

References 17 publications

(30 reference statements)

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“…We pointed out that the straightforward separation of phase leads to a sign problem that is, in a sense, infinitely bad. The method, obviously inspired by the "Lefshetz thimble" approach [5,[22][23][24][25][26][27][28][29][30][31][32] [33], is based on a deformation of the region of integration of the path integral into complex space but, contrary to the Lefshetz thimble approach it does not require a priori knowledge of the position of the critical points, their thimbles and the contribution of each one to the original integral. We test it with success, on a simple quantum mechanical model where the complex Langevin method fails to converge.…”

Section: Discussionmentioning

confidence: 99%

Monte Carlo Study of Real Time Dynamics on the Lattice

et al. 2016

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Monte Carlo studies involving real time dynamics are severely restricted by the sign problem that emerges from highly oscillatory phase of the path integral. In this letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via Monte Carlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced "contraction algorithm" to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and in principle applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm.

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Section: Discussionmentioning

confidence: 99%

Monte Carlo Study of Real Time Dynamics on the Lattice

et al. 2016

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“…The thimble of the critical point φ c is defined as the set of points that, under the (downward) flow defined by the equations 5) and approach φ c asymptotically. We can gain some insight into eq.…”

Section: Jhep05(2016)053mentioning

confidence: 99%

“…That is why our previous calculation in [14], performed with a fairly large value of T , was constrained to the region of the tangent space mapped onto the main thimble, failed to include the contributions corresponding to the remaining thimbles and produced some wrong results corresponding to the contribution of only one thimble (typically, at larger couplings). 5 Let us clarify the way the one and multi-thimble results are obtained with the contraction method. There are two limits to be considered: the T → ∞ and the infinite statistics limit N → ∞.…”

Section: Jhep05(2016)053mentioning

confidence: 99%

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Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

Alexandru

Başar

Bedaque

et al. 2016

J. High Energ. Phys.

162

157

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We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action ("Lefschetz thimble"). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.

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“…Recently there has been progress in implementing a new method [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The basic idea is to complexify the real variables of the path integral and deform the region of integration to a region where the integrand is real and positive.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo algorithm for simulating fermions on Lefschetz thimbles

2016

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A possible solution of the notorious sign problem preventing direct Monte Carlo calculations for systems with non-zero chemical potential is to deform the integration region in the complex plane to a Lefschetz thimble. We investigate this approach for a simple fermionic model. We introduce an easy to implement Monte Carlo algorithm to sample the dominant thimble. Our algorithm relies only on the integration of the gradient flow in the numerically stable direction, which gives it a distinct advantage over the other proposed algorithms. We demonstrate the stability and e ciency of the algorithm by applying it to an exactly solvable fermionic model and compare our results with the analytical ones. We report a very good agreement for a certain region in the parameter space where the dominant contribution comes from a single thimble, including a region where standard methods su↵er from a severe sign problem. However, we find that there are also regions in the parameter space where the contribution from multiple thimbles is important, even in the continuum limit.2

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Relativistic Bose gas on a Lefschetz thimble

Abstract: In these proceedings, we summarize the Lefschetz thimble approach to the sign problem of Quantum Field Theories. In particular, we review its motivations, and we summarize the results of the application of two different algorithms to two test models.

Cited by 6 publications

References 17 publications

Monte Carlo Study of Real Time Dynamics on the Lattice

Monte Carlo Study of Real Time Dynamics on the Lattice

Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

Monte Carlo algorithm for simulating fermions on Lefschetz thimbles

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