Machine learning to alleviate Hubbard-model sign problems

Wynen, Jan-Lukas; Berkowitz, Evan; Krieg, Stefan; Luu, Thomas; Ostmeyer, Johann

doi:10.1103/physrevb.103.125153

Cited by 34 publications

(27 citation statements)

References 80 publications

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“…It is unclear to what extent the holomorphic gradient flow algorithm adopted here will remain efficient. Some other techniques may be needed, such as the tempered thimbles, the learnifolds, and the path optimization algorithms reviewed in [20] and further developed in, e.g., [74,75,76,77].…”

Section: Discussionmentioning

confidence: 99%

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Jia

2021

Preprint

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Evaluating gravitational path integrals in the Lorentzian has been a longstanding challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity.In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.

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Section: Discussionmentioning

confidence: 99%

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Jia

2021

Preprint

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“…While much of the focus of LMC efforts continues to be on Lattice QCD, the algorithms and methods so developed are now rapidly finding their place among the large variety of Hamiltonian theories in condensed matter physics. Recently, preliminary studies of nanotubes [10,11] have appeared and treatments of the Fermion sign problem based on formal developments [12,13] have made promising progress towards first-principles treatments of fullerenes [14] and doped systems [15]. Moreover, our understanding of these algorithms' ergodicity properties [16] and computational scaling [17,18] has also recently been placed on a firm footing.…”

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confidence: 99%

“…We have used these ensembles to compute the Mott gap ∆ as a function of U and β to locate a quantum critical point (QCP) at T = 0 [5] using finite-size scaling (FSS) [23][24][25]. The single-particle gap ∆ was found to open at U/κ = 3.834 (14). Though this is widely expected to coincide with a semimetal-AFMI transition, we did not characterize the nature of the transition in Ref.…”

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confidence: 99%

Semimetal–Mott insulator quantum phase transition of the Hubbard model on the honeycomb lattice

et al. 2020

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The Hubbard model on the honeycomb lattice undergoes a quantum phase transition from a semimetallic to a Mott insulating phase and from a disordered to an anti-ferromagnetically phase. We show that these transitions occur simultaneously and we calculate the critical coupling 𝑈 𝑐 = 3.835(14) as well as the critical exponents 𝜈 = 1.181(43) and 𝛽 = 0.898(37) which are expected to fall into the 𝑆𝑈 (2) Gross-Neveu universality class. For this we employ Hybrid Monte Carlo simulations, extrapolate the single particle gap and the spin structure factors to the thermodynamic and continuous time limits, and perform a data collapse fit. We also determine the zero temperature values of single particle gap and staggered magnetisation on both sides of the phase transition.

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“…The complex Langevin method is a numerical approach used to circumvent the numerical sign problem arising in the computation of ensemble averages with complex Boltzmann weights. Such issues may appear in the real-time quantum field theories [17,8], coupled quantum systems with chemical potentials such as the Hubbard model [48,49] and the quantum chromodynamics at finite density [32,7], and also the superstring theory [10,36]. In these applications, one usually encounters strong oscillations in high-dimensional functions, leading to significant cancellations when integrating the functions.…”

Section: Introductionmentioning

confidence: 99%

Regularization of Complex Langevin Method

Cai,

Kuang,

Tan

2021

Preprint

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The complex Langevin method, a numerical method used to compute the ensemble average with a complex partition function, often suffers from runaway instability. We study the regularization of the complex Langevin method via augmenting the action with a stabilization term. Since the regularization introduces biases to the numerical result, two approaches, named 2R and 3R methods, are introduced to recover the unbiased result. The 2R method supplements the regularization with regression to estimate the unregularized ensemble average, and the 3R method reduces the computational cost by coupling the regularization with a reweighting strategy before regression. Both methods can be generalized to the SU(n) theory and are assessed from several perspectives. Several numerical experiments in the lattice field theory are carried out to show the effectiveness of our approaches.

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Machine learning to alleviate Hubbard-model sign problems

Cited by 34 publications

References 80 publications

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Semimetal–Mott insulator quantum phase transition of the Hubbard model on the honeycomb lattice

Regularization of Complex Langevin Method

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