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arXiv:2307.06785 · Physical review. B./Physical review. B

Fermionic sign problem minimization by constant path integral contour shifts

Christoph Gäntgen, Evan Berkowitz, Thomas Luu, Johann Ostmeyer, Marcel Rodekamp

Published May 20, 2024• 8 citations
arXivHTMLPDF

Abstract

The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of ${\mathrm{C}}_{60}$ fullerenes (buckyballs) that are successful over a wide parameter range.

DOI
10.1103/physrevb.109.195158

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arXiv:2003.01549•Mar 2, 2020•0 citations
The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of C60{\mathrm{C}}_{60}C60​ fullerenes (buckyballs) that are successful over a wide parameter range.
We apply constant imaginary offsets to the path integral for a reduction of the sign problem in the Hubbard model. These simple transformations enhance the quality of results from HMC calculations without compromising the speed of the algorithm. This method enables us to efficiently calculate systems that are otherwise inaccessible due to a severe sign problem. To support this claim, we present observables of the C20 and C60 fullerenes. Furthermore, we demonstrate that at a certain offset, the sign problem is completely lifted in the limit of large chemical potential.
The numerical sign problem is a hindrance for various areas of research that rely on stochastic sampling methods. It occurs when the sign or phase of the weight function in a path integral is not positive-definite. This leads to cancellations in the Monte Carlo sampling, which can significantly slow down convergence and render the simulation infeasible. Calculations of the Hubbard model are affected by the sign problem when the chemical potential is non-zero or when the spatial lattice is not bipartite. The overall objective of this thesis is the reduction of the sign problem in the Hubbard model with methods that may be applied to other models as well. The approach is to deform the integration contour into complex space and find a transformation that efficiently improves the distribution of the complex phase. The first publication of this cumulative thesis explores the simplest case of such a transformation, which is a constant shift of the real plane in imaginary direction. This transformation showed surprising potential and had the numerical advantage of not requiring the calculation of a Jacobian determinant, as opposed to general transformations. However, it was not obvious which offset would yield the best results. We present an algorithm that numerically approaches the optimal offset iteratively. The drawback of this method is that it relies on Markov Chain Monte Carlo sampling, which comes at significant computational cost and is affected by the sign problem itself, leading to reduced reliability. Therefore, we also present two analytic approximations that can be used directly or as a starting point for the iterative algorithm. To test the method, we calculate single particle correlation functions and the charge expectation value of carbon nanostructures, such as graphene sheets and fullerenes. The second publication introduces a refinement of the iterative optimization algorithm, increasing the stability and convergence speed. This enhancement is achieved by fitting a Gaussian curve to the average phase and its first two derivatives. We again present the physical observables, now including a tetrahedron and a larger honeycomb lattice. Additionally we extract single particle energies from the correlation functions, achieving a good agreement with the exactly solvable tetrahedron. The third publication applies a group equivariant neural network as transformation. The goal was to surpass the limitations of the constant shift while keeping the model as efficient as possible. For this purpose, we encoded the physical symmetries into the model architecture. Results show the potential to reduce the sign problem further than the constant shift. A comparison with dense neural networks shows more efficient use of parameters and training data. Tests with transfer learning lead to mixed results, indicating situational applicability. Overall these developments allow for more efficient calculations of sign problem affected systems.
A bstract We suggest an approach for simulating theories with a sign problem that relies on optimisation of complex integration contours that are not restricted to lie along Lefschetz thimbles. To that end we consider the toy model of a one-dimensional Bose gas with chemical potential. We identify the main contribution to the sign problem in this case as coming from a nearest neighbour interaction and approximately cancel it by an explicit deformation of the integration contour. We extend the obtained expressions to more general ones, depending on a small set of parameters. We find the optimal values of these parameters on a small lattice and study their range of validity. We also identify precursors for the onset of the sign problem. A fast method of evaluating the Jacobian related to the contour deformation is proposed and its numerical stability is examined. For a particular choice of lattice parameters, we find that our approach increases the lattice size at which the sign problem becomes serious from L ≈ 32 to L ≈ 700. The efficient evaluation of the Jacobian ( O ( L ) for a sweep) results in running times that are of the order of a few minutes on a standard laptop.
This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in DDD dimension, with n=D+1n=D+1n=D+1 fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.
This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.
Nowadays the term 'sign problem' is used to identify two different problems. The ideas to overcome the first type of the 'sign problem' of strongly oscillating complex valued imtegrand in the Feynman path integrals comes from Picard-Lefschetz theory and a complex version of Morse theory. The main idea is to select Lefschetz thimbles as the cycle approaching the critical point at the path-integration, where the imaginary part of the complex action stays constant. Since the imaginary part of the action is constant on each thimble, the sign problem disappears and the integral can be calculated much more effectively. Here based on the Metropolis -- Hastings algorithm a new method of calculations of the integral of the strongly oscillating integrands has been prosed. Some simple test calculation and comparison with available analytical results have been carried out. The second type of the 'sign problem' arises at studies Fermi systems by path integral approach and is caused by the requirement of antisymmetrization of the real valued matrix elements of the density matrix. An explicit analytical expression for effective pair pseudopotential in phase space has been discussed in Wigner formulation of quantum mechanics. Obtained pseudopotential allow to account for Fermi statistical effects as realizes the Pauli blocking of fermions due to the repulsion between identical fermions, which prevents their occupation of same phase space cell. To test this approach, calculations of the momentum distribution function of the ideal system of Fermi particles have been presented over a wide range of momentum and degeneracy parameter.