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arXiv:2407.06896 · International Symposium on Lattice Field Theory (LATTICE)

Reducing the Sign Problem with simple Contour Deformation

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

Published May 5, 2024• 1 citation
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Abstract

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.

DOI
10.22323/1.453.0007

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Marcel Rodekamp, Christoph Gäntgen

Many fascinating systems suffer from a severe (complex action) sign problem preventing us from calculating them with Markov Chain Monte Carlo simulations. One promising method to alleviate the sign problem is the transformation of the integration domain towards Lefschetz Thimbles. Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian. In this proceedings we present a new preliminary Neural Network architecture based on complex-valued affine coupling layers. This network performs such a transformation efficiently, ultimately allowing simulation of systems with a severe sign problem. We test this method within the Hubbard Model at finite chemical potential, modelling strongly correlated electrons on a spatial lattice of ions.

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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 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.
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.
We study the sign problem in the Hubbard model on the hexagonal lattice away from half-filling using the Lefschetz thimbles method. We identify the saddle points, reduce their amount, and perform quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour in complex space. Finally, the results are compared against exact diagonalization (ED). We show that the sign problem can be substantially weakened, even in the regime with low temperature and large chemical potential, where standard QMC techniques exhibit an exponential decay of the average sign.
Many fascinating systems suffer from a severe (complex action) sign problem preventing us from calculating them with Markov Chain Monte Carlo simulations. One promising method to alleviate the sign problem is the transformation of the integration domain towards Lefschetz Thimbles. Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian. In this proceedings we present a new preliminary Neural Network architecture based on complex-valued affine coupling layers. This network performs such a transformation efficiently, ultimately allowing simulation of systems with a severe sign problem. We test this method within the Hubbard Model at finite chemical potential, modelling strongly correlated electrons on a spatial lattice of ions.
Many fascinating systems suffer from a severe (complex action) sign problem preventing us from calculating them with Markov Chain Monte Carlo simulations. One promising method to alleviate the sign problem is the transformation of the integration domain towards Lefschetz Thimbles. Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian. In this proceedings we present a new preliminary Neural Network architecture based on complex-valued affine coupling layers. This network performs such a transformation efficiently, ultimately allowing simulation of systems with a severe sign problem. We test this method within the Hubbard Model at finite chemical potential, modelling strongly correlated electrons on a spatial lattice of ions.