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arXiv:2509.07881 · Physical review. D/Physical review. D.

Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models

Attila Pásztor, Dávid Pesznyák

Published Nov 10, 2025• 1 citation
arXivHTMLPDF

Abstract

We perform a detailed analysis of the fermionic sign problem in a series of one-dimensional integrals, that are achieved as extreme (one-site) limits of genuine physics models. Altogether we studied a Hubbard-like, a Gross-Neveu-like, a Thirring-like, and a Chern-Simons-like integral. We compare the Lefschetz-thimble structure for these integrals with contours obtained with the holomorphic flow equations at different flow times and with numerically optimized continuous integration contours, defined by a maximal value of the expectation values of the phases. With the holomorphic flow equation, we perform the large flow-time limit, so that the average phase corresponds to its value on the thimbles. In all of these integrals we observe that the convergence to this value is not monotonic, meaning that there is an optimal flow time where the sign problem is weaker than it is on the thimbles. Furthermore, we find that for all of these toy models, numerical optimization can find continuous contours on which the sign problem is considerably weaker than it is both on the thimbles and at flowed integration contours at the optimal flow time.

DOI
10.1103/fywf-77b4

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We perform a detailed analysis of the fermionic sign problem in a series of one-dimensional integrals, that are achieved as extreme (one-site) limits of genuine physics models. Altogether we studied a Hubbard-like, a Gross-Neveu-like, a Thirring-like, and a Chern-Simons-like integral. We compare the Lefschetz-thimble structure for these integrals with contours obtained with the holomorphic flow equations at different flow times and with numerically optimized continuous integration contours, defined by a maximal value of the expectation values of the phases. With the holomorphic flow equation, we perform the large flow-time limit, so that the average phase corresponds to its value on the thimbles. In all of these integrals we observe that the convergence to this value is not monotonic, meaning that there is an optimal flow time where the sign problem is weaker than it is on the thimbles. Furthermore, we find that for all of these toy models, numerical optimization can find continuous contours on which the sign problem is considerably weaker than it is both on the thimbles and at flowed integration contours at the optimal flow time.
The Lefschetz-thimble approach to path integrals is applied to a one-site model of electrons, i.e., the one-site Hubbard model. Since the one-site Hubbard model shows a non-analytic behavior at the zero temperature and its path integral expression has the sign problem, this toy model is a good testing ground for an idea or a technique to attack the sign problem. Semiclassical analysis using complex saddle points unveils the significance of interference among multiple Lefschetz thimbles to reproduce the non-analytic behavior by using the path integral. If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities. Analyzing such singular behaviors semiclassically, we propose a criterion to identify the necessary number of Lefschetz thimbles. We argue that this interference of multiple saddle points is a key issue to understand the sign problem of the finite-density quantum chromodynamics.
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.
The concept of Lefschetz thimble decomposition is one of the most promising possible modifications of Quantum Monte Carlo (QMC) algorithms aimed at alleviating the sign problem which appears in many interesting physical situations, e.g. in the Hubbard model away from half filling. In this approach one utilizes the fact that the integral over real variables with an integrand containing a complex fluctuating phase is equivalent to the sum of integrals over special manifolds in complex space ("Lefschetz thimbles"), each of them having a fixed complex phase factor. Thus, the sign problem can be reduced if the resulting sum contains terms with only a few different phases. We explore the complexity of the sign problem for the few-site Hubbard model on square lattice combining a semi-analytical study of saddle points and thimbles in small lattices with several steps in Euclidean time with results of test QMC calculations. We check different variants of conventional Hubbard-Stratonovich transformation based on Gaussian integrals. On the basis of our analysis we reveal a regime with minimal number of relevant thimbles in the vicinity of half filling. In this regime we found only two relevant thimbles for the few-site lattices studied in the paper. There is also indirect evidence of the existence of this regime in more realistic systems with large number of Euclidean time slices. In addition, we derive a new non-Gaussian representation of the interaction term, where the number of relevant Lefschetz is also reduced in comparison with conventional Gaussian Hubbard-Stratonovich transformation.
In this paper, we propose a framework for studying the properties of the Lefschetz thimbles decomposition for lattice fermion models approaching the thermodynamic limit. The proposed set of algorithms includes the Schur complement solver and the exact computation of the derivatives of the fermion determinant. It allows us to solve the gradient flow equations taking into account the fermion determinant exactly, with high performance. Being able to do so, we can find both real and complex saddle points and describe the structure of the Lefschetz thimbles decomposition for large enough lattices which allows us to extrapolate our results to the thermodynamic limit. We describe the algorithms for a general lattice fermion model, with emphasis on two widely used types of lattice discretizations for relativistic fermions (staggered and Wilson fermions), as well as on interacting tight-binding models for condensed matter systems. As an example, we apply these algorithms to the Hubbard model on a hexagonal lattice. Several technical improvements allow us to deal with lattice volumes as large as $12\ifmmode\times\else\texttimes\fi{}12$ with ${N}_{\ensuremath{\tau}}=256$ steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low-temperature, and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) transformation were studied, and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we provide evidence for the existence of an optimal regime for the hexagonal lattice Hubbard model, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the gradient flow to deform the integration contour into the complex plane. These calculations were made on small volumes (Ns=8{N}_{s}=8Ns​=8 sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit an exponential decay of the average sign. The results are compared versus exact diagonalization, and we demonstrate the importance of choosing an optimal form for the HS transformation for the Hubbard model to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS (Blankenbecler, Scalapino, and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.