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arXiv:1509.07146 · New Journal of Physics

Lefschetz-thimble analysis of the sign problem in one-site fermion model

Yuya Tanizaki, Yoshimasa Hidaka, Tomoya Hayata

Published Mar 1, 2016• 91 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/18/3/033002

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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.
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.
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.
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 major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.
To each complex saddle point of an action, one can attach a Lefschetz thimble on which the imaginary part of the action is constant. Cauchy theorem states that summation over a set of thimbles produces the exact result. This reorganization of the path integral, is an appealing starting point for various approximations: In the realm of auxiliary quantum Monte Carlo methods it provides a framework to alleviate the negative sign problem. Here, we suggest to constrain the integration to the \textit{dominant} thimbles: the thimbles attached to the saddle points with the largest statistical weight. For the Hubbard model, in a formulation where the the Hubbard Stratonovitch field couples to the charge, this provides a \textit{symmetry} consistent approximation to the physics of the Hubbard model: constraining the integration domain does not explicitly break a symmetry. We can test this approach for the Hubbard model at half-filling on a bipartite lattice. The paper builds on the previously developed instanton gas approach, where an exhaustive saddle point approximation was constructed. We present results, showing that the dominant thimbles approximation provides results that are in remarkable agreement with the exact results for various fermionic observables including spin and charge order parameters and single electron spectral functions. We discuss implications of our results for simulations away of half filling.
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.