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The sign problem and the Lefschetz thimbles in two dimensional Hubbard model

S. N. Valgushev

Published Jun 1, 2019• 1 citation
OpenAlex

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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.

arXiv:1509.07146•New Journal of Physics•Mar 1, 2016•91 citations
The tempered Lefschetz thimble method (TLTM) is a parallel-tempering algorithm towards solving the numerical sign problem. It tames both the sign and ergodicity problems simultaneously by tempering the system with the flow time of continuous deformations of the integration region. In this article, after reviewing the basics of the TLTM, we explain a new algorithm within the TLTM that enables us to estimate the expectation values precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate the effectiveness of the algorithm, we apply the TLTM to the quantum Monte Carlo simulation of the Hubbard model away from half filling on a two-dimensional lattice of small size, and show that the obtained numerical results agree nicely with exact values.
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.