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

TL;DR

This work shows that the sign problem in the one-site Hubbard model is deeply linked to interference among multiple complex Lefschetz thimbles. By performing a semiclassical analysis of the complex saddle points, the authors connect thimble structure to the fermion spectrum and Lee–Yang zeros, revealing why a single thimble is insufficient and how the number of required thimbles grows with inverse temperature. A practical criterion for the necessary number of thimbles is proposed, and numerical results demonstrate that including up to five thimbles can reproduce the exact density plateaus at low temperature, while too few thimbles yield unphysical results. The study argues that thimble interference may play a crucial role in finite-density QCD Silver Blaze phenomena, offering insights for tackling the sign problem in more complex theories.

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.

Figures (3)

• Figure 1: Behaviors of Morse's downward flow equation for $\beta U=30$, $U=1$, and $\mu/U=2$ (a) [$\mu/U=0$ (b)]. Star-shape black points show singular points of logarithm, and red blobs show complex saddle points $z_{\sigma}$.
• Figure 2: Behaviors of the number density $n=-\mathrm{i}\langle\varphi\rangle/U$ as a function of $\mu/U$ at $\beta U=30$. The solid black line shows the exact solution. Other lines, solid red one, dashed green one, and dotted blue one, show the result of one-, three-, and five-thimble approximations, which integrate over $\mathcal{J}_0$, $\mathcal{J}_{0}\cup\mathcal{J}_{\pm 1}$, and $\mathcal{J}_{0}\cup\mathcal{J}_{\pm1}\cup\mathcal{J}_{\pm2}$, respectively.
• Figure 3: Schematic illustration of the Silver Blaze problem in the finite-density QCD. The baryon number density jumps at $\mu\simeq m_N/3$ (solid blue line). Phase quenched theory shows the early onset of the baryon number density (dashed red line).