Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density

TL;DR

This paper analyzes the Lefschetz thimble structure in a one-dimensional lattice Thirring model at finite density, identifying critical points, thimble contributions, and their behavior across different chemical potentials to aid Monte Carlo simulations.

Contribution

It systematically studies the Lefschetz thimble structure in the lattice Thirring model, revealing how multiple thimbles contribute across the crossover region and their role in the continuum and low-temperature limits.

Findings

At small and large chemical potentials, a single thimble dominates.

Multiple thimbles contribute significantly in the crossover region.

Adding multi thimble contributions recovers the rapid crossover behavior.

Abstract

We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multi thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multiple thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.