# Lefschetz thimbles in fermionic effective models with repulsive vector-field

## TL;DR

This paper addresses the auxiliary sign problem in fermionic models with vector fields using Lefschetz thimbles and proposes a new method to handle singularities from multivalued functions during complexified path integration.

## Contribution

It introduces a novel prescription for fixing gradient flow trajectories on the same Riemann sheet to improve Lefschetz thimble construction in fermionic models.

## Key findings

- The auxiliary sign problem can be mitigated with Lefschetz thimbles.
- A new method fixes flow trajectories on the same Riemann sheet.
- The approach handles singularities from multivalued functions effectively.

## Abstract

We discuss two problems in complexified auxiliary fields in fermionic effective models, the auxiliary sign problem associated with the repulsive vector-field and the choice of the cut for the scalar field appearing from the logarithmic function. In the fermionic effective models with attractive scalar and repulsive vector-type interaction, the auxiliary scalar and vector fields appear in the path integral after the bosonization of fermion ilinears. When we make the path integral well-defined by the Wick rotation of the vector field, the oscillating Boltzmann weight appears in the partition function. This "auxiliary" sign problem can be solved by using the Lefschetz-thimble path-integral method, where the integration path is constructed in the complex plane. Another serious obstacle in the numerical construction of Lefschetz thimbles is caused by singular points and cuts induced by multivalued functions of the complexified scalar field in the momentum integration. We propose a new prescription which fixes gradient flow trajectories on the same Riemann sheet in the flow evolution by performing the momentum integration in the complex domain.