Lefschetz thimbles in fermionic effective models with repulsive vector-field

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 bilinears. 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.

Evading the model sign problem in the PNJL model with repulsive vector-type interaction via path optimization

“…There are many approaches to the sign problem as discussed in the previous and present lattice meetings. We here concentrate on the complexified variable methods, the complex Langevin method (CLM) [8,9], the Lefschetz thimble method (LTM) [10,11,12], and the path optimization method [7,13,14]. In CLM, we can sample configurations by solving the complex Langevin equation.…”

“…However, it should be noted that CLM is not guaranteed to give correct results, when the distribution of the drift term (derivative of the action) does not fall off exponentially or faster at large values [9]. Since there are many zeros of the Fermion determinant in the Nambu-Jona-Lasinio model in the complexified auxiliary fields [11], the drift term distribution would have a power-law tail. In LTM, by solving the holomorphic flow equations, one can obtain the integral path (manifold) referred to as a thimble on which the imaginary part of the action is constant.…”

Evading the model sign problem in the PNJL model with repulsive vector-type interaction via path optimization

“…There are many approaches to the sign problem as discussed in the previous and present lattice meetings. We here concentrate on the complexified variable methods, the complex Langevin method (CLM) [8,9], the Lefschetz thimble method (LTM) [10,11,12], and the path optimization method [7,13,14]. In CLM, we can sample configurations by solving the complex Langevin equation.…”

Path optimization for U(1) gauge theory with complexified parameters

“…The path optimization method is based on the standard path integral formulation with the complexification of dynamical variables [15,16]; the actual procedure is performed as follows:…”