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.
Highlights
The sign problem appearing in the path integral is a serious obstacle to perform precise nonperturbative computations in various quantum systems: The Boltzmann weight in the partition function oscillates and it induces the serious cancellation to the numerical integration process
We have investigated the auxiliary sign problem which arises when the fermionic theory has the repulsive vectorfield in variables of integration after the bosonization procedure: The Boltzmann weight in the partition function should oscillate by the repulsive vector-field when we make the path-integral well-defined
If fermionic effective models do not have the sign problem and those path-integral are ill-defined after the simple bosonization procedure, the Wick rotation of the repulsive vector-field cures the illness
Summary
The sign problem appearing in the path integral is a serious obstacle to perform precise nonperturbative computations in various quantum systems: The Boltzmann weight in the partition function oscillates and it induces the serious cancellation to the numerical integration process. The sign problem attracts much more attention recently in the lattice simulation of Quantum Chromodynamics (QCD) at finite density It is caused by the combination of the gluon field ( A μ ) and the real quark chemical potential (μ) in the fermion determinant of the Boltzmann weight; see Ref. The sign problem comes back, when the repulsive vectorcurrent interaction is included in the NJL model and the auxiliary vector field is Wick rotated to make the path integral well-defined. This type of the sign problem which we call the auxiliary sign problem in this paper has not been discussed in the four-dimensional space-time, previously. The most convenient form of the flow equations is given as
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