The $\theta$-term, CP$^{N-1}$ Model and the Inversion Approach in the Imaginary $\theta$ Method

TL;DR

This paper explores the behavior of the CP^{N-1} lattice field theory with a theta-term using imaginary theta methods and an inversion approach, revealing step-like phenomena and discussing their physical significance.

Contribution

It investigates the role of the inversion approach in the imaginary theta method within the weak coupling region of the CP^{N-1} model, providing new insights into topological charge behavior.

Findings

Step-like behavior observed in the x-h relation in weak coupling.

The inversion approach effectively analyzes the weak coupling region.

Discussion on the physical meaning of the step-like behavior.

Abstract

The weak coupling region of CP$^{N-1}$ lattice field theory with the $\theta$-term is investigated. Both the usual real theta method and the imaginary theta method are studied. The latter was first proposed by Bhanot and David. Azcoiti et al. proposed an inversion approach based on the imaginary theta method. The role of the inversion approach is investigated in this paper. A wide range of values of $h=-{\rm Im} \theta$ is studied, where $\theta $ denotes the magnitude of the topological term. Step-like behavior in the $x$-$h$ relation (where $x=Q/V$, $Q$ is the topological charge, and $V$ is the two dimensional volume) is found in the weak coupling region. The physical meaning of the position of the step-like behavior is discussed. The inversion approach is applied to weak coupling regions.