CP(N-1) Models with Theta Term and Fixed Point Action

TL;DR

This study calculates the topological charge distribution and free energy in lattice CP(N-1) models using a fixed point action, revealing scaling behavior for N=4 but not for N=2, and investigates instanton dynamics through effective power analysis.

Contribution

It introduces a fixed point action approach to analyze topological properties and instanton dynamics in CP(N-1) models, highlighting differences between N=2 and N=4.

Findings

Scaling behavior observed for N=4 in P(Q) and F(θ)

No scaling behavior observed for N=2 as expected

Similar instanton dynamics behavior in simulations and theoretical models

Abstract

The topological charge distribution P(Q) is calculated for lattice ${\rm CP}^{N-1}$ models. In order to suppress lattice cut-off effects we employ a fixed point (FP) action. Through transformation of P(Q) we calculate the free energy $F(\theta)$ as a function of the $\theta$ parameter. For N=4, scaling behavior is observed for P(Q), $F(\theta)$ as well as the correlation lengths $\xi(Q)$. For N=2, however, scaling behavior is not observed as expected. For comparison, we also make a calculation for the ${\rm CP}^{3}$ model with standard action. We furthermore pay special attention to the behavior of P(Q) in order to investigate the dynamics of instantons. For that purpose, we carefully look at behavior of $\gamma_{\it eff}$, which is an effective power of P(Q)($\sim \exp(-CQ^{\gamma_{\it eff}})$), and reflects the local behavior of P(Q) as a function of Q. We study $\gamma_{\it eff}$ for two cases, one of which is the dilute gas approximation based on the Poisson distribution of instantons and the other is the Debye-H\"uckel approximation of instanton quarks. In both cases we find similar behavior to the one observed in numerical simulations.