Theta-vacuum and large N limit in CP^{N-1} sigma models

TL;DR

This paper investigates the theta dependence of vacuum energy in CP^{N-1} models, revealing two distinct regimes influenced by volume and N, with implications for instanton physics and phase transitions at large N.

Contribution

It provides a detailed analysis of theta dependence in CP^{N-1} models using semiclassical, 1/N expansion, and nodal structure approaches, clarifying instanton roles and phase transition phenomena.

Findings

Identifies two regimes with different theta dependence: dilute instanton gas and large N resonance.

Shows 1/N expansion aligns with instanton physics at finite volume.

Reveals a phase transition at 1/N = 0 depending on the order of limits taken.

Abstract

The theta dependence of the vacuum energy density in CP^{N-1} models is re-analysed in the semiclassical approach, the 1/N expansion and arguments based on the nodal structure of vacuum wavefunctionals. The 1/N expansion is shown not to be in contradiction with instanton physics at finite (spacetime) volume V. The interplay of large volume V and large N parameter gives rise to two regimes with different theta dependence, one behaving as a dilute instanton gas and the other dominated by the traditional large N picture, where instantons reappear as resonances of the one-loop effective action, even in the absence of regular instantonic solutions. The realms of the two regimes are given in terms of the mass gap m by m^2 V << N and m^2 V >> N, respectively. The small volume regime m^2 V << N is relevant for physical effects associated to the physics of the boundary, like the leading role of edge states in the quantum Hall effect, which, however, do not play any role in the thermodynamic limit at large N. Depending on the order in which the limits N -> \infty and V -> \infty are taken, two different theories are obtained; this is the hallmark of a phase transition at 1/N = 0.