Pathologies of the large-N limit for RP^{N-1}, CP^{N-1}, QP^{N-1} and mixed isovector/isotensor sigma-models

TL;DR

This paper analyzes the phase diagrams of large-N sigma-models, revealing phase transition pathologies at infinite N and providing insights into their origins and implications for finite N.

Contribution

It computes the large-N phase diagram for RP^{N-1}, CP^{N-1}, and QP^{N-1} sigma-models, identifying forbidden phase transitions and clarifying their origin.

Findings

Large-N limit shows phase transitions forbidden at finite N

Complex zeros of the partition function approach the real axis as N increases

New correlation inequalities and results on zeros of hypergeometric functions

Abstract

We compute the phase diagram in the N\to\infty limit for lattice RP^{N-1}, CP^{N-1} and QP^{N-1} sigma-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=\infty limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N\to\infty. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component sigma-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.