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

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.

LaTeX, 88 pages, 33 figures

References in corpus (3)

• The nature of the continuum limit in the 2D $R{P}^2$ gauge model
• The Difference between Abelian and Non-Abelian Models: Fact and Fancy
• Gauged O(n) spin models in one dimension

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• First-order transitions for very nonlinear sigma models
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