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

摘要

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 generallyfor mixed isovector/isotensor models. We show that the N=\infty limit exhibitsphase transitions that are forbidden for any finite N. We clarify the origin ofthese 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 thereal axis as N\to\infty. We conjecture the correct phase diagram for finite Nas a function of the spatial dimension d. Along the way, we prove some newcorrelation inequalities for a class of N-component sigma-models, and we obtainsome new results concerning the complex zeros of confluent hypergeometricfunctions.