The large N limit of SU(N) integrals in lattice models

Abstract

The standard U (N ) and SU (N ) integrals are calculated in the large N limit. Our main finding is that for an important class of integrals this limit is different for two groups. We describe the critical behaviour of SU (N ) models and discuss implications of our results for the large N behaviour of SU (N ) lattice gauge theories at finite temperatures and non-zero baryon chemical potential. The key ingredients of our approach are 1) expansion of the integrals into a sum over irreducible representations and 2) calculation of sums over partitions of r of products of dimensions of two different representations of a symmetric group S r .

The Polyakov loop models in the large N limit: Phase diagram at finite density

“…1 for various values of b. When b decreases the line tends to the critical line of the one-site model [14].…”

“…Using the integral representation for the deltas in the last expression, one can perform the invariant integration with the help of methods of Ref. [14]. We find…”

“…In the region ρ 0 > 1/2, corresponding to large values of ferromagnetic couplings in S g (ρ) and/or large values of α, the solution can be found with the help of the following expansion for the function P (u, r) in (13) (see Appendix of Ref. [14])…”

“…In Ref. [14] we have shown that when µ is non-vanishing the SU(N) integrals essentially differ from U(N) ones. Therefore, the analysis of [10] should be re-examined for SU(N) model, and this is a main goal of the present paper.…”

“…Such dual forms are free of the sign problem and can be reliably studied via Monte-Carlo simulations [7,8]. Furthermore, these models can be used to investigate different aspects of the large N behavior of finite-temperature LGTs [9,10,11,12,13,14].…”