2026/06/08 by Denjoe O'Connor, Sanjaye Ramgoolam
#hep-th #cond-mat.stat-mech #math-ph #math.CO #math.MP #math.RT
We consider the SO(d) and O(d) invariant sectors of the bosonic d-matrix harmonic oscillator with U(N) gauge symmetry. The micro-canonical degeneracy Z( N , d , k ) for fixed energy k is expressed as a pairing between an N-dependent vector and a d-dependent vector in the space of partitions of the integer k. This pairing formula is derived by counting invariant words in multi-matrix variables Xij,a, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group Sk, Schur-Weyl duality and harmonic analysis on the homogeneous space U(d)/SO(d). Analytic formulae for large N and k with k ≤ N are obtained using group integrals over U(N) and SO(d) (or O(d)). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value k\rm crit, due to finite N modifications to the counting, thus forming what we denote as a characteristic caloric fold in the E versus T curve. Data from the pairing formula is well fitted by k\rm crit ∼ N2 \over 4 for small values of d. A derivation of this large N formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large N,d limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.