Symmetry Breaking and Phase Transitions in Random Non-Commutative Geometries and Related Random-Matrix Ensembles

Ensembles of random fuzzy non-commutative geometries may be described in terms of finite ((N^2)-dimensional) Dirac operators and a probability measure. Dirac operators of type ((p,q)) are defined in terms of commutators and anti-commutators of (2^{p+q-1}) hermitian matrices (H_k) and tensor products with a representation of a Clifford algebra. Ensembles based on this idea have recently been used as a toy model for quantum gravity, and they are interesting random-matrix ensembles in their own right. We provide a complete theoretical picture of crossovers, phase transitions, and symmetry breaking in the (N \to \infty ) limit of 1-parameter families of quartic Barrett-Glaser ensembles in the one-matrix cases ((1,0)) and ((0,1)) that depend on one coupling constant (g). Our theoretical results are in full agreement with previous and new Monte-Carlo simulations.

💡 Research Summary

The paper investigates random fuzzy non‑commutative geometries by focusing on the simplest Dirac‑operator constructions, the (1,0) and (0,1) cases, which are built from a single Hermitian N×N matrix H. The Dirac operators are defined as D⁺ = H⊗I + I⊗Hᵀ and D⁻ = H⊗I – I⊗Hᵀ, with the additional trace‑free condition tr H = 0 imposed for the (0,1) geometry. Barrett and Glaser introduced a one‑parameter family of actions
S_g(H) = N⁻² tr D⁴ + g N⁻² tr D²,
which, after expanding in terms of H, yields quartic and quadratic trace terms together with extra contributions proportional to (tr H)² and tr H tr H³ in the (1,0) case. The parameter g controls the shape of the effective potential: for g < 0 the potential acquires a Mexican‑hat form, suggesting the possibility of spontaneous symmetry breaking.

By diagonalising H, the authors rewrite the matrix model as a one‑dimensional Coulomb gas of eigenvalues λ_i with logarithmic repulsion and a single‑particle potential V_g(λ) = 2λ⁴ + 2gλ². Additional two‑body interactions U_{±,g}(λ,λ′) appear, differing between the two geometries: for (1,0) the interaction contains terms 2gλλ′ + 4λλ′³ + 4λ³λ′ + 6λ²λ′², while for (0,1) it reduces to 6λ²λ′². The total energy functional is
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