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

TL;DR

This work addresses phase transitions and symmetry breaking in two one-parameter random-matrix ensembles modeling finite non-commutative geometries via Dirac operators in the large-$N$ limit. By combining Coulomb-gas descriptions with the Riemann-Hilbert method, it derives equilibrium measures for the (0,1) and (1,0) geometries, revealing a third-order 1-cut to 2-cut crossover at $g_{-,cr}=-4\sqrt{2}$ in (0,1) and a first-order symmetry-breaking transition at $g_{+\mathrm{cr}}\approx -3.187$ in (1,0$, including explicit forms where available. The results show strong agreement with Monte Carlo simulations at finite $N$, correct prior analytical errors, and clarify how the Dirac-spectrum and its convolution behave across these transitions. These findings illuminate spectral properties of finite non-commutative geometries and provide a principled framework for studying similar multi-matrix systems in quantum gravity-inspired random-matrix models.

Abstract

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.

Figures (6)

• Figure 1: Equilibrium density (equivalent to expected density of states) for the $(0,1)$ case for various values of the coupling constant $g$. The dashed black curve is the density at the critical value $g=-4\sqrt{2}$.
• Figure 2: Left: Dependence of the second moment $m_2$ on the coupling constant $g$ for the $(0,1)$ case (blue curve: dependence for $g>-4\sqrt{2}$; orange curve: dependence for $g<-4\sqrt{2}$; dashed black curve: linear behavior $-\frac{g}{8}$ which coincides with the orange curve for $g<-4\sqrt{2}$. Right: Support of the equilibrium density for the $(0,1)$ case. For $g>-4\sqrt{2}$ the interval $[-b,b]$ for the corresponding 1-cut solutions is drawn in blue. For $g>-4\sqrt{2}$ the two symmetric intervals $[-b,-a]$ and $[a,b]$ for the corresponding 2-cut solutions are drawn in orange.
• Figure 3: Free energy of various candidates for the equilibrium density of states in $(1,0)$ geometries. Blue: the explicitly known symmetric 1-cut solution. Orange: the explicitly known symmetric 2-cut solution. Green: the numerically found broken symmetry 2-cut solution. In the $(0,1)$ case only the symmetric solutions (orange and blue) exist and they match at $g_{-, \mathrm{cr}}=-4\sqrt{2}$. In the $(1,0)$ case the additional broken-symmetry solution minimizes the free energy for $g < g_{+,\mathrm{cr}}$. The inset graph shows where the free energies of the symmetric 1-cut solution and the broken symmetry 2-cut solutions cross (the broken symmetry solution exists beyond the crossing which cannot be seen in the large graph due to the width of the curves which hides the bifurcation point). The critical value can be read off as $g_{+,\mathrm{cr}}\approx -3.187$.
• Figure 4: Equilibrium densities in the $(1,0)$ geometries for various values of the coupling constant above and below the critical value.
• Figure 5: Left: Support of the equilibrium density in the $(1,0)$ case where the support for the 1-cut densities for $g> g_{+,\mathrm{cr}}$ is shown in blue and the support of the broken symmetry 2-cut densities for $g> g_{+,\mathrm{cr}}$ is shown in green. Right: Dependence of the moments $m_2$ (large graph), $m_1$ (upper inset), and $m_3$ (lower inset) on the coupling constant. The critical value $g_{+,\mathrm{cr}}$ is indicated by the vertical dashed lines. The jumps of all moments at the critical value indicate a 1st order phase transition.