Pinpointing Triple Point of Noncommutative Matrix Model with Curvature

TL;DR

This paper investigates how adding a curvature term to a Hermitian matrix model influences its phase structure, potentially enhancing renormalizability and revealing new multi-cut phases through analytical and numerical methods.

Contribution

It introduces a curvature-modified matrix model inspired by noncommutative geometry, analyzing its phase diagram and phase transitions both analytically and via simulations.

Findings

Curvature term shifts the triple point in phase diagram.

Curvature suppresses the striped phase at large N.

Possible discovery of a new multi-cut phase at finite matrix size.

Abstract

We study a Hermitian matrix model with a quartic potential, modified by a curvature term $\mathrm{tr}(R\Phi^2)$, where $R$ is a fixed external matrix. Inspired by the truncated Heisenberg algebra formulation of the Grosse--Wulkenhaar model, this term breaks unitary invariance and, through perturbative expansion, induces an effective multitrace matrix model. We analyze the resulting action both analytically and numerically, including Hamiltonian Monte Carlo simulations, focusing on two features closely tied to renormalizability: the shift of the triple point and the suppression of the noncommutative striped phase. Our findings show that the curvature term drives the phase structure toward renormalizable behavior by removing the striped phase in the large-$N$ limit, while also unexpectedly revealing a possible novel multi-cut phase observed at the level of finite matrix size.