Pinpointing Triple Point of Noncommutative Matrix Model with Curvature

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.

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