Pinpointing Triple Point of Noncommutative Matrix Model with Curvature

We study a Hermitian matrix model with a quartic potential, modified by a curvature term $\mathrm{tr}(RΦ^2)$, 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.

💡 Research Summary

The paper investigates a Hermitian matrix model with a quartic potential that is modified by a curvature term tr(R Φ²), where R is a fixed external matrix. This term, inspired by the truncated Heisenberg algebra formulation of the Grosse–Wulkenhaar (GW) model, explicitly breaks unitary invariance and generates multitrace interactions when the angular degrees of freedom are integrated out. The authors focus on two phenomena that are closely linked to the renormalizability of non‑commutative field theories: the shift of the triple point in the phase diagram and the suppression of the non‑commutative striped phase.

First, the GW model is rewritten in matrix form via a Weyl transform, introducing background matrices X and Y that approximate the non‑commutative plane at large N. The curvature matrix R is diagonal with entries proportional to the integer labels of the matrix size, reproducing the harmonic oscillator term of the original GW action. By discarding the kinetic operator K and keeping only the potential terms, the authors obtain a simplified action S = N tr