Pinpointing Triple Point of Noncommutative Matrix Model with Curvature

TL;DR

This work analyzes a noncommutative Grosse--Wulkenhaar matrix model augmented by a curvature term that generates multitrace corrections and breaks unitary invariance. Through a perturbative expansion up to $O(g_r^6)$ and Hamiltonian Monte Carlo simulations, the authors derive analytic transition lines (notably the S1/S2 and S2/A1 lines) and show that the curvature term shifts the triple point by $\delta g_2^{\text{tp}}$ and suppresses the noncommutative striped phase, steering the model toward renormalizable behavior. The study also reveals a potential novel multi-cut eigenvalue phase at finite $N$, with a detailed analysis of the eigenvalue distribution via the resolvent and moments. The results provide a mechanistic link between stripe suppression and renormalizability in noncommutative matrix models and lay groundwork for extending these insights to NC gauge theories and other curvature-induced multitrace settings.

Abstract

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.

Figures (13)

• Figure 1: Eigenvalue distribution $\rho(\lambda)$ in the matrix GW model for representative phases (${N=24}$). The left and the right distributions correspond to the disordered and ordered vacua, respectively, both of which also appear in commutative models. The central orange distribution, the so-called striped (or matrix) vacuum, is unique to the noncommutative model and features eigenvalues of both signs. This phase breaks translational symmetry and leads to a spatial modulation of magnetization.
• Figure 2: Phase diagrams of the $N=24$ matrix GW model without (left panel) and with (right panel) the curvature term. Darker regions correspond to lower values of the specific heat, $\operatorname{Var} S_\text{\tiny GW}^\text{\tiny M}/N^2$, while lighter regions indicate higher values. The bright stripes mark the phase transition lines, with phase labels (abbreviations) indicated in the plot. Note that the triple point in the right-hand panel is shifted by $\delta g_2^\text{tp} \approx 16 g_r = 16 \times 0.2 = 3.2$ relative to the left-hand panel.
• Figure 3: Phase transition lines obtained from the sixth-order effective action. Black dots indicate HMC simulation results for the action $S$ at ${g_r = 0.02}$, extrapolated to the ${N \to \infty}$ limit. The dashed line denotes the S2/A1 transition, which is absent in this model.
• Figure 4: Comparison of the turning points predicted by the second-, fourth-, and sixth-order approximations to the exact S1/S2 transition line \ref{['S1-S2-line']} of the model.
• Figure 5: Densely distributed dots indicate the numerically confirmed S1-phase solutions of the $O(g_r^6)$ effective action \ref{['Seff']} at ${g_r = 0.02}$. The solid line represents the corresponding $O(g_r^6)$ analytical S1/S2 transition line \ref{['S1-S2-line']}.