Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

TL;DR

This paper investigates the phase diagram of a non-commutative matrix field model, revealing how curvature influences phase transitions and helps resolve UV/IR mixing issues in the Grosse-Wulkenhaar model.

Contribution

It demonstrates the critical role of curvature in the phase structure of the Grosse-Wulkenhaar model and its impact on the triple point and UV/IR mixing.

Findings

Curvature term shifts the triple point away from the origin.

Turning off curvature causes the triple point to collapse to the origin.

The stripe phase moves to infinity, alleviating UV/IR mixing problems.

Abstract

We construct and analyze the phase diagram of a self-interacting matrix field in two dimensions coupled to the curvature of the non-commutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar's. The curvature term proves crucial for the diagram's structure: when turned off, the triple point collapses into the origin as matrices grow larger; when turned on, the triple point recedes from the origin proportionally to the coupling strength and the matrix size. The coupling attenuation that turns the Grosse-Wulkenhaar model into a renormalizable version of the $\phi^4_\star$-model cannot stop the triple point recession. As a result, the stripe phase escapes to infinity, removing the problems with UV/IR mixing.