Phase diagram and dispersion relation of the non-commutative \lambda \phi^{4} model in d=3

TL;DR

This study non-perturbatively analyzes the phase diagram and dispersion relation of a three-dimensional non-commutative bbb5b5b5b4 model, revealing complex phases, UV/IR mixing, and confirming the existence of a striped phase in the continuum limit.

Contribution

It provides the first non-perturbative numerical analysis of the phase structure and dispersion relation of the non-commutative bbb5b5b5b4 bbb5b5b5b4 model in three dimensions, including the existence of a striped phase.

Findings

Identification of a phase diagram with uniform and striped phases

Extraction of a non-perturbative dispersion relation confirming UV/IR mixing

Evidence for a striped phase in the continuum limit

Abstract

We present a non-perturbative study of the \lambda \phi^{4} model in a three dimensional Euclidean space, where the two spatial coordinates are non-commutative. Our results are obtained from numerical simulations of the lattice model, after its mapping onto a dimensionally reduced, twisted Hermitian matrix model. In this way we first reveal the explicit phase diagram of the non-commutative \lambda \phi^{4} lattice model. We observe that the ordered regime splits into a phase of uniform order and a phase of two stripes of opposite sign, and more complicated patterns. Next we discuss the behavior of the spatial and temporal correlators. From the latter we extract the dispersion relation, which allows us to introduce a dimensionful lattice spacing. To extrapolate to zero lattice spacing and infinite volume we perform a double scaling limit, which keeps the non-commutativity tensor constant. The dispersion relation in the disordered phase stabilizes in this limit, which represents a non-perturbative renormalization. In particular this confirms the existence of a striped phase in the continuum limit, in accordance with a conjecture by Gubser and Sondhi. The extrapolated dispersion relation also exhibits UV/IR mixing as a non-perturbative effect. Finally we add some observations about a Nambu-Goldstone mode in the striped phase, and about the corresponding model in d=2.