Renormalization Group Equations and the Lifshitz Point In Noncommutative Landau-Ginsburg Theory

TL;DR

This paper performs a one-loop RG analysis of noncommutative Landau-Ginsburg theory, revealing a Lifshitz point, a noncommutative Wilson-Fisher fixed point, and novel effects of UV-IR mixing on critical behavior.

Contribution

It introduces a modern Wilsonian RG approach to noncommutative Landau-Ginsburg theory, identifying a Lifshitz point and analyzing UV-IR mixing effects at one loop.

Findings

Discovery of a noncommutative Wilson-Fisher fixed point.

Identification of a Lifshitz point with a first-order phase transition.

Observation of a negative $ heta$-dependent anomalous dimension.

Abstract

A one-loop renormalization group (RG) analysis is performed for noncommutative Landau-Ginsburg theory in an arbitrary dimension. We adopt a modern version of the Wilsonian RG approach, in which a shell integration in momentum space bypasses the potential IR singularities due to UV-IR mixing. The momentum-dependent trigonometric factors in interaction vertices, characteristic of noncommutative geometry, are marginal under RG transformations, and their marginality is preserved at one loop. A negative $\Theta$-dependent anomalous dimension is discovered as a novel effect of the UV-IR mixing. We also found a noncommutative Wilson-Fisher (NCWF) fixed point in less than four dimensions. At large noncommutativity, a momentum space instability is induced by quantum fluctuations, and a consequential first-order phase transition is identified together with a Lifshitz point in the phase diagram. In the vicinity of the Lifshitz point, we introduce two critical exponents $\nu_m$ and $\beta_k$, whose values are determined to be 1/4 and 1/2, respectively, at mean-field level.