Phase structure of non-commutative scalar field theories

Abstract

We investigate the phase structure of non-commutative scalar field theories and find evidence for ordered phases which break translation invariance. A self-consistent oneloop analysis indicates that the transition into these ordered phases is first order. The phase structure and the existence of scaling limits provides an alternative to the structure of counter-terms in determining the renormalizability of non-commutative field theories. On the basis of the existence of a critical point in the closely related planar theory, we argue that there are renormalizable interacting non-commutative scalar field theories in dimensions two and above. We exhibit this renormalization explicitly in the large N limit of a non-commutative O(N) vector model.

Stripes from (non-commutative) stars

“…Our current data do not address this issue -much larger volumes must be simulated in order to determine whether this phase truly survives the large N limit. Nevertheless, we feel that our simulations lend nonperturbative, albeit qualitative support to the analysis of [1] and [2]. However, it is clearly of more interest to do this in higher dimensional theories.…”

“…The possibility of having a ground state which breaks translational invariance was first observed in [1]. The quadratic part of the effective action, obtained from (1) by a one-loop self-consistent Hartree calculation, is…”

“…One of the more spectacular manifestations of this mixing is the appearance of a so-called striped phase which breaks translational invariance in noncommutative φ 4 theory. The prediction of this phase was based on a selfconsistent Hartree treatment of one-loop diagrams [1] or a one-loop renormalization group approach [2]. The purpose of the following note is to test the predictions using the non-perturbative framework of lattice computer simulations and at the same time test the viability of studying noncommutative field theories by numerical means.…”

Non-commutative Chern–Simons for the quantum Hall system and duality

“…Recent work on these interesting phases of the two-dimensional electron gas has revealed that magnetic symmetry plays a crucial role in low-energy dynamics of these states. Interestingly, recent work on non-commutative field theories has suggested that their phase diagrams involve analogs of these phases [41].…”

Fundamentals on the Noncommutative Plane

“…A second possibility is to let θ → 0 so thatθ remains fixed. This parameter is known to be an order parameter in the striping phase transition of scalar φ 4 theory in noncommutative space, [8] and it may have relevance in this system as well.…”