Year: 2005
Type: conference-paper
Abstract: We present a numerical study of the λφ 4 model in three Euclidean dimensions, where the two spatial coordinates are non-commutative (NC). We first show the explicit phase diagram of this model on a lattice. The ordered regime splits into a phase of uniform order and a “striped phase”. Then we discuss the dispersion relation, which allows us to introduce a dimensionful lattice spacing. Thus we can study a double scaling limit to zero lattice spacing and infinite volume, which keeps the non-commutativity parameter constant. The dispersion relation in the disordered phase stabilizes in this limit, which represents a non-perturbative renormalization. From its shape we infer that the striped phase persists in the continuum, and we observe UV/IR mixing as a non-perturbative effect. 1 The Non-Commutative Plane We consider a NC plane given by Hermitian coordinate operators ˆxµ, which obey [ˆxµ, ˆxν] = iθǫµν. (1) We impose a (fuzzy) lattice structure on this plane by means of the operator identity exp i 2π a ˆxµ = ˆ1. (2) Now the periodicity of the momentum components kµ implies θkµ/2a ∈ Z. (3) Hence the lattice is automatically periodic, say over the lattice volume N × N. Then the noncommutativity parameter corresponds to θ = Na 2 /π. (4) Since the star product does not affect the bilinear terms, λ determines the strength of NC effects. We consider this model in d = 3 with a commutative Euclidean time and a NC plane. Its formulation on a N3 lattice can be mapped onto a matrix model1 of the form
Authors: W. BIETENHOLZ, F. HOFHEINZ, J. NISHIMURA
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1485871323