Field theory on a non‐commutative plane: a non‐perturbative study

Abstract

The 2d gauge theory on the lattice is equivalent to the twisted Eguchi-Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1-and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non-commutative gauge theory, so the observed large N scaling demonstrates the non-perturbative renormalizability of this non-commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter.Next we investigate the 3d λφ 4 model with two non-commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d = 4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non-commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result.We also confirm the recent observation by Ambjørn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry.

The continuum phase diagram of the 2d non-commutative λϕ 4 model

“…That phase, which does not exist in the commutative λφ 4 model, was also discussed based on renormalisation group methods [7] and on the Cornwall-Jackiw-Tomboulis effective action [8,9]. In d = 3, more precisely in the case of a NC plane and a commutative Euclidean time direction, the existence of such a striped phase was observed explicitly in a non-perturbative, numerical study on the lattice [10][11][12], in agreement with the qualitative prediction of ref. [6].…”

“…Such a striped phase was also observed numerically in the 2d lattice model [10][11][12][13], but the survival of that phase in the DSL has never been clarified. One might suspect that it does not survive this limit, i.e.…”

“…The momentum value corresponding to the minimum E stabilises as one approaches a Double Scaling Limit (DSL), i.e. by taking the continuum (lattice spacing a → 0) and thermodynamic (physical volume and number of degrees of freedom tending to infinity) limits, at fixed non-commutativity tensor [10][11][12]. This shows that the striped phase does indeed exist in d = 3, which is a manifestation of the coexistence of UV and IR divergences.…”

“…To a very good accuracy, its location stabilises for N 19, if the phase diagram is plotted as a function of N 3/2 m 2 and N 2 λ. Note that this differs from the 3d case, where the choice of axes making the scaling properties manifest is N 2 m 2 and N 2 λ [10][11][12]. In d = 2, in the limit of very large quartic couplings -in which the JHEP10(2014)056 kinetic term (in the matrix formulation) becomes negligible -the theory reduces to a onematrix model, for which the boundary of the disordered phase is given bym 2 c = −2 √λ [39].…”

“…Also in the 3d case non-exponential correlations were observed within the NC plane [10][11][12]. However, in that case the decay was exponential in the third (commutative) direction, which did provide a correlation length, along with the aforementioned dispersion relation E( p).…”