Frank Hofheinz

Phase diagram and dispersion relation of the noncommutative lambda phi**4 model in d = 3

We present a non-perturbative study of the ?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 hermitean matrix model. In this way we first reveal the explicit phase diagram of the non-commutative ?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.

A non-perturbative study of gauge theory on a non-commutative plane

Abstract:We performa non-perturbativestudyofpuregauge theoryin atwo dimensionalnon-commutative (NC) space. On the lattice, it is equivalent to the twisted Eguchi-Kawaimodel, which we simulated at N ranging from 25 to 515. We observe a clear large-N scalingfor the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakovlines. The 2-point functions agree with a universal wave function renormalization. Based ona Morita equivalence, the large-N double scaling limit corresponds to the continuum limitof NC gauge theory, so the observed large-N scaling demonstrates the non-perturbativerenormalizability of this NC field theory. The area law for the Wilson loops holds at smallphysical area as in commutative 2d planar gauge theory, but at large areas we find anoscillating behavior instead. In that regime the phase of the Wilson loop grows linearlywith the area. This agrees with the Aharonov-Bohm effect in the presence of a constantmagnetic field, identified with the inverse non-commutativity parameter.Keywords: Non-Commutative Geometry, Matrix Models, Lattice Gauge Field Theories,Field Theories in Lower Dimensions.

Non-commutative field theories beyond perturbation theory

We investigate two models in non-commutative (NC) field theory by means of Monte Carlo simulations. Even if we start from the Euclidean lattice formulation, such simulations are only feasible after mapping the systems onto dimensionally reduced matrix models. Using this technique, we measure Wilson loops in 2d NC gauge theory of rank 1. It turns out that they are non-perturbatively renormalizable, and the phase follows an Aharonov-Bohm effect if we identify θ = 1/B. Next we study the 3d λ ϕ4 model with two NC coordinates, where we present new results for the correlators and the dispersion relation. We further reveal the explicit phase diagram. The ordered regime splits into a uniform and a striped phase, as it was qualitatively conjectured before. We also confirm the recent observation by Ambjo rn and Catterall that such stripes occur even in d = 2, although they imply the spontaneous breaking of translation symmetry. However, in d = 3 and d = 2 we observe only patterns of two stripes to be stable in the range of parameters investigated.

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

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 λϕ4 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 Ambjo rn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry.

First Simulation Results for the Photon in a Non-Commutative Space

We present preliminary simulation results for QED in a non-commutative 4d space-time, which is discretized to a fuzzy lattice. Its numerical treatment becomes feasible after its mapping onto a dimensionally reduced twisted Eguchi-Kawai matrix model. In this formulation we investigate the Wilson loops and in particular the Creutz ratios. This is an ongoing project which aims at non-perturbative predictions for the photon, which can be confronted with phenomenology in order to verify the possible existence of non-commutativity in nature.

Numerical results on the non-commutative πθ4 model☆

The UV/IR mixing in the πθ4 model on a non-commutative (NC) space leads to new predictions in perturbation theory, including Hartree-Fock type approximations. Among them there is a changed phase diagram and an unusual behavior of the correlation functions. In particular this mixing leads to a deformation of the dispersion relation. We present numerical results for these effects in d = 3 with two NC coordinates.

The Renormalizability of Gauge Theory on a Non-Commutative Plane

Abstract:We performa non-perturbativestudyofpuregauge theoryin atwo dimensionalnon-commutative (NC) space. On the lattice, it is equivalent to the twisted Eguchi-Kawaimodel, which we simulated at N ranging from 25 to 515. We observe a clear large-N scalingfor the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakovlines. The 2-point functions agree with a universal wave function renormalization. Based ona Morita equivalence, the large-N double scaling limit corresponds to the continuum limitof NC gauge theory, so the observed large-N scaling demonstrates the non-perturbativerenormalizability of this NC field theory. The area law for the Wilson loops holds at smallphysical area as in commutative 2d planar gauge theory, but at large areas we find anoscillating behavior instead. In that regime the phase of the Wilson loop grows linearlywith the area. This agrees with the Aharonov-Bohm effect in the presence of a constantmagnetic field, identified with the inverse non-commutativity parameter.Keywords: Non-Commutative Geometry, Matrix Models, Lattice Gauge Field Theories,Field Theories in Lower Dimensions.

Perfect actions: insights in the lattice fermion problem☆

Abstract A revamping of the standard renormalization group approaches leads to an explicit expression of lattice formulations for arbitrary bilinear fermion actions, which can be further exploited to define new improvement methods and Chiral Gauge Theories on the lattice. In this setup the statement can be made that any fermionic action can be considered to be a Ginsparg-Wilson action or even a Perfect Action, with all associated consequences conserved. Here we present the formalism and a simple example in d = 2.

On the quantum geometry of string theory

Abstract The IKKT or IIB matrix model has been proposed as a non-perturbative definition of type IIB superstring theories. It has the attractive feature that space-time appears dynamically. It is possible that lower dimensional universes dominate the theory, therefore providing a dynamical solution to the reduction of space-time dimensionality. We summarize recent works that show the central role of the phase of the fermion determinant in the possible realization of such a scenario.