Twisted Eguchi-Kawai Research Articles

Numerical stochastic perturbation theory applied to the twisted Eguchi-Kawai model

We present the results of an exploratory study of the numerical stochastic perturbation theory (NSPT) applied to the four dimensional twisted Eguchi-Kawai (TEK) model. We employ a Kramers type algorithm based on the Generalized Hybrid Molecular Dynamics (GHMD) algorithm. We have computed the perturbative expansion of square Wilson loops up to O(g8). The results of the first two coefficients (up to O(g4)) have a high precision and match well with the exact values. The next two coefficients can be determined and even extrapolated to large N, where they should coincide with the corresponding coefficients for ordinary Yang-Mills theory on an infinite lattice. Our analysis shows the behaviour of the probability distribution for each coefficient tending to Gaussian for larger N. The results allow us to establish the requirements to extend this analysis to much higher order.

The two-dimensional twisted reduced principal chiral model revisited

Motivated by our previous study of the Twisted Eguchi-Kawai model for non minimal twists, we re-examined the behaviour of the reduced version of the two dimensional principal chiral model. We show that this single matrix model reproduces the same features as the standard lattice model. In particular, scaling towards the continuum limit, the correct value of the internal energy, the magnetic susceptibility and the mass gap. Given our capacity to reach larger values of N, we use the reduced model to study the nature and properties of its large N phase transition existing at intermediate coupling. We conclude that the transition is of first order.

Large N meson masses from a matrix model

We explain how to compute meson masses in the large $N$ limit using the twisted Eguchi-Kawai model. A very simple formula is derived, and we show how it leads in a fast and efficient way to results which are in fairly good agreement with other determinations. The method is easily extensible to reduced models with dynamical fermions based on the twisted reduction idea.

The SU(∞) twisted gradient flow running coupling

We measure the running of the SU(∞) ’t Hooft coupling by performing a step scaling analysis of the Twisted Eguchi-Kawai (TEK) model, the SU(N) gauge theory on a single site lattice with twisted boundary conditions. The computation relies on the conjecture that finite volume effects for SU(N) gauge theories defined on a 4-dimensional twisted torus are controlled by an effective size parameter $$ \tilde{l}=l\sqrt{N} $$ , with l the torus period. We set the scale for the running coupling in terms of $$ \tilde{l} $$ and use the gradient flow to define a renormalized ’t Hooft coupling $$ \lambda \left(\tilde{l}\right) $$ . In the TEK model, this idea allows the determination of the running of the coupling through a step scaling procedure that uses the rank of the group as a size parameter. The continuum renormalized coupling constant is extracted in the zero lattice spacing limit, which in the TEK model corresponds to the large N limit taken at fixed value of $$ \lambda \left(\tilde{l}\right) $$ . The coupling constant is thus expected to coincide with that of the ordinary pure gauge theory at N = ∞. The idea is shown to work and permits us to follow the evolution of the coupling over a wide range of scales. At weak coupling we find a remarkable agreement with the perturbative two-loop formula for the running coupling.

Large N reduction with the twisted Eguchi-Kawai model

We examine the breaking of Z N symmetry recently reported for the Twisted Eguchi-Kawai model (TEK). We analyse the origin of this behaviour and propose simple modifications of twist and lattice action that could avoid the problem. Our results show no sign of symmetry breaking and allow us to obtain values of t he large N infinite volume string tension in agreement with extrapolations from results based upon straightforward methods.

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.

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

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

Classical Solutions of the TEK Model and Noncommutative Instantons in Two Dimensions

The twisted Eguchi-Kawai (TEK) model provides a non-perturbative definition of noncommutative Yang-Mills theory: the continuum limit is approached at large $N$ by performing suitable double scaling limits, in which non-planar contributions are no longer suppressed. We consider here the two-dimensional case, trying to recover within this framework the exact results recently obtained by means of Morita equivalence. We present a rather explicit construction of classical gauge theories on noncommutative toroidal lattice for general topological charges. After discussing the limiting procedures to recover the theory on the noncommutative torus and on the noncommutative plane, we focus our attention on the classical solutions of the related TEK models. We solve the equations of motion and we find the configurations having finite action in the relevant double scaling limits. They can be explicitly described in terms of twist-eaters and they exactly correspond to the instanton solutions that are seen to dominate the partition function on the noncommutative torus. Fluxons on the noncommutative plane are recovered as well. We also discuss how the highly non-trivial structure of the exact partition function can emerge from a direct matrix model computation. The quantum consistency of the TEK formulation is eventually checked by computing Wilson loops in a particular limit.

Non-commutative Principal Chiral Models

Twisted Eguchi-Kawai reduced chiral models are shown to be formally equivalent to a U(1) non-commutative parent theory. The non-commutative theory describes the vacuum dynamics of the non-commutative charged tachyonic field of a brane system. To make contact with the continuum non-commutative theory, a double scaling large N limit for the reduced model is required. We show a possible limiting procedure, which we propose to investigate numerically. Our numerical results show substantial consistency with the outlined procedure

The Renormalizability of Gauge Theory on a Non-Commutative Plane

We perform a non-perturbative study of pure gauge theory in a two dimensional non-commutative (NC) space. On the lattice, it 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. Based on a Morita equivalence, the large-N double scaling limit corresponds to the continuum limit of NC gauge theory, so the observed large-N scaling demonstrates the non-perturbative renormalizability of this NC 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.

Twisted Eguchi-Kawai Reduced Chiral Models

We study the twisted Eguchi-Kawai (TEK) reduction procedure for large-N unitary matrix lattice models. In particular, we consider the case of two-dimensional principal chiral models, and use numerical Monte Carlo (MC) simulations to check the conjectured equivalence of TEK reduced model and standard lattice model in the large-N limit. The MC results are compared with the large-N limit of lattice principal chiral models to verify the supposed equivalence. The consistency of the TEK reduction procedure is verified in the strong-coupling region, i.e. for $\beta<\beta_c$ where $\beta_c$ is the location of the large-N phase transition. On the other hand, in the weak-coupling regime $\beta>\beta_c$, relevant for the continuum limit, our MC results do not support the equivalence of the large-N limits of the lattice chiral model and the corresponding TEK reduction. The implications for the correspondence between TEK model and noncommutative field theory are also discussed.

The large- N phase transition of lattice SU( N) gauge theories

We investigate the large- N phase transition of lattice SU( N) gauge theories in the Wilson formulation, by performing a Monte Carlo simulation of the twisted Eguchi-Kawai model. A variant of the multicanonical algorithm allows a detailed exploration of the phase transition and a precise determination of the transition temperature.

A Solvable Twisted One-Plaquette Model

We solve a hot twisted Eguchi-Kawai model with only timelike plaquettes in the deconfined phase, by computing the quadratic quantum fluctuations around the classical vacuum. The solution of the model has some novel features: the eigenvalues of the timelike link variable are separated in L bunches, if L is the number of links of the original lattice in the time direction, and each bunch obeys a Wigner semicircular distribution of eigenvalues. This solution becomes unstable at a critical value of the coupling constant, where it is argued that a condensation of classical solutions takes place. This can be inferred by comparison with the heat-kernel model in the Hamiltonian limit, and the related Douglas–Kazakov phase transition in QCD2. As a byproduct of our solution, we can reproduce the dependence of the coupling constant from the parameter describing the asymmetry of the lattice, and compare it to previous results by Karsch.

Extrema of the twisted Eguchi-Kawai action and the finite Heisenberg group

Abstract We show that representations of the finite Heisenberg group are related to fluxons, which are special configurations of the field in the twisted Eguchi-Kawai model. Fluxons correspond to extrema of the action and in particular to the ground state of this model. We construct and classify these configurations in an even-dimensional space using standard methods of group theory. We also derive the necessary and sufficient conditions for the existence and uniqueness of such configurations.

The complete set of twists for the twisted Eguchi-Kawai model

We solve the twist problem of the Eguchi-Kawai model completely and present a large class of new twists. Some of these new twists are useful for economic Monte-Carlo simulations of theSU(N) model for largeN.

The large- N deconfinement phase transition in a partially reduced Eguchi-Kawal model

We construct generalized twisted Eguchi-Kawai models which for large- N reduce space-time to a lattice of arbitrary size. Large- N lattice gauge theory at finite temperature is investigated in a model on a lattice with L 0 time slices and two lattice points in very time slice. We observe the large- N deconfinement phase transition in the weak coupling region. Assuming asymptotic scaling we find a transition temperature T c = (101±4) Λ L .

Monte Carlo renormalization group in the large-N limit.

The use of generalized twisted Eguchi-Kawai models is proposed to determine the ..beta.. function of large-N lattice gauge theory.

On the deconfining transition of SU(∞) gauge theory

Monte Carlo simulations for twisted Eguchi-Kawai models with asymmetric couplings have been used to investigate the large-N deconfining phase transition of pure SU(N) gauge theories. Data with N = 25, 36, 49 and 64 and the variable asymmetry parameter ξ allow us to disentangle these theories' bulk first-order transition from their physical deconfining transition. The SU(64) data is in good agreement with asymptotic freedom and gives a critical temperature of Tc/ΛE = 1 118 ± 6 or Tc/√σ = 0.42 + 0.05, which is close to the SU(3) ratio of deconfining temperature to the string tension. The data also suggest that the transition is first order.

Weak coupling analysis of the finite-temperature twisted Eguchi-Kawai model

We show that with a suitable twist the finite-temperature twisted Eguchi-Kawai model reproduces the planar perturbation expansion of the full Wilson theory in the weak coupling sector.

SU (∞) gauge theories on asymmetric lattices

We study SU (∞) gauge theory on an asymmetric lattice using a generalisation of the twisted Eguchi-Kawai model. We show that it is possible to remove the large- N bulk transition by choosing a sufficiently asymmetric lattice and hence to study the physical deconfinement transition without being affected by the former. Results for N = 16 indicate a strong first order deconfinement transition.