Phase of the Wilson Line

This paper discusses the global $Z(N)$ symmetry of finite-temperature, $SU(N)$, pure Yang-Mills lattice gauge theory and the physics of the phase of the Wilson line expectation value. In the high $T$ phase, $\langle L \rangle$ takes one of $N$ distinct values proportional to the $Nth$ roots of unity in $Z(N)$, and the $Z(N)$ symmetry is broken. Only one of these is consistent with the usual interpretation $\langle L \rangle = e^{-F/T}$. This relation should be generalized to $\langle L \rangle = z e^{-F/T}$ with $z \in Z(N)$ so that it is consistent with the negative or complex values. In the Hamiltonian description, the {\em physical} variables are the group elements on the links of the spatial lattice. In a Lagrangian formulation, there are also group elements on links in the inverse-temperature direction from which the Wilson line is constructed. These are unphysical, auxiliary variables introduced to enforce the Gauss law constraints. The following results are obtained: The relation $\langle L \rangle=ze^{-F/T}$ is derived. The value of $z \in Z(N)$ is determined by the external field that is needed for the infinite-volume limit. There is a single physical, high-temperature phase, which is the same for all $z$. The global $Z(N)$ symmetry is not physical; it acts as the identity on all physical states. In the Hamiltonian formulation, the high-temperature phase is not distinguished by physical broken symmetry but rather by percolating flux.Comment: 24 pages, no figures, Latex/Revtex 3, UCD-94-2

Numerical results from large N reduced QCD_2

Some results in QCD_2 at large N are presented using the reduced model on the lattice. Overlap fermions are used to compute meson propagators.Comment: 3 pages, contribution to Lattice 2002, Bosto

Critical Exponent for the Density of Percolating Flux

This paper is a study of some of the critical properties of a simple model for flux. The model is motivated by gauge theory and is equivalent to the Ising model in three dimensions. The phase with condensed flux is studied. This is the ordered phase of the Ising model and the high temperature, deconfined phase of the gauge theory. The flux picture will be used in this phase. Near the transition, the density is low enough so that flux variables remain useful. There is a finite density of finite flux clusters on both sides of the phase transition. In the deconfined phase, there is also an infinite, percolating network of flux with a density that vanishes as $T \rightarrow T_{c}^{+}$. On both sides of the critical point, the nonanalyticity in the total flux density is characterized by the exponent $(1-\alpha)$. The main result of this paper is a calculation of the critical exponent for the percolating network. The exponent for the density of the percolating cluster is $\zeta = (1-\alpha) - (\varphi-1)$. The specific heat exponent $\alpha$ and the crossover exponent $\varphi$ can be computed in the $\epsilon$-expansion. Since $\zeta < (1-\alpha)$, the variation in the separate densities is much more rapid than that of the total. Flux is moving from the infinite cluster to the finite clusters much more rapidly than the total density is decreasing.Comment: 20 pages, no figures, Latex/Revtex 3, UCD-93-2

Adjoint Wilson Line in SU(2) Lattice Gauge Theory

The behavior of the adjoint Wilson line in finite-temperature, $SU(2)$, lattice gauge theory is discussed. The expectation value of the line and the associated excess free energy reveal the response of the finite-temperature gauge field to the presence of an adjoint source. The value of the adjoint line at the critical point of the deconfining phase transition is highlighted. This is not calculable in weak or strong coupling. It receives contributions from all scales and is nonanalytic at the critical point. We determine the general form of the free energy. It includes a linearly divergent term that is perturbative in the bare coupling and a finite, nonperturbative piece. We use a simple flux tube model to estimate the value of the nonperturbative piece. This provides the normalization needed to estimate the behavior of the line as one moves along the critical curve into the weak coupling region.Comment: 21 pages, no figures, Latex/Revtex 3, UCD-93-1

Topological Excitations in the Thirring model

The quantization of the massless Thirring model in the light-cone using functional methods is considered. The need to compactify the coordinate $x^-$ in the light-cone spacetime implies that the quantum effective action for left-handed fermions contains excitations similar to abelian instantons produced by composite of left-handed fermions. Right-handed fermions don't have a similar effective action. Thus, quantum mechanically, chiral symmetry must be broken as a result of the topological excitations. The conserved charge associated to the topological states is quantized. Different cases with only fermionic excitations or bosonic excitations or both can occur depending on the boundary conditions and the value of the coupling.Comment: title changed, one reference added, accepted in Phys. Lett.

Corrections to Scaling and Critical Amplitudes in SU(2) Lattice Gauge Theory

We calculate the critical amplitudes of the Polyakov loop and its susceptibility at the deconfinement transition of SU(2) gauge theory. To this end we carefully study the corrections to the scaling functions of the observables coming from irrelevant exponents. As a guiding line for determining the critical amplitudes we use envelope equations derived from the finite size scaling formulae for the observables. The equations are then evaluated with new high precision data obtained on N^3 x 4 lattices for N=12,18,26 and 36. We find different correction-to-scaling behaviours above and below the transition. Our result for the universal ratio of the susceptibility amplitudes is C_+/C_-=4.72(11) and agrees perfectly with a recent measurement for the 3d Ising model.Comment: LATTICE98(hightemp

The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments we show that the critical exponent $\nu$ describing the vanishing of the physical mass at the critical point is equal to $\nu_\theta/ d_w$. $d_w$ is the Hausdorff dimension of the walk. $\nu_\theta$ is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case of O(N) models, we show that $\nu_\theta=\varphi$, where $\varphi$ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is $\varphi/\nu$ for O(N) models.Comment: 11 pages (plain TeX

Large N reduction with overlap fermions

We revisit quenched reduction with fermions and explain how some old problems can be avoided using the overlap Dirac operator.Comment: Lattice2002(chiral) 3 pages, no figure

Two dimensional fermions in three dimensional YM

Dirac fermions in the fundamental representation of SU(N) live on the surface of a cylinder embedded in $R^3$ and interact with a three dimensional SU(N) Yang Mills vector potential preserving a global chiral symmetry at finite $N$. As the circumference of the cylinder is varied from small to large, the chiral symmetry gets spontaneously broken in the infinite $N$ limit at a typical bulk scale. Replacing three dimensional YM by four dimensional YM introduces non-trivial renormalization effects.Comment: 21 pages, 7 figures, 5 table

Flavor Twisted Boundary Conditions, Pion Momentum, and the Pion Electromagnetic Form Factor

We investigate the utility of partially twisted boundary conditions in lattice calculations of meson observables. For dynamical simulations, we show that the pion dispersion relation is modified by volume effects. In the isospin limit, we demonstrate that the pion electromagnetic form factor can be computed on the lattice at continuous values of the momentum transfer. Furthermore, the finite volume effects are under theoretical control for extraction of the pion charge radius.Comment: 15 pages, 8 figures, revisions to text, refs adde