Absence of vortex condensation in a two dimensional fermionicXYmodel

Abstract

Motivated by a puzzle in the study of two dimensional lattice Quantum Electrodynamics with staggered fermions, we construct a two dimensional fermionic model with a global U (1) symmetry. Our model can be mapped into a model of closed packed dimers and plaquettes. Although the model has the same symmetries as the XY model, we show numerically that the model lacks the well known Kosterlitz-Thouless phase transition. The model is always in the gapless phase showing the absence of a phase with vortex condensation. In other words the low energy physics is described by a non-compact U (1) field theory. We show that by introducing an even number of layers one can introduce vortex condensation within the model and thus also induce a KT transition.PACS numbers: I. MOTIVATIONTwo dimensional lattice Quantum Electrodynamics continues to be of interest today as a test bed for ideas and algorithms for lattice QCD [1,2,3,4,5]. In this work we focus on the formulation with staggered fermions, and refer to it as LQED2 [6]. In the continuum limit, the theory is expected to describe the two-flavor Schwinger model [7]. With massless fermions the twoflavor Schwinger model contains an SU (2)× SU (2) chiral symmetry. Away from the continuum limit, finite lattice spacing effects in LQED2 break the chiral symmetry to a U (1) subgroup. In the mean field approximation this symmetry is spontaneously broken. However, since in two dimensions strong infrared fluctuations forbid spontaneous symmetry breaking, the mean field result is modified [8,9]: Instead the theory develops critical long range (gapless) correlations, which can be detected through the chiral condensate susceptibilitywhere ψ i and ψ i are the staggered fermion fields at the site i on a square lattice. In general one expects χ ∼ AL 2−η in the gapless phase and χ ∼ B when the theory develops a mass gap, where A and B are constants. In the case of a U (1) symmetric theory, the mass gap can be generated only due to vortex condensation. In LQED2, at strong couplings one finds that Eq. (1) holds with η = 0.5 [10]. On the other hand in the continuum limit, using the result in the two-flavor Schwinger model, one expects η = 1. Since previous studies have shown no evidence of a phase transition from strong to weak gauge couplings one might conjecture that the theory is always in the gapless phase with η varying smoothly from 0.5 to 1.0 as a function of the gauge coupling. As far as we know this has not yet been shown analytically or observed numerically. Although at first glance there does not seem to be anything strange with the above scenario, a closer examination reveals an interesting puzzle. Away from the continuum limit we expect the low energy physics of LQED2 to be described by a U (1) symmetric bosonic field theory since the fermions are confined. Is this a compact or a non-compact U (1) field theory? A compact U (1) theory will contain vortices, just like the XY model, and these vortices can condense. Thus, it can undergo the famous Kosterlitz-Thouless(KT) phase transition, ...

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