"Light From Chaos" in Two Dimensions

Abstract

We perform a Monte-Carlo study of the lattice two-dimensional gauged XY-model. Our results confirm the strong-coupling expansion arguments that for sufficiently small values of the spin-spin coupling the "gauge symmetry breaking" terms decouple and the long-distance physics is that of the unbroken pure gauge theory. We find no evidence for the existence, conjectured earlier, of massless states near a critical value of the spin-spin coupling. We comment on recent remarks in the literature on the use of gauged XY-models in proposed constructions of chiral lattice gauge theories. I. MOTIVATION AND SUMMARYThe gauged two-dimensional lattice XY-model has been studied for quite some time. Its simplicity allows for analytic studies via lattice dualities and strong-coupling expansions [1] as well as for numerical analysis, in either a Hamiltonian [2] or Euclidean (see, e.g., [3]) formulation. The model exhibits many phenomena found in more complicated higher-dimensional theories, notably the presence of both confining and Higgs phases [1,2,4,5].The motivation for this short study stems from our interest in using the gauged XY-model, as well as its nonabelian higher-dimensional analogues [4,6,7], in lattice constructions of chiral gauge theories. The lattice formulation of chiral gauge theories is an outstanding problem with no practical solution yet, despite much recent progress; for reviews, see [8]. While not our topic here, we note that the recent constructions of [9,10] have some attractive features (for an earlier proposal of similar flavor, see [11]). Most importantly, they may offer a way around the difficult and unsolved problem of the explicit nonperturbative construction of the fermion measure for general chiral lattice gauge theories. Ref. [9] aims to achieve this by combining older ideas [12] to use non-gauge strong dynamics to decouple the mirror fermions in a vectorlike theory with the recently discovered exact lattice chirality of the Neuberger-Dirac operator.A concrete two-dimensional realization of the proposal [9] has the gauged XY-model as an essential part and was studied in various limits in [10]. Recently, [13] considered the perturbative spectrum of the models of [9] at nonzero gauge coupling. It was claimed there that the gauge boson is always massive and, therefore, the construction of [9, 10] was argued to be irrelevant to the study of unbroken chiral gauge theories, on account of this fact alone.This brings us to the main topic of this paper. The arguments of [13] do not invoke either the fermions or the strong mirror dynamics (admittedly, complicated and not yet completely understood), but concern only the spectrum of the gauged XY-model, arguing that it only * Electronic address: poppitz@physics.utoronto.ca † Electronic address: ywshang@physics.utoronto.ca has a phase with a massive gauge boson. This conclusion contradicts strong-coupling expansion arguments for the decoupling of the "gauge-breaking" terms for small values of the spin-spin coupling. As these arguments have been made many time...

On the decoupling of mirror fermions

“…In Section 3, we briefly review some theoretical developments that have appeared previously (for a more thorough discussion we refer the reader to the original references [8,46,[48][49][50]). Various exact (independent of the coupling) identities obeyed by the mirror polarization operator on the lattice were derived there and provide essential consistency checks on our computer code.…”

Chiral Lattice Gauge Theories via Mirror–fermion Decoupling: A Mission (Im)possible?

“…When the Yukawa couplings are absent, at y = 0, (38) is just the Schwinger model with a neutral Dirac spectator fermion and a unitary Higgs field of charge-1 and action S κ . We will be interested in the small-κ phase of the theory, when the unitary Higgs field fluctuates strongly and decouples at long distances (see 75 for a recent study). We have included both Majorana (yh) and Dirac (y) gauge invariant Yukawa terms in the mirror action, in order to explicitly break all global symmetries of the mirror fermions (except of the U (1) to be gauged, as explained in Section 3.3).…”

Why is the mission impossible? Decoupling the mirror Ginsparg–Wilson fermions in the lattice models for two-dimensional Abelian chiral gauge theories

“…14 To define the actual operators, one needs the point-splitting procedure because of the fermi statistics. A possible choice is the following: In [73][74][75][76][77][78][79][80], the 345 model is formulated by introducing all possible Dirac-and Majoranatype Yukawa couplings to the XY spin field in order to break the global continuous symmetries of the mirror sector.…”

“…76) where A, B stand for the anti-symmetrized indices A = [ab], B = [cd]. The link field of the U(1) gauge field can be set to unity, U (x, µ) = 1, from the beginning in the weak gauge-coupling limit.The two-point vertex function satisfies the Ward-Takahashi relations,For the gauge-variation η µ…”

“…49) Indeed, this type of the Majorana-Yukawa couplings are used in the formulation of the 345 model by Bhattacharya, Chen, Giedt, Poppitz and Shang in consideration [73][74][75][76][77][78][79][80]. However, again, the limit of the large Majorana mass parameter M is singular.…”