A Novel Non-Perturbative Lattice Regularization of an Anomaly-Free $1 + 1d$ Chiral $SU(2)$ Gauge Theory

Abstract

We present a numerical treatment of a novel non-perturbative lattice regularization of a 1 + 1d SU (2) Chiral Gauge Theory. Our approach follows recent proposals that exploit the newly discovered connection between anomalies and topological (or entangled) states to show how to create a lattice regularization of any anomaly-free chiral gauge theory. In comparison to other methods, our regularization enjoys on-site fermions and gauge action, as well as a physically transparent fermion Hilbert space. We follow the 'mirror fermion' approach, in which we first create a lattice regularization of both the chiral theory and its mirror conjugate and then introduce interactions that gap out only the mirror theory. The connection between topological states and anomalies shows that such interactions exist if the chiral theory is free of all quantum anomalies. Instead of numerically intractable fermion-fermion interactions, we couple the mirror theory to a Higgs field driven into a symmetry-preserving, disordered, gapped phase.

Metallic states beyond Tomonaga-Luttinger liquids in one dimension

“…Following the ideas in Ref. 52, we can also construct a strongly interacting metallic phase of spin-1/2 electrons where the left-movers and right-movers have very different behavior. We will call such metallic phases chiral metallic phases.…”

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

“…Since the 2+1D domain wall fermion is nothing but 2+1D classical formulation of a 2D Topological Insulator [126][127][128] (Chern Insulator/IQHE without time-reversal symmetry; class A in 2D classified by Z), our result here provides the explicit procedure to bridge between the two constructions for 1+1D chiral gauge theories, the 2+1D classical construction of domain wall fermion with boundary interactions to decouple the mirrormodes [114] and the 2D quantum Hamiltonian construction of TI/TSC with gapped boundary phases [109][110][111][112][113]. And the mirror-fermion model in terms of overlap fermions is obtained precisely as the 1+1D low-energy effective local lattice theory, and it can describe directly the gapless/gapped boundary phases.…”

“…On the other hand, this question of decoupling the mirror degrees of freedoms in the 345 model was also studied by Wang and Wen [110] from the point of view of the Hamiltonian construction based on Topological Insulators/Superconductors [109][110][111][112][113]: based on the effective bosonic (bosonized) description of the 2D Chern Insulator by the bulk Chern-Simon gauge theory and the boundary chiral-boson theory with sine-Goldon couplings, it was shown that the boundary phase can be fully gapped in the 345(0) model by the two sine-Goldon couplings required precisely to break the two "would-be anomalous" global U(1) symmetries in the mirror sector. This result suggests that the mirror fermions can be decoupled by the suitable choice of the coupling strengths of the symmetry-breaking interactions.…”

Comments on Symmetric Mass Generation in 2d and 4d

“…In that setting, the goal is to find a way to gap the fermion doublers, leaving the original fermions unscathed, in the hope of finding a route to constructing chiral gauge theories on the lattice. This idea was first proposed by Eichten and Preskill [15] and interesting variants on the theme have been explored by a number of different authors over the years [16][17][18][19][20][21][22][23]. Nearly all of these approaches rely on strong coupling effects at the lattice scale, where irrelevant operators can drive the low energy dynamics.…”