Abstract
A $U(1)$ gauge theory coupled to a Wilson fermion on a $2+1$ dimensional cubic lattice is known to exhibit Chern insulator like topological transitions as a function of the the ratio $M/R$ where $M$ is the fermion mass and $R$ is the Wilson parameter. I show that, with $M$ and $R$ held fixed, a rectangular lattice with anisotropic lattice spacing can exhibit distinct topological phases as a function of the lattice anisotropy. As a consequence, a $2+1$ dimensional lattice theory without any domain wall in the fermion mass can still exhibit chiral edge modes on a $1+1$ dimensional defect across which lattice spacing changes abruptly. Likewise, a domain wall in the fermion mass on a uniform rectangular lattice can exhibit discrete changes in the number and chirality of zero modes as a function of lattice anisotropy. The construction presented in this paper can be generalized to higher dimensional space-time lattices.
Highlights
The domain wall construction of chiral fermions in relativistic quantum field theories (QFT) [1,2] has close parallels with the physics of quantum Hall effect (QHE) [3,4,5,6,7]
The analysis of Wilson fermions on a rectangular lattice in this paper reveals new topological phases that are not accessible on a cubic lattice
One of the most interesting features of the analysis is associated with the edge modes where I find that the number of topologically protected edge modes does not always equal the number of zero mode solutions to the Dirac equation, i.e., the equations of motion can admit zero mode solutions of opposite chirality on a 1 þ 1 dimensional discontinuity
Summary
The domain wall construction of chiral fermions in relativistic quantum field theories (QFT) [1,2] has close parallels with the physics of quantum Hall effect (QHE) [3,4,5,6,7]. The 2 þ 1 dimensional version of this construction involves a fermion with a spatially varying mass of the form mεðx2Þ with εðx2 > 0Þ 1⁄4 1 and εðx2 < 0Þ 1⁄4 −1 coupled to a Uð1Þ gauge field. In the infrared this theory exhibits a chiral zero mode localized on the domain wall at x2 1⁄4 0. The bulk on the other hand exhibits a low energy Chern-Simons theory of level 1 on one side of the wall and level 0 on the other This mimics the physics of QHE, anomalous quantum Hall effect to be precise [8], where the QHE sample is described
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