Flat bands in tight-binding lattices with anisotropic potentials

TL;DR

This paper constructs anti-T symmetric Hamiltonians with flat bands in tight-binding models on Bravais lattices, revealing localization transitions and contrasting with traditional flat band states.

Contribution

It introduces a novel method to realize flat bands with anisotropic potentials and analyzes their localization properties, expanding understanding of flat band phenomena.

Findings

Flat bands are achieved by tuning hoppings and potential shapes.

Eigenstates show localization transition along the potential direction.

Unbounded potentials lead to always localized eigenstates.

Abstract

We consider tight-binding models on Bravais lattices with anisotropic onsite potentials that vary along a given direction and are constant along the transverse one. Inspired by our previous work on flat bands in anti-\(\mathcal{PT}\) symmetric Hamiltonians [Mallick et al., Phys.~Rev.~A 105, L021305 (2022)], we construct an anti-\(\mathcal{PT}\) symmetric Hamiltonians with an \(E=0\) flat band by tuning the hoppings and the shapes of potentials. This construction is illustrated for the square lattice with bounded and unbounded potentials. Unlike flat bands in short-ranged translationally invariant Hamiltonians, we conjecture that the considered \(E=0\) flat bands do not host compact localized states. Instead the flat-band eigenstates exhibit a localization transition along the potential direction upon increasing the potential strength for bounded potentials. For unbounded potentials flat-band eigenstates are always localized irrespective of the potential strength.