Going beyond k.p theory: a general method for obtaining effective Hamiltonians in both high and low symmetry situations

TL;DR

This paper introduces a versatile method for deriving effective Hamiltonians that surpasses traditional k.p theory, applicable to systems with varying symmetry, and demonstrates its effectiveness on diverse low-dimensional materials including graphene variants.

Contribution

The authors develop an exact mapping from two-centre tight-binding to continuum Hamiltonians, enabling analysis of high and low symmetry systems with systematic correction capabilities.

Findings

Derived effective Hamiltonians for various 2D materials.

Identified charge pooling and localized current states in dislocation networks.

Provided a unified framework for describing stacking deformations in bilayer graphene.

Abstract

We provide a method for the generation of effective continuum Hamiltonians that goes beyond the well known k.p method in being equally effective in both high, and low (or no) symmetry situations. Our approach is based on a surprising exact map of the two-centre tight-binding method onto a compact continuum Hamiltonian, with a precise condition given for the hermiticity of the latter object. We apply this method to a broad range of low dimensional systems of both high and low symmetry: graphene, graphdiyne, {\gamma}-graphyne, 6,6,12-graphyne, twist bilayer graphene, and partial dislocation networks in Bernal stacked bilayer graphene. For the single layer systems the method yields Hamiltonians for the ideal lattices, as well as a systematic theory for corrections due to deformation. In the case of bilayer graphene we provide a compact expression for an effective field capable of describing any stacking deformation of the bilayer; twist bilayer graphene, as well as the partial dislocation network in AB stacked graphene, emerge as special cases of this field. For the latter system we find (i) charge pooling on the mosaic of AB and AC segments near the Dirac point and (ii) localized current carrying states on the partials with the current density characterized by both intralayer and interlayer components.