Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models

TL;DR

This paper introduces a generalized transfer matrix formalism for noninteracting tight-binding models, enabling analysis of bulk and edge states, including in cases with non-invertible hopping matrices, and connects topological invariants to edge state windings.

Contribution

It develops a novel transfer matrix approach applicable even when the hopping matrix is non-invertible, extending the analysis of topological properties in lattice models.

Findings

Constructed transfer matrices for various lattice models including Chern insulators and graphene.

Linked edge state windings to bulk Chern numbers via topological invariants.

Described the symplectic nature of transfer matrices and related windings to Maslov indices.

Abstract

We construct a generalized transfer matrix corresponding to noninteracting tight-binding lattice models, which can subsequently be used to compute the bulk bands as well as the edge states. Crucially, our formalism works even in cases where the hopping matrix is non-invertible. Following Hatsugai [PRL 71, 3697 (1993)], we explicitly construct the energy Riemann surfaces associated with the band structure for a specific class of systems which includes systems like Chern insulator, Dirac semimetal and graphene. The edge states can then be interpreted as non-contractible loops, with the winding number equal to the bulk Chern number. For these systems, the transfer matrix is symplectic, and hence we also describe the windings associated with the edge states on $Sp(2, \mathbb{R})$ and interpret the corresponding winding number as a Maslov index.