Knot theory for two-band model of two-dimensional square lattice with high topological numbers

TL;DR

This paper introduces a novel knot theory approach to analyze two-dimensional square lattice models, linking topological invariants to knot linking numbers, and enabling the design of materials with higher topological numbers.

Contribution

It reinterprets the topological invariant as a Gauss linking number and extends the two-band model to realize higher topological numbers.

Findings

P = 0, ±1 topological numbers are derived

Knot theory provides a new perspective on topological invariants

Modified models achieve higher topological numbers, including ±2

Abstract

A knot theory for two-dimensional square lattice is proposed, which sheds light on design of new two-dimensional material with high topological numbers. We consider a two-band model, focusing on the Hall conductance {\sigma}xy = e^2/hbar*P, where P is a topological number, the so-called Pontrjagin index. By re-interpreting the periodic momentum components kx and ky as the string parameters of two entangled knots, we discover that P becomes the Gauss linking number between the knots. This leads to a successful re-derivation of the typical P-evaluations in literature: P = 0;{\pm}1. Furthermore, with the aid of this explicit knot theoretical picture we modify the two-band model to achieve higher topological numbers, P = 0;{\pm}1;{\pm}2.