# The Haldane model under nonuniform strain

## TL;DR

This paper investigates how nonuniform strain affects the Haldane model, revealing strain-induced pseudo-magnetic fields, Landau levels, and edge state modifications, with implications for topological quantum systems.

## Contribution

It provides a detailed analysis of strain effects on the Haldane model, including pseudo-Landau levels, edge state behavior, and topological transition modifications, highlighting differences from graphene.

## Key findings

- Strain induces pseudo-magnetic fields and Landau levels in the Haldane model.
- Strain lifts degeneracy of zeroth pseudo-Landau levels at different valleys.
- Edge state dispersion is affected differently in armchair and zigzag ribbons.

## Abstract

We study the Haldane model under strain using a tight-binding approach, and compare the obtained results with the continuum-limit approximation. As in graphene, nonuniform strain leads to a time-reversal preserving pseudo-magnetic field that induces (pseudo) Landau levels. Unlike a real magnetic field, strain lifts the degeneracy of the zeroth pseudo Landau levels at different valleys. Moreover, for the zigzag edge under uniaxial strain, strain removes the degeneracy within the pseudo-Landau levels by inducing a tilt in their energy dispersion. The latter arises from next-to-leading order corrections to the continuum-limit Hamiltonian, which are absent for a real magnetic field. We show that, for the lowest pseudo-Landau levels in the Haldane model, the dominant contribution to the tilt is different from graphene. In addition, although strain does not strongly modify the dispersion of the edge states, their interplay with the pseudo-Landau levels is different for the armchair and zigzag ribbons. Finally, we study the effect of strain in the band structure of the Haldane model at the critical point of the topological transition, thus shedding light on the interplay between non-trivial topology and strain in quantum anomalous Hall systems.