Topological phases of the dimerized Hofstadter butterfly

TL;DR

This paper investigates how dimerization in a square lattice under a magnetic field creates various topological phases, including quadrupole insulators and Chern insulators, with phase transitions influenced by magnetic flux and filling fractions.

Contribution

It reveals the emergence of multiple topological phases in a dimerized Hofstadter model, including quadrupole and Chern insulators, and characterizes their phase transitions.

Findings

Existence of a quadrupole insulator phase at half-filling with quantized quadrupole moment.

Presence of Chern insulator phases at quarter and lower fillings with various Chern numbers.

Identification of both bulk- and boundary-obstructed topological phase transitions.

Abstract

In this work, we study the topological phases of the dimerized square lattice in the presence of an external magnetic field. The dimerization pattern in the lattice's hopping amplitudes can induce a series of bulk energy gap openings in the Hofstadter spectrum at certain fractional fillings, giving rise to various topological phases. In particular, we show that at $\frac{1}{2}$-filling the topological quadrupole insulator phase with a quadrupole moment quantized to $\frac{e}{2}$ and associated corner-localized mid-gap states exists in certain parameter regime for all magnetic fluxes. At $\frac{1}{4}$ filling, the system can host obstructed atomic limit phases or Chern insulator phases. For those configurations gapped at fillings below $\frac{1}{4}$, the system is in Chern insulator phases of various non-vanishing Chern numbers. Across the phase diagram, both bulk-obstructed and boundary-obstructed topological phase transitions exist in this model.