Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries

TL;DR

This paper introduces a real-space classification of Hofstadter phases using symmetry-protected invariants, revealing how magnetic flux induces reentrant symmetries and topological responses in band structures.

Contribution

It provides a novel real-space topological classification framework for Hofstadter phases, including the role of projective symmetries and Schur multipliers in flux responses.

Findings

Real-space invariants encode bulk and boundary responses.

Reentrant projective symmetries form due to flux periodicity.

Schur multipliers predict Hofstadter responses with zero-flux topology.

Abstract

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the non-perturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this work, we present a real-space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected Real Space Invariants (RSIs) which encode bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology.