Fractional disclination charge and discrete shift in the Hofstadter butterfly

Abstract

In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to non-trivial quantized responses. Here we study a particular invariant, the discrete shift $\mathcal{S}$, for the square lattice Hofstadter model of free fermions. $\mathcal{S}$ is associated with a ${\mathbb{Z}}_M$ classification in the presence of $M$-fold rotational symmetry and charge conservation. $\mathcal{S}$ gives quantized contributions to (i) the fractional charge bound to a lattice disclination, and (ii) the angular momentum of the ground state with an additional, symmetrically inserted magnetic flux. $\mathcal{S}$ forms its own `Hofstadter butterfly', which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for $\mathcal{S}$ in terms of density and flux per plaquette for the Hofstadter bands, and we derive a number of general constraints. We show that bands with the same Chern number may have different values of $\mathcal{S}$, although odd and even Chern number bands always have half-integer and integer values of $\mathcal{S}$ respectively.

6+14 pages. Minor edits to the main text and the appendix

References in corpus (3)

• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Low-energy effective theory in the bulk for transport in a topological phase
• Classification of (2+1)D invertible fermionic topological phases with symmetry

Cited by in corpus (22)

• Topological Phases of Photonic Crystals under Crystalline Symmetries
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Response to polarization and weak topology in Chern insulators
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Symmetry indicators in commensurate magnetic flux
• Effective action approach to the filling anomaly in crystalline topological matter
• Definition and Classification of Fermi Surface Anomalies
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases
• Many-body multipole index and bulk-boundary correspondence
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
• Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux
• Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators
• Disclinations, dislocations, and emanant flux at Dirac criticality
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes
• Detection of 2D SPT Order with Partial Symmetries