Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies

Abstract

We show how to define a quantized many-body charge polarization $\vec{\mathcal{P}}$ for (2+1)D topological phases of matter, even in the presence of non-zero Chern number and magnetic field. For invertible topological states, $\vec{\mathcal{P}}$ is a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, ${\mathbb{Z}}_2$, or ${\mathbb{Z}}_1$ topological invariant in the presence of $M=2$, $3$, $4$, or $6$-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. $\vec{\mathcal{P}}$ manifests in the bulk of the system as (i) a fractional quantized contribution of $\vec{\mathcal{P}}\cdot \vec{b}\text{ mod 1}$ to the charge bound to lattice disclinations and dislocations with Burgers vector $\vec{b}$, (ii) a linear momentum for magnetic flux, and (iii) an oscillatory system size dependent contribution to the effective 1d polarization on a cylinder. We study $\vec{\mathcal{P}}$ in lattice models of spinless free fermions in a magnetic field. We derive predictions from topological field theory, which we match to numerical calculations for the effects (i)-(iii), demonstrating that these can be used to extract $\vec{\mathcal{P}}$ from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point $\text{o}$, there is a topological invariant, the discrete shift ${\mathcal{S}}_{\text{o}}$, such that $\vec{\mathcal{P}}$ specifies the dependence of ${\mathcal{S}}_{\text{o}}$ on $\text{o}$. We derive colored Hofstadter butterflies, corresponding to the quantized value of $\vec{\mathcal{P}}$, which further refine the colored butterflies from the Chern number and discrete shift.

25+16 pages, 10+11 figures, minor edits to the main text and the appendix

References in corpus (5)

• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Low-energy effective theory in the bulk for transport in a topological phase
• Electric polarization in a Chern insulator
• Fractional disclination charge and discrete shift in the Hofstadter butterfly

Cited by in corpus (19)

• Topological Phases of Photonic Crystals under Crystalline Symmetries
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Response to polarization and weak topology in Chern insulators
• 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
• Definition and Classification of Fermi Surface Anomalies
• Filling constraints on translation invariant dipole conserving systems
• Optimally Localized Wannier Functions for 2D Chern Insulators
• Many-body multipole index and bulk-boundary correspondence
• Polarization Jumps across Topological Phase Transitions in Two-dimensional Systems
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• 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
• Polarization-based indices in quantum many-body systems: validity and extension beyond one dimension
• Detection of 2D SPT Order with Partial Symmetries