Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly

Abstract

The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here we show how to extract a set of many-body invariants $\{{\Theta }_{\text{o}}^{\pm }\}$, where $\text{o}$ is a high symmetry point, from partial rotations in (2+1)D invertible fermionic states. Our results apply in the presence of magnetic field and Chern number $C\ne 0$, in contrast to previous work. $\{{\Theta }_{\text{o}}^{\pm }\}$ together with $C$, chiral central charge $c_{-}$, and filling $\nu$ provide a complete many-body characterization of the topological state with symmetry group $G=\text{U}(1){\times }_{\varphi }[{\mathbb{Z}}^2⋊{\mathbb{Z}}_M]$. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where $G$-crossed modular $S,T$ matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization.

arXiv comment6+12 pages, 3+8 figures, minor edits to the main text and the appendix

References in corpus (38)

• Classification of topological quantum matter with symmetries
• Topological entanglement entropy
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Detecting topological order in a ground state wave function
• Periodic table for topological insulators and superconductors
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Topological quantum chemistry
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological classification of crystalline insulators through band structure combinatorics
• Symmetry Protected Topological phases of Quantum Matter
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Magnetic Topological Quantum Chemistry
• Topological characterization of fractional quantum Hall ground states from microscopic Hamiltonians
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Building crystalline topological phases from lower-dimensional states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Momentum polarization: an entanglement measure of topological spin and chiral central charge
• Many-body topological invariants for fermionic symmetry-protected topological phases
• Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Many-body Chern number from statistical correlations of randomized measurements
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Chiral central charge from a single bulk wave function
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Detecting two dimensional symmetry protected topological order in a ground state wave function
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Extraction of many-body Chern number from a single wave function
• Fermionic symmetry fractionalization in (2+1)D
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Rotation Symmetry-Protected Topological Phases of Fermions
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D

Cited by in corpus (12)

• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order
• Disclinations, dislocations, and emanant flux at Dirac criticality
• Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes
• Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux
• Detection of 2D SPT Order with Partial Symmetries