On the relation of the entanglement spectrum to the bulk polarization

Abstract

The bulk polarization is a ${\mathbb{Z}}_2$ topological invariant characterizing non-interacting systems in one dimension with chiral or particle-hole symmetries. We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it. In the symmetric case, the known relation between the bulk polarization and the number of virtual topological edge modes is recovered. We use the bulk polarization to compute Chern numbers in 1D and 2D, which illuminates their known relation to the entanglement spectrum. Furthermore we discuss an alternative bulk polarization that can carry more information about the surface spectrum than the conventional one and can simplify the calculation of Chern numbers.

arXiv comment15 pages, 9 figures

References in corpus (27)

• Quantum Spin Hall Effect in Graphene
• Topological entanglement entropy
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Detecting topological order in a ground state wave function
• Quantized Electric Multipole Insulators
• Entropy and Area
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Calculation of reduced density matrices from correlation functions
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Zoo of quantum-topological phases of matter
• The Quantum-Mechanical Position Operator in Extended Systems
• Fractional Chern Insulator
• Entanglement spectrum of topological insulators and superconductors
• Entanglement entropy and entanglement spectrum of the Kitaev model
• Simplified topological invariants for interacting insulators
• Entanglement entropy and the Berry phase in solid states
• Many-Body Order Parameters for Multipoles in Solids
• Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond
• Trace Index and Spectral Flow in the Entanglement Spectrum
• AIII and BDI Topological Systems at Strong Disorder
• Topological Hamiltonian as an Exact Tool for Topological Invariants
• Flux insertion, entanglement, and quantized responses
• Entanglement spectrum and Wannier center flow of the Hofstadter problem
• Inequivalent Berry phases for the bulk polarization
• Edge States, Entanglement Spectra, and Wannier Functions in Haldane's Honeycomb Lattice Model and its Bilayer Generalization
• Entanglement in 3D Kitaev Spin Liquids

Cited by in corpus (6)

• Polarization and entanglement spectrum in non-hermitian systems
• Quantization of topological indices in critical chains at low temperatures
• Thermodynamics and entanglement entropy of the non-Hermitian SSH model
• Entanglement and particle fluctuations of one-dimensional chiral topological insulators
• Unified Bulk-Entanglement Correspondence in Non-Hermitian Systems
• From Hermitian critical to non-Hermitian point-gapped phases