Invariance of topological indices under Hilbert space truncation

Abstract

We show that the topological index of a wavefunction, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of the resulting state, the invariant index may acquire a different physical interpretation. If the index is symmetry protected, the truncation should preserve the protecting symmetry. We discuss implications of this invariance using paradigmatic integer and fractional Chern insulators, $Z_2$ topological insulators, and Spin-$1$ AKLT and Heisenberg chains, as well as its relation with the notion of bulk entanglement. As a possible application, we propose a partial quantum tomography scheme from which the topological index of a generic multi-component wavefunction can be extracted by measuring only a small subset of wavefunction components, equivalent to the measurement of a bulk entanglement topological index.

arXiv comment11 pages including supplementary. v2: Minor text change, reference correction and update. v3: Text and SM update. To appear in Physical Review Letters

References in corpus (40)

• Topological Insulators
• Topological insulators and superconductors
• Many-Body Physics with Ultracold Gases
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• The density-matrix renormalization group in the age of matrix product states
• Classification of topological quantum matter with symmetries
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Topological Insulator Materials
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Entanglement spectrum of a topological phase in one dimension
• Calculation of reduced density matrices from correlation functions
• Quantum Orders and Symmetric Spin Liquids
• An equivalent expression of Z2 Topological Invariant for band insulators using Non-Abelian Berry's connection
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
• Inversion Symmetric Topological Insulators
• Experimental reconstruction of the Berry curvature in a topological Bloch band
• Robustness of the Spin-Chern number
• Ultracold atoms out of equilibrium
• Time-reversal-symmetry-broken quantum spin Hall effect
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Non-local order in gapless systems: Entanglement Spectrum in Spin Chains
• Tomography of band insulators from quench dynamics
• Spinless Topological Insulators without Time-Reversal Symmetry
• Non-Abelian Quantum Hall Effect in Topological Flat Bands
• Topological aspects of quantum spin Hall effect in graphene: Z${}_2$ topological order and spin Chern number
• Dynamical quantum phase transitions: Role of topological nodes in wavefunction overlaps
• Variational Monte Carlo study of chiral spin liquid in the extended Heisenberg model on the Kagome lattice
• Topological Classification of Gapped Spin Chains :Quantized Berry Phase as a Local Order Parameter
• Symmetry protected fractional Chern insulators and fractional topological insulators
• Entanglement spectrum and Wannier center flow of the Hofstadter problem
• Fractional Chern insulators with strong interactions far exceeding bandgaps
• Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State
• Fractional Chern Insulators from the nth Root of Bandstructure
• Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via Entanglement Entropy and Modular S-matrix
• Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology
• Entanglement Chern number for an extensive partition of a topological ground state
• Entanglement Spectra and Entanglement Thermodynamics of Hofstadter Bilayers
• Adiabatic continuity, wavefunction overlap and topological phase transitions
• Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum

Cited by in corpus (4)

• Non-Hermitian Pseudo-Gaps
• Demonstrating the AKLT spectral gap on 2D degree-3 lattices
• Emergent Non-Hermitian Edge Polarisation in an Hermitian Tight-binding Model
• A New Method of Solving Third Order Non-Linear Ordinary Complex Differential Equation by Generalizing Prelle-Singer Method