Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems

Abstract

This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems, rather than considering a single Hamiltonian, one considers a family of Hamiltonians that depend continuously on some parameters. After discussing the notion of phases of parametrized systems, we formulate a bulk-boundary correspondence for an important bulk quantity, the Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and then in arbitrary dimension. This clarifies the physical interpretation of the higher Berry curvature, which in one spatial dimension is a flow of (ordinary) Berry curvature. In d dimensions, the higher Berry curvature is a flow of (d-1)-dimensional higher Berry curvature. Based on this, we discuss one-dimensional systems that pump Chern number to/from spatial boundaries, resulting in anomalous boundary modes featuring isolated Weyl points. In higher dimensions, there are pumps of the analogous quantized invariants obtained by integrating the higher Berry curvature. We also discuss the consequences for parametrized systems of Kitaev's proposal that invertible phases are classified by a generalized cohomology theory, and emphasize the role of the suspension isomorphism in generating new examples of parametrized systems from known invertible phases. Finally, we present a pair of general quantum pumping constructions, based on physical pictures introduced by Kitaev, which take as input a d-dimensional parametrized system, and produce new (d+1)-dimensional parametrized systems. These constructions are useful for generating examples, and we conjecture that one of the constructions realizes the suspension isomorphism in a generalized cohomology theory of invertible phases.

arXiv comment35 pages, 11 figures. v2: references added

References in corpus (30)

• Topological Insulators
• Topological insulators and superconductors
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Zoo of quantum-topological phases of matter
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Three-Dimensional Topological Insulators
• Symmetry Protected Topological phases of Quantum Matter
• Fractons
• Theta, Time Reversal, and Temperature
• Fermion Path Integrals And Topological Phases
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Fracton Phases of Matter
• Topology and Broken Symmetry in Floquet Systems
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• Reflection positivity and invertible topological phases
• Gauging spatial symmetries and the classification of topological crystalline phases
• Comments on Abelian Higgs Models and Persistent Order
• Walls, Anomalies, and (De)Confinement in Quantum Anti-Ferromagnets
• Symmetry Protected Topological phases and Generalized Cohomology
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Anomalies of Duality Groups and Extended Conformal Manifolds
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Higher-dimensional generalizations of the Berry curvature
• Insensitivity of bulk properties to the twisted boundary condition
• Minimalist approach to the classification of symmetry protected topological phases
• Quantum spin systems on infinite lattices
• Anomalies involving the space of couplings and the Zamolodchikov metric
• A classification of phases of bosonic quantum lattice systems in one dimension
• On adiabatic cycles of quantum spin systems

Cited by in corpus (1)

• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking