Tensor Berry connections and their topological invariants

Abstract

The Berry connection plays a central role in our description of the geometric phase and topological phenomena. In condensed matter, it describes the parallel transport of Bloch states and acts as an effective "electromagnetic" vector potential defined in momentum space. Inspired by developments in mathematical physics, where higher-form (Kalb-Ramond) gauge fields were introduced, we hereby explore the existence of "tensor Berry connections" in quantum matter. Our approach consists in a general construction of effective gauge fields, which we ultimately relate to the components of Bloch states. We apply this formalism to various models of topological matter, and we investigate the topological invariants that result from generalized Berry connections. For instance, we introduce the 2D Zak phase of a tensor Berry connection, which we then relate to the more conventional first Chern number; we also reinterpret the winding number characterizing 3D topological insulators to a Dixmier-Douady invariant, which is associated with the curvature of a tensor connection. Besides, our approach identifies the Berry connection of tensor monopoles, which are found in 4D Weyl-type systems [Palumbo and Goldman, Phys. Rev. Lett. 121, 170401 (2018)]. Our work sheds light on the emergence of gauge fields in condensed-matter physics, with direct consequences on the search for novel topological states in solid-state and quantum-engineered systems.

10 pages, 1 table. Published version

References in corpus (14)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Generalized Global Symmetries
• Topological Quantum Matter with Ultracold Gases in Optical Lattices
• Observation of Dirac Monopoles in a Synthetic Magnetic Field
• An Aharonov-Bohm interferometer for determining Bloch band topology
• Artificial gauge fields in materials and engineered systems
• Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
• Manipulating Topological Edge Spins in One-Dimensional Optical Lattice
• The 6D quantum Hall effect and 3D topological pumps
• Revealing tensor monopoles through quantum-metric measurements
• Topological Classification of Gapped Spin Chains :Quantized Berry Phase as a Local Order Parameter
• Electromagnetic response of quantum Hall systems in dimensions five and six and beyond
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator

Cited by in corpus (29)

• Experimental Observation of Tensor Monopoles with a Superconducting Qudit
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
• Ground-state quantum geometry in superconductor-quantum dot chains
• Transconductance quantization in a topological Josephson tunnel junction circuit
• Four-dimensional semimetals with tensor monopoles: From surface states to topological responses
• Non-Abelian tensor Berry connections in multi-band topological systems
• Higher-form Gauge Symmetries in Multipole Topological Phases
• Tensor Monopoles in superconducting systems
• Topological properties of a non-Hermitian quasi-1D chain with a flat band
• Tensor monopoles and negative magnetoresistance effect in optical lattices
• Link between \emph{Zitterbewegung} and topological phase transition
• Geometry of the parameter space of a quantum system: Classical point of view
• Sequential quantum phase transitions in $J_1$-$J_2$ Heisenberg chains with integer spins ($S>1$): Quantized Berry phase and valence-bond solids
• Bilayer Haldane system: Topological characterization and adiabatic passages connecting Chern phases
• Fermion-fermion duality in 3+1 dimensions
• Higher Gauge Theory
• Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators
• Second Euler number in four dimensional synthetic matter
• 2-Form U(1) Spin Liquids: Classical Model and Quantum Aspects
• A synthetic monopole source of Kalb-Ramond field in diamond
• Topological Phases for Extended Objects: Semiclassical Phase-Space Approach with Tensorial Coordinates
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Singular connection approach to topological phases and resonant optical responses
• An exceptional surface and its topology
• A Study of Berry Connection and Complex Analysis for Topological Characterization
• Nonlinear Odd Viscoelastic Effect
• Universal intrinsic higher-rank spin Hall effect