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

Xueda Wen, Marvin Qi+6 more|arXiv (Cornell University)|Dec 14, 2021

Topological Materials and Phenomena41 references4 citations

TL;DR

This paper establishes a bulk-boundary correspondence for higher Berry curvature in parametrized gapped quantum systems, generalizing the Thouless charge pump to higher dimensions. It introduces the concept of 'flow' of Berry curvature and demonstrates that pumping of topological invariants—such as Chern numbers and Kapustin-Spodyneiko (KS) invariants—leads to anomalous boundary modes with isolated Weyl points, providing a physical interpretation of higher Berry curvature as a flow of lower-dimensional curvature in d-dimensional 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.

Motivation & Objective

• To generalize the Thouless charge pump to higher-dimensional parametrized quantum systems using higher Berry curvature.
• To clarify the physical interpretation of the Kapustin-Spodyneiko (KS) higher Berry curvature as a flow of (d−1)-dimensional curvature in d dimensions.
• To establish a bulk-boundary correspondence linking the flow of higher Berry curvature to the emergence of topologically protected boundary modes.
• To explore the role of the suspension isomorphism in generating new invertible parametrized systems from known topological phases.
• To construct universal quantum pumping mechanisms that generate (d+1)-dimensional parametrized systems from d-dimensional ones, potentially realizing the suspension isomorphism in generalized cohomology theories of invertible phases.

Proposed method

• Formalizes parametrized gapped quantum many-body systems as continuous maps from a parameter space X to the space of gapped Hamiltonians.
• Introduces the concept of 'flow' of Berry curvature, where in d dimensions, the higher Berry curvature is interpreted as a flow of (d−1)-dimensional curvature.
• Applies the Kapustin-Spodyneiko (KS) invariant to quantify topological invariants in parametrized systems, particularly in one and higher dimensions.
• Derives local expressions for higher Berry curvature (F^{(2)}_q, F^{(3)}_{pq}) using imaginary-time-ordered correlation functions of local operators with vanishing ground-state expectation values.
• Constructs two universal quantum pumping mechanisms: one over the suspension SX and one over X×S¹, which generate new (d+1)-dimensional parametrized systems from d-dimensional ones.
• Uses clutching constructions and inverse systems over S³ to explicitly realize and analyze the bulk-boundary correspondence in one-dimensional models.

Experimental results

Research questions

• RQ1 How can the Kapustin-Spodyneiko higher Berry curvature be physically interpreted in terms of curvature flow in parametrized quantum systems?
• RQ2 What is the bulk-boundary correspondence for higher Berry curvature in one-dimensional parametrized systems, and how does it lead to anomalous boundary modes?
• RQ3 How do quantum pumping constructions over SX and X×S¹ generate new (d+1)-dimensional parametrized systems, and do they realize the suspension isomorphism in generalized cohomology theories?
• RQ4 What is the role of locality in the higher Berry curvature, and how does it support the bulk-boundary correspondence in generic systems?
• RQ5 How does the generalized cohomology classification of invertible phases relate to the construction of parametrized systems and their topological invariants?

Key findings

• The higher Berry curvature in d dimensions is physically interpreted as a flow of (d−1)-dimensional curvature, generalizing the 1D notion of Berry curvature flow.
• In one-dimensional systems, pumping of the Chern number via the KS invariant leads to the emergence of isolated Weyl points in the boundary modes, indicating anomalous topological response.
• The bulk-boundary correspondence is explicitly verified in an exactly solvable 1D lattice model over S³, where the KS invariant computed from the bulk matches the topological invariant of the boundary.
• The higher Berry curvature is shown to be local, with dominant contributions arising from regions near the relevant lattice sites, as demonstrated via imaginary-time-ordered correlation functions.
• The two proposed quantum pumping constructions—over SX and X×S¹—generate (d+1)-dimensional parametrized systems from d-dimensional ones, with the latter conjectured to realize the suspension isomorphism in generalized cohomology.
• In higher dimensions, the KS invariant in (d+1)-dimensional systems can be computed from the bulk-boundary correspondence, and the construction yields quantized invariants analogous to Chern numbers in lower dimensions.