[Paper Review] Topological charge conservation for continuous insulators

Guillaume Bal|arXiv (Cornell University)|Jun 15, 2021

Topological Materials and Phenomena4 citations

TL;DR

This paper establishes a topological charge conservation law for continuous topological insulators and superconductors in Euclidean space by defining the topological charge via the Fredholm index of a Hamiltonian augmented with domain walls. It proves that this index equals the quantized line conductivity, generalizing bulk-edge correspondence to higher dimensions and providing a computationally tractable method using the Fedosov-Hörmander formula and Clifford algebra decomposition.

ABSTRACT

This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. A Fedosov-Hörmander formula implementing in Euclidean spaces an Atiyah-Singer index theorem allows for an explicit computation of the index in terms of the symbol of the Fredholm operator. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable degree of a naturally associated map. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. This asymmetry is captured by a line conductivity, a physical observable of the system. We prove that the line conductivity is quantized and given by the index of a second Fredholm operator of Toeplitz type. We also prove a topological charge conservation stating that the two aforementioned indices agree. This result generalizes to higher dimensions and higher-order topological insulators the bulk-edge correspondence of two-dimensional materials. We apply this procedure to evaluate the topological charge of several classical examples of (standard and higher-order) topological insulators and superconductors in one, two, and three spatial dimensions.

Motivation & Objective

• To classify elliptic pseudo-differential Hamiltonians describing topological insulators and superconductors in Euclidean space using domain walls.
• To define the topological charge of a system as the Fredholm index of a confining operator constructed by augmenting the Hamiltonian with domain walls.
• To establish a topological charge conservation law linking the bulk topological invariant (Fredholm index) to the physical observable of line conductivity.
• To generalize the two-dimensional bulk-edge correspondence to higher-dimensional and higher-order topological insulators.
• To provide an explicit, computable formula for the topological charge using the Fedosov-Hörmander index theorem and Clifford algebra structure.

Proposed method

• Construct a Fredholm operator $ F = H_{d-1} - i m(x_d) $ by augmenting the Hamiltonian with a domain wall in the final spatial dimension, ensuring ellipticity and proper growth at infinity.
• Use the Fedosov-Hörmander formula to compute the Fredholm index of $ F $, which defines the topological charge, based on the symbol of the Hamiltonian.
• For Hamiltonians with a Clifford algebra structure, compute the index as the degree of a map derived from the symbol, enabling efficient computation.
• Define a second Fredholm operator of Toeplitz type to capture the line conductivity, a physical observable of asymmetric transport along one-dimensional interfaces.
• Prove that the index of the Toeplitz operator (line conductivity) equals the index of the confining Fredholm operator (topological charge), establishing topological charge conservation.
• Apply the framework to classical examples in 1D, 2D, and 3D, including higher-order topological insulators and regularized models with flat bands or Landau levels.

Experimental results

Research questions

• RQ1 How can the topological charge of a continuous topological insulator be systematically defined and computed in arbitrary spatial dimensions?
• RQ2 What is the relationship between the bulk topological invariant (Fredholm index of a confining operator) and the physical observable of line conductivity?
• RQ3 To what extent does the bulk-edge correspondence generalize to higher-dimensional and higher-order topological insulators?
• RQ4 Under what conditions does the topological charge conservation law fail, and how can such models be regularized to fit the framework?
• RQ5 Can the topological charge be computed efficiently using geometric invariants like the degree of a map when the Hamiltonian admits a Clifford algebra decomposition?

Key findings

• The topological charge of a Hamiltonian is defined as the Fredholm index of a confining operator $ F = H_{d-1} - i m(x_d) $, constructed by adding domain walls to the original Hamiltonian.
• The line conductivity, a physical observable of asymmetric transport along one-dimensional interfaces, is quantized and equals the index of a second Fredholm operator of Toeplitz type.
• The key result is topological charge conservation: the two indices—the bulk topological invariant and the line conductivity—agree exactly, generalizing bulk-edge correspondence to higher dimensions.
• For Hamiltonians with a Clifford algebra structure, the topological charge is given by the degree of a map derived from the symbol, enabling simple and explicit computation.
• The method applies to standard and higher-order topological insulators and superconductors in 1D, 2D, and 3D, including models with flat bands after regularization.
• The theory does not apply to models with essential spectrum degeneracies such as Landau levels or flat bands (e.g., shallow water wave Hamiltonians), but regularized versions can be made compatible.