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A space-adiabatic approach for bulk-defect correspondences in lattice models of topological insulators

Danilo Polo Ojito, Emil Prodan, Tom Stoiber

Abstract

In space-adiabatic approaches one can approximate Hamiltonians that are modulated slowly in space by phase-space functions that depend on position and momentum. In this paper, we establish a rigorous relation between this approach and the operator-theoretic approach for topological insulators with defects, which employs $C^*$-algebras and operator K-theory. Using such tools, we show that by quantizing phase-space functions one can construct lattice Hamiltonians which are gapped at certain spatial limits and carry protected states at defects such as boundaries, hinges, and corners. Moreover, we show that the topological invariants that protect the latter can be computed in terms of the symbol functions. This enables us to compute boundary maps in K-theory that are relevant for bulk-defect correspondences.

A space-adiabatic approach for bulk-defect correspondences in lattice models of topological insulators

Abstract

In space-adiabatic approaches one can approximate Hamiltonians that are modulated slowly in space by phase-space functions that depend on position and momentum. In this paper, we establish a rigorous relation between this approach and the operator-theoretic approach for topological insulators with defects, which employs -algebras and operator K-theory. Using such tools, we show that by quantizing phase-space functions one can construct lattice Hamiltonians which are gapped at certain spatial limits and carry protected states at defects such as boundaries, hinges, and corners. Moreover, we show that the topological invariants that protect the latter can be computed in terms of the symbol functions. This enables us to compute boundary maps in K-theory that are relevant for bulk-defect correspondences.

Paper Structure

This paper contains 12 sections, 24 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Continuous fields for adiabatic quantization
  3. Flat configuration spaces
  4. Connes-Thom isomorphism at work
  5. Chern numbers of adiabatic quantizations
  6. Boundary maps in the adiabatic picture
  7. Applications
  8. Bulk-interface correspondence
  9. Elementary codimension n defect
  10. Corner states
  11. A geometry with multiple corners
  12. K-theory and cyclic cohomology

Key Result

Proposition 1.6

Those K-theoretic classes are by construction related via the boundary maps of the exact sequence eq: 1

Figures (3)

  • Figure 1: Examples of real-space geometries and configuration spaces. Top: The configuration space for an interface between two materials A, B consists of a single one-cell ${\mathbb R}_{AB}$ with two infinite points attached $\Omega={\mathbb R}_{AB}\sqcup\{\infty_A\}\sqcup\{\infty_B\}$. Bottom: The configuration space for a Y-type interface between materias A, B and C, has the cell decomposition shown at the bottom right. The interface is invariant only w.r.t. translations orthogonal to the displayed plane, hence corresponding to a codimension $2$ defect which gives rise to a $2$-cell ${\mathbb R}^2_{ABC}$. The one-cells ${\mathbb R}_{XY}$ each correspond to the asymptotic interfaces of two materials only and the three points $\infty_A, \infty_B, \infty_C$ of the $0$-cell are the different bulk limits.
  • Figure 2: Left: A rational mirror-symmetric cone in real-space as an intersection of two half-spaces (in $d>2$ this is a view from the top). Right: CW-structure of $\Omega$.
  • Figure 3: Visualization of the flat CW-complex $\Omega$ (right) corresponding to infinite-volume limits of a square slab (top view on the left). The exterior square (dotted) is identified with the infinite point of the one-point compactification of $\Omega$ and the loop marks the boundary of ${\mathbb R}^2_{12}$.

Theorems & Definitions (57)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6: PSbook, Sec. 4.3
  • Definition 1.7
  • Example 1.8
  • Proposition 1.9
  • proof
  • ...and 47 more