Topological pumps and adiabatic cycles

TL;DR

This paper explores loops in the Hamiltonian space of trivial insulators, revealing conditions under which boundary gaps must close and identifying these loops as topological pumps of various physical quantities.

Contribution

It introduces a framework for understanding adiabatic cycles in trivial insulators as topological pumps, extending the concept beyond topologically nontrivial insulators.

Findings

Certain loops in Hamiltonian space necessarily close boundary gaps.

These loops can pump charge, fermion parity, or other quantities depending on symmetry and dimension.

The work links trivial insulators' adiabatic cycles to topological pumping phenomena.

Abstract

Topological insulators have gapless states at their boundaries while trivial insulators generically do not. We consider loops in the spaces of Hamiltonians of topologically trivial Bloch insulators, and show that there exist loops for which the boundary gap must necessarily close at some point or points along the loop. We show that some of these loops may be regarded, depending on the symmetry class of the insulator and its physical dimension, as defining pumps of charge, fermion parity, and in more exotic cases of other quantities such as a $Z_2$ parity.