QLunch: An index theorem for quantum charge transport

Speaker: Alex Bols from KU Leuven

Title: An index theorem for quantum charge transport

Abstract:  Consider a gapped groundstate of a local charge conserving Hamiltonian, and some unitary `process' which leaves the groundstate invariant. How much charge is transported during the process? I will present a theorem which says that, under some natural conditions, the transported charge is quantized.

This theorem has a few interesting applications:

- If the process is translation of a translation invariant gapped groundstate, we recover the Lieb-Schultz-Mattis theorem (Charge density in a gapped groundstate is integer).

- The process can be taken to be a cycle of a Thouless pump, thus giving a many-body proof of the fact the Thouless pumps pump integer amounts of charge.

- The Laughlin argument for quantization of the Hall conductance is also a pumping argument, and we indeed obtain a proof of quantization of the Hall conductance in the many-body setting (first proven by Hastings and Michalakis using a Chern number approach).

-Taking time evolution as the unitary process, we obtain Bloch's theorem; the groundstate carries no current. 

- ...? 

The full story can be found here: https://arxiv.org/pdf/1810.07351.pdf