Topological quantization by controlled paths: application to Cooper pairs pumps

TL;DR

This paper explores how topological properties of quantum systems with tunable parameters can be used to achieve quantization of physical quantities, with a focus on Cooper pair pumps and criteria for topological quantization.

Contribution

It introduces the concept of topological quantization by controlled paths and proposes four criteria to determine its applicability in quantum systems.

Findings

Topological singularities in parameter space affect phase accumulation.

Chern indices can be used to characterize eigenvector singularities.

Criteria are proposed to identify when topological quantization is possible.

Abstract

When physical systems are tunable by three classical parameters, level degeneracies may occur at isolated points in parameter space. A topological singularity in the phase of the degenerate eigenvectors exists at these points. When a path encloses such point, the accumulated geometrical phase is sensitive to its presence. Furthermore, surfaces in parameter space enclosing such point can be used to characterize the eigenvector singularities through their Chern indices, which are integers. They can be used to quantize a physical quantity of interest. This quantity changes continuously during an adiabatic evolution along a path in parameter space. Quantization requires to turn this path into a surface with a well defined Chern index. We analyze the conditions necessary to a {\em Topological Quantization by Controlled Paths}. It is applied to Cooper pair pumps. For more general problems, a set of four criteria are proposed to check if topological quantization is possible.