Topology vs localization in synthetic dimensions

TL;DR

This paper investigates the relationship between topology and localization in synthetic dimensions of quantum systems, revealing topological obstructions to localized Wannier functions and proposing a frame-based approach when orthonormality is relaxed.

Contribution

It provides a topological analysis of Bloch functions in synthetic dimensions, identifying Chern class obstructions and introducing a Parseval frame method for spanning energy bands.

Findings

Obstructions to localized Wannier functions are characterized by first two Chern classes.

Relaxing orthonormality allows spanning energy bands with a finite Parseval frame.

The second Chern class plays a key role in 4D topological systems.

Abstract

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g. in optical lattices of ultracold atoms, we discuss here $d$-dimensional periodic, gapped quantum systems for $d \le 4$, with focus on the topology of the occupied energy states. We perform this analysis by asking whether the spectral subspace below the gap can be spanned by smooth and periodic Bloch functions, corresponding to localized Wannier functions in position space. By constructing these Bloch functions inductively in the dimension, we show that if they are required to be orthonormal then in general their existence is obstructed by the first two Chern classes of the underlying Bloch bundle, with the second Chern class characterizing in particular the 4-dimensional situation. If the orthonormality constraint is relaxed, we show how $m$ occupied energy bands can be spanned by a Parseval frame comprising at most $m+2$ Bloch functions.