Fractional domain wall statistics in spin chains with anomalous symmetries

TL;DR

This paper investigates how anomalous symmetries in quantum spin chains influence the exchange statistics of domain wall excitations, revealing fractional statistics linked to the symmetry anomaly and proposing measurement methods.

Contribution

It establishes a direct relation between MPU symmetry anomalies and domain wall exchange statistics, including fractional behavior for $ ext{Z}_2$ symmetries, and introduces measurement protocols.

Findings

Anomalous MPU symmetries lead to fractional domain wall statistics.

Exchange statistics are determined by the symmetry anomaly.

Explicit measurement operators for domain wall statistics are proposed.

Abstract

We study the statistics of domain wall excitations in quantum spin chains. We focus on systems with finite symmetry groups represented by matrix product unitaries (MPUs), i.e. finite depth quantum circuits. Such symmetries can be anomalous, in which case gapped phases which they support must break the symmetry. The lowest lying excitations of those systems are thus domain wall excitations. We investigate the behavior of these domain walls under exchange, and find that they can exhibit non-trivial exchange statistics. This statistics is completely determined by the anomaly of the symmetry, and we provide a direct relation between the known classification of MPU symmetry actions on ground states and the domain wall statistics. Already for the simplest case of a $\mathbb Z_2$ symmetry, we obtain that the presence of an anomalous MPU symmetry gives rise to domain wall excitations which behave neither as bosons nor as fermions, but rather exhibit fractional statistics. Finally, we show that the exchange statistics of domain walls is a physically accessible quantity, by devising explicit measurement operators through which it can be determined.