Fractional topology and multi-period re-quantization in open quantum systems

TL;DR

This paper investigates fractional topological numbers in open quantum systems, showing how they can arise from relaxed symmetry conditions and how they recover integer quantization over multiple periods, with implications for photonic lattices.

Contribution

It introduces a framework for understanding fractional topology in open quantum systems, especially under conditions where traditional quantization is effectively relaxed.

Findings

Fractional winding numbers occur when symmetry conditions are relaxed.

Winding numbers depend continuously on system parameters with discontinuities at purity-gap closings.

Multi-period analysis restores integer quantization of topological invariants.

Abstract

We study fractional topological numbers in open quantum systems described by the Gorin--Kossakowski--Sudarsha--Lindblad master equation. Under symmetry conditions ensuring quantization, we show that single-valued physical states in momentum space give rise to integer winding numbers that remain integer during time evolution. Fractional values arise when this condition is effectively relaxed, such that the topology is evaluated over a restricted sector or exhibits an effective multi-branch structure. In these cases, the winding number is not quantized over the fundamental Brillouin zone and can depend continuously on system parameters, with discontinuities at purity-gap closings. However, when extended over multiple momentum periods, the winding recovers integer quantization. These effects are illustrated in a Su--Schrieffer--Heeger chain with gain and loss and can be probed in long-range hopping photonic lattices with fractional fillings via Bloch state tomography. Our results provide a unified framework for understanding fractional topology in open quantum systems.