Stable measurement-induced Floquet enriched topological order

TL;DR

This paper investigates a Floquet code exhibiting topological order and time-crystal behavior, analyzing phase transitions and stability under perturbations, thereby advancing understanding of measurement-induced topological phases.

Contribution

It introduces a continuous interpolation between Floquet and toric codes, revealing a phase transition characterized by a divergent length scale and demonstrating the stability of Floquet enriched topological order.

Findings

Identified a phase transition with divergent length scale.

Constructed a phase diagram showing stability of Floquet topological order.

Demonstrated the existence of a transition between time-crystal and stationary phases.

Abstract

The Floquet code utilizes a periodic sequence of two-qubit measurements to realize the topological order. After each measurement round, the instantaneous stabilizer group can be mapped to a honeycomb toric code, explaining the topological feature. The code also possesses a time-crystal order - the $e-m$ transmutation after every cycle, breaking the Floquet symmetry of the measurement schedule. This behavior is distinct from the stationary topological order realized in either random circuits or time-independent Hamiltonian. Therefore, the resultant phase belongs to the overlap between the classes of Floquet enriched topological orders and measurement-induced phases. In this work, we construct a continuous path interpolating between the Floquet and toric codes, focusing on the transition between the time-crystal and stationary topological phases. We show that this transition is characterized by a divergent length scale. We also add single-qubit perturbations to the model and obtain a richer two-dimensional parametric phase diagram of the Floquet code, showing the stability of the Floquet enriched topological order.