Operator Space Transport and the Emergence of Boundary Time Crystals

TL;DR

This paper introduces a quantum-compatible framework for analyzing boundary time crystals, revealing their emergence from non-trivial operator space transport and non-Hermitian dynamics.

Contribution

It develops an operator space tensor representation to classify collective spin dynamics and explains BTC emergence through non-reciprocal transport mechanisms.

Findings

BTCs result from non-reciprocal operator space transport.

The framework unifies different dynamical regimes in a single model.

BTC oscillations are insensitive to initial conditions due to delocalized eigenmodes.

Abstract

Boundary time crystals (BTCs) are prominent examples of continuous time crystals in collective spin systems governed by Lindbladian evolution. To date, their analysis has mostly relied on semiclassical and numerical approaches. Here, we develop a fully quantum-compatible framework to classify collective spin dynamics and show that BTC behavior emerges from the absence of non-trivial weak symmetries of the Liouvillian. To this end, we introduce an irreducible tensor representation of operator space, in which the Lindbladian dynamics maps onto a non-Hermitian hopping problem. Within this picture, the dynamics corresponds to the transport of operator weight across tensor sectors. This mapping allows an identification of distinct dynamical regimes, including collective precession, pure relaxation, and the BTC phase, within a single unified framework. We show that BTCs arise from non-reciprocal transport in operator space, which delocalizes Liouvillian eigenmodes across multiple tensor sectors. This non-reciprocal transport provides a microscopic mechanism for the insensitivity to initial conditions of BTC oscillations. More broadly, our results establish operator space transport as a perspective for understanding dissipative many-body dynamics and highlights connections to non-Hermitian phenomena.