Analytical solution of boundary time crystals via the superspin basis

TL;DR

This paper derives an analytical Liouvillian framework for boundary time crystals in dissipative spin systems, revealing their long-time dynamics, symmetry breaking, and conditions for genuine BTC phases.

Contribution

It introduces a superspin basis to analytically describe the extreme boundary time crystal regime, providing explicit eigenvalues and insights into the dynamics.

Findings

Analytical expressions for Liouvillian eigenvalues in the BTC regime

Demonstration of spontaneous breaking of time-translation symmetry

Identification of models supporting genuine BTC phases

Abstract

Boundary time crystals (BTCs) in dissipative collective spin systems have been extensively studied using numerical, mean-field, and perturbative approaches. However, an explicit Liouvillian description governing the long-time dynamics deep within the time crystal phase has remained elusive. Here, we derive an effective Liouvillian that analytically captures the extreme BTC regime, where dissipation is parametrically weak and oscillatory order is maximally robust. By introducing a superspin representation of Liouville space, we obtain closed-form expressions for the Liouvillian eigenvalues to first order in the dissipation strength, providing direct access to decay rates, oscillation frequencies, and their thermodynamic scaling. Applying this framework to the canonical BTC model we analytically recover spontaneous breaking of continuous time-translation symmetry and persistent oscillations in the thermodynamic limit. In contrast, we show that other dissipative spin models exhibit only single-frequency oscillatory dynamics and therefore do not support genuine BTC phases. Our results establish a controlled analytical framework for the long-time dynamics in the extreme BTC regime.