Analytical solution of boundary time crystals via the superspin basis

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.

Figures (8)

• Figure 1: The Liouvillian spectrum for the BTC model given in Eq. (\ref{['eq:nori_model']}), showing a comparison between the numerically exact results (red circles) and the analytical results obtained via the superspin method (black crosses). Here $N=3$, $\Omega_x=1$ and $\Gamma=0.1$.
• Figure 2: An illustration of the thermodynamic limit of $g(\bar{s})$ given in Eq. (\ref{['dos_eqn']}) for $\Gamma=0.1$. We observe that the sectors become densely populated near $\bar{s}=0$.
• Figure 3: Liouvillian spectra for the BTC model of Eq. (\ref{['eq:nori_model']}) in the case of $N=10$ and $\Omega_x=1$. On the left we show the spectrum for $\Gamma=0.1$ and on the right for $\Gamma=2$. The numerically computed eigenvalues are represented via red circles.
• Figure 4: The Liouvillian spectrum for model A, showing a comparison between the numerically exact results (red circles) and the analytical results obtained via the superspin method (black crosses). Here $N=3$, $\Omega_x=1$ and $\Gamma=0.1$.
• Figure 5: The Liouvillian spectrum for model B, showing a comparison between the numerically exact results (red circles) and the analytical results obtained via the superspin method (black crosses). Here $N=3$, $\Omega_x=1$ and $\Gamma=0.1$.