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arXiv:2507.06998

Analytical solution of boundary time crystals via the superspin basis

Dominik Nemeth, Alessandro Principi, Ahsan Nazir

Published Jul 9, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2507.06998

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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.
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Boundary time crystals (BTCs) provide a route to persistent symmetry-breaking oscillations in open quantum systems without an external clock. We study finite-size BTC-like dynamics in collectively driven and dissipative spin ensembles using a hardware-motivated spin–cavity description and an autonomous collective-spin Liouvillian. A unified diagnostic pipeline identifies late-time oscillatory pockets, quantifies their metrological resource through the long-time averaged quantum Fisher information (QFI), and tests their compatibility with complex Liouvillian slow modes. In the autonomous regime, the QFI displays near-Heisenberg scaling over the accessible sizes. The corresponding Liouvillian modes are oscillatory, but their real decay rates do not show reliable finite-size closing; the spectrum is therefore used as a consistency check rather than proof of a thermodynamic BTC phase. Local dephasing simultaneously suppresses the oscillatory response and metrological gain. The results establish a finite-size connection among autonomous oscillations, slow modes, and persistent metrological resources while leaving the thermodynamic gap-closing question open.
Quantum synchronization (QS) in dissipative systems is often inferred from smooth phase locking, leaving open whether its breakdown constitutes a genuine nonequilibrium transition. Here we introduce a Liouvillian framework that classifies driven-dissipative dynamics by the structure of the undriven dissipative background and show that QS breaks down via a Hopf-type dynamical phase transition into a boundary time crystal (BTC). The character of this transition is determined by the background attractor: systems with a self-sustained oscillator (SSO) support robust non-resonant BTCs, whereas those with a polar fixed point (PFP) sustain BTCs only at resonance and lose them under detuning. We identify sharp dynamical and spectral signatures of the QS-BTC transition and thereby establish, within U(1)-symmetric collective-spin Lindbladians driven by a single coherent tone, a background-based allowed/forbidden criterion that unifies QS, its breakdown, and time-crystalline order within a single Liouvillian framework.
Boundary time crystals are a class of exotic dissipative quantum phases that spontaneously break continuous time-translation symmetry in the thermodynamic limit of open quantum systems. In finite-size systems, the long-time evolution of boundary time crystals exhibits decaying oscillations that cannot be captured by widely used mean-field theory. To address this issue, we develop an effective approach called the stroboscopic rotating wave approximation, which provides a well-approximated state for the long-time evolution of boundary time crystals in the strongly driven regime. In this approach, the order parameter exhibits both a long-time decaying envelope governed by an effective Lindblad superoperator and short-time oscillations dominated by a reduced quantum dynamical semigroup. Our results reveal that the competition among dephasing processes along three directions induces persistent oscillations, marking the emergence of the boundary-time-crystal phase. We obtain the analytical expressions for the steady-state density operator, the oscillation period, and the decay rate of the order parameter when the coherent energy splitting exceeds the dissipation rate. Our work provides a beyond-mean-field tool for studying the dynamics of periodically driven open quantum systems and understanding the formation of time crystals.
Boundary time crystals (BTCs) are paradigmatic examples of continuous time crystals that emerge in collective, dissipative quantum systems. Their microscopic origin, however, remains obscured by the complexity of Lindbladian many-body dynamics and has largely been investigated through semiclassical and numerical approaches. Here, we develop an exact operator space framework for collective spin Lindbladians based on irreducible tensor representations. We map the Lindbladian dynamics onto a local non-Hermitian hopping problem, where weak symmetries impose selection rules on operator transport, while the hopping amplitudes and their relative strengths determine the resulting dynamical regimes. We identify the microscopic ingredients that distinguish models supporting conventional collective precession from those supporting BTC dynamics. Our approach further reveals how non-unital dissipation can provide a source for the traceless operator subspace, providing a microscopic mechanism for the reduced sensitivity of BTC dynamics to initial conditions beyond mean-field descriptions. More broadly, our results establish operator space transport as a general perspective for understanding dissipative many-body dynamics and its connections to non-Hermitian phenomena.
Boundary time crystals (BTCs) break time-translation symmetry and exhibit long-lived, robust oscillations insensitive to initial conditions. We show that collective spin BTCs can admit emergent topological winding numbers in operator space. Expanding the density operator in a spherical tensor basis, we map the Lindblad dynamics onto an effective local hopping problem, where collective degrees of freedom label sites of an emergent two-dimensional operator space lattice and identify topological obstructions that enforce the delocalization of operator modes on the lattice. The resulting spectral delocalization provides a natural explanation for the robust oscillatory dynamics observed in BTCs. When combined with non-reciprocal transport of operator weight across operator space, this mechanism moreover also leads to the universality of long-time dynamics across a broad class of initial states. Our results frame BTC dynamics as a form of topologically constrained operator space transport and suggest a close connection to non-Hermitian skin-effects.