Closed Quantum Systems Defy Expectations by Maintaining Localised Synchronisation Despite Disorder

Synchronization in closed quantum systems represents a fascinating example of ergodicity breaking, challenging the expectation that strongly interacting systems will thermalise. Nicolas Loizeau and Berislav Buča, both from the Niels Bohr Institute at the University of Copenhagen, investigate this phenomenon in a disordered Heisenberg spin chain, revealing how spatial synchronization emerges even in the absence of external driving. Their work demonstrates that strong random disorder fragments the system’s global dynamical symmetry into localised patches of synchronised spins. Importantly, the researchers show that weak disorder can be understood using Krylov space perturbation theory, preserving coherent oscillations through only second-order corrections to the frequency of the dynamical symmetry.

This research offers a novel understanding of non-trivial dynamics in closed quantum systems and provides an example of transient dynamical symmetry at stronger disorder levels, potentially informing future studies of many-body localisation and quantum chaos. Researchers have uncovered a surprising phenomenon in quantum systems: the emergence of synchronized behaviour even within completely isolated environments. Strongly interacting quantum systems typically relax towards thermal equilibrium, yet certain symmetries can allow them to evade this fate. This work demonstrates that synchronization, a collective, rhythmic behaviour, can arise not in the open systems previously studied, but within a closed, disordered Heisenberg spin chain, a fundamental model in quantum magnetism. The research reveals that strong random disorder doesn’t simply destroy order, but instead fragments a global symmetry into a collection of localized dynamical symmetries, each oscillating at a unique frequency. This discovery challenges conventional understanding of ergodicity breaking, where systems fail to explore all possible states, and offers a new perspective on this concept by linking the disorder inherent in many-body localization with the non-stationary dynamics of dynamical symmetries. By investigating synchronization within a closed system, the team identified a mechanism where spins lock into locally synchronized patches, effectively creating miniature, independent oscillators within the chain. Utilising Krylov space perturbation theory, they show that weak disorder subtly alters the system’s inherent frequencies, preserving coherent oscillations, though at stronger disorder levels, this perturbation introduces a finite lifetime to the symmetry, resulting in a transient, yet observable, synchronized state. The implications of this work extend beyond fundamental quantum mechanics, potentially informing our understanding of collective behaviour in diverse systems, from neural networks to materials science. Understanding synchronization in spin systems could pave the way for creating highly homogeneous and time-dependent magnetic field sources, with potential applications in improving the resolution of magnetic resonance imaging (MRI). A Lanczos algorithm underpinned the investigation of dynamical symmetries within Liouvillian Krylov space, a technique increasingly employed to examine integrability and quantum chaos. This method begins by constructing the Krylov space for an observable, O0, under a given Hamiltonian, H, iteratively building an orthonormal basis of operators by recursively applying the Liouvillian, L, to O0 while ensuring orthonormalization at each step. Initially, O1 is calculated as [H, O0] divided by b1, where b1 represents the norm of LO0. Subsequent operators, On, are then generated for n greater than 2, involving further application of the Liouvillian and normalization to maintain orthogonality. The resulting orthonormal ‘Krylov-basis’ {On} and associated ‘Lanczos coefficients’ bn are then used to construct the Liouvillian itself, represented as a tridiagonal matr. To facilitate semi-analytical treatment, a simplified ‘Saw model’ was introduced, featuring an alternating +w, −w perturbation added to b1 = 2 + O(w2), b2 = 4√3w + O(w2), and b3 = 2r55/3 + O(w2). Notably, b2 exhibits a first-order contribution in w, vanishing entirely in the absence of disorder, effectively disconnecting sectors of the Krylov chain. Spatial synchronization emerges in the disordered Heisenberg spin chain, manifesting as locally synchronized patches of spins. Observations of the dynamics under the Hamiltonian reveal that at a disorder parameter of 0.2, spins oscillate in phase, indicating synchronization across the chain. Further analysis demonstrates that at a disorder level of 0.4, this coherent behaviour persists, though with subtle variations in the oscillation patterns. However, at a significantly higher disorder of 0.8, the system fragments into distinct local patches, each approximately five spins in size, oscillating at differing frequencies. These patches exhibit coherent motion within themselves, but lack long-range phase coherence with other regions of the chain. Fourier transforms of the time signals confirm these observations, displaying sharp peaks at consistent frequencies for low disorder, indicative of synchronized oscillations. As disorder increases, these peaks broaden and split into multiple peaks, each corresponding to the oscillation frequency of a specific local patch. Specifically, at a disorder of 0.8, one patch between sites 4 and 10 oscillates at a frequency of approximately 1.5, while another patch between sites 10 and 15 oscillates at a frequency around 1.5. The study reveals that even in the presence of disorder, the system initially attempts to preserve a coherent oscillation, but increasing disorder modifies this symmetry, giving it a finite lifetime. This transient dynamical symmetry provides an example of how disorder can induce synchronization in a closed system, despite the expectation that such systems should typically thermalize. The research demonstrates that the natural thermalization dynamics of a disordered system can dissipate non-synchronized modes, leaving only synchronized ones in the long-time limit.

Scientists have long been captivated by the puzzle of synchronisation, how disparate parts of a system can lock into coherent, collective behaviour. This work offers a significant step forward in understanding this phenomenon, not in the familiar realm of coupled oscillators, but within the more complex and chaotic environment of disordered quantum systems. The challenge has always been to reconcile the tendency of strongly interacting systems to descend into thermal equilibrium with the surprising emergence of order, as seen in instances of synchronisation. While the Krylov space perturbation theory provides a powerful analytical tool, its applicability may be limited to weaker disorder regimes. Future work must explore whether these findings hold in more complex, higher-dimensional systems, and whether the transient nature of the synchronisation can be overcome. The next frontier likely lies in bridging the gap between these theoretical models and experimental realisations, perhaps using engineered quantum materials to directly observe and manipulate these subtle forms of collective behaviour. 👉 More information 🗞 Krylov space perturbation theory for quantum synchronization in closed systems 🧠 ArXiv: https://arxiv.org/abs/2602.11431 Tags: