Krylov space perturbation theory for quantum synchronization in closed systems

TL;DR

The paper addresses ergodicity breaking and synchronization in a closed disordered quantum system, using a disordered Heisenberg spin chain to show emergence of spatially localized synchronization. It develops Krylov-space perturbation theory for the Liouvillian, showing that weak disorder couples the dynamical-symmetry sector $S^+$ to the rest of Krylov space, producing only a second-order correction to the frequency $\omega_{S^+}$, while strong disorder yields a finite lifetime for $S^+$. A tractable Saw model provides analytic control, exact first few Lanczos coefficients, and a clear demonstration that synchronization is robust with disorder, with local patches oscillating at distinct frequencies and a decay rate that matches perturbation theory. Overall, the work introduces a Krylov-space framework to study perturbed closed quantum dynamics, revealing how global dynamical symmetries fragment into local ones and offering insights into emergent subsystems in quantum theory.

Abstract

Strongly interacting quantum many-body systems are expected to thermalize, however, some evade thermalization due to symmetries. Quantum synchronization provides one such example of ergodicity breaking, but previous studies have focused on open systems. Here, motivated by the problem of ergodicity breaking in closed systems and the study of non-trivial dynamics, we investigate synchronization in a closed disordered Heisenberg spin chain. In the presence of large random disorder, strongly breaking the permutation symmetry of the system, we observe the emergence of spatial synchronization, where spins lock into locally synchronized patches. This behavior can be interpreted as a fragmentation of the global dynamical symmetry $S^+$ into a collection of local dynamical symmetries, each characterized by a distinct frequency. In the weak-disorder regime, still without permutation symmetry, we show that the synchronization mechanism can be understood perturbatively within Krylov space. In the absence of disorder, the Krylov space associated with the dynamical symmetry $S^+$ is two-dimensional. Introducing disorder couples this subspace to the remainder of the Krylov space. This coupling leads only to a second-order correction to the frequency of the dynamical symmetry, thereby preserving coherent oscillations despite the presence of small disorder. At stronger disorder, the perturbation modifies $S^+$ so that it acquires a finite lifetime, providing an example of a transient dynamical symmetry.

Figures (7)

• Figure 1: (A) A well know example of classical synchronization: when a set of metronomes are coupled together, they end up in phase. (B) Here we are studding a quantum analog. We consider a spin chain with interactions and a disordered magnetic field. We observe that the spins organize in local dynamical symmetries (colors) oscillating at different frequencies.
• Figure 2: Time evolution of $\langle Z_i\rangle$ for a single random realization of Hamiltonian \ref{['eq:H']}. The vertical axis is the spacial site index. The right plots show the Fourier transform of the time signal. At low disorder, the spins synchronize. At larger disorder (w=0.8), we observe the formation of local patches of about 5 spins oscillating at different frequencies.
• Figure 3: Time evolution of $\langle Z_i\rangle$ in the Saw model \ref{['eq:Hsaw']}. The green lines on the Fourier plot indicate the two magnetic fields strengths $2(1\pm w)$. We observe strong synchronization: only one frequency $\omega=2$ is visible.
• Figure 4: In the absence of disorder, the $b_2=0$ and the left of the Krylov chain is inaccessible to the dynamics. When a disorder is introduced, $b_2\sim w^2$ couple the first and second sites of the Krylov chain to the rest of the chain. This can be treated as a perturbation of the Liouvillian. Following ref Loizeau2025, we also open the Krylov chain to the right in order to simulate the thermodynamic limit.
• Figure 5: Decay rate of the dynamical symmetry v.s disorder $w$ for the Saw model. The colors indicate the exact value and the black crosses indicate the second order perturbation theory result. The inset shows the full spectrum of the Krylov chain for different $w$. Only the perturbed $S^+$ dynamical symmetry is shown on the main plot. This corresponds to the central top dots on the inset. Cf. Ref. Loizeau2025 for more details on the method to extract the full Liouvillian spectrum.