Solving short- and long-time dynamics of closed quantum many-body systems is one of the main challenges of both atomic and condensed matter physics. For locally interacting closed systems, the dynamics of local observables can always be expanded into (pseudolocal) eigenmodes of the Liouvillian, so-called dynamical symmetries. They come in two classes: transient operators, which decay in time and perpetual operators, which either oscillate forever or stay the same (conservation laws). These operators provide a full characterization of the dynamics of the system. Deriving these operators, apart from a very limited class of models, has not been possible. Here, we present a method to numerically and analytically derive some of these dynamical symmetries in infinite closed systems by introducing a naturally emergent open boundary condition on the Krylov chain. This boundary condition defines a partitioning of the Krylov space into system and environment degrees of freedom, where nonlocal operators make up an effective bath for the local operators. We demonstrate the practicality of the method on some numerical examples and derive analytical results in two idealized cases. Our approach lets us directly relate the operator growth hypothesis to thermalization and exponential decay of observables in chaotic systems and provides a powerful approach for computing notably challenging many-body dynamics.
Solving short- and long-time dynamics of closed quantum many-body systems is one of the main challenges of both atomic and condensed matter physics. For locally interacting closed systems, the dynamics of local observables can always be expanded into (pseudolocal) eigenmodes of the Liouvillian, so-called dynamical symmetries. They come in two classes: transient operators, which decay in time and perpetual operators, which either oscillate forever or stay the same (conservation laws). These operators provide a full characterization of the dynamics of the system. Deriving these operators, apart from a very limited class of models, has not been possible. Here, we present a method to numerically and analytically derive some of these dynamical symmetries in infinite closed systems by introducing a naturally emergent open boundary condition on the Krylov chain. This boundary condition defines a partitioning of the Krylov space into system and environment degrees of freedom, where nonlocal operators make up an effective bath for the local operators. We demonstrate the practicality of the method on some numerical examples and derive analytical results in two idealized cases. Our approach lets us directly relate the operator growth hypothesis to thermalization and exponential decay of observables in chaotic systems and provides a powerful approach for computing notably challenging many-body dynamics.
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.
We study symmetry-resolved Krylov complexity in finite-dimensional chaotic quantum many-body systems. When both the Hamiltonian and the initial operator commute with a conserved charge, the operator dynamics decomposes into independent symmetry sectors, each with its own Krylov chain. We show that, after saturation, the unresolved Krylov complexity is additive over symmetry sectors. In the absence of additional Liouvillian degeneracies, the late-time contribution of a sector with Hilbert-space dimension dq is controlled by dq(dq−1), leading to a dimension-weighted equipartition that approaches the simple large-sector scaling dq2/∑q′dq′2. This late-time rule differs from the early-time weighted-average discussed in the literature and is governed instead by the dimensions of the accessible operator spaces. We support the analytic prediction with numerical studies of the real and complex SYK models, a chaotic bosonic spin model, and the mixed-field Ising chain. Our results show that resolving exact symmetries is essential for interpreting the saturation value of Krylov complexity as a diagnostic of chaotic operator growth.
In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.
We introduce Krylovedgespectroscopy, a many-body operator-space protocol for detecting and classifying one-dimensional symmetry-protected topological phases from local boundary dynamics. A Hermitian boundary operator generates a semi-infinite Krylov hopping chain whose boundary weight obeys an exact zero-frequency normalizability criterion. Open-periodic, boundary-bulk, and symmetry-preserving boundary-perturbation tests identify protected boundary memory, while a finite-depth leakage residual certifies when an explicitly reconstructed operator is already near zero frequency. Classification minimizes the normalized commutator over symmetry-resolved boundary operators in fixed charge sectors. For bosonic ZN×ZN phases, the recovered endpoint charge gives the cohomology label. The method requires no explicit ground-state wavefunction, entanglement spectrum, or guessed dressed edge or string operator. For finite-range Hamiltonians, locality eliminates full-system exact diagonalization and guarantees thermodynamic-limit convergence at fixed Krylov depth. We demonstrate the protocol in cluster, clock, Haldane, and trivial spin-1 chains. In the exactly solvable cluster chain, the transition between the gapped topological and trivial phases manifests as a localization-delocalization transition of the Krylov edge mode on the Krylov chain. This transition occurs precisely at the bulk gap closing, and its localization-length exponent coincides with the Ising correlation-length exponent. Through operator Krylov dynamics, our work turns local boundary evolution into a direct spectroscopy of many-body topology.
We consider finite-dimensional many-body quantum systems described by time-independent Hamiltonians and Markovian master equations, and present a systematic method for constructing smaller-dimensional, reduced models that e x a c t l y reproduce the time evolution of a set of initial conditions or observables of interest. Our approach exploits Krylov operator spaces and their extension to operator algebras, and may be used to obtain reduced linear models of minimal dimension, well-suited for simulation on classical computers, or reduced quantum models that preserve the structural constraints of physically admissible quantum dynamics, as required for simulation on quantum computers. Notably, we prove that the reduced quantum-dynamical generator is still in Lindblad form. By introducing a new type of observable-dependent symmetries , we show that our method provides a non-trivial generalization of techniques that leverage symmetries, unlocking new reduction opportunities. We quantitatively benchmark our method on paradigmatic open many-body systems of relevance to condensed-matter and quantum-information physics. In particular, we demonstrate how our reduced models can quantitatively describe decoherence dynamics in central-spin systems coupled to structured environments, magnetization transport in boundary-driven dissipative spin chains, and unwanted error dynamics on information encoded in a noiseless quantum code.
Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. Although these diagnostics were originally motivated by questions of chaos and integrability, quadratic fermionic Hamiltonians are often expected to exhibit trivial Lanczos structure. Here we show that, in the long-range Kitaev chain, Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. We introduce the Krylovstaggeringparameter for the Lanczos coefficients. In the short-range Kitaev chain with balanced hopping and pairing, we derive analytically for arbitrary system size (valid in the thermodynamic limit) and show that this quantity is exactly constant and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase. Away from that limit, long-range couplings and pairing-hopping imbalance deform the simple flat structure and analytical control is lost, nevertheless, we show that the sign pattern of the diagnostic still tracks whether the lowest excitation gap is controlled by boundary modes or by bulk excitations. These results are enabled by an exact single-particle operator Lanczos algorithm, as derived in this work, which reduces the recursion from exponentially large operator space to a finite-dimensional linear problem and achieves machine precision for chains of hundreds of sites. Krylov diagnostics thus emerge as practical probes of boundary-versus-bulk low-energy physics in topological superconductors with broken U(1) symmetry and algebraically decaying couplings.