Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterizing the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have nonmonotonic behavior as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution.
Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterizing the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have nonmonotonic behavior as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution.
Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are a prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an order parameter for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum-mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs while introducing an operator-independent diagnostic for quantum chaos, offering more universal insights and a deeper understanding of the general properties of quantum chaotic systems.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
The growth of simple operators is essential for the emergence of chaotic dynamics and quantum thermalization. Recent studies have proposed different measures, including the out-of-time-order correlator and Krylov complexity. It is established that the out-of-time-order correlator serves as the signature of quantum many-body chaos, while the Krylov complexity provides its upper bound. However, there exist nonchaotic systems in which Krylov complexity grows exponentially, indicating that the Krylov complexity itself is not a witness of many-body chaos. In this Letter, we introduce the missing ingredient called the Krylov metric K_{mn}, which probes the size of the Krylov basis. We propose that the universal criteria for fast scramblers include (i) the exponential growth of Krylov complexity, (ii) the diagonal elements K_{nn}∼n^{h} with h∈(0,1], and (iii) the negligibility of off-diagonal elements K_{mn} with m≠n. We further show that h=ϰ/2α is a ratio between the quantum Lyapunov exponent ϰ and the Krylov exponent α. This proposal is supported by both generic arguments and explicit examples, including solvable Sachdev-Ye-Kitaev models, Luttinger liquids, and many-body localized systems. Our results provide a refined understanding of how chaotic dynamics emerge from the Krylov space perspective.
Krylov complexity is a measure of operator complexity that exhibits universal behavior and bounds a large class of other measures. In this paper, we generalize Krylov complexity from a closed system to an open system coupled to a Markovian bath, where Lindbladian evolution replaces Hamiltonian evolution. We show that Krylov complexity in open systems can be mapped to a non-Hermitian tight-binding model in a half-infinite chain. We discuss the properties of the non-Hermitian terms and show that the strengths of the non-Hermitian terms increase linearly with the increase of the Krylov basis index n. Such a non-Hermitian tight-binding model can exhibit localized edge modes that determine the long-time behavior of Krylov complexity. Hence, the growth of Krylov complexity is suppressed by dissipation, and at long times, Krylov complexity saturates at a finite value much smaller than that of a closed system with the same Hamiltonian. Our conclusions are supported by numerical results on several models, such as the Sachdev-Ye-Kitaev model and the interacting fermion model. Our work provides insights for discussing complexity, chaos, and holography for open quantum systems.
Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large N theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to N=60 Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, λ, across a wide range in temperatures, including in the regime where λ approaches the universal bound, λ=2π/β.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.