Assessing the saturation of Krylov complexity as a measure of chaos

Español, Bernardo L.; Wisniacki, Diego A.

doi:10.1103/physreve.107.024217

Cited by 13 publications

(1 citation statement)

References 26 publications

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“…The Krylov dimension, in this case, is of the order K ∼ 4000. For the closed case, we get an exact agreement with the results of [10] (see also [45]).…”

Section: Closed Systemssupporting

confidence: 82%

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Bhattacharya¹,

Nandy²,

Nath³

et al. 2023

Preprint

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Continuing the previous initiatives [1,2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and chaotic systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

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“…The Krylov dimension, in this case, is of the order K ∼ 4000. For the closed case, we get an exact agreement with the results of [10] (see also [45]).…”

Section: Closed Systemssupporting

confidence: 82%

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Bhattacharya¹,

Nandy²,

Nath³

et al. 2023

Preprint

View full text Add to dashboard Cite

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Universal chaotic dynamics from Krylov space

Erdmenger,

Jian,

Xian

2023

J. High Energ. Phys.

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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.

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KPZ scaling from the Krylov space

Gorsky,

Nechaev,

Valov

2024

J. High Energ. Phys.

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Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems, such as Heisenberg spin chains, has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp–plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling, or equivalently, the diffusion-superdiffusion transition at the critical Euclidean time $$ {t}_E^{\ast } $$ t E ∗ = ccrK, for the Krylov chain of finite length K, and ccr = O(1). In particular, we find a scaling ~ K1/3 for fluctuations in the one-point correlation function and a dynamical scaling ~ K−2/3 associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is interesting outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise term of order O (1/N) is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

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Assessing the saturation of Krylov complexity as a measure of chaos

Cited by 13 publications

References 26 publications

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Universal chaotic dynamics from Krylov space

KPZ scaling from the Krylov space

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