Krylov complexity in saddle-dominated scrambling

Bhattacharjee, Budhaditya; Cao, Xiangyu; Nandy, Pratik; Pathak, Tanay

doi:10.1007/jhep05(2022)174

Cited by 80 publications

(30 citation statements)

References 77 publications

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“…where C K middle is the K-complexity of the eigenvector |ω middle ) 8 and by definition 0 ≤ C K middle ≤ K. Hence it is found that for a flat operator…”

Section: A2 Role Of the One-point Function In Xxzmentioning

confidence: 98%

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Krylov complexity from integrability to chaos

Rabinovici

Sánchez-Garrido

Shir

et al. 2022

J. High Energ. Phys.

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We apply a notion of quantum complexity, called “Krylov complexity”, to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of the late-time bound to the amount of disorder present in the Lanczos sequence, by mapping the complexity evolution to an auxiliary off-diagonal Anderson hopping model. We compare the late-time saturation of K-complexity in the chaotic phase with that of random matrix ensembles and find that the chaotic system indeed approaches the RMT behavior in the appropriate symmetry class. We investigate the dependence of the results on the two key ingredients of K-complexity: the dynamics of the Hamiltonian and the character of the operator whose time dependence is followed.

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“…where C K middle is the K-complexity of the eigenvector |ω middle ) 8 and by definition 0 ≤ C K middle ≤ K. Hence it is found that for a flat operator…”

Section: A2 Role Of the One-point Function In Xxzmentioning

confidence: 98%

“…A mathematically precise definition of the complexity of time evolution under a given Hamiltonian is given by Krylov complexity [3][4][5] or 'K-complexity' for short. To date, several aspects of Krylov complexity have been studied in various setups and systems, for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Krylov complexity from integrability to chaos

Rabinovici

Sánchez-Garrido

Shir

et al. 2022

J. High Energ. Phys.

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“…Further tests, numerical calculations, applications and generalizations include chaotic Ising chains and systems with many-body localization [40,41], a variety of exemplary systems, including 1d and 2d Ising models as well as 1d Heisenberg models [42], operator growth in conformal field theory [43,44], lattice systems with local interactions under Euclidean time evolution [45], the emergence of bulk Poincaré symmetry in systems exhibiting large-N factorization [46], topological phases of matter [47], integrable models with saddle-point dominated scrambling [48], cosmological Krylov complexity [49], a condition for the irreversibility of operator growth [50], complexity in field theory [51], and a fundamental and universal limit to the growth of the Krylov complexity [52].…”

Section: Introductionmentioning

confidence: 99%

Krylov complexity and orthogonal polynomials

Mück,

Yang

2022

Preprint

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Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.

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“…Said hypothesis concerns the asymptotic growth of the Lanczos coefficients and it basically states that the coefficients should eventually attain linear growth [with a logarithmic correction in one dimension]. The hypothesis is backed up by analytical as well as numerical evidence [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Stability of exponentially damped oscillations under perturbations of the Mori-Chain

Heveling

Wang²,

Bartsch³

et al. 2022

J. Phys. Commun.

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There is an abundance of evidence that some relaxation dynamics, e.g., exponential decays, are much more common in nature than others. Recently, there have been attempts to trace this domi- nance back to a certain stability of the prevalent dynamics versus generic Hamiltonian perturbations. In the paper at hand, we tackle this stability issue from yet another angle, namely in the framework of the recursion method. We investigate the behavior of various relaxation dynamics with respect to alterations of the so-called Lanczos coefficients. All considered scenarios are set up in order to comply with the “universal operator growth hypothesis”. Our numerical experiments suggest the existence of stability in a larger class of relaxation dynamics consisting of exponentially damped oscillations. Further, we propose a criterion to identify “pathological” perturbations that lead to uncommon dynamics.

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Krylov complexity in saddle-dominated scrambling

Cited by 80 publications

References 77 publications

Krylov complexity from integrability to chaos

Krylov complexity from integrability to chaos

Krylov complexity and orthogonal polynomials

Stability of exponentially damped oscillations under perturbations of the Mori-Chain

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