Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2508.05444 · Journal of High Energy Physics

Krylov exponents and power spectra for maximal quantum chaos: an EFT approach

Saskia Demulder, Maria Knysh, Andrew Rolph

Published Apr 9, 2026• 0 citations
arXivHTMLPDF

Abstract

A bstract We examine the effective field theory (EFT) of maximal chaos through the lens of Krylov complexity and the Universal Operator Growth Hypothesis. We test the relationship between two measures of quantum chaos: out-of-time-ordered correlators (OTOCs) and Krylov complexity. In the EFT, a shift symmetry of the hydrodynamic modes enforces the maximal Lyapunov exponent in OTOCs, λ L = 2 πT , while simultaneously constraining thermal two-point autocorrelators. We solve these constraints on the autocorrelator, and calculate the Lanczos coefficients and Krylov exponents for several examples, finding both λ K = λ L and λ K = λ L /2. This demonstrates that, within the EFT, the shift symmetry alone is insufficient to enforce maximal Krylov exponents even when the Lyapunov exponent is maximal. In particular, this result suggests a tension with the conjectured bound λ L ≤ λ K ≤ 2 πT . Finally, we identify autocorrelator solutions whose power spectra closely resemble the so-called thermal product formula seen in holographic systems.

DOI
10.1007/jhep04(2026)059

Related papers

Krylov complexity in conformal field theory

Anatoly Y. Dymarsky, Michael Smolkin

Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.

arXiv:2104.09514•Physical review. D/Physical review. D.•Oct 1, 2021•167 citations

Dissecting Quantum Many-Body Chaos in the Krylov Space

Liangyu Chen, Baoyuan Mu, Huajia Wang +1 more

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.

arXiv:2404.08207•Physical Review Letters•May 15, 2025•12 citations

Chaos and operator growth in 2d CFT

Surbhi Khetrapal

Abstract We study the out-of-time-ordered correlator (OTOC) in a zero temperature 2d large-c CFT under evolution by a Liouvillian composed of the Virasoro generators. A bound was conjectured in [1] on the growth of the OTOC set by the Krylov complexity which is a measure of operator growth. The latter grows as an exponential of time with exponent 2α, which sets an upper bound on the Lyapunov exponent, ΛL≤ 2α. We find that for a two dimensional zero temperature CFT, the OTOC decays exponentially with a Lyapunov exponent which saturates this bound. We show that these Virasoro generators form the modular Hamiltonian of the CFT with half space traced out. Therefore, evolution by this modular Hamiltonian gives rise to thermal dynamics in a zero temperature CFT. Leveraging the thermal dynamics of the system, we derive this bound in a zero temperature CFT using the analyticity and boundedness properties of the OTOC.

arXiv:2210.15860•Journal of High Energy Physics•Mar 23, 2023•19 citations

Krylov complexity in saddle-dominated scrambling

Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy +1 more

A bstract In semi-classical systems, the exponential growth of the out-of-time-order correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.

arXiv:2203.03534•Journal of High Energy Physics•May 25, 2022•138 citations

Scaling relations of spectral form factor and Krylov complexity at finite temperature

Chengming Tan, Zhiyang Wei, Ren Zhang

In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and the spectral form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite-temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients b_{n}, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in Parker et al. [Phys. Rev. X 9, 041017 (2019)2160-330810.1103/PhysRevX.9.041017]. This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope α of the b_{n} is bounded by πk_{B}T, where k_{B} is the Boltzmann constant and T is the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by g in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of g decreases as well. Considering that α also represents the operator growth rate, we establish a quantitative relationship between the ergodicity indicator and the Lanczos coefficients' slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite-temperature effects on the Krylov complexity, the SFF, and the connection between ergodicity and operator growth.

Physical review. E•Jan 17, 2025•3 citations

Krylov complexity and chaos in deformed Sachdev-Ye-Kitaev models

Shira Chapman, Saskia Demulder, Damián A. Galante +2 more

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.

arXiv:2407.09604•Physical review. B./Physical review. B•Jan 22, 2025•24 citations
A bstract We examine the effective field theory (EFT) of maximal chaos through the lens of Krylov complexity and the Universal Operator Growth Hypothesis. We test the relationship between two measures of quantum chaos: out-of-time-ordered correlators (OTOCs) and Krylov complexity. In the EFT, a shift symmetry of the hydrodynamic modes enforces the maximal Lyapunov exponent in OTOCs, λ L = 2 πT , while simultaneously constraining thermal two-point autocorrelators. We solve these constraints on the autocorrelator, and calculate the Lanczos coefficients and Krylov exponents for several examples, finding both λ K = λ L and λ K = λ L /2. This demonstrates that, within the EFT, the shift symmetry alone is insufficient to enforce maximal Krylov exponents even when the Lyapunov exponent is maximal. In particular, this result suggests a tension with the conjectured bound λ L ≤ λ K ≤ 2 πT . Finally, we identify autocorrelator solutions whose power spectra closely resemble the so-called thermal product formula seen in holographic systems.
Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.
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.
Abstract We study the out-of-time-ordered correlator (OTOC) in a zero temperature 2d large-c CFT under evolution by a Liouvillian composed of the Virasoro generators. A bound was conjectured in [1] on the growth of the OTOC set by the Krylov complexity which is a measure of operator growth. The latter grows as an exponential of time with exponent 2α, which sets an upper bound on the Lyapunov exponent, ΛL≤ 2α. We find that for a two dimensional zero temperature CFT, the OTOC decays exponentially with a Lyapunov exponent which saturates this bound. We show that these Virasoro generators form the modular Hamiltonian of the CFT with half space traced out. Therefore, evolution by this modular Hamiltonian gives rise to thermal dynamics in a zero temperature CFT. Leveraging the thermal dynamics of the system, we derive this bound in a zero temperature CFT using the analyticity and boundedness properties of the OTOC.
A bstract In semi-classical systems, the exponential growth of the out-of-time-order correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.
In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and the spectral form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite-temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients b_{n}, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in Parker et al. [Phys. Rev. X 9, 041017 (2019)2160-330810.1103/PhysRevX.9.041017]. This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope α of the b_{n} is bounded by πk_{B}T, where k_{B} is the Boltzmann constant and T is the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by g in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of g decreases as well. Considering that α also represents the operator growth rate, we establish a quantitative relationship between the ergodicity indicator and the Lanczos coefficients' slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite-temperature effects on the Krylov complexity, the SFF, and the connection between ergodicity and operator growth.
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.