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Universal chaotic dynamics from Krylov space

Johanna Erdmenger, Shao-Kai Jian, Zhuo-Yu Xian

arXiv:2303.12151v3hep-thcond-mat.stat-mechcond-mat.str-elnlin.CD

TL;DR

The paper asks which features of Krylov-space evolution distinguish chaotic from non-chaotic systems. It derives an Ehrenfest relation and studies transition probabilities, finding that a long ramp—not linear complexity growth—marks chaotic behavior and accompanies the Krylov-complexity peak.

  • Problem

    The paper examines which features of Krylov complexity and wavefunction evolution in Krylov space characterize quantum chaos.

    Source: §1.1 · p. 3

  • Method

    The authors derive a generalized Ehrenfest theorem for Krylov complexity and analyze transition probabilities in the Krylov basis for chaotic and non-chaotic systems.

    Source: §1.1 · p. 3

  • Results

    A long rise-slope-ramp-plateau pattern in transition probability characterizes chaotic behavior and explains the Krylov-complexity peak, while linear growth also occurs in non-chaotic systems.

    Source: §7 · p. 41

  • Takeaways & Limitations

    Transition probability in Krylov subspaces can reveal a linear ramp associated with chaos, unlike global Krylov complexity alone.

    Source: §7 · p. 41

    First-order continuum polynomials can violate polynomial degree, orthogonality, completeness, and normalization; the second-order formalism resolves their complex values.

    Source: §2.3.1 · p. 15

Abstract

from arXiv · show

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.

1 Introduction

The paper tests which features of Krylov-space evolution characterize chaos, focusing on the maximally entangled state and its relation to spectral statistics. It finds that early linear growth is not diagnostic, while a long ramp in transition probability distinguishes chaotic behavior and accounts for the complexity peak.

1.2 Chaos and Krylov space

  • Quantum chaos is associated with correlated energy levels, including level repulsion and spectral rigidity; the SFF's ramp reflects spectral rigidity and can diagnose chaos.Source: §1.2 · p. 4
  • For the maximally entangled state, Krylov complexity is determined by the Hamiltonian spectrum and tracks the SFF's quadratic growth, linear growth, peak, and saturation.The transitions occur at times close to those in the SFF.Source: §1.2 · p. 7

1.3 Organization of the paper

  • The generalized Ehrenfest theorem relates early-time Krylov-complexity growth to the spectral form factor, but linear growth also occurs in non-chaotic systems.For Gaussian matrix ensemble Lanczos coefficients, the growth is determined by the SFF slope.Source: §1.1 · p. 3, §1.3 · p. 7
  • In chaotic systems, the transition probability shows a rise-slope-ramp-plateau pattern with an exponentially long ramp, which directly accounts for the Krylov-complexity peak.The pattern is also reported for subspace observables in Krylov space.Source: §1.3 · p. 8
  • For uncorrelated, non-chaotic spectra, the transition probability lacks a ramp and the complexity peak is absent, although linear growth persists.Source: §1.3 · p. 8
  • In SYK4, transition probability has a rise-slope-ramp-plateau pattern; in SYK2, the ramp is short at small n and absent at large n.Source: §1.3 · p. 8

2 Krylov state complexity

The section constructs Krylov space by orthogonalizing repeated Liouvillian actions on a reference state, then tracks evolved states in that basis. It derives an Ehrenfest relation, develops continuum treatments, and shows that for a maximally entangled reference state the complexity depends only on the Hamiltonian spectrum.

2.1 Krylov space

  • Krylov space is formed from repeated Liouvillian actions on a reference state, with Gram-Schmidt orthogonalization producing its orthogonal basis.The Krylov space has dimension L, the index at which the generated sequence first becomes linearly dependent.Source: §2.1 · p. 9

2.2 Krylov complexity of evolving states

  • The evolved target state is e^-τL|0⟩, with τ = β + it; its Krylov-chain amplitudes define the state’s complexity and transition probabilities.The survival probability is the n = 0 transition probability, |ϕ0(τ)|² = |S(τ)|².Source: §2.2 · p. 11
  • The Ehrenfest theorem relates the second derivative of Krylov complexity to the expectation value of the gradient of squared Lanczos coefficients.The authors describe it as a classical equation of motion for Krylov complexity.Source: §2.2 · p. 12
  • At low temperature, Krylov complexity approaches a constant plus oscillations at the gap E1 − E0; long-time transition probabilities retain equal-energy terms when oscillating phases average away.The stated plateau-time setup assumes no degeneracy.Source: §2.2 · pp. 12–13

2.3 Continuum limit

  • The continuum limit replaces the discrete Krylov index with a continuous coordinate and approximates differences by derivatives, simplifying the equations under a smoothness assumption.The authors note that this smoothness assumption requires clarification in a second-order formalism.Source: §2.3 · p. 13, §2.3.1 · p. 13

2.3.1 First-order formalism

  • The first-order continuum formalism yields forward propagation, but its smoothness assumption breaks discrete parity symmetry and its polynomials can lose polynomial form, orthogonality, normalization, and reality.The continuum approximation applies only once the wavefunction spreads; the backward propagation is needed after reflection from the chain endpoint.Source: §2.3.1 · pp. 14–15

2.3.2 Second-order formalism

  • The second-order formalism treats even and odd sectors separately when the spectrum has even parity; their recurrences decouple, so the sectors need not join smoothly.The continuum treatment assumes continuity within each parity sector and imposes boundary conditions separately.Source: §2.3.2 · pp. 15–16

2.4 Krylov complexity for the TFD state

  • For a maximally entangled reference state, Krylov complexity depends only on the spectrum, and uniform d-fold degeneracy gives Krylov-space dimension L = D/d with energy overlaps 1/L.The Lanczos algorithm is unaffected by the choice of unitary operators used to define the maximally entangled state.Source: §2.4 · pp. 17–18

3 Krylov complexity at early times

For maximally entangled states, the large-dimension Lanczos-coefficient limit supports an Ehrenfest-theorem relation between Krylov complexity and the spectral form factor (SFF). The results distinguish SFF-related behavior from linear growth and saturation, which are not necessarily chaos-specific.

3 Krylov complexity at early times

  • In a GUE realization, a_n fluctuates around zero while b_n decreases from 1 to 0; real-time profiles show a weakening shock wave and lengthening tail, whereas imaginary-time profiles localize at the ground state.Lanczos-coefficient fluctuations strengthen with increasing n.Source: §3 · pp. 18–19

3.1 Lanczos coefficients in the large dimension limit

  • For maximally entangled states, the large-L Lanczos-coefficient limit is a fixed point of the Liouvillian tridiagonalization algorithm and gives the semicircle law.The Liouvillian is factorized rather than a random Hamiltonian on the double-copy Hilbert space.Source: §3.1 · pp. 20–21

3.2 Krylov complexity from spectral form factor

  • Under the Lanczos-coefficient approximation, the second derivative of Krylov complexity is given by the SFF, while spectral and Krylov complexities agree through linear growth but deviate at late times.The derivation neglects statistical correlations between the Lanczos coefficients and the wave function.Source: §3.2 · p. 22
  • The disconnected SFF term contributes the slope, while its connected term contributes the ramp and plateau.The ramp contribution applies for t < 2π⟨ρ(E)⟩ in the energy window; the plateau contribution appears as that window shrinks to zero.Source: §3.2 · p. 23

3.2.1 Infinite temperature limit

  • At β = 0, Krylov complexity grows linearly as 16t/3π, but its saturation at (L − 1)/2 is attributed to the discrete spectrum and is not necessarily related to chaos.The spectral complexity instead saturates at L − 1, and its saturation is related to chaos.Source: §3.2.1 · p. 24

3.2.2 Low temperature limit

  • At low temperature, spectral complexity and the change in numerical Krylov complexity share quadratic-to-linear growth but approach different late-time saturation values.The initial Krylov complexity also grows with inverse temperature β.Source: §3.2.2 · p. 25

4 Krylov complexity and chaos at late times

Late-time Krylov-complexity growth and saturation are not reliable chaos markers: growth depends mainly on Lanczos-coefficient behavior, and saturation follows spectral discreteness. Instead, chaotic spectra show a long ramp in Krylov-basis transition probabilities, linked to chaos and the late-time complexity peak.

4 Krylov complexity and chaos at late times

  • Linear Krylov-complexity growth is mainly set by the small-n dependence of Lanczos coefficients, while saturation follows spectral discreteness and also occurs for uncorrelated spectra.Numerical Krylov and spectral complexities agree only at early times in the Gaussian ensembles.Source: §4 · pp. 25–26
  • In chaotic spectra, transition probabilities show a rise-slope-ramp-plateau pattern with a long ramp, and that ramp is responsible for the late-time Krylov-complexity peak.Source: §4 · p. 26

4.1 Numerical simulation

  • The transition probability has rise-slope-ramp-plateau dynamics, ramping up for n≲L/2 and down for n≳L/2; its ramp and plateau timing vary with n.For n≲L/2, the plateau is near 2L; for n≳L/2, it is greater than 2L.Source: §4.1 · p. 27

4.2 A first look at late times

  • After saturation at β=0, transition probabilities are not self-averaging, whereas Krylov complexity is self-averaging.Source: §4.2 · p. 30
  • For β=0 and n≪L, the connected transition probability ramps as t/(2L²) to 1/L at 2L; the approximation becomes unreliable as polynomial degree approaches L.The derivation neglects effects of high-frequency polynomial oscillations and polynomial-spectrum correlations.Source: §4.2 · p. 31

4.3 Krylov complexity in the continuum limit

  • 0.75L at t≈0.7L: continuum-limit Krylov complexity peaks there, then nearly saturates at 0.5L for t>L; the disconnected part dominates growth and peak, while the connected part contributes to saturation.Source: §4.3 · p. 34

4.4 The peak of Krylov complexity

  • The complexity peak is the weighted combination of large-n transition-probability peaks in the rise-slope region; mismatched dip times also explain why linear growth before the peak is not a chaos signature.Source: §4.4 · p. 35

5 Non-chaotic spectrum

An uncorrelated spectrum with a semicircle-law density reproduces early linear Krylov-complexity growth, but its transition probability lacks a long ramp and its complexity has no late-time peak.

5 Non-chaotic spectrum

  • The uncorrelated spectrum is sampled from GUE energies individually; its Lanczos coefficients approach the semicircle-law limit on average but fluctuate strongly, especially at large n.The ensemble retains the semicircle-law density while removing level correlations.Source: §5 · p. 35
  • For n not too large, the transition probability exhibits rise-slope-plateau behavior without a ramp and has a universal plateau value 1/L; its plateau time is highly dependent.Source: §5 · p. 36
  • As n approaches L, the transition-probability peak between rise and slope decays or disappears, consistent with probability conservation.For n ≲ L, the probability is bounded by 1 − n/L while rising.Source: §5 · p. 37
  • The early linear complexity growth matches the RMT rate, so linear growth for a maximally entangled reference state does not distinguish spectral chaos.The early-time calculation uses the spectral form factor in the Ehrenfest theorem.Source: §5 · p. 37
  • At t ≳ L, non-chaotic Krylov complexity slows and approaches a plateau from below without a peak, unlike the GUE result.The missing peak follows the absence of a ramp in the transition probability.Source: §5 · p. 37

6 The SYK model

The paper extends its transition-probability analysis to SYK models, finding a long ramp and complexity peak for SYK4, but no such peak for SYK2.

6 The SYK model

  • The SYKq model describes N Majorana fermions with q-fermion random coupling, and the paper reports that universal rise-slope-ramp-plateau behavior also appears in SYK models with quartic or higher fermion interactions.The value of J is specified so that the first Lanczos coefficient is b1 = 1.Source: §6 · p. 38
  • For β = 0, transition probability exhibits rise-slope-ramp-plateau behavior, with ramp ∼t/(2L^2), common plateau time tp ≈2L and value 1/L; Krylov complexity peaks after linear growth before the plateau.Source: §6 · p. 40
  • For β = 0, the early-time SYK4 complexity grows linearly in time because the first term in its stated expression dominates when K(t; β) ≪ L.The transition-probability ramp is associated with a complexity peak before the plateau.Source: §6 · p. 40
  • For SYK2, the transition probability has a negligible ramp at finite n and an n-dependent plateau time, while its common plateau value is 1/L; Krylov complexity has no peak.SYK2 is integrable and has weak level correlations and repulsion.Source: Figure 13 · p. 39, §6 · p. 40

7 Conclusion and outlook

The conclusion identifies transition-probability ramps, rather than linear Krylov-complexity growth, as a chaos-related signal and connects them to complexity peaks. It also notes approximation limits and outlines further directions.

7 Conclusion and outlook

  • The generalized Ehrenfest theorem describes the early linear-growth region in both chaotic and non-chaotic systems, so linear growth alone cannot discriminate them.Its random-matrix derivation assumes statistical correlations between the wavefunction and spectrum can be neglected.Source: §7 · pp. 40–41
  • The transition amplitude exhibits a universal rise-slope-ramp-plateau pattern, and its linear ramp is a robust indicator of chaotic systems, unlike global Krylov complexity alone.The authors relate this signal to spectral rigidity and the spectral form factor.Source: §7 · p. 41
  • The ramp in transition probability explains the presence or absence of late-time Krylov-complexity peaks in chaotic and non-chaotic systems, respectively.The paper argues that observables defined on subspaces of Krylov space can reveal the ramp behavior.Source: §7 · p. 41
  • The predicted Krylov-complexity saturation value is inaccurate because the approximation neglects Lanczos-coefficient fluctuations; incorporating correlations and treating sites n ∼ L remain open directions.The authors also suggest investigating further quantum-mechanical bounds on Krylov-chain observables.Source: §7 · p. 41
  • The connected Krylov-complexity contribution shapes peak and saturation structure at Hilbert-space timescales, potentially relevant to finite-N holographic black-hole evolution.The analytical expression for TFD-state Krylov complexity may serve as a starting point for constructing its gravity dual.Source: §7 · p. 42

A Imaginary time evolution

  • The imaginary-time continuum treatment transforms the wavefunction and coordinate to obtain a Schrödinger equation and a corresponding Ehrenfest theorem for real time.The transformed coordinate obeys dy = dx/(ϵb).Source: A Imaginary time evolution · pp. 42–43

B Spectral complexity for Gaussian ensembles

The appendix extends spectral-complexity calculations to Gaussian ensembles and describes how ensemble-averaged Krylov probabilities are approximated when exact averaging is difficult.

B Spectral complexity for Gaussian ensembles

  • The appendix generalizes spectral-complexity calculations to GOE and GSE using ensemble-specific two-point kernels; only the GUE integral is analytic, while other cases are computed numerically.The density is taken from the semicircle law, and the analysis focuses on β = 0.Source: B Spectral complexity for Gaussian… · pp. 43–45
  • The disconnected spectral-complexity contribution is universal across the three Gaussian ensembles, while the connected part depends strongly on the kernel.The connected expression uses a narrow kernel to simplify the energy integrals.Source: B Spectral complexity for Gaussian… · p. 44
  • At early times t ≪ L, the connected spectral-complexity contribution scales as t^2/L and is negligible relative to the disconnected part.Source: B Spectral complexity for Gaussian… · p. 45
  • At late times t ≳ L, spectral complexity can slow, stop, or rebound across ensembles; the GUE approaches log t rather than a constant because of kernel analyticity properties.For kernels analytic in the upper half-plane, the spectral complexity instead saturates to a time-independent value.Source: B Spectral complexity for Gaussian… · p. 45

C.1 Formal Krylov approach in ensemble average

  • The formal construction obtains the Krylov basis by orthogonalizing a Vandermonde matrix built from a non-degenerate spectrum, then defines transition probabilities and complexity from the resulting basis.The ensemble-average expressions use the evolution and complexity matrices.Source: §C.1 · pp. 45–46
  • Because exact ensemble averaging involves up to 2n-point functions, the formal Krylov calculation approximates spectrum-dependent polynomials by their limiting forms and accounts for confinement.The approximation uses limiting Lanczos coefficients.Source: §C.1 · p. 47

C.2 General linear ramp from Gaussian ensemble

The Gaussian-ensemble calculation relates the connected transition-probability generating function to spectral-density fluctuations and yields a linear ramp. The argument uses the semicircle saddle and a finite-domain Fourier transform.

C.2 General linear ramp from Gaussian ensemble

  • The derivation expands the Gaussian-ensemble spectral-density action around its semicircle saddle and retains its quadratic fluctuation term.The connected generating function is related to the two-point function of spectral-density fluctuations.Source: §C.2 · p. 47
  • A finite spectral interval maps by Fourier transformation to discrete frequencies s_n = 2πn/Λ.The spectrum lies in [−Λ/2, Λ/2], and n is a natural number.Source: §C.2 · p. 47
  • The connected part of the generating function develops a linear ramp in time.The derivation shifts the time arguments by small amounts relative to the observation time.Source: §C.2 · p. 48

C.3 Confinement of polynomials

The polynomial confinement approximation accounts for correlations between a polynomial’s energy argument and the spectrum. It also links the shrinking energy support of Krylov states to reduced entanglement.

C.3 Confinement of polynomials

  • The transition-probability ensemble average depends on multipoint spectral-density correlations because each polynomial is determined by the full spectrum.The polynomial is fixed by the Krylov-basis orthogonality relation, equivalently the Lanczos algorithm.Source: §C.3 · p. 48
  • For GUE levels, ψ_n(E_p) nearly vanishes outside |E_p| ≲ 2√(1 − n/L), reflecting confinement to a shrinking spectral interval.The confinement is attributed to the decrease of the Lanczos coefficient b_n; setting it to zero approximately splits the Krylov matrix into two blocks.Source: §C.3 · p. 49
  • The confinement property decreases entanglement in Krylov states from the maximally entangled initial state.For 1 ≪ n ≪ L the decrease is slow, then becomes rapid as n approaches L and the supporting subspace shrinks quickly.Source: §C.3 · pp. 50–51

C.4 Approximate the reduced spectral density and polynomials

The section approximates the reduced spectral density and Krylov polynomials using a semicircle law and two polynomial constructions. Their accuracy depends on the regime and on how the spectrum is cut.

C.4 Approximate the reduced spectral density and polynomials

  • The reduced spectral density of the remaining Krylov block is semicircular, and its approximation matches the numerical level statistics well.The density is normalized to L using an additional factor 1/(1 − n/L).Source: §C.4 · p. 51
  • The continuum-limit polynomial is localized near a characteristic curve, unlike the numerical wavefunction’s long tails, so it is used only for the connected transition-probability part.Its tails would give incorrect slope behavior in the disconnected part.Source: §C.4 · p. 52
  • The deformed Chebyshev construction adjusts the polynomial degree to match the characteristic curve and approaches degree n at small n and zero at n = L.It modifies the Chebyshev polynomial to account for the shrinking spectral domain.Source: §C.4 · pp. 52–53
  • The approximate polynomials reproduce the weighted spectral density’s moving average and oscillation frequency, but have sharper edges than the numerical statistics.The hard spectral cut produces cliff-like approximate edges versus smooth numerical edges; the normalization remains a good approximation.Source: §C.4 · p. 53

C.5 More on chaos in transition probabilities

The section approximates transition probabilities by combining polynomial and spectral-density approximations. For n ≲ L/2, it accounts for rise-slope, ramp-up, and plateau behavior, while larger n remains difficult to describe analytically.

C.5 More on chaos in transition probabilities

  • The approximation incorporates finite polynomial degree and confinement, but neglects correlations between spectra in the polynomials.The authors state that this correlation contribution cannot be calculated in their treatment.Source: §C.5 · pp. 53, 55
  • The real-time behavior has a sharp rise and long slope, with the second term contributing to a plateau whose value is 1/L.The analytical rise-slope behaviors match numerical results well until n approaches L, while the ramp-up behaviors are smaller than the numerical results.Source: §C.5 · pp. 56–57
  • The ramp-up term arises from the oscillating polynomial product, which shifts time scales; the two triangular contributions separate for n > 0 and create a step.The resulting ramp region is discussed for n ≲ L/2.Source: §C.5 · pp. 56–57
  • For n ≳ L/2, strong Lanczos-coefficient and polynomial fluctuations prevent an analytical account of the rise-slope and ramp-down behavior.The transition probability also fluctuates strongly at large n in a single realization.Source: §C.5 · p. 57
  • The authors suggest that phase fluctuations cancel slope oscillations across realizations, leaving a moving average that resembles the numerical ramp-down.This is presented as a possible explanation for n ≳ L/2.Source: §C.5 · pp. 57–58

D Constant Lanczos coefficients

The constant-Lanczos example illustrates how the Ehrenfest theorem relates Krylov-complexity growth to endpoint probabilities. At short times, the evolution is effectively described by the first term of Eq. (D.2).

D Constant Lanczos coefficients

  • The example uses dimensionless spectrum, Lanczos coefficients, and time τ = β + it, with b_0 = b_L = 0 at the chain endpoints.The interior Lanczos coefficients are constant.Source: D Constant Lanczos coefficients · p. 58
  • With constant interior Lanczos coefficients and vanishing endpoint coefficients, Krylov-complexity growth is driven by probability at the two chain endpoints.The other terms in the Ehrenfest theorem vanish in this case, apart from the endpoint contributions.Source: D Constant Lanczos coefficients · p. 58
  • At n ≪ L, Eq. (D.1) describes flat Lanczos coefficients, and short-time evolution is effectively governed by the first term of Eq. (D.2).This short-time description applies to the wave function and Krylov complexity in a general system.Source: D Constant Lanczos coefficients · p. 58

D.1 Eigenstate

For the eigenstate case, the orthogonal normalized polynomials are Chebyshev polynomials of the second kind. The appendix gives expressions for low-temperature convergence and the long-time plateau.

D.1 Eigenstate

  • The orthogonal normalized polynomials are Chebyshev polynomials of the second kind.Source: §D.1 · p. 59
  • The eigenenergy and measure are specified for this eigenstate construction.Source: §D.1 · p. 59
  • At low temperature, the complexity converges to an expression describing the β ≫ L limit.The cited passage says the tedious sub-leading term is not shown.Source: §D.1 · p. 59
  • At long-time average, the complexity has a plateau value given by the expression referenced in Eq. (2.28).Source: §D.1 · p. 59

D.2 Large dimension limit

The large-dimension analysis gives a typical square-to-linear-to-plateau growth of Krylov complexity at finite temperature and compares the formulas with numerical results. The finite-dimension treatment uses a continuous Schrödinger equation to study behavior near the plateau time.

D.2 Large dimension limit

  • In the large-L limit, the wave function is expressed using an integral, and the Krylov amplitudes become nonzero for all n.The analysis uses modified Bessel functions to calculate survival amplitude and complexity along imaginary time.Source: §D.2 · p. 59
  • Along real time, the analysis integrates the complexity change ΔK(t; β) and obtains a linear-growth approximation for β = 0.The approximation is stated as 16/(3π) times t.Source: §D.2 · pp. 59–60
  • At finite temperature, real-time Krylov complexity shows typical square-to-linear-to-plateau growth, and the analytical formulas are compared with numerical results in Fig. 19.The figure plots complexity against inverse temperature and time for L = 1024, with dots for numerical results and solid curves for analytical approximations.Source: §D.2 · p. 60, Figure 19 · p. 60

D.3 Finite dimension effects

  • To examine behavior near plateau time at large but finite L, the analysis adopts a continuous Schrödinger equation with a potential and boundary conditions at y = 0 and y = L.The setup releases the Krylov-space bound L in favor of ˜L, which is eventually taken to infinity.Source: §D.3 · p. 60
  • With flat Lanczos coefficients, Krylov complexity is the expectation value of position y, and its value is obtained for general β and t.Source: §D.3 · p. 61
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