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Krylov complexity from integrability to chaos

E. Rabinovici, A. Sánchez-Garrido, R. Shir, J. Sonner

arXiv:2207.07701v2hep-thcond-mat.str-elquant-ph

TL;DR

The paper asks how Krylov complexity distinguishes integrable from chaotic quantum dynamics and studies this question using integrability-breaking deformations of the XXZ chain. It relates late-time saturation to Lanczos-sequence disorder and compares chaotic results with random-matrix ensembles. The chaotic chain approaches the random-matrix behavior of the appropriate symmetry class.

  • Problem

    The paper investigates how K-complexity can distinguish integrable from chaotic quantum systems and how its behavior depends on Hamiltonian dynamics and operator choice.

    Source: §1 · pp. 2–3

  • Method

    The study analyzes deformed XXZ chains, maps Krylov dynamics to an off-diagonal Anderson problem, and compares results with Gaussian random-matrix ensembles.

    Source: §2 · p. 5, §5 · pp. 11, 13

  • Results

    The chaotic deformed XXZ chain approaches the K-complexity saturation behavior of the appropriate random-matrix universality class.

    Source: §1 · p. 3, §5.3 · p. 17

  • Takeaways & Limitations

    Late-time K-complexity saturation is a probe of integrable versus chaotic behavior, with lower values in integrable systems and increasing values toward chaos.

    Source: §2 · p. 5, §6 · p. 17

Abstract

from arXiv · show

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.

1 Introduction

The paper uses Krylov complexity to study how operator dynamics distinguish integrable from chaotic quantum systems. It interpolates between these regimes with deformed XXZ chains and compares chaotic behavior with random-matrix ensembles.

1 Introduction

  • Krylov complexity quantifies how Heisenberg evolution spreads an initial operator through successively generated commutator directions.The construction uses an orthogonal operator basis adapted to the Hamiltonian dynamics.Source: §1 · pp. 2–3
  • The study introduces integrability-breaking deformations of the XXZ chain to bridge integrable and chaotic behavior while tracking late-time K-complexity.The paper focuses on finite systems away from the thermodynamic limit.Source: §1 · p. 3
  • Integrable systems explore Krylov space less efficiently, while maximally chaotic systems have exponentially large late-time complexity saturation values.Interacting integrable systems show lower saturation through localization effects, whereas free systems typically scale linearly or polynomially.Source: §1 · p. 3
  • Chaotic spin-chain dynamics agree at late times with random-matrix behavior in the appropriate symmetry class.The comparison includes random-matrix ensembles with and without time-reversal symmetry.Source: §1 · pp. 3–4

2 Review of K-complexity and its late-time saturation value

Krylov complexity is constructed from a Lanczos-generated operator basis and measures the average position of Heisenberg evolution on the resulting Krylov chain. Its late-time saturation depends on both operator spreading and the operator’s trace, which can suppress universal chaotic behavior.

2 Review of K-complexity and its late-time saturation value

  • The Lanczos procedure iteratively commutes and orthonormalizes operators, producing a Krylov chain with Lanczos coefficients and dimension K.The Krylov-space dimension obeys K ≤ D^2 − D + 1, and the bound is saturated in the studied cases.Source: §2 · p. 5
  • Heisenberg evolution becomes wavefunction dynamics on the Krylov chain, with K-complexity given by the time-dependent average position of that wavefunction.The amplitudes remain normalized by unitarity and are initialized at the first Krylov site.Source: §2 · p. 5

2.1 Effect of operator’s trace on late-time saturation value of K-complexity

  • A nonzero operator trace can make the n = 0 late-time probability disproportionately large and thereby pull the K-complexity saturation downward.Universal behavior is therefore studied using zero-one-point-function or explicitly traceless operators.Source: §2.1 · p. 7
  • For zero-one-point-function operators, the two-point function equals its connected part, while ETH predicts a plateau at 1/D.This choice isolates universal spectral effects from disconnected contributions.Source: §2.1 · pp. 7–8

2 Review of K-complexity and its late-time saturation value

  • For chaotic systems, approximately uniform late-time transition probabilities across K Krylov sites imply a saturation value approaching K.The late-time behavior is connected to the Liouvillian spectrum and the energy differences coupled by the operator.Source: §2 · p. 6, §2.1 · p. 8

3 XXZ and its integrability breaking

The XXZ chain supplies an integrable starting point, while local and next-nearest-neighbor deformations interpolate toward chaos. Consecutive level-spacing ratios diagnose this transition, which occurs as deformation strength increases but can weaken again at larger couplings.

3 XXZ and its integrability breaking

  • The XXZ spin chain is integrable and exhibits Poisson level-spacing statistics.The study adds local and next-nearest-neighbor terms that break integrability.Source: §3 · p. 8
  • The analysis uses consecutive level-spacing-ratio distributions and their mean to distinguish Poissonian integrable statistics from GOE-like chaotic statistics.The normalized ratio is compared with analytical Poisson and Wigner-ensemble predictions.Source: §3 · p. 8

3.1 Choice of sector and local operator

  • The calculations fix symmetry sectors and use a local operator preserving total-spin and parity symmetries, with its trace removed before Lanczos analysis.Odd-length chains and centered impurities maintain the relevant sector for the integrability-breaking term.Source: §3.1 · p. 9

3.2 r-statistics for XXZ with integrability breaking terms

  • Increasing integrability-breaking strength drives the XXZ statistics from integrable toward chaotic behavior, but sufficiently larger strength makes the system less chaotic again.Figure 2 compares both ratio distributions and their means with Poisson and GOE benchmarks.Source: §3.2 · pp. 9–10, Figure 2 · p. 10

4 K-complexity and integrability-chaos transition

The interpolating XXZ models connect Lanczos-sequence disorder with late-time K-complexity saturation. As integrability breaking increases, Lanczos disorder generally decreases and K-complexity saturation generally rises.

4 K-complexity and integrability-chaos transition

  • K-complexity evolution maps to an off-diagonal Anderson problem, where greater Lanczos disorder localizes Krylov-space eigenvectors and lowers saturation.Reduced disorder is associated with less localization and higher saturation values.Source: §4 · p. 10
  • The standard deviation of log(b_n/b_n+1) generally decreases as integrability breaking strengthens, indicating less disorder in the Lanczos sequence.The distribution has mean approximately zero.Source: §4 · p. 11, Figure 3 · p. 11
  • K-complexity saturation generally increases with the integrability-breaking coefficient across the interpolating Hamiltonians.The paper reports this relationship consistently in Figures 4 and 5 and also observes corresponding changes in time-dependent profiles.Source: §4 · p. 11

5 RMT results and dependence on universality class

RMT comparisons show that late-time K-complexity depends on both Hamiltonian universality class and operator structure, while ETH operators introduce a continuous Thouless-energy dependence. The chaotic deformed XXZ chain agrees with the appropriate GOE-plus-real-operator RMT prediction.

5.1 Influence of the structure of the seed operator

  • An operator constant in the energy basis has late-time saturation ∼K/2 regardless of the Hamiltonian spectrum, showing that operator structure can dominate spectral statistics.Typical RMT operators instead use independently random real or complex matrix elements in the energy basis.Source: §5.1 · pp. 13–14
  • K-complexity saturation depends on both the Hamiltonian’s universality class and whether the operator breaks time reversal.The RMT reference uses GOE real-symmetric or GUE complex-Hermitian Hamiltonians.Source: §5 · p. 13, §5.1 · p. 14, Figure 7 · p. 15
  • For GOE Hamiltonians, random operators saturate above sparse operators, with complex random operators highest; for GUE, the two random operators are quantitatively similar and slightly exceed sparse operators.The GUE Hamiltonian already breaks time reversal, making the real and complex random-operator cases similar.Source: §5.1 · p. 14, Figure 7 · p. 15

5.2 Deviations from the RMT Ansatz: ETH operators

  • ETH-operator saturation increases continuously with Thouless energy, from the lower pure-RMT-Ansatz limit toward larger values.The Thouless energy is normalized by the total bandwidth Λ to remove system-dependent spectral scaling.Source: §5.2 · pp. 15–17, Figure 8 · p. 16

5.3 ETH in the deformed XXZ

  • In the deformed XXZ chain, increasing integrability breaking changes the spectrum from Poissonian to Wigner-Dyson statistics and the operator from non-ETH to ETH behavior.The integrable case has non-Gaussian fluctuations, whereas ε_d = 0.94 satisfies the ETH Ansatz.Source: §5.3 · p. 17
  • The chaotic deformed XXZ chain reaches approximately 0.4K at E_T/Λ ∼ 0.05, matching the GOE+real RMT prediction.At ε_d = 0.94, the Hamiltonian is chaotic and the operator satisfies ETH with E_T/Λ ∼ 0.05.Source: §5.3 · p. 17

6 Discussion

The study finds that late-time K-complexity distinguishes chaotic from integrable behavior, while operator one-point functions and unresolved Krylov localization qualify its interpretation.

6 Discussion

  • Increasing the integrability-breaking coefficient increases late-time K-complexity, while integrable systems have quantitatively lower saturation values than chaotic systems.The chaotic regime shows reasonable agreement with random-matrix predictions in the corresponding universality class.Source: §6 · p. 17
  • The late-time saturation is argued to be exponentially large generically in chaotic systems but lower in strongly coupled integrable systems without exact energy degeneracies.Source: §6 · p. 17
  • Nonzero operator one-point functions can affect late-time K-complexity similarly to the disconnected contribution controlling correlator plateaus.Source: §6 · p. 17
  • For zero-one-point-function operators, ETH differences between integrable and chaotic regimes are subtle: both show similar Thouless energies, but chaotic off-diagonal fluctuations are more Gaussian.The fitted scale is ET /Λ ∼0.055 in both regimes.Source: Figure 9 · p. 18
  • Analytical understanding of the observed phenomena remains open, including establishing Krylov-chain localization and extending the framework to time-dependent Hamiltonians.The discussion also identifies holographic incorporation as a future direction.Source: §6 · pp. 17, 19

A Connected part of autocorrelation function and saturation value of K-complexity

The appendix analyzes how subtracting an operator’s one-point function isolates connected autocorrelation behavior and changes the late-time Krylov transition probabilities.

A Connected part of autocorrelation function and saturation value of K-complexity

  • The analysis assumes normalized hermitian operators, no exact energy degeneracies, and Gaussian-ensemble random variables with order-one matrix elements.The ETH/RMT operator Ansatz specifies the energy-basis matrix elements used in the estimates.Source: A Connected part of autocorrelation… · pp. 19–20
  • The long-time average of the full two-point function is dominated by the squared one-point function, reflecting late-time disconnection.Source: A Connected part of autocorrelation… · p. 20
  • Under the RMT operator Ansatz, the connected two-point function has a long-time average of order 1/D, while its transition probability plateaus at 1/D^2 ∼e^-2S.The initial connected correlator and transition probability remain order one.Source: A Connected part of autocorrelation… · pp. 21, 23
  • Subtracting the one-point function produces a traceless operator whose connected two-point function equals the full correlator of the redefined operator.Source: A Connected part of autocorrelation… · pp. 20–22
  • For operators with nonzero trace, the transition probability Q00 can remain order one and pull the K-complexity saturation downward.Its size is controlled by the operator one-point function rather than universal autocorrelation behavior.Source: A Connected part of autocorrelation… · p. 22

A.1 Example: Hopping vs number operator in cSYK4

In cSYK4, a nonzero one-point function makes the number operator’s late-time transition probability concentrate at Q00, lowering K-complexity saturation, while traceless operators approach the uniform chaotic profile.

A.1 Example: Hopping vs number operator in cSYK4

  • The number operator’s constrained one-point function is distributed across sites at fixed filling and controls an order-one Q00, thereby lowering late-time K-complexity.Subtracting the one-point function produces an operator whose Lanczos sequence is tied to connected two-point-function moments.Source: §A.1 · p. 23
  • The long-time average of the two-point function is dominated by the squared one-point function, consistent qualitatively with large-N factorization.Establishing a late-time plateau additionally requires controlling the long-time variance.Source: §A.1 · p. 23
  • For the full number operator, Q00 remains order one, whereas the remaining Q0n values are more uniformly distributed but reduced by normalization.This behavior is illustrated for cSYK4 in Figure 10.Source: Figure 10 · p. 24, §A.1 · p. 24
  • Hopping and subtracted number operators have zero one-point function, yielding more uniform transition probabilities and K-complexity saturation closer to K/2.For ETH operators, the connected Q00 is exponentially suppressed, so it does not compete with the other transition probabilities.Source: §A.1 · p. 24

A.2 Role of the one-point function in XXZ

In XXZ, subtracting the one-point function does not qualitatively change K-complexity saturation because the connected two-point function remains unsuppressed in the integrable system.

A.2 Role of the one-point function in XXZ

  • Fixed-magnetization sectors constrain the on-site Pauli one-point functions, which need not vanish in integrable XXZ.Unlike the chaotic cSYK4 case, the individual expectation values are not assumed to be equal.Source: §A.2 · pp. 24–25
  • Subtracting the one-point function from XXZ on-site Pauli operators does not drastically alter K-complexity saturation.Figure 11 attributes the remaining undersaturation to the decaying transition-probability profile associated with Krylov-chain localization.Source: Figure 11 · p. 25, §A.2 · p. 25

B Profile of transition probability for flat operator

For a flat operator, the transition-probability profile and K-complexity can be analyzed directly in the Krylov basis, showing a saturation near K/2 for both GOE and Poissonian Hamiltonians.

B Profile of transition probability for flat operator

  • A flat operator is defined by constant matrix elements and normalized Krylov-basis weights, allowing its transition probabilities to be computed from Lanczos coefficients.The analysis uses the odd-dimensional Krylov basis and the middle Liouvillian eigenvector’s zero eigenvalue.Source: B Profile of transition probability… · p. 26
  • The Krylov-basis expansion relates transition probabilities Q0n to the Lanczos coefficients and the Liouvillian eigenvector components.Normalization identities and the middle eigenvector constrain the resulting profile and its complexity estimate.Source: B Profile of transition probability… · pp. 26–27
  • The flat-operator complexity estimate uses K = D^2 − D + 1 and bounds the middle eigenvector’s individual K-complexity between 0 and K.For sufficiently large D, the resulting asymptotic value approaches the reported K/2 behavior.Source: B Profile of transition probability… · p. 27
  • Flat operators with GOE and Poissonian Hamiltonians both exhibit K-complexity saturation values close to K/2 at D = 32.The comparison indicates that this flat-operator saturation is largely independent of whether the Hamiltonian has GOE or Poissonian statistics.Source: Figure 12 · p. 28, B Profile of transition probability… · p. 28
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