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Spread complexity in saddle-dominated scrambling

Kyoung-Bum Huh, Hyun-Sik Jeong, Juan F. Pedraza

arXiv:2312.12593v3hep-thnlin.CDquant-ph

TL;DR

The paper investigates whether spread complexity can distinguish genuine quantum chaos from saddle-dominated scrambling, where unstable saddles produce misleading exponential-growth signatures. Using the Lanczos algorithm in integrable LMG and inverted harmonic oscillator models, it finds chaotic-like ramp-peak-slope-plateau behavior and relates spread complexity to the spectral form factor and Krylov-space transition probability. Analytic checks confirm the Ehrenfest theorem and quadratic early-time growth, while the authors conclude that additional physical input is needed for unambiguous chaos diagnosis.

  • Problem

    Exponential OTOC growth and Krylov operator complexity can appear in integrable systems with unstable saddles, motivating a more discriminating probe of quantum chaos.

    Source: §1 · pp. 3, 5

  • Method

    The paper applies the Lanczos algorithm to spread complexity of thermofield-double states in the integrable LMG model and inverted harmonic oscillator.

    Source: §2.1 · p. 6, §5 · p. 20

  • Results

    Spread complexity exhibits the chaotic-like ramp-peak-slope-plateau structure, while its late-time behavior relates directly to the spectral form factor and Krylov-space transition probability.

    Source: §5 · pp. 20–21

  • Takeaways & Limitations

    Spread complexity is a valuable probe, but its chaotic-looking signatures in saddle-dominated integrable systems require additional physical input for reliable chaos diagnosis.

    Source: §5 · p. 21

    Unstable saddle points constrain spread complexity’s ability to distinguish saddle-dominated scrambling from genuine quantum chaos.

    Source: §3.1 · p. 12, §5 · p. 21

Abstract

from arXiv · show

Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within \emph{integrable} systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of \emph{chaotic} systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity.

1 Introduction

The introduction frames the challenge of distinguishing genuine quantum chaos from saddle-dominated scrambling and motivates spread complexity as a state-based probe. It positions this study in integrable LMG and inverted-harmonic-oscillator models, testing spread complexity’s chaotic signatures and its relations to other diagnostics.

Motivation

  • Exponential OTOC growth can arise near unstable saddle points in integrable systems, so it is insufficient alone to identify quantum chaos.The LMG model and inverted harmonic oscillator are presented as toy models of saddle-dominated scrambling.Source: §1 · p. 3

Spread complexity

  • Spread complexity quantifies the minimum wave-function spread over Hilbert-space bases, uniquely attained in the Krylov basis.It uses state Lanczos coefficients and differs from operator complexity in involving two coefficient types, (a_n, b_n).Source: §1 · pp. 3–4
  • In chaotic systems, thermofield-double spread complexity is conjectured to show a linear ramp, peak, downward slope, and plateau, while the peak diminishes in integrable systems.This pattern is described as analogous to the spectral form factor’s slope, dip, ramp, and plateau.Source: §1 · p. 4

Scope and contribution

  • The study tests spread complexity in the integrable LMG model and inverted harmonic oscillator, both featuring unstable saddle points and saddle-dominated scrambling.The models are selected as representative quantum-mechanical examples and compared with prior OTOC analyses.Source: §1 · p. 4

Motivation

  • The paper asks whether spread complexity can distinguish saddle-dominated scrambling from genuine quantum chaos more effectively than OTOCs and Krylov operator complexity.Prior work found exponential Krylov operator complexity in these integrable models, motivating this comparison.Source: §1 · pp. 4–5

Scope and contribution

  • The paper validates the ramp-peak-slope-plateau proposal, compares spread complexity with the spectral form factor, and studies transition probability as a signature of the peak.It also derives early-time behavior analytically and examines the Ehrenfest theorem of spread complexity.Source: §1 · p. 5

2 Preliminaries

This section defines spread complexity through the Krylov basis and Lanczos algorithm, derives its Ehrenfest relation and quadratic early-time growth, and tests these results in chaotic and integrable billiards.

2.1 Spread complexity and Lanczos algorithm: a quick review

  • The Lanczos algorithm generates orthonormal Krylov states and coefficients a_n and b_n, making the Hamiltonian tridiagonal in the Krylov basis.The Hessenberg form provides an equivalent route to the same coefficients and complexity outcomes in the paper’s examples.Source: §2.1 · pp. 6–7
  • Spread complexity measures the average depth of a time-evolving state in the Krylov basis, obtained by minimizing wave-function spread over Hilbert-space bases.The Krylov basis uniquely realizes this minimum, and the thermofield double state is the state studied here.Source: §2.1 · p. 7
  • The paper investigates whether the peak and plateau structure previously associated with chaotic systems also appears in integrable systems with saddle-dominated scrambling.The billiard example compares stadium billiards, treated as chaotic, with circle billiards, treated as integrable.Source: §2.1 · p. 7, §2.3 · pp. 9–10

2.2 More on spread complexity: Ehrenfest theorem and early time growth

  • The Ehrenfest theorem expresses spread-complexity evolution through the Liouvillian, Lanczos coefficients, and transition amplitudes, and is stated to hold for any system by construction.The paper numerically verifies this assertion.Source: §2.2 · p. 8
  • Quadratic early-time growth follows analytically from the Schrödinger equation, with the leading behavior determined by b_1 rather than a_n.The same functional form is reported for Krylov operator complexity.Source: §2.2 · p. 8

2.3 Example: quantum billiard system (chaotic vs. integrable system)

  • At β = 0, the chaotic stadium billiard exhibits the characteristic peak structure, while finite β suppresses spread complexity and eliminates that peak.The finite-temperature effect agrees with observations reported for the Sachdev-Ye-Kitaev model and random matrix theory.Source: §2.3 · pp. 9–10, Figure 2 · p. 10
  • At early times, numerical billiard results agree well with the analytic expression (2.14).The comparison uses data at β = 0, 0.001, and 0.003 for both chaotic and integrable billiards.Source: §2.3 · p. 10, Figure 3 · p. 10, Figure 2 · p. 10

3 Spread complexity in the Lipkin-Meshkov-Glick model

The LMG model is an integrable finite-dimensional spin system with an unstable saddle, where spread complexity develops several features associated with chaotic systems. The analysis connects these features to Lanczos dynamics, spectral form factors, transition probabilities, temperature, and analytic relations.

3.1 The LMG model

  • The LMG model has an unstable saddle at (x_s, y_s, z_s) = (1, 0, 0), and exponential growth near it is not by itself evidence of chaos.The model is analyzed as an integrable system exhibiting saddle-dominated scrambling.Source: §3.1 · pp. 11–12, §3 · p. 11

3.2.1 Lanczos coefficients

  • The Lanczos coefficient b_n tends to vanish near n = 2S + 1, suggesting that the Krylov basis spans the full Hilbert space even in this integrable model.The coefficient a_n instead shows model-dependent non-monotonic oscillations at small n.Source: §3.2.1 · p. 12, Figure 4 · p. 13

3.2.2 Spread complexity

  • Spread complexity exhibits a ramp-peak-slope structure followed by a plateau, resembling conjectured chaotic-system behavior and potentially failing to distinguish saddle-dominated scrambling from generic chaos.The authors suggest extending the corresponding conjecture beyond quantum chaos to include saddle-dominated scrambling.Source: §3.2.2 · p. 13, Figure 5 · p. 13
  • The peak and saturated spread-complexity values depend on spin S, with fitting curves used for C_max and C(t = ∞).The saturated value is evaluated numerically at t = 2000.Source: §3.2.2 · p. 14, Figure 6 · p. 14
  • Spread complexity grows quadratically at early times, agrees closely with the analytic result, and satisfies the proposed Ehrenfest-theorem relation involving its second time derivative and Lanczos data.The numerical verification is shown by comparing the two sides of the theorem.Source: Figure 7 · p. 14, §3.2.2 · p. 15, Figure 8 · p. 15
  • In the LMG model, spread complexity resembles the spectral form factor: its linear growth may track the SFF slope, its peak may follow a clear SFF ramp, and its plateau is associated with the SFF plateau.The relation also uses the transition probability, with |ψ_0|^2 corresponding to the SFF; these late-time relations hold independently of chaotic behavior for the maximally entangled reference state.Source: §3.2.2 · pp. 15–17
  • Finite inverse temperature β suppresses spread complexity and removes its characteristic peak.The temperature dependence is illustrated by comparing β = 0, 3, and 10.Source: §3.2.2 · p. 17, Figure 11 · p. 17

4 Spread complexity in the inverted harmonic oscillator

The inverted harmonic oscillator is classically regular but has an unstable potential maximum, making it a model of saddle-dominated scrambling. Its spread complexity shows peak-like behavior and relationships with analytical growth, the SFF, transition probability, and temperature.

4.1 The inverted harmonic oscillator model

  • The classically regular inverted harmonic oscillator contains an unstable potential maximum whose curvature is set by λ, while the x^4 term ensures well-defined energies.The model uses λ = 2, g = 1/50 and truncates at i = 200.Source: §4.1 · p. 18

4.2 Spread complexity in the inverted harmonic oscillator

  • The Lanczos coefficients show b_n tending to zero near system size, while a_n varies more monotonically than in the LMG model.These coefficients are presented in Fig. 13.Source: §4.2 · p. 19, Figure 13 · p. 19
  • The spread complexity develops a peak-like feature associated with conjectured chaotic behavior, extending the proposed conjecture to saddle-dominated scrambling.The authors frame this as a possible refinement rather than a definitive chaos diagnosis.Source: §4.2 · p. 19, Figure 14 · p. 19
  • Numerical spread-complexity results agree with analytic early-time expressions and validate the Ehrenfest theorem for complexity.Figure 15 compares numerical and analytic early-time behavior and the two sides of the Ehrenfest relation.Source: §4.2 · p. 19, Figure 15 · p. 20
  • The spread-complexity transition timescale aligns with the SFF, whose transition probability is |ψ_0|^2, while a distinct SFF ramp contributes to the spread-complexity peak and controls late-time behavior.These relationships are shown in Fig. 16 and include the late-time expression (3.12).Source: §4.2 · pp. 19–20, Figure 16 · p. 20
  • Increasing β suppresses spread complexity and diminishes its characteristic peak.The temperature dependence is illustrated for β = 0, 0.05, and 0.15 in Fig. 17.Source: §4.2 · p. 20

5 Conclusions and outlook

The study finds that spread complexity in integrable systems with saddle-dominated scrambling can mimic chaotic-system signatures. The results connect its dynamics to the SFF and transition probability while motivating additional tests across models.

5 Conclusions and outlook

  • The LMG model and inverted harmonic oscillator exhibit a ramp-peak-slope-plateau structure in spread complexity, despite being used as integrable saddle-dominated systems.The conclusion states that this resemblance may hinder separation of saddle-dominated scrambling from genuine quantum chaos.Source: §5 · pp. 20–21
  • Spread complexity and the SFF can share transition timescales in integrable systems with saddle points, while SFF late-time behavior is linked to transition probability in Krylov space.A clear ramp in transition probability can produce a spread-complexity peak in these systems.Source: §5 · p. 21
  • Analytic comparisons confirm the Ehrenfest theorem for complexity and show quadratic early-time spread-complexity growth whose rate corresponds to a Lanczos coefficient.The numerical results agree with the derived early-time expression (2.14).Source: §5 · p. 21
  • Further studies of models such as the Dicke and Feingold-Peres systems could test whether the observed peak is generic across saddle-dominated scrambling.The Feingold-Peres model is noted to display saddle-dominated scrambling despite being classically chaotic.Source: §5 · pp. 21–22
  • The paper identifies Krylov complexity as an unambiguous complexity definition relevant to ongoing questions about black-hole interiors and spacetime emergence.It presents broader exploration across quantum-mechanical settings as future work.Source: §5 · p. 22
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