Complexity transition in the Dicke model of light-matter interaction

Source: arXiv:2607.21583 · Published 2026-07-23 · By Yicheng Zhang, Erhai Zhao

TL;DR

This paper addresses identifying distinct dynamical complexity regimes and sharp transitions in the far-from-equilibrium quantum dynamics of the Dicke model, a fundamental light-matter interaction system. It explores whether tuning the spin-boson coupling strength g induces non-analytic transitions in complexity measures like Krylov complexity (CK) and Krylov entropy (SK), which quantify operator growth and wavepacket spread in Krylov space. The authors present numerical evidence for a well-defined complexity transition at a critical coupling g_c, separating two qualitatively distinct dynamical regimes: a "regular" regime with localized, oscillatory Krylov wavepackets and a "chaotic" regime with extended, delocalized wavepackets exhibiting chaotic features and spaghettification analogies. They characterize these regimes and transitions through scaling analyses of CK with system size N, jumps in SK, spectral statistics, and semiclassical Lyapunov exponents. The intricate dynamics are analyzed via an effective single-particle tight-binding model in Krylov space with linear Lanczos coefficients mapping to a Rindler spacetime picture, offering a synthetic gravity analogy for the transition. This represents the first observation of sharp, nonanalytic complexity transitions in a canonical physically motivated quantum many-body system without random disorder. The results deepen understanding of complexity growth, chaos onset, and quantum phase-like transitions in far-from-equilibrium many-body dynamics with broadly applicable methodology.

Key findings

• Krylov complexity CK shows a nonmonotonic and sharp transition near the critical coupling g_c ~ 0.5 at resonance, with CK first decreasing smoothly then sharply rising accompanied by large fluctuations (Fig.1a).
• At the transition, CK scales as N^{3/2} approaching g_c from below, distinct from scaling ~ N^{5/2} for g > g_c, indicating a separate complexity phase (Fig.1c).
• Krylov entropy SK jumps sharply at g_c, consistent with a first-order-like transition, with SK/log(N) increasing markedly across the transition (Fig.1d).
• Traditional DPT probes like time-averaged magnetization <S_z> and boson number <n_b> are nearly flat for g < g_c but change after g_c without symmetry breaking, suggesting the complexity transition is not tied to equilibrium phase transitions (Fig.3a).
• Spectral statistics shift from Poissonian (integrable) behavior below g_c to Wigner-Dyson (chaotic) above g_c in energy windows near initial energy, aligning with CK/SK transitions (Fig.3c).
• Semiclassical Lyapunov exponents become nonzero near g_c, confirming onset of chaos coincident with the complexity transition, though with some parameter sensitivity (Fig.3d).
• The Lanczos coefficients (a_k, b_k) increase roughly linearly with k, generating a Krylov space tight-binding Hamiltonian analogous to a particle in (1+1)D Rindler spacetime with synthetic gravity, explaining the wavepacket “bouncing” (regular) versus “spaghettification” (chaotic) regimes (Section on wave mechanics).
• Disorder in Lanczos coefficients limits wavepacket spreading and shapes the complexity transition; removing disorder yields monotonic growth of complexity without transition (Fig.2 and discussion).

Methodology — deep read

1. Threat Model & Assumptions: The authors focus on the unitary quench dynamics of the Dicke model, describing N two-level atoms coupled globally to one bosonic cavity mode. The key parameter is the dimensionless spin-boson coupling strength g̃, which tunes the system between integrable, regular dynamics (low g̃) and chaotic regimes (high g̃). They assume perfect unitary evolution without noise or decoherence suitable for cold atom/ion trap systems. The adversary concept does not apply here.

2. Data: The analysis is numerical and theoretical. Simulations are performed for system sizes N = 20, 30, 40 with initial states as product states of spin coherent states (polar angle θ0) and photon vacuum (α0 = 0) or other coherent states. The main data are time-evolved wavefunctions analyzed via Lanczos recursion extracting Krylov basis parameters. Observables include Krylov complexity CK, Krylov entropy SK, spectral statistics within finite energy windows, semiclassical Lyapunov exponents, and standard magnetization/boson number averages. Time averages are performed over intervals such as gt ∈ [50, 200].

3. Architecture/Algorithm: The central tool is Krylov complexity, defined by recursively applying the Hamiltonian to initial states and constructing an orthonormal Krylov basis via the Lanczos algorithm. This yields a 1D tight-binding chain Hamiltonian in Krylov space with site energies a_k and hoppings b_k. The time evolution of the initial state corresponds to a single-particle wavepacket initially localized at k=0 propagating on this Krylov lattice. The probability distribution over sites p_k(t) defines CK(t) as the expectation value of position k and SK(t) as the Shannon entropy of p_k(t). They analyze the dynamics of CK, SK as functions of g̃ and system size N.

4. Training Regime: Not applicable as this is not a machine learning paper. Numerical evolution of the wavefunction uses exact diagonalization or Krylov-subspace techniques tailored to the infinite Hilbert space truncated appropriately. Parameters like δ=Ω (resonance) and initial states θ0=0, α0=0 are chosen.

5. Evaluation Protocol: The protocol includes scanning g̃ across integrable to chaotic regimes to observe CK, SK transitions. Detailed scaling analyses of CK with N test critical exponents. Time averaging smooths fluctuations. Comparisons to traditional measures (magnetization averages, entanglement entropy SvN, level spacing statistics r, Lyapunov exponents λL) serve as baselines. The spectral statistics are restricted to microcanonical windows near initial state energy to ensure relevance. Various initial states and parameter ratios Ω/δ are also analyzed for robustness. The Lanczos coefficients a_k, b_k are fitted and analyzed both as measured and smoothed or linearized models to isolate disorder effects.

6. Reproducibility: Complete code or dataset release is not mentioned, but Lanczos recursion and Krylov complexity are standard numerical techniques. The Dicke model parameters and initial states are openly specified. Supplemental materials (not included here) presumably provide additional details and numerical data. The methodology should be reproducible by researchers with access to similar computational tools and knowledge.

Concrete Example Walkthrough: For N=20 spins at resonance Ω=δ, starting from an initial spin coherent state pointing along z and vacuum photon state (θ0=0, α0=0), the Hamiltonian is recursively applied to generate the Krylov basis states. The Lanczos coefficients (a_k, b_k) form an ascending linear potential with hopping amplitudes increasing linearly, analogous to a particle in Rindler spacetime with acceleration parameter η increasing with g̃. At low g̃ < g_c ≈ 0.5, the wavepacket oscillates and remains localized, showing small CK and SK values. As g̃ approaches g_c, CK sharply transitions with scaling ~N^{3/2}, and for g̃ > g_c, CK increases rapidly with scaling ~N^{5/2} and wavepacket delocalizes via “spaghettification,” with large SK indicating extended operator growth. Concurrent broadening of spectral level spacings, rising entanglement entropy and positive semiclassical Lyapunov exponents mark transition to chaos. Disorder in Lanczos coefficients modulates wavepacket spreading and confines it below g_c but allows extended chaotic dynamics above it, confirming the physical picture of the complexity transition.

Technical innovations

• Mapping Krylov complexity transition in the Dicke model to a single-particle tight-binding problem with linear Lanczos coefficients interpreted as synthetic Rindler spacetime, linking operator growth to analogue gravitational dynamics.
• Discovery of a sharp nonanalytic transition in Krylov complexity and entropy at critical coupling g_c separating regular oscillatory and chaotic spreading Krylov regimes, demonstrated numerically for the first time in a standard, disorder-free many-body model.
• Identification of distinct power-law scaling regimes for Krylov complexity near the transition, with CK ~ N^{3/2} below transition and CK ~ N^{5/2} above, differing from previously studied spin chains with random disorder.
• Use of Lanczos coefficient disorder analysis to explain wavepacket confinement and transition mechanisms, showing how microscopic disorder controls complexity growth and localization in Krylov space.

Baselines vs proposed

• Poisson level spacing (integrable phase): r ~ 0.386 vs chaotic phase (Wigner-Dyson): r ~ 0.536 above g_c
• Scaling of CK near transition: CK ~ N^{3/2} at g < g_c vs CK ~ N^{5/2} at g > g_c
• Spin-boson entanglement entropy SvN ~ 0.1 log(N+1) below g_c vs ~0.9 log(N+1) above g_c
• Lyapunov exponent λ_L ~ 0 below g_c vs λ_L > 0 for g > g_c

Limitations

• Finite system sizes (N=20 to 40) limit precision of critical scaling exponents and transition sharpness; infinite size extrapolation not performed.
• No explicit adversarial or noise robustness analysis since focus is fundamental quantum dynamics, limiting direct bot-defense security insight.
• The infinite Hilbert space is truncated numerically, possibly affecting quantitative accuracy at very large Krylov index k.
• Only unitary, closed-system dynamics considered; open system effects or experimental imperfections could alter observed transitions.
• The analogy to Rindler spacetime and synthetic gravity is based on approximate linear fits to Lanczos coefficients; exact mapping and generality remain to be fully proven.
• Reproducibility depends on numerical expertise since complete code release is not stated.

Open questions / follow-ons

• How universal is the complexity transition across other quantum many-body models beyond the Dicke model, especially systems with different symmetries or spatial dimensions?
• What are the effects of finite temperature, dissipation, or experimental noise on the existence and observability of the complexity transition?
• Can Krylov complexity and its transition be directly measured or approximated experimentally in cavity QED, trapped ion arrays, or cold atoms platforms?
• How does the synthetic gravity analogy generalize to non-linear or disordered Lanczos coefficient landscapes in other complex quantum systems?

Why it matters for bot defense

While this work is fundamentally focused on quantum many-body physics far from equilibrium, its approach—using Krylov complexity as a sharp probe to detect transitions between qualitatively distinct dynamical regimes—offers conceptual tools for bot-defense researchers investigating complexity growth in computational systems. The identification of distinct scaling regimes and non-analytic changes in complexity measures suggests new ways to characterize transitions from simple to chaotic behaviors. Although the Dicke model and Krylov framework are specialized, similar recursive operator growth or basis-expansion techniques could be adapted to analyze computational complexity or bot behavior transitions in CAPTCHA or bot-defense algorithms. Moreover, the emphasis on wavepacket spreading and disorder-induced confinement may inspire analogous models of adversarial perturbations or complexity traps. Bot-defense engineers might consider how complexity phase diagrams and transitions could reveal tipping points distinguishing legitimate user behavior from bots or automated attacks, even if the underlying physics differs. However, direct application requires significant adaptation, and experimental signatures would differ. These results primarily enrich the theoretical toolkit available for understanding complexity growth and chaotic transitions, which underly many ML and security domains.

Cite

@article{arxiv2607_21583,
  title={ Complexity transition in the Dicke model of light-matter interaction },
  author={ Yicheng Zhang and Erhai Zhao },
  journal={arXiv preprint arXiv:2607.21583},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.21583}
}

Read the full paper

• arXiv:2607.21583 (abstract)
• arXiv:2607.21583 (PDF)