Krylov Complexity Phase Transitions

• Krylov Complexity Phase Transitions are qualitative changes in operator spreading as system parameters tune across a critical value.
• They leverage the Lanczos algorithm to construct a Krylov basis, revealing distinct dynamical regimes such as ergodic, fractal, and localized phases.
• KCPT offers a robust diagnostic of chaotic versus integrable behavior, complementing traditional measures like entanglement and spectral statistics.

Krylov complexity phase transitions (KCPT) refer to qualitative changes in the dynamics of operator growth and complexity, as quantified by Krylov complexity, as a system parameter is tuned across a critical value. Krylov complexity—typically defined as the expectation value of the position operator in a Krylov basis constructed via the Lanczos algorithm—serves as a canonical probe of operator spreading and quantum information scrambling in quantum many-body systems. The paper of KCPT aims to characterize and classify transitions between dynamical regimes such as integrable, chaotic, localized, multifractal, or topologically distinct phases, where the late-time or scaling behavior of Krylov complexity (and associated observables) provides unique diagnostic power that complements traditional measures such as entanglement and level statistics.

1. Formalism of Krylov Complexity and Operator Growth

Krylov complexity ($K(t)$ or $C(t)$) quantifies the "distance" a time-evolved operator or state has traveled along a Krylov chain generated by iteratively acting with the Liouvillian or Hamiltonian. For an operator $O$, the Krylov basis is defined by repeated commutation with the Hamiltonian, $O_{n+1} = [H, O_n] - b_{n-1}^2 O_{n-1}$, with orthonormalization at each step. The time-evolved operator is written as

$O(t) = \sum_{n=0}^\infty i^n \phi_n(t) O_n,$

where the coefficients $\phi_n(t)$ satisfy a tridiagonal recursion set by the Lanczos coefficients $b_n$. Krylov complexity is given by

$K(t) = \sum_{n=0}^\infty n\, |\phi_n(t)|^2.$

This measure is basis-independent (within the Krylov construction), does not require an arbitrary cost function, and is sensitive to the full operator or state evolution.

Crucially, Krylov complexity is intimately related to operator growth and scrambling. In the context of chaotic quantum systems, the early-time behavior is often exponential, $K(t) \sim e^{2\alpha t}$, set by the linear growth of $b_n \sim \alpha n$; this rate can upper-bound the Lyapunov exponent. At late times, $K(t)$ typically saturates to a maximal value reflecting the Hilbert space dimension.

2. KCPT and Dynamical Phase Boundaries: Ergodic, Fractal, Localized Regimes

KCPTs emerge most naturally in quantum systems admitting a tunable parameter (e.g., disorder strength, interaction, percolation probability, Trotter step, or non-Hermitian deformation) such that the qualitative nature of operator spreading—ergodic, multifractal, localized, or otherwise—can be controlled.

A paradigmatic setting is the Rosenzweig–Porter (RP) random matrix ensemble (Bhattacharjee et al., 10 Jul 2024):

• Ergodic phase ($\gamma \leq 1$): $b_n^2 \sim 1-n/N$; Krylov basis delocalizes, operator wavefunction spreads ballistically, and Krylov complexity shows a rapid linear or exponential increase followed by a prominent peak before plateauing.
• Fractal phase ($1 < \gamma \leq 2$): $b_n^2$ decreases more sharply, Krylov basis exhibits multifractality, and the time evolution of $K(t)$ reflects intermediate ramp and less pronounced peak.
• Localized phase ($\gamma > 2$): $b_n \to 0$, operator wavefunction localizes in Krylov space, and $K(t)$ saturates quickly to a small value with no peak, mirroring Anderson localization.

The ergodic–fractal and fractal–localized transitions are sharply characterized by the scaling of the inverse participation ratio (IPR) in Krylov space and the phenomenological $q$-logarithm ansatz for $b_n^2$: $b_n^2 = -p\,\ln_q(x), \quad x = n/N,$ where $q$ interpolates from 0 (ergodic) to 1 (localized). The scaling exponent $\mathcal{D}_2$, extracted from Krylov IPR, takes values 1 (ergodic), $2-\gamma$ (fractal), and 0 (localized), directly signaling phase transitions (Bhattacharjee et al., 10 Jul 2024).

Similar phase structure is observed in models of fermions on graphs, where operator growth and Krylov complexity scaling with system size $N$ differentiate phases that are indistinct under entanglement entropy:

• For free fermions on regular graphs, $D_{\mathrm{Krylov}} \sim N$ ($d=2$) versus $N^2$ ($d=3$);
• For interacting fermions, $D \sim 4^{N^\alpha}$ ($d=2$) with $0.38 \le \alpha \le 0.59$, $D \sim 4^N$ ($d=3$); revealing "complexity-enriched" phases inaccessible to conventional entanglement analysis (Xia et al., 11 Apr 2024).

3. Critical Behavior and Scaling Theory in KCPT

Critical exponents and scaling laws associated with KCPT have been analyzed in quantum percolation models (QPM), where Krylov dimension $D$ serves as an order parameter: $D(p, L) = L^{\gamma/\nu} \mathcal{F}(L/\xi),$ with $\xi \sim |p-p_c|^{-\nu}$ a correlation length. For non-interacting fermions, the KCPT coincides quantitatively with the classical percolation transition $p_{c,k} = p_c$ and exhibits divergence as $D \sim |p-p_{c,k}|^{-\delta}$, with the exponent $\delta = (4-\tau)/\sigma$ inherited from classical percolation criticality. In contrast, for interacting systems, KCPT is shifted to lower $p_{c,k} < p_c$ due to an exponential scaling $d(s) \sim 4^s$ for cluster size $s$, an effect analogous to a quantum Griffiths phase where rare-region contributions dominate complexity (Xia et al., 29 Jul 2025).

In random matrix settings, the height and presence of a "Krylov complexity peak" (KCP) serves as a quantitative order parameter: $\Delta C = C(t_p) - C(t \to \infty),$ nonzero in chaotic (Wigner–Dyson statistics) regimes and vanishing in integrable (Poisson statistics) regimes (Baggioli et al., 24 Jul 2024, Huh et al., 6 Dec 2024). This peak is robustly correlated with the Brody parameter $b$ describing the level statistics crossover, as shown for GOE/GUE–Poisson transitions (Huh et al., 6 Dec 2024).

4. KCPT in Non-Hermitian and Floquet Systems

KCPT probes dynamical phase transitions in non-Hermitian systems, where new phases arise due to $\mathcal{PT}$ symmetry breaking and non-Hermitian disorder:

• In non-Hermitian quantum spin chains, localization and "reciprocity breaking" in Krylov space (captured by the metric $R_K^{(d)} = \frac{1}{d}\sum_n \cos\theta_n$ with $\theta_n = \arg(b_n c_n)$) sharply distinguishes chaotic and non-chaotic regimes (Zhou et al., 27 Jan 2025).
• For non-Hermitian Su-Schrieffer-Heeger models, phase transitions (real$\to$complex$\to$purely imaginary spectra) are detected as discontinuities in the derivatives of stationary Krylov spread complexity and as crossover in dynamical time scales extracted from "Krylov fidelity" evolution (Medina-Guerra et al., 24 Mar 2025).

In Floquet and quantum circuit contexts, KCPT appears as either a smooth crossover (in chaotic systems) or a nonanalytic transition (in integrable/free systems) between regimes governed by Trotter step size. The onset of maximally ergodic operators with vanishing autocorrelations signals a transition to maximal linear growth of Krylov complexity (Suchsland et al., 2023).

5. Interpretation, Universality, and Experimental Protocols

KCPT provides a parameter-free, operator-independent diagnostic of chaotic–integrable transitions that complements (and sometimes exceeds) the resolving power of spectral statistics, OTOCs, and entanglement. The emergence (or suppression) of a KCP, scaling of late-time saturation, or signatures in the variance of Lanczos/Arnoldi coefficients serve as precise markers of dynamical phase boundaries.

Experimental protocols for KCPT include Trotterized evolution and Gram–Schmidt orthogonalization schemes to construct the Krylov basis and measure coefficients via quantum swap or Hadamard testing, allowing the explicit measurement of Krylov complexity on near-term quantum devices, including in cold atom platforms, Rydberg arrays, and superconducting qubit systems (Xia et al., 29 Jul 2025). OTOCs provide an indirect experimental probe of Krylov complexity growth rates, since the Lyapunov exponent is bounded by twice the asymptotic Lanczos slope (Xia et al., 11 Apr 2024).

6. Relation to Operator Spreading, Topology, and Gravity Duals

KCPT is sensitive to a wide range of operator growth phenomena:

• In topological systems such as the Kitaev chain, Krylov (spread) complexity sharply indicates quantum critical points and topologically nontrivial phases via plateaux and singularities in its derivatives (Caputa et al., 2022).
• In field theoretic and holographic models, the transition from oscillatory to exponential growth in K-complexity aligns with confinement/deconfinement and heating/non-heating transitions, providing a nonlocal order parameter not captured by entanglement or mass gaps alone (Anegawa et al., 9 Jan 2024, Malvimat et al., 24 Feb 2024).
• In AdS/CFT and JT gravity, Krylov complexity is linked to geometric lengths (e.g., wormhole length), further connecting complexity phase transitions to geometric transitions in dual gravity (Rabinovici et al., 8 Jul 2025).

7. Open Questions and Outlook

Research in KCPT is ongoing in several directions:

• Establishing universal scaling laws and critical exponents for KCPT across a broader class of models, including higher dimensions, open/lindbladian systems, and time-dependent drives.
• Determining the precise mapping between operator growth rates (Lanczos coefficient slope), Lyapunov exponents, and spectral properties in field theory.
• Understanding the "complexity algebra" and geometrical duals in higher-dimensional and interacting quantum field theories (Rabinovici et al., 8 Jul 2025).
• Extending the instanton path-integral framework to compute late-time plateaux and classify nonperturbative effects in operator complexity (Beetar et al., 17 Jul 2025).
• Investigating the interplay between KCPT, many-body localization, and measurement-induced transitions, especially in disordered or nonunitary systems.

KCPT establishes Krylov complexity as a powerful, quantitative, and experimentally accessible tool for characterizing phase transitions in dynamical complexity and operator growth, with broad applicability in many-body quantum physics, quantum information, and quantum gravity.