Krylov complexity as an order parameter for quantum chaotic-integrable transitions

This paper asks whether a specific feature of Krylov complexity—its “Krylov complexity peak” (KCP)—can serve as an order parameter for the transition between quantum chaotic and integrable behavior in many-body systems. The question matters because, while quantum chaos is commonly diagnosed using late-time spectral statistics (e.g., random-matrix-theory level repulsion) and early-time dynamical probes such as out-of-time-order correlators (OTOCs), these diagnostics can be operator-dependent, time-scale dependent, or difficult to access experimentally. Krylov complexity offers an alternative, potentially more universal characterization: it tracks how a time-evolving state spreads over a Krylov basis generated by repeated action of the Hamiltonian. Recent work suggested that in chaotic systems the Krylov state complexity exhibits a characteristic four-stage evolution—an initial ramp, a peak that overshoots a late-time plateau, then a decline into a plateau—while integrable systems lack the overshoot/peak. Building on this, the authors propose that the presence and height of the peak is a hallmark of quantum chaos.

Methodologically, the authors study two representative models that are well understood to interpolate between chaotic and integrable regimes: (1) the mass-deformed Sachdev–Ye–Kitaev (SYK) model and (2) the sparse SYK model. In both cases they compute Krylov state complexity for a thermofield double (TFD) initial state. The mass-deformed SYK Hamiltonian contains random quartic interactions plus an additional random quadratic “mass” term with strength $\kappa$. Increasing $\kappa$ drives the system from a chaotic regime ($\kappa \approx 0$) toward an integrable regime (large $\kappa$). The sparse SYK Hamiltonian contains random quartic interactions but with a sparsity parameter $p$ (equivalently the number of nonzero terms per row $k\sim p{N}^4$); decreasing $p$ (or $k$) drives the system toward integrability.

Krylov complexity is computed by constructing a Krylov basis $\{|K_n⟩\}$ using the Lanczos algorithm, which yields Lanczos coefficients $\{a_n,b_n\}$. In the Krylov basis the Hamiltonian becomes tridiagonal, and the time-dependent Krylov wavefunction components ${\psi }_n(t)$ satisfy a discrete Schrödinger equation. The Krylov state complexity is then defined as $C(t)={\sum }_{n}n|{\psi }_n(t){|}^2$. The initial state is the TFD state at inverse temperature $\beta$, which is built from the energy eigenstates of the Hamiltonian on two copies of the Hilbert space.

To connect Krylov complexity to more standard chaos probes, the authors also compute the spectral form factor (SFF), defined as $\mathrm{SFF}(t)=|Z(\beta +it){|}^2/|Z(\beta ){|}^2$, which in chaotic systems shows a slope–dip–ramp–plateau structure. They use a known identity at infinite temperature relating the long-time average of the SFF to the late-time Krylov complexity plateau: ${\lim }_{T\to \infty }\frac{1}{T}{\int }_0^T\mathrm{SFF}(t)dt=\frac{1}{1+2C(t\to \infty )}$ (with the appropriate system-size mapping). This provides a consistency check and motivates comparing the disappearance of the SFF ramp with the disappearance of the KCP.

For numerical implementation, the authors use system size $N=26$ (dimension $d$ of the relevant symmetry sector scales exponentially with $N$) and set the SYK coupling $J=1$. They argue that $N=26$ is sufficiently large for convergence of the Krylov-complexity features. They also validate numerical consistency via normalization and Ehrenfest-theorem checks in an appendix.

The key result is that the KCP height behaves like an order parameter. They define an order parameter as the difference between the peak value and the late-time plateau value: $$\Delta C:=C(t=t_{\mathrm{peak}})-C(t\to \infty ).$$ By construction, $\Delta C\ne 0$ indicates a chaotic system with a peak overshoot, while $\Delta C=0$ indicates integrability where the peak vanishes.

In the mass-deformed SYK model at infinite temperature $\beta =0$, the normalized complexity $C(t)/d$ shows the four-stage chaotic evolution for small $\kappa$: a linear ramp up to a peak at $t=t_{\mathrm{peak}}$, followed by a decline into a $\kappa$-independent plateau with $C(t\to \infty )/d\approx 1/2$. As $\kappa$ increases, the peak occurs earlier and eventually disappears. Quantitatively, $\Delta C$ decreases smoothly from about $\Delta C\approx 0.1$ near $\kappa \to 0$ to $\Delta C=0$ at large $\kappa$. The critical $\kappa$ where $\Delta C$ vanishes matches the previously reported chaotic–integrable transition value ${\kappa }_c\approx 66$ from spectral statistics.

At finite temperature, the authors find three systematic effects: (I) in the chaotic regime, $\Delta C$ becomes temperature-dependent and decreases as $\beta$ increases; (II) the transition point shifts to smaller $\kappa$; and (III) the transition becomes a smooth crossover rather than a sharp critical point. They further construct a phase diagram and fit the critical line with an empirical exponential dependence: $${\kappa }_c(\beta )\approx {\kappa }_c(0)e^{-2\pi \beta }.$$ Using ${\kappa }_c(0)\approx 66$, they estimate that for $\kappa =1$ the critical inverse temperature is ${\beta }_c\approx 6.58$. This prediction is reported to match the observed disappearance of the SFF ramp in their SFF plots.

In the sparse SYK model at $\beta =0$, the authors again observe that decreasing sparsity (increasing $p$ or $k$) restores the chaotic four-stage Krylov evolution with a peak, while increasing sparsity suppression (smaller $k$) reduces the peak height and alters the late-time saturation value. They explain the saturation deviation at small $k$ by emergent discrete symmetries leading to exact degeneracies, which suppress the expected late-time Krylov plateau. The KCP order parameter $\Delta C$ transitions to zero at a critical $k_c\approx 1$, consistent with earlier level-statistics-based results (e.g., $r$-parameter statistics). Importantly, unlike the mass-deformed case, the critical $k_c$ is approximately temperature-independent across the tested $\beta$ values.

They corroborate the KCP transitions with the SFF: in both models, the SFF ramp disappears around the same parameter values where $\Delta C$ vanishes. They also discuss an alternative candidate order parameter based on the depth of the SFF dip, but find it is not robust in the $\beta \to 0$ limit (the dip feature persists in the SYK2 limit), whereas the KCP remains a clearer diagnostic.

The authors acknowledge limitations implicitly through their methodology choices. The study is numerical and restricted to relatively small system size $N=26$, though they argue finite-size effects are minimal for the convergence of Krylov-complexity features. The order parameter is defined using the TFD state; while the paper emphasizes operator-independence in the sense of not relying on a specific OTOC operator, it still depends on the choice of initial state ensemble (TFD) and temperature $\beta$. They also note that the KCP’s universality beyond the tested SYK variants remains to be established, and they highlight potential counterexamples such as integrable systems with saddle-dominated scrambling (e.g., Lipkin–Meshkov–Glick) or mixed-phase-space systems.

Practically, the work suggests that measuring or computing Krylov complexity (or its peak height) could provide a relatively universal chaos diagnostic that does not require selecting a particular OTOC operator and can be compared directly to spectral form factor behavior. This is relevant for theoretical studies of thermalization and chaos in strongly correlated systems, and potentially for holography-inspired contexts where TFD states and complexity measures are natural. In addition, because the KCP is framed as an order parameter, it offers a systematic way to map chaotic–integrable phase boundaries in model Hamiltonians.

Overall, the paper’s core contribution is the proposal and demonstration that the Krylov complexity peak height, quantified by $\Delta C$, vanishes in integrable regimes and shows critical/crossover behavior across chaotic–integrable transitions in two SYK-based models, aligning with spectral statistics and OTOC-based expectations while offering an operator-independent diagnostic.