Krylov Distribution

TL;DR

This work introduces the Krylov distribution $\mathcal{D}(ξ)$ as a static, resolvent-based diagnostic that tracks how inverse-energy weight from the resolvent-dressed state $|\psi(ξ)\rangle=(H-ξ)^{-1}|ψ_0\rangle$ distributes along the Krylov basis generated from $|ψ_0\rangle$. By analyzing the Krylov resolvent amplitudes $ψ_n(ξ)=\langle n|(H-ξ)^{-1}|ψ_0\rangle$ and their recursion, the authors identify three universal regimes in the thermodynamic limit: (i) exponential localization and saturating $\mathcal{D}(ξ)$ outside spectral support; (ii) extensive growth $\mathcal{D}(ξ)\sim N/2$ inside continuous spectra; and (iii) sublinear or logarithmic scaling near spectral edges and quantum critical points, with specific edge scalings including $N^{2/3}$ for square-root edges. They provide exact results for three solvable models—constant Lanczos coefficients, a displaced harmonic oscillator, and an $SU(1,1)$ chain—to illustrate how spectral structure and Lanczos coefficient growth control Krylov-space geometry, including resonant peaks, power-law tails, and logarithmic growth. They also explore the Krylov distribution in the Ising model, showing that $\mathcal{D}(ξ)$ is sensitive to spectral organization and to initial-state choice, thereby offering a complementary diagnostic to spectral functions and dynamics. Finally, the paper connects $\mathcal{D}(ξ)$ to fidelity susceptibility and the quantum geometric tensor, proposing a unified framework that links static response, spectral structure, and Krylov-space geometry with potential experimental relevance in quantum simulators.

Abstract

We introduce the Krylov distribution $\mathcal{D}(ξ)$, a static Krylov-space diagnostic that characterizes how inverse-energy response is organized in Hilbert space. The central object is the resolvent-dressed state $(H-ξ)^{-1}|ψ_0\rangle$, whose decomposition in the Krylov basis generated from a reference state defines a normalized distribution over Krylov levels. Unlike conventional spectral functions, which resolve response solely along the energy axis, the Krylov distribution captures how the resolvent explores the dynamically accessible subspace as the spectral parameter $ξ$ is varied. Using asymptotic analysis, exact results in solvable models, and numerical studies of an interacting spin chain, we identify three universal regimes: saturation outside the spectral support, extensive growth within continuous spectra, and sublinear or logarithmic scaling near spectral edges and quantum critical points. We further show that fidelity susceptibility and the quantum geometric tensor admit natural decompositions in terms of Krylov-resolved resolvent amplitudes.

Figures (7)

• Figure 1: Krylov distribution $\mathcal{D}(\ell)$ for the quadratic Hamiltonian, computed numerically from the exact resolvent amplitudes Eq. \ref{['eq:psi_quad_exact']} with a Krylov cutoff $N=25$. Sharp peaks appear at the discrete eigenvalues $E_m$, reflecting the pole structure of the resolvent. At resonance, the Krylov distribution spreads over many layers, with peak heights growing linearly with $m$ according to $\mathcal{D}(E_m) = m + \gamma^2$. In contrast, off-resonant values of $\ell$ exhibit strong Krylov-space localization, with $\mathcal{D}(\ell)$ remaining of order unity.
• Figure 2: Krylov distribution $\mathcal{D}(\ell)$ as a function of the Krylov cutoff $N$ for the quadratic Hamiltonian, computed numerically from Eq. \ref{['eq:psi_quad_exact']} for several fixed values of $\ell$. The distribution grows approximately linearly at small $N$ and saturates at larger $N$, reflecting the finite support of the resolvent-dressed state in Krylov space. Oscillatory features originate from near-resonant enhancement when $\xi$ approaches an eigenvalue; choosing half-integer values of $\ell$ suppresses these oscillations and yields smoother scaling behavior.
• Figure 3: Krylov distribution $\mathcal{D}(\xi)$ for the $SU(1,1)$ chain with $h=2$ and Krylov cutoff $N=25$, plotted as a function of the spectral parameter $\xi$. The curves correspond to different values of the representation parameter $\alpha$: $\alpha=1$ (blue), $\alpha=2$ (red), and $\alpha=3$ (green), and are computed numerically from the exact resolvent amplitudes in Eq. \ref{['exact-SU11']}. The smooth, non-resonant dependence on $\xi$ reflects the absolutely continuous spectrum associated with the non-compact $SU(1,1)$ algebra and the absence of Krylov-space localization.
• Figure 4: Krylov distribution $\mathcal{D}(\xi)$ for the $SU(1,1)$ chain with $h=2$ and $\alpha=1$, plotted as a function of the Krylov cutoff $N$ for several fixed values of the spectral parameter $\xi$ (indicated in the inset). The growth of $\mathcal{D}(\xi)$ is sublinear and consistent with the asymptotic $N/\ln N$ scaling expected for this model, illustrating the slow but delocalized exploration of Krylov space induced by the underlying non-compact algebraic structure.
• Figure 5: Krylov distribution $\mathcal{D}(\xi)$ for the Ising chain with $L=9$ and $g=-1.05$ for different product initial states: (left) $|X+\rangle$, (center) $|Y+\rangle$, and (right) $|Z+\rangle$. Blue curves correspond to the integrable case ($h=0$), while brown curves correspond to the chaotic case ($h=0.5$).