Krylov Distribution

TL;DR

The paper introduces the Krylov distribution as a new diagnostic tool to analyze how inverse-energy responses are organized in Hilbert space, revealing universal regimes and connecting to quantum geometric quantities.

Contribution

It defines the Krylov distribution based on the resolvent-dressed state, providing a novel perspective beyond conventional spectral functions, with analytical and numerical analysis of universal behaviors.

Findings

Identifies three universal regimes: saturation, extensive growth, and sublinear scaling.

Connects Krylov distribution to fidelity susceptibility and quantum geometric tensor.

Demonstrates the approach in solvable models and an interacting spin chain.

Abstract

We introduce the Krylov distribution $\mathcal{D}(\xi)$, a static Krylov-space diagnostic that characterizes how inverse-energy response is organized in Hilbert space. The central object is the resolvent-dressed state $(H-\xi)^{-1}|\psi_0\rangle$, 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 $\xi$ 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.