Krylov Distribution

This paper introduces the Krylov distribution, a novel static diagnostic that characterizes the organization of inverse-energy response within the dynamically accessible Krylov subspace, revealing universal scaling regimes across spectral supports and connecting fidelity susceptibility to Krylov-resolved resolvent amplitudes.

The Gist

Imagine you are standing in a vast, dark library (the Hilbert Space) filled with millions of books (quantum states). You have a specific book in your hand, your Reference State ($|\psi_0\rangle$).

Usually, physicists study how this book changes over time. They ask: "If I let time pass, how does this book spread out to other shelves? Does it stay on the first shelf, or does it run wild to the back of the library?" This is called Krylov Complexity, and it's like watching a drop of ink spread in water.

This paper introduces a new way to look at the library. Instead of watching time pass, they ask: "If I ask a very specific question about energy, how does the library respond?"

They use a tool called the Resolvent (think of it as a magical magnifying glass tuned to a specific energy level, $\xi$). When you shine this light on your starting book, it doesn't just show you that book; it highlights a whole chain of related books nearby, weighted by how "close" they are in energy.

The authors call the result of this process the Krylov Distribution. Here is the simple breakdown of what they found:

1. The Setup: The Library Ladder

To organize this library, they build a special ladder (the Krylov Basis).

• Rung 0: Your starting book.
• Rung 1: The books most directly related to it.
• Rung 2: The books related to those, and so on.

The "Krylov Distribution" is simply a measurement of how high up this ladder the magical light reaches.

• If the light only illuminates the bottom few rungs, the distribution is low.
• If the light spreads all the way to the top of the ladder, the distribution is high.

2. The Three Universal Regimes (The Three Scenarios)

The authors discovered that the height of this light depends entirely on where you tune your magnifying glass (the energy $\xi$) relative to the library's "catalog" (the spectrum).

Scenario A: The "Gap" (Outside the Library)

Imagine tuning your magnifying glass to an energy level that doesn't exist in the library (like looking for a book about "flying pigs" in a library of only history books).

• What happens: The light barely reaches the first few rungs of the ladder. It fades away quickly.
• The Result: The distribution stays low and flat.
• Analogy: It's like shouting in a soundproof room. The sound dies out immediately because there's nothing to resonate with.

Scenario B: The "Crowded Aisle" (Inside the Spectrum)

Now, tune your glass to an energy level where many books exist (a crowded aisle).

• What happens: The light doesn't fade. It spreads out evenly, illuminating rungs all the way up the ladder. The higher the ladder, the more rungs get lit.
• The Result: The distribution grows linearly (it gets bigger and bigger as the library gets bigger).
• Analogy: It's like shouting in a busy market. The sound bounces off everything, filling the whole space. The "response" is extensive and widespread.

Scenario C: The "Edge" or "Critical Point" (The Border)

What if you tune your glass right to the edge of the library, or to a special, chaotic spot where the books are arranged strangely (a quantum critical point)?

• What happens: The light spreads, but not as fast as in the crowded aisle. It climbs the ladder, but it slows down as it goes higher.
• The Result: The distribution grows, but slowly (sublinearly or logarithmically).
• Analogy: It's like shouting at the edge of a canyon. The sound travels far, but it gets muffled and distorted by the unique shape of the walls.

3. Why This Matters

This new tool is powerful because it connects two things that usually seem separate:

1. Spectral Structure: The "shape" of the energy levels (are there gaps? is it continuous?).
2. Static Response: How the system reacts to a gentle nudge (like changing a magnetic field).

The authors show that by looking at how high the light climbs the ladder, you can instantly tell if the system is:

• Gapped (has a safety gap, light stays low).
• Chaotic (light spreads wildly).
• Critical (light spreads in a special, slow way).

4. The "Fidelity" Connection

Finally, they show that this ladder-climbing measurement is directly related to Fidelity Susceptibility.

• Think of it this way: If you slightly change the rules of the library (a parameter change), how much does your starting book change?
• The "Krylov Distribution" tells you exactly where in the library that change happens. Does it happen right next to you (low rung), or does it require a massive reorganization of the whole library (high rung)?

Summary

In simple terms, this paper invents a new ruler to measure quantum systems.

• Instead of watching how a system moves over time (like a runner), they measure how a system stretches when you poke it with a specific energy question.
• If the system is "safe" (has a gap), it barely stretches.
• If the system is "fluid" (continuous spectrum), it stretches all the way.
• If the system is "critical" (on the edge of a phase change), it stretches in a unique, slow pattern.

This helps physicists understand the hidden geometry of quantum matter without needing to simulate complex time-evolution, offering a fresh, static snapshot of how quantum information is organized.

Technical Summary

1. Problem Statement

In quantum many-body physics, understanding how quantum states explore Hilbert space is central to phenomena like thermalization, quantum chaos, and criticality. While Krylov complexity ($C(t)$) has been established as a powerful dynamical diagnostic tracking the spreading of a state $|\psi(t)\rangle = e^{-iHt}|\psi_0\rangle$ along the Krylov chain, it is inherently time-dependent.

Many physically significant phenomena (e.g., adiabatic deformations, geometric phases, inverse-gap physics, and breakdown of adiabaticity near critical points) are static or quasi-static responses to external parameters. These are naturally encoded in the resolvent operator $R(\xi) = (H - \xi)^{-1}$ rather than the time-evolution operator. The authors identify a gap in the literature: there is no established framework to characterize how inverse-energy response (weighted by $1/(E-\xi)$) explores the Krylov subspace. Specifically, how does the resolvent-dressed state distribute itself across Krylov levels as the spectral parameter $\xi$ varies?

2. Methodology

The authors introduce a new static diagnostic called the Krylov Distribution, $D(\xi)$, formulated within the Krylov subspace generated by a reference state $|\psi_0\rangle$.

• Krylov Construction: Starting from $|\psi_0\rangle$, the Lanczos algorithm generates an orthonormal basis $\{|n\rangle\}$ where the Hamiltonian $H$ takes a tridiagonal (Jacobi) form defined by Lanczos coefficients $a_n$ and $b_n$.

• Resolvent-Dressed State: They define the state $|\psi(\xi)\rangle = (H - \xi)^{-1}|\psi_0\rangle$.

• Krylov Resolvent Amplitudes: The projection of this state onto the Krylov basis yields amplitudes $\psi_n(\xi) = \langle n | (H - \xi)^{-1} | \psi_0 \rangle$. These satisfy a static recursion relation analogous to the Schrödinger equation but with $\xi$ as the energy parameter.

• Definition of $D(\xi)$:

  1. A normalized probability distribution is defined: $P_n(\xi) = \frac{|\psi_n(\xi)|^2}{\sum_\ell |\psi_\ell(\xi)|^2}$.
  2. The Krylov Distribution is the first moment of this distribution:
     $$D(\xi) = \sum_n n P_n(\xi)$$
     This quantity measures the average "depth" or spread of the resolvent-dressed state along the Krylov chain.

• Analytical Tools: The authors employ asymptotic analysis of orthogonal polynomials (associated with the Jacobi operator), spectral theory, and exact solutions for specific models. They also perform numerical simulations on interacting spin chains.

3. Key Contributions

1. Conceptual Framework: Elevating the resolvent-dressed state from an auxiliary tool for computing spectral functions to a primary diagnostic object. $D(\xi)$ serves as the static analogue to Krylov complexity, replacing time $t$ with the spectral parameter $\xi$.
2. Universal Regimes: Identification of three distinct, universal scaling behaviors for $D(\xi)$ in the thermodynamic limit (large Krylov dimension $N$), determined by the position of $\xi$ relative to the spectrum:
   • Outside the Spectrum (Gapped): Exponential localization of amplitudes leads to saturation of $D(\xi)$ to an $O(1)$ constant.
   • Inside the Continuous Spectrum: Delocalization leads to extensive growth, $D(\xi) \sim N/2$.
   • At Spectral Edges/Critical Points: Sublinear or logarithmic scaling ($D(\xi) \sim N^{2/3}$, $\log N$, etc.), reflecting singularities in the density of states and the asymptotic structure of Lanczos coefficients.
3. Connection to Quantum Geometry: The authors demonstrate that Fidelity Susceptibility and the Quantum Geometric Tensor admit natural decompositions in terms of Krylov-resolved resolvent amplitudes, extending geometric response theory to arbitrary reference states.
4. Exact Solvable Models: Derivation of closed-form results for three paradigmatic models to validate the theoretical scaling laws.

4. Key Results

A. Universal Scaling Regimes

• Case I: $\xi$ outside the spectrum.
  The amplitudes decay exponentially: $|\psi_n(\xi)| \sim e^{-n\kappa}$.
  Result: $D(\xi)$ saturates to a finite value independent of system size. This indicates that inverse-energy processes are dominated by states near the spectral edge.
• Case II: $\xi$ inside the continuous spectrum.
  The amplitudes are oscillatory and delocalized on average: $|\psi_n(\xi)|^2 \sim O(1)$.
  Result: $D(\xi) \sim N/2$. The resolvent explores the entire dynamically accessible subspace uniformly.
• Case III: Spectral Edges and Critical Points.
  • Regular Band Edges (Square-root DOS): The density of states $\rho(E) \sim (E-E^*)^{1/2}$. The amplitudes follow Airy function asymptotics, leading to $|\psi_n|^2 \sim n^{-4/3}$.
    Result: $D(\xi) \sim N^{2/3}$ (Sublinear growth).
  • Quantum Critical Points: For gapless systems with constant DOS ($\alpha=0$), $|\psi_n|^2 \sim n^{-2}$.
    Result: $D(\xi) \sim \log N$.

B. Exact Solvable Models

1. Constant Lanczos Coefficients ($b_n = b$): Models a bounded continuous spectrum. Confirms the transition from exponential saturation (outside) to linear growth (inside) and $N^{2/3}$ scaling at the edge.
2. Displaced Harmonic Oscillator ($b_n \sim \sqrt{n}$): Models a discrete, unbounded spectrum.
   • Off-resonance: Strong localization ($D(\xi) \sim O(1)$).
   • On-resonance (at eigenvalues $E_m$): Sharp peaks where $D(E_m) = m + \gamma^2$. The distribution spreads linearly with the eigenvalue index $m$.
3. SU(1, 1) Chain ($b_n \sim \alpha n$): Models maximally chaotic systems (like SYK) with continuous spectra.
   • Exhibits power-law tails $|\psi_n|^2 \sim n^{-1}$.
   • Result: $D(\xi) \sim N / \ln N$. This signifies power-law delocalization and the absence of spectral gaps.

C. Numerical Studies (Mixed-Field Ising Model)

• Applied to a spin-1/2 chain with transverse ($g$) and longitudinal ($h$) fields.
• Integrable vs. Chaotic:
  • Product States: The integrable regime ($h=0$) shows larger oscillation amplitudes and higher $D(\xi)$ values compared to the chaotic regime ($h \neq 0$).
  • Gibbs/Random States: Ensemble averaging washes out fine-grained spectral differences, making $D(\xi)$ more universal but less sensitive to integrability.
• Conclusion: $D(\xi)$ is a sensitive probe of spectral structure but its ability to distinguish chaos depends on the reference state's spectral resolution.

5. Significance

• New Diagnostic Tool: Provides a static, energy-resolved probe of Hilbert space structure that complements dynamical measures like Krylov complexity. It is particularly useful for studying systems where time evolution is difficult to access or where static response is the primary physical quantity of interest.
• Bridge to Geometry: By linking the Krylov distribution to Fidelity Susceptibility and the Quantum Geometric Tensor, the work unifies spectral analysis with quantum information geometry, offering a way to decompose geometric response into "Krylov layers."
• Universality Classes: The scaling of $D(\xi)$ provides a quantitative method to identify spectral singularities (gaps, edges, critical points) and universality classes based on the growth of Lanczos coefficients.
• Experimental Relevance: While measuring the full distribution is challenging, the connection to fidelity susceptibility suggests that aspects of $D(\xi)$ could be accessible in quantum simulators via existing measurement protocols.

In summary, the paper establishes the Krylov Distribution as a fundamental static diagnostic that reveals how inverse-energy responses organize themselves within the emergent geometry of Krylov space, offering deep insights into spectral structure, criticality, and quantum chaos.