The resonant level model from a Krylov perspective: Lanczos coefficients in a quadratic model

This paper demonstrates that in the quadratic resonant level model, the growth behavior of Lanczos coefficients can be arbitrarily tuned by adjusting the coupling to the hybridization band, thereby proving that these coefficients are inadequate as a universal criterion for distinguishing between integrable and chaotic systems or for predicting physical behavior like autocorrelation decay.

The Gist

The Big Picture: Measuring Chaos with a "Ruler"

Imagine you are trying to figure out if a room is chaotic (like a mosh pit) or orderly (like a library). In the world of quantum physics, scientists have developed a special "ruler" called Lanczos coefficients to measure this.

The prevailing theory (the "Operator Growth Hypothesis") says:

• If the system is chaotic: The numbers on this ruler grow linearly (like a straight line going up: 1, 2, 3, 4...).
• If the system is orderly (integrable): The numbers stay small, grow slowly (like a square root), or stop growing entirely.

Scientists have been using this ruler to classify how complex a system is. If the numbers shoot up fast, they assume the system is chaotic. If they stay flat, they assume it's simple.

The Experiment: A Simple Toy Model

The authors of this paper decided to test this ruler on a very simple, perfectly predictable system called the Resonant Level Model.

• The Setup: Imagine a single "impurity" (a guest) sitting in a room full of "fermions" (a crowd of people). The guest can talk to the crowd.
• The Twist: The rules of the game are quadratic. In physics-speak, this means the system is "easy" and "solvable." It is the opposite of chaotic. It's like a perfectly tuned piano; if you hit a key, you know exactly what note comes out. There is no chaos here.

The researchers asked: "If we use our 'chaos ruler' on this perfectly simple system, what will it say?"

The Surprise: The Ruler is Broken

They changed how the "guest" talks to the "crowd" (the coupling) in four different ways. Here is what happened:

1. Case A (Box Coupling): The ruler showed numbers growing like a square root (slow growth).
2. Case B (Semicircle Coupling): The ruler showed numbers that were perfectly constant (flat).
3. Case C (Gaussian Coupling): The ruler showed numbers growing like a square root.
4. Case D (Hyperbolic Secant Coupling): The ruler showed numbers growing linearly (1, 2, 3, 4...).

The Shock: In Case D, the ruler screamed "CHAOS!" because the numbers were growing in a straight line. But the system was NOT chaotic. It was the same simple, solvable model they started with.

The Analogy: The Echo Chamber

Think of the system as an echo chamber.

• The "Chaos" Theory says: "If the echo gets louder and more complex over time, the room must be a wild, chaotic cave."
• This Paper says: "Wait a minute. We built a perfect, quiet, rectangular room (the simple model). But if we change the shape of the walls just slightly, the echo can sound like it's getting louder and more complex, even though the room is still perfectly quiet and simple."

The authors showed that by simply changing the "shape of the walls" (the coupling profile), they could make the "echo" (the Lanczos coefficients) look like it was growing linearly, even though nothing chaotic was actually happening.

The "Magic Trick"

The paper goes even further. They proved that you can actually design the coupling to make the ruler show any pattern you want.

• Want the numbers to look like a straight line? You can tune the system to do that.
• Want them to look like a curve? You can do that too.
• Want them to be random? You can do that.

This means the "ruler" isn't telling you about the nature of the system (chaotic vs. simple); it's just telling you about the shape of the connection between the parts.

The Real-World Result: It Doesn't Matter

Finally, they looked at what the system actually did (the dynamics). They watched how the "guest" moved over time.

• Even though the "ruler" showed four completely different patterns (flat, square root, linear), the actual movement of the guest looked exactly the same in the long run.
• In the "wide-band limit" (when the crowd is very large and dense), the guest's behavior decayed exponentially (faded away) in all four cases.

The Conclusion

The main takeaway is a warning to physicists: Don't trust the Lanczos coefficients blindly.

Just because the numbers on the "chaos ruler" are growing fast doesn't mean the system is chaotic. You can get "chaotic-looking" numbers from a perfectly simple, non-chaotic system just by tweaking how the parts connect.

In short: The paper proves that this popular tool for measuring chaos is flawed because it can be fooled by simple, non-chaotic systems. It's like trying to judge how wild a party is by looking at the price of the drinks; sometimes, a quiet party can have expensive drinks, and a wild party can have cheap ones. You need to look at the actual dancing, not just the price tag.

Technical Summary

1. Problem Statement

The paper addresses a central question in modern quantum many-body physics regarding the Operator Growth Hypothesis (OGH). The OGH posits that in chaotic quantum systems, the Lanczos coefficients ($b_n$) of local observables exhibit asymptotically linear growth, whereas integrable or non-chaotic systems typically show bounded or sublinear (e.g., square-root) growth.

The authors challenge the reliability of using the asymptotic behavior of Lanczos coefficients as a definitive classifier for chaoticity or integrability. Specifically, they investigate whether a system that is exactly solvable (quadratic) and lacks "operator growth" (in the sense of expanding into higher-order many-body operators) can still exhibit diverse Lanczos coefficient structures (constant, square-root, or linear growth) solely by tuning the coupling to the environment.

2. Methodology

The study employs the Recursion Method (Lanczos algorithm) within the Krylov subspace formalism.

• Model: The authors analyze the Resonant Level Model (RLM), a quadratic Hamiltonian describing a single non-interacting impurity coupled to a continuum of fermionic bath modes:
  $$H = \sum_k \epsilon_k c_k^\dagger c_k + \epsilon_d d^\dagger d + \sum_k V_k (c_k^\dagger d + d^\dagger c_k)$$
  Since the model is quadratic, it is exactly solvable, and single-particle operators remain confined to the single-particle sector under time evolution (no operator growth).

• Observables: The focus is on the Majorana fermion operators of the impurity, specifically $\gamma_1 = d^\dagger + d$.

• Analytical Approach:

  1. Instead of iteratively computing Lanczos coefficients via the standard algorithm, the authors derive an integrodifferential equation for the autocorrelation function $C(t)$.
  2. They establish a direct link between the memory kernel $K(t)$ (the Fourier transform of the coupling density $J(\omega)$) and the Lanczos coefficients via the continued fraction representation of the Laplace transform of $C(t)$.
  3. They analytically derive closed-form expressions for the Lanczos coefficients for four distinct coupling density profiles $J(\omega)$:
     • (i) Hard-edge box (finite bandwidth).
     • (ii) Semi-circle distribution.
     • (iii) Gaussian distribution.
     • (iv) Hyperbolic secant (sech) distribution.

• Existence Argument: The authors invoke Bochner's theorem to argue that for any desired set of Lanczos coefficients (that correspond to a valid autocorrelation function), there exists a corresponding positive, symmetric coupling density $J(\omega)$ that generates them.

3. Key Contributions

• Decoupling Coefficient Structure from Integrability: The paper demonstrates that the structure of Lanczos coefficients is not an intrinsic indicator of the system's chaoticity. A purely quadratic (integrable) model can exhibit linear growth of $b_n$, a feature previously associated with chaotic systems.
• Arbitrary Coefficient Engineering: The authors prove that by suitably choosing the coupling profile $J(\omega)$, one can engineer essentially arbitrary behaviors for the Lanczos coefficients within the same quadratic model.
• Operator Growth vs. Coefficient Growth: The study clarifies the distinction between "operator growth" (expansion into a larger Hilbert space of many-body operators) and the growth of Lanczos coefficients. In this model, operators do not grow (they remain single-particle), yet the Lanczos coefficients can grow linearly.

4. Results

The authors derived explicit forms for the Lanczos coefficients ($b_n$) for the four coupling cases (with $\mu$ as a scaling factor and $\alpha$ as the bandwidth parameter):

Case	Coupling Profile $J(\omega)$	Kernel $K(t)$	Lanczos Coefficient Behavior ($b_n$)
(i)	Hard-edge Box	$\frac{\sin(\alpha t)}{\alpha t}$	Approximately constant (for large $n$)
(ii)	Semi-circle	$\frac{J_1(2\alpha t)}{\alpha t}$	Exactly constant
(iii)	Gaussian	$e^{-(\alpha t)^2/2}$	Square-root growth ($\sim \sqrt{n}$)
(iv)	Hyperbolic Secant	$\text{sech}(\alpha t)$	Linear growth ($\sim \alpha n$)

Dynamical Implications:

• Despite the drastic differences in the structure of $b_n$ (ranging from constant to linear), the physical dynamics of the autocorrelation function $C(t)$ are remarkably similar in the wide-band limit ($\alpha \to \infty$).
• In this limit, all cases exhibit exponential decay ($C(t) \propto e^{-\Gamma t}$), characteristic of Markovian relaxation.
• This confirms that the asymptotic growth of Lanczos coefficients does not necessarily dictate distinct physical behaviors (e.g., chaos vs. integrability) in the thermodynamic limit.

5. Significance

• Critique of OGH as a Universal Classifier: The findings suggest that the linear growth of Lanczos coefficients is not a sufficient condition to label a system as chaotic. It can arise in exactly solvable, non-interacting systems simply due to the spectral properties of the bath coupling.
• Refining Krylov Complexity: The paper highlights that Krylov complexity and Lanczos coefficients must be interpreted with caution. The "growth" of coefficients can be an artifact of the chosen basis and coupling rather than a signature of quantum chaos or scrambling.
• Future Directions: The work opens avenues for studying more complex models (e.g., adding Hubbard interactions to the RLM) to see if the distinction between coefficient growth and operator growth persists in truly interacting, potentially chaotic systems. It also suggests that the "simplicity" of dynamics (exponential decay) can coexist with complex Lanczos structures.

In conclusion, the paper provides a rigorous counter-example to the idea that Lanczos coefficient growth is a robust diagnostic for quantum chaos, demonstrating that in quadratic systems, the coefficient structure is entirely tunable via the coupling profile and does not correlate with the presence of operator growth or chaotic dynamics.