Universal properties of the many-body Lanczos algorithm at finite size

This paper conjectures and numerically verifies that the large-$n$ scaling of ratios between consecutive Lanczos coefficients in finite-size many-body systems is universally determined by the hydrodynamic tails of local operator autocorrelation functions and the presence of zero-modes.

The Gist

Imagine you are trying to predict how a complex machine will behave after you've been running it for a very, very long time. Maybe it's a giant clockwork toy, or a quantum computer, or even a pot of boiling water. You want to know: Will it eventually settle down into a calm, predictable state (thermalize), or will it keep wobbling and remembering exactly how you started it?

This paper is about a specific mathematical tool called the Lanczos Algorithm, which physicists use to try to answer that question. Think of this algorithm as a "magnifying glass" that breaks down the complex motion of a quantum system into a simple, step-by-step ladder.

Here is the breakdown of the paper's discovery, explained through simple analogies:

1. The Problem: The "Finite Size" Trap

In the real world, we can't simulate infinite systems. We can only simulate systems of a certain size (like a grid of 10x10 atoms).

• The Old View: Scientists used to think that if you looked at the "ladder" (the Lanczos coefficients) too far up the steps, the finite size of your simulation would mess everything up. They thought the ladder would just become random noise once you got past a certain point.
• The New Discovery: This paper says, "Wait a minute!" Even in these small, finite systems, the ladder isn't just random noise. It actually follows a hidden, universal pattern that tells us exactly how the system will behave in the long run.

2. The Three Scenarios (The Three Conjectures)

The authors propose that the shape of this "ladder" at the very top depends on what kind of physics the system is doing. They identify three distinct scenarios:

Scenario A: The "Hydrodynamic" Flow (The River)

• The Analogy: Imagine a river flowing. If you drop a leaf in, it drifts downstream. Eventually, the leaf spreads out, and the river reaches a steady flow.
• The Physics: In most chaotic systems, energy and information spread out like a fluid (hydrodynamics).
• The Ladder Pattern: The paper predicts that for these systems, the steps on the ladder eventually settle into a specific, steady rhythm. The "height" of the steps tells you how fast the system forgets its past. If the steps follow this specific pattern, the system will eventually settle into a calm, thermal state, but it will leave a tiny, permanent "fingerprint" (a plateau) that depends on the size of the system.

Scenario B: The "Vanishing" Act (The Ghost)

• The Analogy: Imagine a magician dropping a ball, and instead of bouncing, it just disappears into thin air.
• The Physics: Sometimes, a specific part of the system doesn't interact with the main flow of energy. It's like a ghost that doesn't touch the river.
• The Ladder Pattern: In this case, the ladder steps don't settle into a rhythm; they start to wobble wildly or shrink so fast that the "plateau" (the memory of the start) disappears completely. The system forgets everything instantly.

Scenario C: The "Immortal" Memory (The Frozen Statue)

• The Analogy: Imagine a statue that never melts, no matter how hot the sun gets. It remembers its shape forever.
• The Physics: This happens in special systems with "Zero Modes." These are special parts of the system that are completely isolated from the chaos. They never thermalize; they keep their memory forever.
• The Ladder Pattern: Here, the ladder steps oscillate perfectly back and forth between two values, like a pendulum that never stops. This perfect oscillation is the mathematical signature of a system that never forgets.

3. The "Universal" Secret

The most exciting part of this paper is the idea of Universality.
Usually, in physics, every system is different. A system of magnets behaves differently than a system of electrons.

• The Breakthrough: The authors found that once you get past the "beginning" of the ladder (the first few steps), all these different systems start to look the same. They all fall into one of the three patterns above.
• Why it matters: This means that even if you are simulating a tiny, imperfect system on a computer, you can look at the "tail end" of the Lanczos ladder and predict with high confidence whether the real, infinite system will thermalize, vanish, or stay frozen.

4. The "Noise" Issue (Appendix A)

The authors also had to deal with a technical headache. When you run these calculations on a computer, tiny rounding errors (like a calculator getting slightly confused) usually make the results garbage after a while.

• The Fix: They proved that even with these computer errors, the universal pattern still shines through. It's like trying to hear a specific song in a noisy room; even if the room is loud, if you know the melody, you can still recognize the tune. This gives them confidence that their results are real physics, not just computer glitches.

Summary

Think of the Lanczos algorithm as a crystal ball.

• Before: People thought the crystal ball only worked for the first few seconds of the future.
• Now: This paper shows that if you look deep enough into the crystal ball (at the large steps of the ladder), you can see a clear, universal code.
• The Code: It tells you if the system is a River (flows and settles), a Ghost (disappears), or a Statue (remembers forever).

This is a big deal because it allows scientists to use small, manageable computer simulations to understand the deep, long-term behavior of complex quantum materials, which is crucial for building future quantum computers and understanding exotic states of matter.

Technical Summary

1. Problem Statement

The Lanczos Algorithm (LA) is a powerful numerical tool for studying quantum many-body dynamics, particularly through the analysis of Lanczos coefficients ($b_n$). Recent theoretical developments (e.g., the Universal Operator Growth Hypothesis) suggest that the behavior of these coefficients reveals universal features of chaotic vs. integrable systems.

However, a critical limitation exists in current applications:

• Finite-Size Effects: Numerical simulations are restricted to finite lattice sizes ($L$). While the initial coefficients ($n < n^*$) often display universal behavior independent of $L$, the coefficients at large $n$ ($n > n^*$) are heavily affected by finite-size effects, typically manifesting as a "plateau" or fluctuation.
• The Gap: Previous studies focused on the infinite-system limit or the early-time regime ($n < n^*$). There was no established theoretical framework connecting the finite-size plateau of autocorrelation functions to the scaling of Lanczos coefficients in the large-$n$ regime ($n > n^*$).
• Goal: The authors aim to identify universal scaling laws for Lanczos coefficients in finite-size systems and establish a rigorous link between the late-time behavior of autocorrelation functions and the asymptotic behavior of the coefficients $b_n$.

2. Methodology

The authors employ a combination of analytical derivation and extensive numerical simulations.

• Theoretical Framework:

  • They analyze the autocorrelation function of a local observable $O$ at infinite temperature: $C_L(t) = \langle O(t)O \rangle - \langle O(t) \rangle \langle O \rangle$.
  • They utilize the Lanczos basis to represent the Liouvillian superoperator $\mathcal{L} = [H, \cdot]$ as a tridiagonal matrix with coefficients $b_n$.
  • They derive an exact formula relating the infinite-time limit of the autocorrelation function, $C_L(\infty)$, to the Lanczos coefficients:
    $$C_L(\infty) = \frac{1}{1 + \sum_{n=1}^{\infty} \prod_{m=1}^{n} \left(\frac{b_{2m-1}}{b_{2m}}\right)^2}$$
  • They define a rate $\Gamma_n(L) = -\ln(b_n/b_{n+1})^2$ (for odd $n$) to characterize the convergence of the series in the denominator.

• Numerical Techniques:

  • Models: They simulate various 1D and 2D models, including non-integrable Ising models with transverse/longitudinal fields, long-range interacting Ising models, and models with approximate zero modes.
  • Algorithms: To overcome the numerical instability of the standard Lanczos algorithm (which loses orthogonality at high $n$), they primarily use full Gram-Schmidt orthogonalization. This requires significant memory (up to 600 GB RAM) but ensures the orthonormality of the Krylov basis.
  • Validation: They compare results from the full orthogonalization method against the standard 3-step Lanczos algorithm and Exact Diagonalization (ED) to confirm that numerical instabilities do not alter the universal scaling laws.

3. Key Contributions: The Threefold Conjecture

The central contribution of the paper is a threefold conjecture linking the hydrodynamic properties of the system to the scaling of $\Gamma_n(L)$ for $n > n^*$ (where $n^*$ is the crossover scale, typically proportional to system size $L$).

Conjecture 1: Hydrodynamic Behavior (Non-zero Plateau)

• Context: Generic chaotic systems where the observable overlaps with conserved charges (e.g., energy).
• Prediction: The autocorrelation function decays algebraically to a finite plateau $C_L(\infty) \sim L^{-md}$, where $m$ is the order of the Hamiltonian power overlapping with $O$, and $d$ is the spatial dimension.
• Scaling Law: The rate $\Gamma_n(L)$ approaches a positive constant for $n > n^*$:
  $$\Gamma_n(L) \sim L^{-(md+1)}$$
  This ensures the denominator in the $C_L(\infty)$ formula converges to a finite value scaling with $L$.

Conjecture 2: Vanishing Late-Time Plateau

• Context: Observables that do not overlap with conserved charges (e.g., odd under time reversal).
• Prediction: $C_L(\infty)$ vanishes (or is exponentially small) as $L \to \infty$.
• Scaling Law: The rate $\Gamma_n(L)$ either converges to 0 sufficiently fast or becomes negative:
  $$\Gamma_n(L) \leq \frac{\alpha}{n}, \quad \text{with } 0 < \alpha < 2$$
  This causes the denominator in the $C_L(\infty)$ formula to diverge, forcing the plateau to zero.

Conjecture 3: Strong Zero Modes

• Context: Systems with exact or approximate conserved local operators (Strong Zero Modes).
• Prediction: $C_L(\infty)$ remains non-zero and independent of system size in the thermodynamic limit.
• Scaling Law: The rate satisfies:
  $$\Gamma_n \geq \frac{\alpha}{n}, \quad \text{with } \alpha > 2$$
  For exact zero modes, the Lanczos chain terminates, effectively making $\Gamma_n \to \infty$ beyond the termination point, guaranteeing a non-zero plateau.

4. Results

The authors support their conjectures with numerical benchmarks across several models:

1. Short-Range 1D Ising Model:

   • For $\sigma^z_1$ (overlaps with $H$, $m=1$), the cumulative product of ratios decays exponentially. The extracted rate scales as $L^{-2}$, matching Conjecture 1 ($m=1, d=1 \implies -(1+1) = -2$).
   • For $\sigma^z_1 \sigma^z_2$ (overlaps with $H^2$, $m=2$), the rate scales as $L^{-3}$, consistent with the conjecture.
   • For $\sigma^y_1$ (no overlap with conserved charges), the cumulative product increases, indicating $\Gamma_n$ is negative, consistent with Conjecture 2.

2. Approximate Zero Modes:

   • In a model with an approximate boundary zero mode, the cumulative product saturates to a finite value, implying $\Gamma_n \approx 0$. This distinguishes it from the vanishing plateau case (Conjecture 2) where the product grows.

3. Long-Range and 2D Systems:

   • Long-Range 1D: The scaling of the crossover point $n^*$ is modified (likely $n^* \sim L^a$ with $a < 1$), but the algebraic decay of the rate $\Gamma(L)$ still holds, supporting the universality of the conjecture.
   • 2D Systems: Despite computational limits on system size, the data shows that the rate $\Gamma_n(L)$ decreases as the lattice size increases, consistent with the predicted scaling $L^{-3}$ for $d=2, m=1$.

4. Numerical Stability Check:

   • Comparisons between standard Lanczos and full Gram-Schmidt methods show that while the coefficients differ at very high $n$ due to numerical noise, the universal scaling laws and the resulting autocorrelation functions remain robust.

5. Significance

• Bridging Theory and Simulation: The paper resolves a major bottleneck in applying Lanczos-based methods to finite-size simulations. It provides a concrete method to extract universal hydrodynamic information (like diffusion constants or decay exponents) from the finite-size plateau of Lanczos coefficients.
• Universality: It demonstrates that finite-size effects are not merely noise but carry universal information about the system's transport properties and conserved quantities.
• Diagnostic Tool: The scaling of $\Gamma_n(L)$ serves as a diagnostic tool to distinguish between different physical regimes:
  • Hydrodynamic transport (Conjecture 1).
  • Non-hydrodynamic decay (Conjecture 2).
  • Topological/Zero-mode protection (Conjecture 3).
• Future Directions: The work opens avenues for studying operator complexity, Floquet systems, and non-equilibrium dynamics in finite systems using the Lanczos algorithm, moving beyond the limitations of infinite-system assumptions.

In summary, the authors successfully generalize the Universal Operator Growth Hypothesis to finite-size systems, establishing a rigorous quantitative link between the late-time behavior of autocorrelation functions and the asymptotic scaling of Lanczos coefficients.