Estimate of equilibration times of quantum correlation functions in the thermodynamic limit based on Lanczos coefficients

This paper proposes and validates a scheme to estimate the equilibration times of local observables in quantum chaotic systems using Lanczos coefficients, demonstrating that these times are finite and physically realistic even in the thermodynamic limit.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Question: How Long Until Things "Settle Down"?

Imagine you drop a drop of red ink into a glass of clear water. At first, you see a distinct red blob. But over time, the ink spreads out until the whole glass is a uniform, pale pink. The water has reached equilibrium.

In the world of quantum physics (the very small world of atoms and particles), scientists have long known that systems eventually reach this "settled down" state. But there is a nagging question: How long does it actually take?

For some systems, the math suggests it could take longer than the age of the universe to settle. But we know from real life that things settle down quickly. Why? This paper tries to answer that question.

The Problem with the Old Tools

To figure out how long it takes for a quantum system to settle, scientists usually look at a "memory" of the system. They ask: "If I poke the system now, how long until it forgets the poke?"

Traditionally, to calculate this, you need to know the behavior of the system for an infinite amount of time. It's like trying to predict the weather for the next 10,000 years just to know if you need an umbrella tomorrow. It's computationally impossible for complex systems.

The New Tool: The "Lanczos Ladder"

The authors of this paper propose a clever shortcut. Instead of looking at the whole infinite timeline, they use a mathematical tool called the Lanczos algorithm.

Think of the Lanczos algorithm as building a ladder to climb up the complexity of the system.

• Each rung of the ladder is a number called a Lanczos coefficient.
• The first few rungs are easy to calculate, even for huge systems.
• The higher you go, the harder it gets.

The paper's main discovery is this: You don't need to climb the whole ladder.

If the rungs of the ladder start to grow in a smooth, predictable pattern (like steps getting slightly taller in a regular rhythm), you can stop after just the first few rungs. You can then use a simple formula to guess the height of the entire ladder.

The "Smoothness" Metaphor

Imagine you are trying to guess the shape of a mountain range by looking at the first few hills.

• Scenario A (Smooth): The hills get taller in a very steady, smooth way. If you see this pattern, you can confidently predict the rest of the mountain. You know the peak isn't going to suddenly spike up or crash down.
• Scenario B (Jagged): The hills are jagged, jagged, and unpredictable. One is tall, the next is tiny, the next is a cliff. In this case, looking at the first few hills tells you nothing about the rest of the mountain.

The authors found that in chaotic quantum systems (which are the most common in nature), the "hills" (the Lanczos coefficients) usually become smooth very quickly.

The "Magic" Result

Because these coefficients become smooth so fast, the authors' method allows them to estimate the equilibration time (how long it takes to settle) using only a handful of numbers.

• The Old Way: "We need to simulate the universe for a billion years to be sure."
• The New Way: "We only need to look at the first 10 or 20 numbers, see that they are smooth, and we can calculate the answer in seconds."

Their calculations show that for most physical systems, the time it takes to reach equilibrium is very short—much shorter than the age of the universe. It happens on a "realistic" timescale, like seconds or minutes, not eons.

Why This Matters

This is a big deal for two reasons:

1. It solves a mystery: It explains why the world around us settles down quickly, even though the math for complex systems looks scary and infinite.
2. It saves time: Scientists don't need to run massive, expensive computer simulations for years. They can just check if the "ladder rungs" are smooth, and if they are, they can trust a quick estimate.

Summary Analogy

Imagine you are trying to guess the total distance of a marathon.

• The Hard Way: Run the whole 26 miles and measure it.
• The Paper's Way: Run the first 100 meters. If your stride is smooth and steady, you can mathematically predict the total distance with high accuracy without ever running the rest of the race.

The paper proves that in the chaotic quantum world, our "stride" (the Lanczos coefficients) is usually very smooth, allowing us to predict the finish line almost instantly.

Technical Summary

Here is a detailed technical summary of the paper "Estimate of equilibration times of quantum correlation functions in the thermodynamic limit based on Lanczos coefficients" by Wang, Füllgraf, and Gemmer.

1. Problem Statement

The paper addresses a fundamental open question in quantum many-body physics: Why do physical systems equilibrate on realistic time scales (much shorter than the age of the universe), despite the theoretical possibility of extremely long equilibration times?

While the Eigenstate Thermalization Hypothesis (ETH) explains that non-integrable systems thermalize, it does not rigorously explain the timescale of this process. Existing analytical bounds often rely on assumptions violated in typical physical settings, and numerical studies are limited by the exponential growth of Hilbert space, preventing direct simulation of the thermodynamic limit. The authors aim to develop a method to estimate the equilibration time ($T_{eq}$) in the thermodynamic limit using only a finite number of Lanczos coefficients.

2. Methodology

A. Definition of Equilibration Time

The authors define the equilibration time $T_{eq}$ based on the autocorrelation function $C(t)$ of a local observable $O$:
$$T_{eq} := \int_0^\infty |C(t) - C(\infty)|^2 dt$$
Assuming $C(\infty) = 0$ (infinite temperature), this simplifies to the area under the curve of $C^2(t)$. This definition is chosen because it captures the relevant thermalization timescale even in cases of rapid oscillation, unlike the standard integral of $C(t)$.

B. The Recursion Method and Lanczos Coefficients

The study utilizes the recursion method within the Liouville space (Hilbert space of operators).

1. Original Dynamics: The autocorrelation $C(t)$ is determined by the Lanczos coefficients $b_n$ generated by applying the Liouvillian superoperator $\mathcal{L}$ to a seed operator $|O)$.
2. Squared Dynamics: To compute $T_{eq}$ (the integral of $C^2(t)$), the authors map the problem to a product space with a new Liouvillian $\mathcal{L}' = \mathcal{L} \otimes \mathbb{1} + \mathbb{1} \otimes \mathcal{L}$ and a seed operator $|Q_0) = |O) \otimes |O)$.
3. New Coefficients: This generates a new set of Lanczos coefficients, denoted $B_N$, which uniquely determine $C^2(t)$.

C. The Estimation Scheme

The core innovation is a scheme to estimate $T_{eq}$ using only the first $R$ coefficients of the original sequence ($b_n$) to approximate the infinite sequence $B_N$.

1. Relation: The authors establish that for smooth $b_n$, the coefficients $B_N$ for the squared dynamics relate to the original coefficients via the approximation:
   $$B_N \approx 2b_{N/2}$$
2. Linear Extrapolation: Since only a finite number of $b_n$ (typically $n_{max} \approx 20-40$) are numerically accessible, the authors assume that for $N > R$, the coefficients $B_N$ grow linearly:
   $$B_N = \alpha_R N + \beta_R$$
   where $\alpha_R$ and $\beta_R$ are derived from the last two known exact coefficients.
3. Analytical Continuation: Using the continued fraction representation of the Laplace transform of $C^2(t)$, they derive an analytical expression for the tail of the product formula for $T_{eq}$ (denoted $\tilde{p}_R$) under the linear growth assumption. This allows the calculation of an estimated equilibration time $T_{eq}^{rc}$ using only the first $R$ coefficients.

3. Key Contributions

1. Direct Estimation Scheme: The paper proposes a practical, computationally cheap method to estimate equilibration times in the thermodynamic limit without simulating the full time evolution of large systems.
2. Smoothness Criterion: The authors identify that the convergence and accuracy of this method depend critically on the smoothness of the Lanczos coefficients. If $b_n$ grows smoothly (eventually linearly), the method converges rapidly.
3. Theoretical Justification: They provide analytical arguments (via a stochastic process model on the product space) showing that if the original coefficients $b_n$ are smooth, the Krylov vectors for the squared dynamics concentrate on the "forefront" (the outermost counter-diagonal), justifying the approximation $B_N \approx 2b_{N/2}$.
4. Link to Chaos: The study reinforces the connection between quantum chaos and operator growth, noting that chaotic systems typically exhibit smooth, linearly growing Lanczos coefficients.

4. Numerical Results

The authors tested their method on several models:

• Ising Ladder: Energy difference operator.
• Tilted Field Ising (TFI) Chain: Energy density wave operators with different wavenumbers ($q=\pi, \pi/12$).
• Heisenberg XXZ Model: Spin density wave operators (in End Matter).

Findings:

• Convergence: For cases where the Lanczos coefficients $b_n$ are smooth (indicated by a small "smoothness indicator" $\delta_S < 0.2$), the estimated time $T_{eq}^{rc}$ converges quickly (within $R \approx 5-10$ coefficients) and matches the equilibration time obtained from direct dynamical simulations ($T_{eq}^{typ}$) with high accuracy.
• Failure in Non-Smooth Cases: In cases where $b_n$ exhibits significant fluctuations (large $\delta_S$, e.g., near integrable points or specific parameter regimes), the estimate fails to converge, and the approximation $B_N \approx 2b_{N/2}$ breaks down.
• Chaoticity Correlation: The cases with poor convergence correspond to systems with lower average gap ratios ($\langle r \rangle$), indicating less chaotic behavior. Highly chaotic systems ($\langle r \rangle \approx 0.53$) consistently show smooth coefficients and accurate estimates.

5. Significance and Conclusion

• Feasibility of Realistic Timescales: The results imply that for generic chaotic quantum systems, the equilibration time is finite and can be estimated using only the first few Lanczos coefficients. This supports the view that equilibration occurs on a timescale much shorter than the Poincaré recurrence time or the age of the universe.
• Thermodynamic Limit Accessibility: The method bypasses the need for large-scale time evolution simulations, allowing for the estimation of equilibration properties in the thermodynamic limit using data from small, numerically tractable systems.
• Future Directions: The authors suggest applying this framework to disordered systems, higher-dimensional models (e.g., Fermi-Hubbard), and time-dependent (Floquet) systems to estimate relaxation to non-equilibrium steady states.

In summary, the paper provides a robust theoretical and numerical framework linking the smoothness of Lanczos coefficients to the convergence of equilibration time estimates, offering a powerful tool to analyze thermalization in complex quantum many-body systems.