Ruelle-Pollicott resonances of diffusive U(1)-invariant qubit circuits

This paper investigates Ruelle-Pollicott resonances in translationally invariant, magnetization-conserving qubit circuits to demonstrate how the quasi-momentum dependence of the leading eigenvalue characterizes diffusive transport and extracts the diffusion constant, while also conjecturing a continuum of subleading eigenvalues responsible for non-exponential hydrodynamic tails.

The Gist

The Big Picture: Listening to the "Hum" of a Quantum System

Imagine you have a giant, chaotic room filled with thousands of bouncing balls (these are our qubits). If you throw a ball in, it bounces around, hits others, and eventually, the energy spreads out until the whole room is vibrating evenly. This is what physicists call thermalization or diffusion.

For a long time, scientists have tried to figure out how fast this spreading happens and what rules govern it. Usually, they watch the balls directly. But in the quantum world, watching every single ball is impossible because there are too many of them, and the math gets incredibly messy.

This paper introduces a new way to listen to the room instead of watching the balls. The authors propose that every quantum system has a unique "hum" or a set of musical notes it naturally sings as it settles down. These notes are called Ruelle-Pollicott (RP) resonances.

The Main Characters

1. The Qubit Circuit: Think of this as a specific set of rules for how the balls bounce. In this paper, the authors use a "brickwall" pattern where groups of three balls interact in a specific way.
2. The Conserved Quantity (Magnetization): Imagine that while the balls bounce around wildly, the total number of red balls in the room never changes. They just move from one side to the other. In physics, this is called a "conserved quantity" (specifically, magnetization). Because this total amount is fixed, the system can't just forget its past; it has to move this "redness" around, which creates diffusion.
3. The Truncated Propagator: This is the authors' special tool. Imagine you are trying to hear the hum of the room, but you only have a microphone that can pick up sounds from a small corner. You can't hear the whole room at once. So, you build a mathematical model that only looks at a small group of balls and how they interact. You then slowly make your "listening window" bigger and bigger.

The Discovery: The "Diffusion Tune"

The authors found something amazing by listening to these "notes" (the eigenvalues of their mathematical model) at different frequencies (called quasi-momentum, or $k$).

1. The Slow, Smooth Slide (Small $k$)

When they looked at the "lowest notes" (small $k$), they found a very specific pattern. The pitch of the note changed in a perfect, smooth curve that looked like a parabola (a U-shape).

• The Analogy: Imagine a slide. If you are at the very top (zero momentum), the system doesn't decay at all because the "redness" is spread out evenly everywhere. As you move slightly down the slide (small $k$), the system starts to relax, but it does so in a very predictable way.
• The Result: The shape of this slide tells them exactly how fast the "redness" spreads. This speed is called the Diffusion Constant. By just looking at the shape of this mathematical curve, they could calculate the diffusion constant without ever having to simulate the balls moving for a long time. It's like knowing how fast a car is driving just by looking at the curve of its tire tracks.

2. The Sharp Drop-Off (Large $k$)

When they looked at the "high notes" (large $k$), the smooth slide disappeared. The notes dropped off sharply.

• The Analogy: This is like a high-pitched squeak that dies out instantly. These high-frequency ripples in the system don't care about the slow spreading of the "redness." They just vanish quickly on their own. This tells us that for fast, local changes, the system forgets its past very rapidly.

The Hidden "Ghost" Notes (The Continuum)

Here is the most exciting part. The authors suspect that below the main "slide" (the diffusion tune), there isn't just a few more notes, but a continuous fog of sounds.

• The Analogy: Imagine the main note is a clear piano key. But underneath it, there is a thick fog of sound that isn't a single note, but a blend of many.
• Why it matters: This "fog" is responsible for the weird, slow tails you see in nature. Sometimes, things don't just fade away exponentially (like a dying lightbulb); they fade away like a power law (like a slow, lingering echo). The authors believe this "fog" of resonances is what causes those long, slow tails in the data.

Why This Matters

1. It's a Shortcut: Instead of running massive, slow simulations to see how heat or magnetism spreads, you can just look at the "notes" of the system's operator math. It's like diagnosing an engine by listening to its idle sound rather than taking it apart.
2. It Works for Chaos: Even though the system is chaotic and unpredictable in the short term, this method reveals the hidden, predictable order (diffusion) that emerges in the long term.
3. Universal Rules: The authors believe this works for almost any system that has one "conserved thing" (like energy or charge), not just the specific quantum circuits they tested.

Summary in a Nutshell

The authors built a mathematical "stethoscope" to listen to the heartbeat of a chaotic quantum system. They discovered that the system sings a specific song when it diffuses. By analyzing the pitch of this song, they can instantly calculate how fast the system spreads out energy or magnetism. They also suspect there is a hidden "chorus" of ghostly notes underneath the main song that explains why some things fade away slowly and strangely.

This is a powerful new way to understand how the quantum world settles down, turning complex chaos into a simple, readable tune.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing transport properties and the long-time dynamics of many-body quantum systems, specifically those with a single conserved quantity (U(1) symmetry, e.g., magnetization).

• Context: In chaotic quantum systems, correlation functions typically decay exponentially, governed by a relaxation rate. However, in systems with conserved quantities, transport often follows hydrodynamic laws (e.g., diffusion), leading to power-law tails or specific scaling behaviors that are difficult to extract from standard correlation function simulations, especially in the thermodynamic limit.
• Gap: Existing methods to study quantum chaos and transport (e.g., spectral quantifiers, tensor networks) often struggle with the infinite system size limit or require complex extrapolations. The authors aim to adapt the concept of Ruelle-Pollicott (RP) resonances—originally from classical chaotic systems—to translationally invariant quantum circuits to directly extract transport exponents and diffusion constants.

2. Methodology

The authors employ a quasi-momentum-resolved truncated propagator method applied to the Heisenberg evolution of operators.

• Truncated Propagator: Instead of simulating the full unitary evolution (which preserves information indefinitely), the authors project the dynamics onto a subspace of local operators with a finite support size $r$. This introduces a non-unitary "truncation" that mimics dissipation.
• Quasi-Momentum Resolution: The system is translationally invariant. The authors define a basis of extensive observables with good quasi-momentum $k$. The propagator is block-diagonalized by $k$, allowing the study of dynamics for specific momentum modes.
• RP Resonances: The eigenvalues $\lambda(k)$ of this truncated propagator are interpreted as RP resonances. In the limit of infinite support ($r \to \infty$), the leading eigenvalue $\lambda_1(k)$ converges to a value inside the unit circle (for non-conserved quantities) or to 1 (for conserved quantities).
• Model System: The study focuses on 1D qubit circuits with 3-site gates arranged in a 3-layer brickwall geometry. These circuits conserve total magnetization ($M = \sum \sigma^z_j$) but are otherwise chaotic (non-integrable).
• Numerical Techniques:
  • Arnoldi Iteration: Used to find the leading eigenvalues of the truncated propagator for large support sizes ($r \approx 13$).
  • Domain Wall Quench: Used as a benchmark to independently calculate the diffusion constant $D$ via Time-Evolving Block Decimation (TEBD).
  • Weak Dephasing (Lindblad): Used in the appendix to compare the truncated propagator method against a weak dissipation approach, highlighting the robustness of their method.

3. Key Contributions

1. Generalization of RP Resonances: The authors successfully extend the RP resonance formalism to translationally invariant quantum circuits with arbitrary geometry and conserved quantities, resolving the spectrum by quasi-momentum $k$.
2. Extraction of Transport Constants: They demonstrate that the $k$-dependence of the leading RP resonance $\lambda_1(k)$ directly encodes transport properties. Specifically, for diffusive systems, the relation $|\lambda_1(k)| \approx e^{-Dk^2}$ holds for small $k$, allowing the direct numerical extraction of the diffusion constant $D$.
3. Conjecture of RP Continuum: The paper proposes the existence of a continuum of eigenvalues below the leading diffusive resonance. This continuum is argued to govern non-exponential decays, such as power-law hydrodynamic tails and stretched-exponential decays, which discrete eigenvalues cannot capture.
4. Role of Conserved Quantity Powers: The authors explain that subleading eigenvalues near 1 in the $k=0$ sector arise from the powers of the conserved magnetization ($M^2, M^3, \dots$), which are "almost" conserved under truncation, converging to 1 as $1/r$.

4. Key Results

A. Small Quasi-Momenta ($k \to 0$) and Diffusion

• In systems with a conserved magnetization, the leading eigenvalue at $k=0$ is exactly 1 (corresponding to the conserved quantity).
• For small non-zero $k$, the leading eigenvalue follows the scaling:
  $$|\lambda_1(k)| = e^{-Dk^2}$$
• By fitting the numerical data for small $k$, the authors extract the transport dynamical exponent $z \approx 2$ (confirming diffusion) and the spin diffusion constant $D$.
• The extracted $D$ matches values obtained from independent domain-wall quench simulations (TEBD) and Green-Kubo formulas within $\sim 1\%$ precision.

B. Large Quasi-Momenta ($k$ far from 0)

• For large $k$, the diffusive behavior breaks down due to eigenvalue crossings. The leading resonance becomes independent of transport and governs the exponential decay of generic correlation functions.
• The decay rate is determined by the largest eigenvalue across all $k$, which is not necessarily at $k=0$. This challenges the assumption that the slowest decay always occurs for translationally invariant observables.

C. Subleading Eigenvalues and Hydrodynamic Tails

• Powers of Conserved Quantities: The authors show that eigenvalues close to 1 (but $<1$) in the $k=0$ sector correspond to the truncated powers of magnetization ($M^n$). Their convergence to 1 follows a power law $\sim 1/r$.
• RP Continuum: The authors conjecture that the non-exponential decay (hydrodynamic tails, e.g., $t^{-1/2}$ for magnetization correlation) is not governed by a discrete eigenvalue but by a continuum of eigenvalues filling the gap between 0 and the leading resonance.
• Extensive Current: The correlation function of the extensive spin current is shown to decay as a power law in finite systems, but the thermodynamic limit behavior remains an open question, potentially involving subleading corrections to diffusion.

D. Comparison with Other Methods

• The truncated propagator method is shown to be more robust and easier to implement in the thermodynamic limit compared to the Lindblad weak dephasing method. The Lindblad approach requires careful double-limit extrapolation ($N \to \infty$ and noise $\gamma \to 0$) and can suffer from finite-size effects that obscure the true resonance.

5. Significance

• New Tool for Transport: This work provides a powerful, direct method to calculate diffusion constants and transport exponents in the thermodynamic limit without simulating long-time dynamics of specific initial states.
• Understanding Chaos and Conservation: It clarifies the spectral structure of chaotic systems with conservation laws, distinguishing between transport-related resonances (diffusive) and generic chaotic decay.
• Hydrodynamic Tails: The conjecture of an RP continuum offers a spectral explanation for power-law hydrodynamic tails, bridging the gap between discrete spectral methods and continuous hydrodynamic theories.
• Applicability: The method is expected to be generic for any system with exactly one U(1) conserved quantity, applicable to both quantum circuits and potentially Hamiltonian systems (with further development).

In summary, the paper establishes the quasi-momentum-resolved truncated propagator as a rigorous and efficient tool for characterizing diffusive transport and the spectral properties of chaotic quantum many-body systems, revealing a rich structure of resonances and continua that govern both exponential and power-law decay.