Domain-wall melting in all-to-all QSSEP from random-matrix theory

This paper utilizes random matrix theory to demonstrate that the real-time dynamics of a domain wall melting in an all-to-all quantum simple exclusion process (charged SYK$_2$) are governed by a Jacobi process, yielding explicit thermodynamic limit expressions for entanglement entropy and proving the exact coincidence of quantum and classical full-counting statistics for charge.

The Gist

The Big Picture: Melting a Wall of Particles

Imagine you have a long, narrow hallway with many spots on the floor. On the left side of the hallway, every spot is occupied by a person (a particle). On the right side, the hallway is completely empty. This is your "domain wall"—a sharp line separating the crowded side from the empty side.

Now, imagine the rules of the hallway change. Suddenly, everyone is allowed to hop to any other spot in the hallway, not just the ones next to them. Furthermore, the rules for hopping are random and change constantly, like a chaotic game of musical chairs where the music changes speed and direction unpredictably.

The scientists in this paper wanted to understand what happens to this crowd over time as the "wall" melts and the people spread out. Specifically, they wanted to know two things:

1. How mixed up does the crowd get? (This is called "entanglement" in physics).
2. How much does the number of people in a specific section of the hallway fluctuate? (This is called "full-counting statistics").

The Secret Weapon: A Mathematical Crystal Ball

Usually, predicting the behavior of a chaotic quantum system (where particles act like waves and probabilities) is incredibly hard. It's like trying to predict the exact path of every single drop of water in a hurricane.

However, the authors discovered a clever shortcut. They realized that if you look at the "correlation matrix" (a mathematical map showing how likely particles are to be near each other), the numbers inside this map behave exactly like a specific type of random process known as a Jacobi process.

The Analogy:
Think of the particles' positions not as individual people, but as a set of floating bubbles in a jar. As time passes, these bubbles bounce around, merge, and split in a very specific, predictable way governed by the laws of Random Matrix Theory. Instead of tracking every single particle, the authors tracked the "bubbles" (the eigenvalues of the matrix). Because these bubbles follow a known mathematical dance, the authors could write down exact formulas for how the whole system evolves, rather than just guessing.

The Two Main Discoveries

1. The Entanglement Recipe

The first thing they calculated was how "mixed up" the system becomes. In quantum physics, when particles interact, they become entangled, meaning you can't describe one without describing the other.

• The Result: They found a precise formula for how this "mixing" grows over time.
• The Metaphor: Imagine dropping a drop of ink into a glass of water. At first, the ink is a sharp blob. Over time, it spreads until the water is a uniform gray. The authors calculated exactly how fast the ink spreads and what the final shade of gray looks like. They found that the system reaches a steady state where it is "maximally mixed" within the rules of the game (conserving the total number of particles), but it doesn't become perfectly random in every possible way.

2. The Quantum vs. Classical Surprise

The second, and perhaps most surprising, discovery was a comparison between the Quantum world (where particles are fuzzy waves) and the Classical world (where particles are distinct, solid balls).

• The Setup: They compared their quantum "melting wall" to a classical version where people hop randomly but follow standard probability rules (no quantum weirdness).
• The Result: In the limit of a very large system (a huge hallway), the quantum fluctuations and the classical fluctuations turned out to be identical.
• The Metaphor: Imagine two groups of people trying to guess how many people are in the left half of the room. One group is playing by quantum rules (fuzzy, probabilistic), and the other by classical rules (strict, random). The authors found that if the room is big enough, both groups get the exact same answer, at every single moment in time. There is no "quantum correction" needed; the quantum noise averages out to look exactly like classical noise.

Why This Matters (According to the Paper)

• A New Tool: The paper shows that you can use advanced math from Random Matrix Theory (usually used for things like nuclear physics or number theory) to solve difficult problems in quantum transport. It's like using a telescope to look at a microscopic bug and finding it works perfectly.
• Exact Answers: Instead of using approximations or computer simulations that get messy, they provided exact, closed-form mathematical formulas.
• The Bridge: They proved that for this specific type of chaotic transport, the complex quantum world behaves exactly like a simpler classical world, at least when looking at particle numbers. This is rare, as quantum systems usually behave very differently from classical ones.

Summary

The authors studied a chaotic quantum system where particles hop randomly between all possible locations. By realizing that the mathematical "map" of the system follows a known random process (the Jacobi process), they derived exact formulas for how the system mixes and how particle numbers fluctuate. Their biggest finding is that, surprisingly, the quantum fluctuations in this system are indistinguishable from classical fluctuations in a large system, both at the start and at the end of the process.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of the Quantum Symmetric Simple Exclusion Process (QSSEP) with all-to-all hoppings (also known as the charged SYK$_2$ model). Specifically, the authors study the "melting" of an initial domain wall configuration, where the left half of a system of $L$ fermionic sites is initially occupied ($M$ particles) and the right half is empty.

The central challenge addressed is the characterization of coherent quantum fluctuations in this system. While the average dynamics of linear observables in QSSEP maps to the classical Symmetric Simple Exclusion Process (SSEP), quantities that are non-linear in the density matrix—such as entanglement entropy and full-counting statistics (FCS) of particle number—are governed by quantum coherence and are notoriously difficult to compute analytically. Previous approaches relied on replica methods, which become intractable for higher-order moments.

2. Methodology

The authors develop a novel analytic approach that bypasses the replica trick by exploiting Random Matrix Theory (RMT) and the specific structure of the QSSEP Hamiltonian.

• Gaussian State Formalism: Since the QSSEP Hamiltonian is quadratic in fermionic operators, the system remains in a Gaussian state throughout the evolution. The state is fully characterized by its correlation matrix $\Gamma(t)$.
• Mapping to a Jacobi Process:
  • The authors focus on the reduced correlation matrix $\Gamma_\ell(t)$ of a subsystem of size $\ell$.
  • They demonstrate that the non-zero eigenvalues $\{\lambda_j(t)\}$ of this reduced matrix evolve according to a Jacobi process.
  • By analyzing the stochastic differential equation (SDE) governing the unitary evolution of the single-particle basis, they derive a closed system of SDEs for the eigenvalues $\lambda_j(t)$. This system corresponds to the Jacobi ensemble in RMT.
• Thermodynamic Limit Analysis:
  • In the thermodynamic limit ($L, M, \ell \to \infty$ with fixed ratios $\theta = \ell/L$ and $\zeta = M/\ell$), the Jacobi process converges to a free Jacobi process.
  • The authors utilize known results from RMT (specifically regarding the moments of the Jacobi process) to derive a closed differential equation for the moments $m_n(t) = \mathbb{E}[\frac{1}{M}\text{tr}(J_t^n)]$.
• Self-Averaging: They prove that spectral observables (like entanglement entropy and FCS) are self-averaging, meaning their variance scales as $O(L^{-2})$, allowing the replacement of stochastic averages with deterministic limits in the thermodynamic limit.

3. Key Contributions

1. Direct RMT Mapping: The paper establishes a direct link between the real-time dynamics of the QSSEP correlation matrix eigenvalues and the Jacobi process, providing a closed SDE description without needing replica tricks.
2. Exact Analytic Solutions: The authors derive fully explicit, exact analytic expressions for:
   • The time evolution of all eigenvalue moments.
   • The von Neumann entanglement entropy density.
   • The charge full-counting statistics (cumulant generating function).
3. Quantum-Classical Equivalence: A major theoretical breakthrough is the demonstration that, in the thermodynamic limit, the full-counting statistics of the quantum QSSEP coincides exactly with that of the classical all-to-all SSEP at all times, not just in the steady state.

4. Key Results

A. Entanglement Dynamics

For a bipartition where $\ell = M = L/2$ (half-chain), the authors derive an exact formula for the time-dependent entanglement entropy density $s(t)$:
$$s(t) = 2\log(2) - 1 - \sum_{n=1}^\infty \frac{1}{n(2n-1)} \sum_{k=1}^n (-2)^{-2k} \binom{2n}{2k} {}_2F_1\left(\frac{2k+1}{2}, 2k-2n; 4k+1; 2\right) e^{-kt} L_{k-1}^{(1)}(2kt)$$
where $L_n^{(1)}$ are Laguerre polynomials and ${}_2F_1$ is the hypergeometric function.

• Steady State: As $t \to \infty$, the entropy saturates at $s(\infty) = 2\log(2) - 1$. This value is strictly less than the maximum possible entropy $\log(2)$, indicating that the steady state is maximally mixed within the subspace of fixed particle number, but not globally maximally entangled.
• Spectral Distribution: The steady-state eigenvalue distribution follows the arcsine law $\rho_\infty(\lambda) = [\pi\sqrt{\lambda(1-\lambda)}]^{-1}$.

B. Charge Full-Counting Statistics (FCS)

The cumulant generating function $F(\alpha, t)$ for the particle number in the subsystem is derived as:
$$F(\alpha, t) = F_{ss}(\alpha) - 2 \sum_{k \ge 1} e^{-kt} L_{k-1}^{(1)}(2kt) \left(-\frac{\alpha}{4}\right)^k {}_2F_1\left(k+\frac{1}{2}, k; 2k+1; \alpha\right)$$
where $F_{ss}(\alpha) = 2\log\left(\frac{1+\sqrt{1+\alpha}}{2}\right)$ is the stationary value.

C. Quantum vs. Classical Comparison

• Steady State: The stationary FCS of the quantum QSSEP matches the classical SSEP on a complete graph.
• Finite-Time Dynamics: The authors derive evolution equations for both the quantum and classical generating functions. They show that these equations coincide up to finite-size corrections of order $O(L^{-1})$.
• Conclusion: In the thermodynamic limit, the quantum and classical full-counting statistics are identical at all finite times. This implies that for transport observables in this specific all-to-all geometry, coherent quantum fluctuations do not alter the large-deviation statistics of particle transport compared to the classical stochastic process.

5. Significance

• New Analytic Tool: The work provides a powerful alternative to the replica method for studying non-linear observables in random quantum circuits and SYK-like models. By mapping the problem to a Jacobi process, it unlocks exact solutions for all moments.
• Insight into Quantum Transport: The finding that quantum and classical FCS coincide in the thermodynamic limit for the all-to-all QSSEP is profound. It suggests that for certain transport protocols, the "quantumness" (coherence) does not modify the statistical distribution of transported charge, even though it drives the entanglement growth.
• Benchmarking: The derived formulas serve as exact benchmarks for numerical simulations of quantum many-body systems and validate the use of classical stochastic models for predicting quantum transport statistics in highly connected systems.
• Limitations and Future Work: The authors note that this equivalence does not extend to entanglement entropy (which has no classical counterpart) and that finite-size corrections ($O(L^{-1})$) do distinguish the quantum and classical dynamics. Future work is suggested to explore whether this equivalence holds for different initial states or lattice connectivities.

In summary, the paper successfully bridges Random Matrix Theory and quantum non-equilibrium statistical mechanics to provide exact, time-resolved solutions for entanglement and particle fluctuations in a solvable quantum transport model, revealing a surprising deep connection between quantum and classical transport statistics.