Inhomogeneous quenches and GHD in the $ν= 1$ QSSEP model

This paper extends the framework of quantum generalized hydrodynamics to the stochastic $\nu=1$ QSSEP model, successfully deriving the evolution of local quasiparticle occupations and entanglement statistics for inhomogeneous quenches through a combination of conformal field theory techniques and numerical validation.

The Gist

The Big Picture: A Quantum Crowd in the Rain

Imagine a huge, one-dimensional hallway filled with invisible, ghost-like particles (fermions). In a normal quantum world, these particles move in a very orderly, predictable way, like dancers following a strict choreography. They zip back and forth at high speeds, carrying information and "entanglement" (a spooky connection between particles) with them.

But in this paper, the authors are studying a very different scenario. They imagine this hallway is being pelted by random rain. Every time a particle tries to hop to the next spot, the "wind" (the noise) pushes it in a random direction. It's like trying to walk through a crowded room while someone is randomly shoving you left or right every second.

The paper asks: If you start with a messy, uneven crowd of these particles, how do they spread out over time, and how does their "spooky connection" (entanglement) grow when the whole system is being randomly jostled?

The Two Experiments

The researchers set up two classic "what if" scenarios to test this:

1. The Melting Ice Wall (Domain Wall Melting)

• The Setup: Imagine the left half of the hallway is packed shoulder-to-shoulder with particles (density = 1), and the right half is completely empty.
• The Action: At time zero, you remove the invisible wall. In a normal world, the particles would rush into the empty space like a wave, moving in straight lines.
• The Twist: Because of the "random rain," the particles don't move in straight lines. They perform a drunken walk (Brownian motion). Instead of a sharp wavefront, the crowd spreads out like a drop of ink diffusing in water.

2. The Released Trap (Free Expansion)

• The Setup: Imagine the particles are trapped in a bowl on the left side of the hallway by a steep hill (a potential well). They are squished together but not as tightly as in the first experiment.
• The Action: Suddenly, the hill disappears. The particles are free to run into the empty hallway.
• The Twist: Again, the random rain makes their path chaotic. They don't just run; they stumble and drift.

The Secret Weapon: "Quantum Generalized Hydrodynamics" (QGHD)

To solve this, the authors used a powerful mathematical tool called Generalized Hydrodynamics (GHD).

• The Analogy: Think of GHD as a way to describe a crowd without tracking every single person. Instead of saying "Bob moved here, Alice moved there," you say, "The density of people in this area is 50%."
• The Innovation: Usually, GHD works for orderly systems. The authors had to invent a new version, QGHD, to handle the "drunken walk" of the particles. They realized that even though the particles are moving randomly, you can still describe them as "quasiparticles" (ghostly surrogates) that are doing a random walk.

The Big Discovery: How "Spooky Connections" Grow

The most exciting part of the paper is what happens to Entanglement Entropy.

• What is it? Imagine two groups of people. If they are "entangled," knowing the state of one person instantly tells you something about the other, no matter how far apart they are. It's a measure of how much the two sides of the hallway are "connected."
• The Normal World: In a calm, orderly quantum system, this connection grows linearly (fast and steady). It's like a fire spreading quickly down a dry forest.
• The Rainy World (This Paper): Because the particles are stumbling around randomly, the connection grows much slower. It grows logarithmically (very slowly, like a snail).
  • The Math Magic: The authors found that the "speed" of this growth is cut in half compared to the orderly world. The random noise acts like a brake, slowing down the spread of quantum information.

The "Fermi Contour" Map

To get these results, the authors drew a mental map called a Fermi Contour.

• The Analogy: Imagine a map where the X-axis is "where you are" and the Y-axis is "how fast you are moving."
• The Map: In a calm system, this map is a straight line. In this rainy system, the line gets wiggly and distorted by the random wind.
• The Trick: The authors realized that for every single "storm" (a specific realization of the random noise), the particles follow a specific wiggly path. They calculated the entanglement for one specific storm, and then they averaged the results of millions of different storms.
• The Result: Even though every single storm is chaotic and unpredictable, when you average them all together, a beautiful, smooth pattern emerges. The chaos cancels itself out to reveal a predictable law of diffusion.

Why Does This Matter?

1. Real-World Noise: Real quantum computers and experiments are never perfect; they always have noise. This paper gives us a blueprint for how quantum information behaves when things are messy and imperfect.
2. New Math: It proves that we can use "hydrodynamics" (fluid dynamics) to describe quantum systems that are being shaken by random noise. It's like finding a way to predict the flow of a river even when it's raining heavily and the wind is blowing.
3. The "Self-Averaging" Surprise: The paper shows that even though the quantum world is probabilistic, if you wait long enough and look at the average, the randomness disappears, and the system behaves in a very predictable, "self-correcting" way.

Summary in One Sentence

The authors figured out how to predict the spread of quantum connections in a system where particles are constantly being randomly pushed around, discovering that this "quantum rain" slows down the spread of information to a crawl, turning a fast sprint into a slow, diffusive drift.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of the $\nu = 1$ Quantum Symmetric Simple Exclusion Process (QSSEP). This is a one-dimensional system of free fermions characterized by:

• Stochastic Dynamics: The hopping amplitudes between sites are random in time (modeled by complex Itô increments) but uniform in space.
• Inhomogeneous Initial States: The study focuses on two paradigmatic "quench" protocols where the system starts out of equilibrium:
  1. Domain-Wall Melting: A semi-infinite chain where the left half is fully occupied ($\rho=1$) and the right half is empty.
  2. Free Expansion: A finite gas of free fermions confined by a potential, which is suddenly released to expand into a larger region.

The central challenge is to understand how entanglement entropy spreads in this stochastic setting. While Generalized Hydrodynamics (GHD) has been successfully applied to deterministic integrable systems, extending it to stochastic quantum systems to capture both conserved charges and quantum entanglement statistics remains an open problem.

2. Methodology

The authors employ a hybrid theoretical framework combining Quantum Generalized Hydrodynamics (QGHD) with Stochastic Calculus and Conformal Field Theory (CFT).

A. Stochastic Hydrodynamics of Occupation Functions

• Quasiparticle Description: The system is described by a local Fermi occupation function $n_k(x,t)$.
• Stochastic Evolution: Unlike deterministic integrable systems where quasiparticles move ballistically ($x - v_k t$), in the $\nu=1$ QSSEP, the quasiparticle velocity is stochastic. The authors derive that $n_k(x,t)$ evolves according to a stochastic differential equation (SDE) in the Itô sense:
  $$dn_k(x, t) = 2D \partial^2_x n_k(x, t) dt - \partial_x n_k(x, t) d\xi_k$$
  where $d\xi_k$ represents the stochastic velocity increment.
• Diffusive Limit: Upon averaging over noise realizations, the occupation function obeys a standard diffusion equation ($\partial_t \langle n_k \rangle = D \partial^2_x \langle n_k \rangle$), contrasting with the Euler-type equations of deterministic GHD.

B. Quantum Generalized Hydrodynamics (QGHD) for Entanglement

• Fluctuations: The semiclassical Yang-Yang entropy vanishes for these protocols because the occupation function remains binary (0 or 1) in any single noise realization. To capture entanglement, the authors incorporate quantum fluctuations of the occupation function at the Fermi contour (the boundary between filled and empty regions in phase space).
• Luttinger Liquid Mapping: These fluctuations are mapped to a massless free boson field (a Luttinger liquid) living on the Fermi contour $\Gamma_t$.
• CFT Twist Fields: The entanglement entropy for a specific noise realization is calculated using the correlation functions of twist fields inserted at the Fermi points (where the Fermi contour intersects the subsystem cut).
  • For a single noise realization, the Rényi entropy is determined by the geometry of the Fermi contour $\Gamma_t$.
  • The Fermi contour evolves stochastically, parameterized by random variables $\rho$ (amplitude) and $\phi$ (phase) derived from the complex Brownian motion of the hopping.

C. Averaging and Numerical Verification

• Full Counting Statistics: The final physical prediction for the average entanglement entropy is obtained by averaging the QGHD prediction over the probability distribution of all possible Fermi contours generated by the stochastic dynamics.
• Numerical Validation: The authors perform exact numerical simulations on finite lattices ($L \sim 160-400$). Since the initial states are Gaussian and the Hamiltonian is quadratic, the time-evolved state remains Gaussian. They compute the two-point correlation matrix $G_{ij}(t)$ stochastically and derive the entanglement entropy directly, averaging over $\sim 10^3$ noise realizations.

3. Key Contributions

1. Extension of GHD to Stochastic Systems: This work constitutes the first application of Quantum Generalized Hydrodynamics to a stochastic quantum system. It demonstrates that the framework can be successfully extended beyond unitary, integrable dynamics to include stochastic effects.
2. Exact Entanglement Statistics: The authors derive the exact asymptotic behavior of the entanglement entropy and its full counting statistics (variance) in the hydrodynamic limit.
3. Diffusive vs. Ballistic Scaling: They establish a clear distinction between deterministic integrable quenches (ballistic spreading, $S \sim \frac{1}{6} \log t$) and stochastic quenches (diffusive spreading, $S \sim \frac{1}{12} \log t$ for domain walls).
4. Handling Multiple Fermi Points: In the free expansion protocol, the stochastic dynamics can generate four Fermi points (split Fermi seas) for a single noise realization, a scenario absent in the deterministic domain-wall case. The authors provide the CFT machinery to handle these multi-point correlations.

4. Key Results

A. Domain-Wall Melting

• Average Entanglement: The average entanglement entropy for a half-system cut ($\ell=0$) grows logarithmically:
  $$\langle S_{\ell=0}(t) \rangle \simeq \frac{1}{12} \log t + C$$
  The coefficient is half that of the deterministic case ($1/6$), reflecting the diffusive nature of the transport.
• Scaling Function: The spatial profile of the entanglement entropy follows a diffusive scaling form $S(\ell/\sqrt{t})$.
• Self-Averaging: The relative fluctuations of the entanglement entropy vanish as $t \to \infty$ ($\sigma_S / \langle S \rangle \to 0$), meaning the average value represents the typical behavior in the hydrodynamic limit.

B. Free Expansion

• Complex Dynamics: The initial confinement potential leads to a non-trivial phase $\phi$ in the stochastic velocity. This causes particles to reflect off boundaries and populate momentum modes outside the initial range, leading to scenarios with four Fermi points.
• Logarithmic Growth: In the hard-wall limit ($\beta \to \infty$), the average half-system entanglement also grows logarithmically:
  $$\langle S_{\ell=0}(t) \rangle \simeq \frac{1}{8} \log t + \text{const.}$$
  The coefficient ($1/8$) differs from the domain-wall case due to the specific initial density profile and boundary effects.
• Boundary Effects: The authors identify that entanglement evolves near the system boundaries ($x = -L/2$) due to the non-zero phase $\phi$, a feature not present in deterministic free expansion.

5. Significance

• Theoretical Bridge: The paper bridges the gap between macroscopic fluctuation theory (classical diffusion) and quantum entanglement dynamics in noisy systems.
• Universality: It suggests that the "diffusive" reduction of the entanglement growth coefficient is a universal feature of stochastic free-fermion systems with spatially homogeneous noise.
• Future Applications: The framework is presented as a versatile tool that can be extended to:
  • Interacting systems (e.g., noisy XXZ chains).
  • Symmetry-resolved entanglement.
  • Junctions of different ground states.
  • Thermal initial states.

In summary, the paper provides a rigorous, analytically solvable, and numerically verified description of how quantum entanglement spreads in a stochastic environment, establishing a new benchmark for non-equilibrium statistical mechanics in open or noisy quantum systems.