Spacetime picture for entanglement generation in noisy fermion chains

This paper establishes a spacetime picture for entanglement generation in noisy fermion chains by mapping the calculation of Renyi entropies to an SO(2N) Heisenberg spin chain, revealing a semiclassical domain-wall dynamics that transitions from diffusive to ballistic spreading upon the addition of weak interactions.

The Gist

The Big Picture: Entanglement as a Spacetime Movie

Imagine you have a long line of people (a chain of particles) holding hands. At the start, they are only holding hands with their immediate neighbors. This is a state of low entanglement.

Now, imagine the wind blows randomly through the line, causing people to randomly swap hands with their neighbors. Over time, the whole line becomes a tangled mess of connections. This is entanglement generation.

The physicists in this paper wanted to understand how this tangle grows over time. Specifically, they looked at two scenarios:

1. The Free Case: People swap hands randomly, but they don't talk to each other or influence each other's decisions (non-interacting).
2. The Interacting Case: People swap hands, but they also whisper instructions to each other, changing how they move (weakly interacting).

Their goal was to create a "movie" (a spacetime picture) of how this tangle spreads.

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Part 1: The Free Case (The Diffusive Walk)

In the "Free" scenario, the people are like drunkards walking randomly. They don't coordinate.

The Analogy: The Melting Ice Cream Cone
Imagine the boundary between the "tangled" part of the line and the "untangled" part is a sharp line drawn in the sand.

• In the beginning: The line is razor-sharp.
• As time passes: Because the people are moving randomly (diffusively), the sharp line starts to blur. It doesn't stay a straight wall; it smears out like a melting ice cream cone or a drop of ink spreading in water.

The "Two Ghosts" Discovery
The most surprising thing the authors found is that to describe this blurring, you need two invisible ghosts (fields) moving in opposite directions:

1. Ghost A moves forward in time, smearing the line out.
2. Ghost B moves backward in time, also smearing the line out.

They interact with each other, but they are distinct. The "tangle" (entanglement) is essentially the cost of keeping these two ghosts connected while they diffuse.

The Result:
Because the line smears out slowly (like diffusion), the amount of tangle grows slowly. If you wait time $t$, the tangle grows by the square root of time ($\sqrt{t}$). It's a slow, lazy spread.

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Part 2: The Interacting Case (The Ballistic Wall)

Now, imagine we add a rule: The people can whisper to each other. This is "weak interaction."

The Analogy: The Traffic Jam
In the free case, people moved randomly. But now, if someone tries to move, they push their neighbor, who pushes the next one. They start moving as a coordinated unit.

The Change in the Movie:
When the authors added these interactions to their "movie," the smooth, melting ice cream cone suddenly stopped melting.

• Instead of a blurry, spreading wall, the boundary became a sharp, rigid wall again.
• This wall doesn't just sit there; it marches forward at a constant speed.

The Result:
Because the wall is sharp and moves at a constant speed, the tangle grows much faster. If you wait time $t$, the tangle grows linearly with time ($t$). This is called ballistic spreading (like a bullet flying straight).

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The "Crossover": From Drunkards to Soldiers

The paper's main achievement is showing the transition between these two worlds.

• Short Time: When you first start the experiment, the interactions haven't had time to organize the chaos. The system looks like the "Free" case (the blurry, diffusing wall).
• Long Time: Eventually, the interactions take over. The blurry wall stops spreading and sharpens up into a marching wall.

The authors calculated exactly how long it takes for this switch to happen and how thick the blurry wall gets before it sharpens. They found a specific "crossover length" ($l_{int}$).

• If you look at a small piece of the chain (smaller than this length), it looks like the free, diffusive case.
• If you look at a large piece (larger than this length), it looks like the interacting, ballistic case.

Why Does This Matter?

1. New Map for Quantum Chaos: For a long time, physicists had a great map for how entanglement grows in strongly interacting systems (the "Entanglement Membrane" theory). But they didn't have a good map for free systems (where particles don't talk). This paper fills that gap, showing that free systems are like smooth, diffusing fields, while interacting systems are like sharp, marching walls.
2. The "Two Ghosts" Trick: The method they used (involving the two fields moving in opposite time directions) is a powerful new tool. It allows them to turn a complicated quantum problem into a classical physics problem that is much easier to solve.
3. Universality: It shows that even a tiny bit of interaction can completely change the rules of the game, switching the system from a slow, diffusive spread to a fast, ballistic spread.

Summary in One Sentence

The paper reveals that in a noisy quantum system, entanglement usually spreads slowly like a melting ice cream (diffusion), but if you add even a tiny bit of interaction, it suddenly snaps into a fast, marching wall (ballistic motion), and they have mapped out exactly how that transition happens.

Technical Summary

1. Problem Statement

The paper addresses the dynamics of entanglement generation in random free-fermion systems (specifically 1D chains of Majorana modes with random nearest-neighbor hoppings). While the entanglement dynamics of interacting random unitary circuits are well-understood via a "spacetime membrane" picture (where entanglement grows ballistically, $S \sim t$, due to a discrete replica symmetry), the behavior of free fermions is less intuitive.

The authors aim to:

1. Develop a spacetime picture for entanglement generation in non-interacting (free) fermion systems.
2. Determine the universality class of entanglement growth in the absence of interactions.
3. Analyze the crossover to interacting behavior when weak random interactions are introduced, specifically how the transition from free to interacting dynamics alters the spacetime structure of entanglement.

2. Methodology

The authors employ a replica formalism combined with a coherent state path integral approach.

• Model: A 1D chain of Majorana fermions with a time-dependent Hamiltonian $H(t) = -i \sum_i \eta_i(t) \gamma_i \gamma_{i+1}$, where $\eta_i(t)$ are independent Gaussian noise terms. This represents a noisy version of the Kitaev chain.
• Replica Trick: To calculate the $N$-th Rényi entropy $S_N = -\frac{1}{N-1} \ln \text{Tr}(\rho_A^N)$, they consider $2N$ replicas of the system (N forward, N backward). The quantity of interest is the averaged purity $\overline{\text{Tr}(\rho_A^2)}$ (corresponding to $N=2$).
• Disorder Averaging: Averaging over the Gaussian noise transforms the unitary circuit evolution into a deterministic evolution in imaginary time.
• Mapping to Spin Chain: The authors demonstrate that the disorder-averaged replicated system maps exactly to an SO(2N) ferromagnetic Heisenberg spin chain evolving in imaginary time.
  • For $N=2$ (purity), the symmetry is SO(4), which reduces to two decoupled SU(2) chains due to chirality conservation. The relevant sector is a single SU(2) ferromagnet.
• Semiclassical Approximation: For large times ($\Delta^2 t \gg 1$), they apply a saddle-point approximation to the spin coherent state path integral. This treats the spin configuration as a classical field $z(x, \tau)$ evolving in spacetime.

3. Key Contributions and Results

A. The Free-Fermion Spacetime Picture (Diffusive Regime)

In the free-fermion limit, the effective theory possesses a continuous SO(2N) symmetry. This leads to a distinct spacetime picture compared to interacting systems:

• Two Independent Fields: The saddle-point solution requires two independent classical fields, $z(x, \tau)$ and $\bar{z}(x, \tau)$, which evolve in opposite time directions.
  • $z$ relaxes diffusively forward in time.
  • $\bar{z}$ relaxes diffusively backward in time.
• Smooth Domain Walls: The boundary conditions imposed by the entanglement cut (subsystem A vs. B) create a domain wall. In the free case, this domain wall is smooth and relaxes diffusively.
• Entanglement Scaling: The action cost of this smooth domain wall scales as $\sqrt{t}$. Consequently, the average purity decays as $\overline{\text{Tr}(\rho_A^2)} \sim e^{-\kappa \sqrt{t}}$, implying the entanglement entropy grows as:
  $$S_2 \sim \sqrt{t}$$
  This indicates diffusive spreading of information, contrasting with the ballistic spreading ($S \sim t$) seen in generic interacting systems.

B. Effect of Weak Interactions (Crossover to Ballistic)

The authors introduce weak random 4-Majorana interactions to the Hamiltonian.

• Symmetry Breaking: These interactions explicitly break the continuous SO(2N) symmetry down to a discrete symmetry (specifically a dihedral group $D_4$ for $N=2$).
• Finite Width Domain Wall: The anisotropy introduced by the interaction penalizes spin directions away from specific axes. The domain wall can no longer relax to infinite width; instead, it relaxes to a finite width $l_{int} \sim \Delta / \Delta_I$ (where $\Delta_I$ is the interaction strength).
• Crossover Behavior:
  • Short times ($t \ll t_{crossover}$): The system behaves like the free case (diffusive, $\sqrt{t}$ growth).
  • Long times ($t \gg t_{crossover}$): The domain wall appears sharp on scales larger than $l_{int}$. The action becomes extensive in time, leading to ballistic growth of entanglement:
    $$S_2 \sim t$$
    This recovers the "entanglement membrane" picture typical of chaotic interacting systems.

C. Numerical Verification

The authors performed direct numerical simulations of the noisy Majorana chain (tracking correlation functions rather than the full wavefunction).

• The simulations confirm the saddle-point predictions for the free case, showing the $\sqrt{t}$ growth.
• They verify that the statistical fluctuations of the entropy $S_2$ are small, meaning the average of the entropy $\overline{S_2}$ coincides with the entropy of the average purity $\check{S}_2 = -\ln \overline{\text{Tr}(\rho_A^2)}$ to leading order. This is a unique feature of free fermions, unlike interacting circuits where these quantities differ.

4. Significance

• New Universality Class: The paper identifies a distinct universality class for entanglement dynamics in free fermion systems, characterized by diffusive information spreading and a continuous replica symmetry, contrasting with the ballistic spreading and discrete symmetry of interacting systems.
• Spacetime Picture: It provides a concrete, semiclassical spacetime picture for free fermions involving smooth, relaxing domain walls and two independent time-evolving fields, offering a conceptual alternative to the quasiparticle picture often used for non-random free systems.
• RG Flow: The work explicitly describes the Renormalization Group (RG) flow between the Gaussian (free) and interacting universality classes, showing how weak interactions drive the system from diffusive to ballistic entanglement growth.
• Analytical Tool: The mapping to the SO(2N) Heisenberg model and the subsequent saddle-point analysis provide a powerful analytical framework for studying entanglement in noisy quantum systems without relying on heavy numerical diagonalization.

In summary, the paper establishes that while interacting random circuits generate entanglement via sharp "membranes" moving ballistically, free fermion circuits generate entanglement via smooth, diffusing "domain walls," and that weak interactions induce a crossover between these two fundamental dynamical regimes.