Quantum many-body scars in random unitary circuits

This paper constructs an analytically tractable random unitary circuit hosting a single quantum many-body scar and reveals that, while thermodynamically irrelevant for local observables, the scar induces a sharp transition in entanglement dynamics driven by fluctuating interfaces that is invisible to local measurements.

The Gist

The Big Picture: A Quantum Party That Won't End

Imagine a huge, chaotic party (a quantum system) where everyone is dancing wildly. In a normal party, if you start with a specific formation (like everyone holding hands in a circle), the chaos eventually breaks it apart. Everyone mixes, the music changes, and the room reaches a state of "thermal equilibrium"—a messy, random crowd where no specific pattern remains. This is how most quantum systems work; they "thermalize."

However, sometimes, a few people in the crowd refuse to mix. They keep dancing in a perfect circle, ignoring the chaos around them. In physics, these stubborn, non-mixing patterns are called Quantum Many-Body Scars.

For a long time, scientists thought these scars were like fragile glass statues: if you touched them even slightly, they would shatter and the system would return to the messy, thermal state.

This paper asks a simple question: What happens if we poke these scars? Do they break immediately, or do they linger? And does the "messiness" of the party look the same to everyone, or are there secret things happening that only a special camera can see?

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The Experiment: A Digital Playground

The authors built a simplified, mathematical "playground" (a random unitary circuit) to simulate this.

• The Setup: Imagine a line of people (particles). Most of the time, they swap places randomly (chaos).
• The Scar: One specific person (or state) is special. If everyone is in the "zero" state (standing still), they stay standing still. This is the "scar."
• The Twist: The rest of the system is designed to be completely chaotic, scrambling everything else.

They wanted to see what happens when they introduce a tiny bit of "noise" (a perturbation) to this special standing-still state.

Finding 1: The "Melting Ice" Effect (Local Observables)

When the scientists poked the scar with a small disturbance (represented by a variable $\lambda$), they watched how the "standing still" state behaved.

• The Analogy: Imagine a block of ice (the scar) sitting in a warm room (the chaotic system). Even if the room is only slightly warm, the ice will eventually melt.
• The Result: They found that the scar does melt. No matter how small the poke is, given enough time, the system forgets the scar and becomes a completely random, thermal mess.
• The Speed: The rate at which it melts depends on how hard you poke it. If you poke it gently, it melts slowly (like $O(\lambda^2)$). If you poke it hard, it melts fast.
• The Takeaway: To a normal observer looking at just one person in the crowd, the scar is thermodynamically irrelevant. It's gone. The system looks like a standard, chaotic mess.

Finding 2: The "Ghost in the Machine" (Entanglement)

Here is where it gets surprising. While the "normal" observers (looking at local particles) saw the scar disappear, the scientists looked at something deeper: Entanglement.

• The Analogy: Imagine the party guests are holding invisible strings connecting them to each other.
  • In a normal chaotic party, these strings get tangled in a very specific, predictable way.
  • In a scarred party, the strings behave differently.
• The Result: Even though the "ice" melted and the local particles looked random, the entanglement (the invisible strings) remembered the scar!
• The Transition: The authors discovered a sharp "switch."
  • If the poke is very small, the entanglement grows in one way (keeping a memory of the scar).
  • If the poke crosses a certain threshold, the entanglement suddenly switches to a different behavior (forgetting the scar).
• The Takeaway: This is a phase transition that local measurements cannot see. It's like a building that looks normal from the outside, but inside, the foundation has shifted. The scar leaves a "sharp fingerprint" in the quantum connections, even after it's gone from the local view.

Finding 3: The "Random Walker" Picture

To explain how the scar melts, the authors used a picture of fluctuating interfaces.

• The Analogy: Imagine the scar is a line of soldiers standing in a row. The chaotic system is a crowd of people pushing from the sides.
• The Mechanism: The boundary between the "soldiers" (the scar) and the "crowd" (chaos) doesn't stay still. It acts like a drunk person walking on a tightrope (a random walker).
  • The crowd pushes the boundary forward, slowly eating away the soldiers.
  • Sometimes the boundary wiggles back, but on average, it moves forward, erasing the scar.
• The Math: They calculated exactly how fast this "drunk walker" moves and how much it wiggles. This simple picture perfectly predicted how the system thermalizes.

Why This Matters

1. Fragility Confirmed: It confirms that quantum scars are indeed fragile. Without special protection (like conservation laws), they will eventually succumb to chaos.
2. Hidden Complexity: It proves that "thermalization" isn't a single thing. A system can look thermal to a local observer but still hold onto quantum secrets (entanglement) that reveal a different story.
3. New Tools: The authors created a new mathematical model (a random circuit with a scar) that is easy to solve. This gives physicists a "test tube" to study how quantum information is lost and how scars behave without needing supercomputers.

Summary in One Sentence

Even though a quantum "scar" (a stubborn pattern) eventually gets wiped out by chaos and looks like a normal mess to local eyes, it leaves a hidden, sharp signature in the system's deep quantum connections, creating a sudden switch in behavior that only the most sensitive quantum cameras can detect.

Technical Summary

1. Problem Statement

Quantum many-body scars (QMBS) are rare eigenstates that violate the Eigenstate Thermalization Hypothesis (ETH), allowing systems to sustain non-thermal stationary states without the protection of local conservation laws (unlike integrable systems). While scars have been observed in specific models (e.g., Rydberg atom arrays) and are known to be fragile under perturbations, a rigorous theoretical framework describing their large-scale dynamical behavior and thermalization mechanisms has been lacking.

Key open questions addressed by this work include:

• How does a single scar thermalize when coupled to a thermal bath (or infinite-temperature state) in a generic setting?
• What is the nature of the transition between scar-dominated dynamics and thermalization under global perturbations?
• Do scars leave detectable signatures in entanglement dynamics that are invisible to local observables?

2. Methodology

The authors construct an analytically tractable random unitary circuit in one dimension (brick-wall geometry) that hosts a single quantum scar.

• Model Construction:

  • The system consists of $L$ sites with local Hilbert space dimension $q$.
  • The local two-site unitary gate $u$ is defined as:
    $$u = \begin{pmatrix} 1 & 0 \\ 0 & U_{\text{Haar}} \end{pmatrix}$$
    where $U_{\text{Haar}}$ is a Haar-random unitary acting on the orthogonal subspace (dimension $q^2-1$).
  • The Scar: The state $|00\dots0\rangle$ is a stationary state (the scar) because the gate acts as the identity on $|00\rangle$.
  • Thermalization: The Haar block ensures rapid scrambling and thermalization toward the infinite-temperature state in the orthogonal sector.
  • This construction combines the Shiraishi-Mori approach (embedding a scar via specific algebraic structure) with random circuit theory.

• Analytical Tools:

  • 1-Replica Model: Used to study the averaged evolution of local observables. The dynamics are mapped to a stochastic process of "interfaces" (domain walls) between the scar state ($\circ$) and the thermal state ($\bullet$).
  • 2-Replica Model: Used to study entanglement (specifically the second Rényi entropy) and higher-order correlators (OTOCs). This involves a 7-dimensional local generator acting on $(H \otimes H^*)^{\otimes 2}$.
  • Mapping to Random Walks: The interface dynamics are shown to behave as driven random walkers with specific hopping probabilities, belonging to the compact directed percolation universality class.

3. Key Contributions and Results

A. Thermalization Mechanism (Local Observables)

• Interface Dynamics: The authors derive that the boundary between the scar region and the thermal region behaves as a random walker with drift velocity $v = \frac{q-1}{q+1}$ and diffusion coefficient $D = \frac{4q}{(1+q)^2}$.
• Instability: For any non-zero perturbation, the infinite-temperature state progressively erases the scar. The system thermalizes to the infinite-temperature state in the thermodynamic limit.
• Relaxation Rate: For a globally perturbed initial state $|\psi\rangle = (\sqrt{1-\lambda^2}|0\rangle + \lambda|1\rangle)^{\otimes L}$, the local order parameter relaxes to the thermal value with a rate proportional to $\lambda^2$.
  $$\langle O(t) \rangle - \langle O \rangle_{\text{thermal}} \sim \exp(-2v\lambda^2 t)$$
  This $O(\lambda^2)$ scaling is reminiscent of Fermi's Golden Rule and confirms the fragility of the scar under global perturbations.

B. Entanglement Dynamics (Non-Local Observables)

• The Surprise: While local observables cannot distinguish the scar from the thermal state once perturbed ($\lambda \neq 0$), the entanglement dynamics exhibit a sharp phase transition.
• Entanglement Transition: The authors identify a critical perturbation strength $\lambda^* = \sqrt{1 - q^{-1/4}}$.
  • For $\lambda < \lambda^*$: The system retains memory of the scar structure in its entanglement. The saturation value of the half-chain Rényi entropy $S_2$ is determined by the overlap with the scar.
  • For $\lambda > \lambda^*$: The system behaves as a fully thermal random circuit.
• Analytical Prediction: The saturation value of the entropy density in the thermodynamic limit is given by:
  $$\frac{S_2}{L} = \min\left\{ -2\log(1-\lambda^2), \frac{1}{2}\log q \right\}$$
  This result implies a qualitative change in non-local correlations that is invisible to any local measurement.

C. Higher-Order Correlators (OTOC)

• The Out-of-Time-Order Correlator (OTOC) is computed to probe quantum ergodicity.
• The OTOC relaxes to a plateau value consistent with quantum ergodicity (free independence of algebras), confirming that the system becomes ergodic at late times, even if the entanglement structure retains scars for specific perturbation strengths.

4. Significance and Implications

1. Thermodynamic Irrelevance vs. Entanglement Sensitivity: The paper establishes a fundamental distinction: scars are thermodynamically irrelevant for local observables (they thermalize), but they leave a "sharp fingerprint" in entanglement dynamics. This challenges the notion that thermalization implies a complete loss of all quantum memory.
2. Universal Mechanism: The "fluctuating interface" picture provides a universal mechanism for scar-mediated large-scale phenomena, distinct from standard hydrodynamics. It suggests that scars can be described by stochastic processes (random walkers) even in random circuits.
3. Analytical Tractability: By using random unitary circuits, the authors provide the first rigorous, analytically solvable model for the thermalization of a single scar, bridging the gap between specific integrable-like models and generic chaotic systems.
4. New Phase Transition: The discovery of an entanglement transition driven by perturbation strength, which is undetectable by local order parameters, opens new avenues for characterizing quantum phases and the stability of non-thermal states.

Conclusion

Capizzi and Ferté demonstrate that while quantum many-body scars are unstable and eventually lead to thermalization of local observables, they induce a robust, non-trivial transition in entanglement dynamics. This work provides a rigorous "membrane picture" for scar thermalization and highlights that entanglement measures are essential probes for detecting the remnants of non-thermal physics in chaotic quantum systems.