Quantum scarring enhances non-Markovianity of subsystem dynamics

This paper provides numerical evidence that quantum many-body scars act as a microscopic mechanism to enable and enhance non-Markovian dynamics in subsystems, as demonstrated by the PXP model where deformations and initial states that strengthen or weaken scarring signatures correspondingly increase or decrease information backflows and memory retention in subsystem evolution.

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. Usually, if you start the party with a specific song or a specific group of people, the energy eventually spreads out, everyone mixes, and the party settles into a boring, uniform hum. This is called thermalization—the system "forgets" how it started and just becomes a hot, messy equilibrium.

However, sometimes, a few special guests (called Quantum Scars) show up. These guests are weirdly organized. Even though everyone else is dancing chaotically, these special guests keep doing the exact same dance move over and over again, refusing to mix in. Because of them, the party never fully settles down; it keeps "remembering" the start.

This paper asks a specific question: If we only watch a small corner of the party (a "subsystem"), does it also remember the start, or does it forget?

The authors found that when those special "Scar" guests are present, the small corner of the party doesn't just remember the start; it actively pulls information back from the rest of the room. In physics terms, the small part of the system becomes non-Markovian (it has a memory). When the special guests are gone, the small part forgets everything quickly and behaves like a normal, forgetful system.

---

Key Concepts Explained with Analogies

1. The "Scar" vs. The "Thermal" Crowd

• The Thermal Crowd: Imagine a crowd of people in a room. If you shout "Jump!" once, everyone jumps, then they all start chatting randomly. After a while, you can't tell who jumped when. The room has "thermalized."
• The Quantum Scar: Imagine that hidden in that crowd are 10 people wearing matching red hats. Every time the music hits a certain beat, only those 10 people jump in perfect unison, while the rest of the crowd keeps chatting. They are "scars" because they leave a permanent, repeating pattern in the chaos. They don't forget the beat.

2. Markovian vs. Non-Markovian (The "Amnesia" vs. The "Echo")

To understand the main discovery, we need to look at a small group of people (a subsystem) within the big room.

• Markovian (Amnesia): Imagine you are standing in a corner of the room. You shout a secret to the crowd. In a Markovian world, the crowd swallows your secret instantly and never gives it back. Your future state depends only on what you are doing right now. You have no memory of what you shouted five minutes ago. This is like a sponge soaking up water; once it's wet, it stays wet, and the water never flows back out.
• Non-Markovian (The Echo): Now, imagine you shout a secret, and the crowd swirls around, but then, a few seconds later, the crowd pushes the secret back to you. You suddenly remember what you shouted. Your future state depends on what happened in the past. This is an "information backflow." The system is "non-Markovian" because it retains a memory of its history.

3. The Experiment: The PXP Model

The authors used a specific mathematical model called the PXP model (think of it as a specific set of rules for how the "party" dances).

• The Setup: They started the party with a specific pattern (like a checkerboard of people standing up and sitting down).
• The Deformations: They tweaked the rules of the party in two ways:
  1. Enhancing the Scars (PXPZ): They changed the rules to make the "Red Hat" dancers (the scars) even more organized and persistent.
  2. Erasing the Scars (PXPXP): They changed the rules to break the Red Hats' pattern, forcing them to mix in with the chaotic crowd.

4. The Discovery

The authors watched a small group of people (a few spins) in the corner of the room and measured how much they "remembered" their past states.

• When Scars were Enhanced: The small group showed strong memory. They kept getting information pushed back to them from the rest of the room. The "echo" was loud and clear. The system was highly non-Markovian.
• When Scars were Erased: The small group quickly forgot everything. The information flowed out and never came back. The system became Markovian (forgetful).
• Different Starting Points: They also tried starting the party with different patterns. If they started with a pattern that matched the "Red Hats" (the scars), the memory was strong. If they started with a random pattern that didn't match the scars, the memory was weak.

The "Why" Behind It

Why does this happen?
When the "Scar" states exist, they act like a closed loop or a private club inside the big chaotic room. The small group of people (the subsystem) gets stuck in this loop with the rest of the system. Because the loop is so tight and organized, the information can't escape into the chaos and get lost. Instead, it bounces back and forth.

Think of it like a whispering gallery in a cathedral. If you whisper in a normal room, the sound dies out (Markovian). But in a whispering gallery (the Scar), the sound bounces off the walls and comes back to you clearly (Non-Markovian). The "Scar" states create a whispering gallery for quantum information.

Summary of Findings

1. Scars create memory: The presence of Quantum Many-Body Scars makes small parts of a system retain memory of their past.
2. Enhancing Scars = Stronger Memory: If you tweak the system to make the scars stronger, the "memory effect" (non-Markovianity) gets stronger.
3. Erasing Scars = No Memory: If you tweak the system to destroy the scars, the memory effect disappears, and the system behaves like a normal, forgetful thermal system.
4. It's a finer detail: This "memory" is a more subtle, detailed version of the big "revivals" (where the whole system remembers its start). It shows that even a tiny piece of the system is holding onto the past because of these special states.

In short: Quantum scars act like anchors that stop a system from forgetting. They force the system to keep replaying its history, creating a "memory" that can be measured even in the smallest parts of the system.

Technical Summary

1. Problem Statement

The paper addresses the intersection of two distinct phenomena in non-equilibrium quantum many-body systems:

• Quantum Many-Body Scars (QMBS): Special non-thermal eigenstates embedded in the spectrum of otherwise thermalizing systems. When a system is quenched from an initial state with significant overlap with these scars, it exhibits weak ergodicity breaking, leading to persistent oscillations and slow thermalization (e.g., in the PXP model).
• Subsystem Non-Markovianity: In a closed quantum system, any subsystem acts as an open quantum system interacting with the rest of the system (the environment). The dynamics of this subsystem can be Markovian (memory-less, monotonic information loss) or non-Markovian (memory-retaining, characterized by information backflow from the environment).

The Core Question: Is there a microscopic link between the presence of quantum scars and the non-Markovian nature of the dynamics of small subsystems? Specifically, does the "scarring" mechanism, which prevents global thermalization, also enhance the retention of memory (non-Markovianity) in local subsystems?

2. Methodology

The author employs numerical simulations using the Time-Evolving Block Decimation (TEBD) algorithm with Matrix Product States (MPS) to study the dynamics of the PXP model and its deformations.

• Models Studied:
  • Base PXP Model: A kinetically constrained spin chain known to host weak scars.
  • PXPZ Model: A deformation of PXP with specific quasi-local terms ($r=2,3$) that enhance scarring, leading to nearly perfect fidelity revivals and strong $SU(2)$ algebraic structures.
  • PXPXP Model: A deformation that erases scars, restoring ergodicity and thermalization.
• Initial States:
  • Néel State ($\mathbb{Z}_2$): High overlap with scarred states (non-thermalizing dynamics).
  • 3-CDW State ($\mathbb{Z}_3$): Moderate overlap (weakened scarring).
  • Ferromagnetic State: Low overlap (thermalizing dynamics).
• Metric for Non-Markovianity:
  • The study focuses on small subsystems (1 to 4 spins) located in the bulk of the chain.
  • It utilizes the Trace Distance ($T_d$) between temporally separated reduced density matrices ($\rho(t)$ and $\rho(t+\delta)$).
  • Criterion: In a Markovian process, the trace distance must be non-increasing (contractive) under CPTP maps. Non-Markovianity is identified by the violation of this contractivity (i.e., an increase in trace distance), which signifies information backflow from the environment to the subsystem.
  • A quantitative degree of non-Markovianity ($\mathcal{D}$) is calculated by summing the positive slopes (revivals) of the trace distance over time.

3. Key Contributions

1. Establishing a Microscopic Link: The paper provides numerical evidence that quantum scarring is a direct microscopic ingredient that enables and enhances the non-Markovianity of subsystem dynamics.
2. Refined Memory Characterization: It distinguishes between global memory (full system fidelity revivals) and local memory. The retention of memory in transient subsystem states is shown to be a "finer" form of memory effect than global fidelity revivals.
3. Systematic Deformation Analysis: By tuning deformations that either enhance or destroy scars, the author demonstrates a direct correlation:
   • Scarring Enhancement $\rightarrow$ Enhanced Non-Markovianity.
   • Scarring Erasure (Thermalization) $\rightarrow$ Diminished Non-Markovianity.
4. Initial State Dependence: The study confirms that the degree of subsystem non-Markovianity correlates with the initial state's overlap with scarred states, regardless of the specific model deformation.

4. Key Results

A. Effect of Model Deformations (PXP vs. PXPZ vs. PXPXP)

• PXPZ (Enhanced Scars): Exhibits strong, persistent oscillations in the trace distance of subsystems. The degree of non-Markovianity ($\mathcal{D}$) is significantly higher compared to the base PXP model. The information backflow is robust, indicating the subsystem retains memory of its past states for long durations.
• PXPXP (Erased Scars): As the deformation strength ($g$) increases, the system approaches ergodicity. Consequently, the non-monotonicity of the trace distance diminishes, and the dynamics become effectively Markovian (information backflow vanishes).
• Periodicity: The time periods of the trace distance revivals in subsystems match the periods of global fidelity revivals (approx. 4.76 for PXP, 4.52 for PXPZ), linking the energy gaps of the scarred subspace to the timescales of local memory retention.

B. Effect of Initial States

• Néel State ($\mathbb{Z}_2$): Produces the strongest non-Markovianity due to high overlap with scars.
• 3-CDW State ($\mathbb{Z}_3$): Shows weaker non-Markovianity than the Néel state but stronger than thermal states. Interestingly, for this state, smaller subsystems (1-2 spins) exhibit more non-Markovianity than larger ones, a reversal of the trend seen in the Néel case.
• Ferromagnetic State: Exhibits very low (near-zero) non-Markovianity, consistent with rapid thermalization and memory loss.

C. Subsystem Configuration

• The study found that non-contiguous subsystems (spins separated by odd numbers of sites) exhibit stronger non-Markovianity than contiguous or even-separated spins. This suggests that breaking the unit-cell structure of the initial state enhances the interface between the subsystem and its environment, facilitating information backflow.

D. Classical vs. Quantum Memory

• The paper investigates a "classical" counterpart using the Total Variation Distance (TVD) of eigenvalues of the reduced density matrices. It finds systematic non-monotonicities in the TVD as well, suggesting that the memory effects have a classical component, though the full quantum nature requires further investigation.

5. Significance and Implications

• New Perspective on Scars: The work reframes quantum scarring not just as a global phenomenon (fidelity revivals) but as a local phenomenon that fundamentally alters the open-system dynamics of subsystems.
• Thermalization Barrier: It reinforces the idea that non-Markovianity (information backflow) is a mechanism that impedes thermalization. The "scars" effectively trap information within the subsystem-environment loop, preventing the monotonic flow of information required for equilibration.
• Experimental Relevance: Since current ultracold atom experiments can resolve individual spins, the findings regarding small subsystems are directly testable. The paper suggests that measuring local information backflow could be a robust signature of quantum scarring.
• Theoretical Bridge: It bridges the gap between the theory of closed quantum systems (scars, ETH) and open quantum systems (non-Markovianity, information backflow), offering a unified framework to understand non-thermalizing dynamics.

In conclusion, the paper demonstrates that quantum scarring acts as a mechanism for memory retention in local subsystems. By restricting dynamics to a scarred subspace, the system prevents the subsystem from fully entangling with the rest of the Hilbert space, thereby maintaining non-Markovian dynamics and delaying thermalization.