Probing the ergodicity breaking transition via violations of random matrix theoretic predictions for local observables

This paper demonstrates that violations of random matrix theory predictions for local observables, specifically regarding quantum Fisher information dynamics and fluctuation-dissipation relations, can serve as effective witnesses for detecting ergodicity-breaking transitions in quantum many-body systems across integrability, many-body localization, and quantum many-body scars.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: The Great Party Analogy

Imagine a massive, chaotic party (a Quantum Many-Body System) with thousands of guests.

• Ergodic (The Normal Party): In a normal party, if you start in the corner, you eventually wander through the whole room, talk to everyone, and mix with the crowd. You forget where you started. In physics, this is called thermalization. The system "forgets" its initial state and settles into a predictable, average behavior.
• Non-Ergodic (The Broken Party): Sometimes, the party gets stuck. Maybe everyone is glued to their chairs (disorder), or there are secret clubs that only certain people can join (symmetries). In these cases, the guests never mix properly. They stay trapped in small groups, remembering exactly where they started. This is ergodicity breaking.

The Problem: Usually, to tell if a party is "broken," physicists have to look at the entire room at once (global quantities). This is like trying to count every single person in the room to see if they are mixing. It's hard to do in real experiments.

The Solution: This paper proposes a clever trick. Instead of watching the whole room, we just watch one specific guest (a "probe" or "local observable"). If the party is working normally, this one guest will behave in a very specific, predictable way. If the party is broken, this guest will act strangely.

The authors used two "rules of thumb" from a branch of math called Random Matrix Theory (RMT) to predict how this one guest should act. When the rules are broken, we know the system has stopped mixing.

---

The Two "Rules of Thumb" (The Probes)

The paper tests two specific behaviors of our "probe guest" to see if the party is chaotic or stuck.

1. The "Curiosity Meter" (Quantum Fisher Information - QFI)

Imagine the guest has a "curiosity meter" that measures how much they are learning about the room over time.

• The Rule: In a normal, chaotic party, the guest's curiosity should grow linearly (a straight line) for a while before slowing down. It's like someone walking at a steady pace through the crowd.
• The Break: If the party is broken (ergodicity broken), the guest gets stuck. Their curiosity stops growing in a straight line and just grows in a slow, curved way (quadratically).
• The Metaphor: Imagine a runner. In a chaotic race, they run at a steady speed. In a broken race, they are stuck in mud, and their progress slows down drastically. If the runner stops running in a straight line, we know the track is broken.

2. The "Wobble vs. Decay" Test (Fluctuation-Dissipation Relation)

Imagine the guest is trying to stay calm while the party gets wild.

• The Rule: In a normal party, if you push the guest (a disturbance), they will wobble a bit and then settle down. The amount they wobble is mathematically linked to how fast they calm down. It's a perfect balance.
• The Break: If the party is broken, this balance is lost. The guest might wobble wildly and never settle, or they might not wobble at all. The math that usually connects their "wobble" to their "calming speed" stops working.
• The Metaphor: Think of a pendulum in a fluid. If the fluid is normal, the pendulum slows down at a predictable rate based on how much it swings. If the fluid turns into solid jelly (broken ergodicity), the pendulum gets stuck, and the rules of how it slows down no longer apply.

---

The Three "Broken Party" Scenarios Tested

The authors tested these two rules in three different ways the party can get stuck:

1. The "Too Quiet" Party (Integrable Systems)

• What happened: They took a system that was supposed to be chaotic and made the interactions between guests very weak.
• The Result: The guests barely talked to each other. The "Curiosity Meter" stopped growing linearly, and the "Wobble" rules broke.
• Takeaway: If the guests don't interact enough, the system stays stuck in its initial state.

2. The "Glued Chairs" Party (Many-Body Localization - MBL)

• What happened: They added "disorder" (random noise) to the system, like putting random obstacles or glue on the floor.
• The Result: At low disorder, the guests could still move around (linear growth). But once the disorder got strong enough (a critical point), the guests got stuck. The "Curiosity Meter" stopped its linear growth, and the "Wobble" rules failed.
• Takeaway: Too much chaos (disorder) actually freezes the system, preventing it from mixing.

3. The "Secret Club" Party (Quantum Many-Body Scars - QMBS)

• What happened: This is the weirdest one. The system looked chaotic, but it had a hidden structure (like a secret club or a specific dance routine).
• The Result: If the guest started in a "normal" spot, they mixed fine. But if they started in a "special" spot (a scarred state), they kept doing the same dance over and over, never mixing with the crowd. The "Curiosity Meter" didn't grow linearly; it just oscillated.
• Takeaway: Even in a chaotic system, special starting positions can trap the system in a loop, breaking the rules.

---

Why This Matters

Before this paper, scientists had to look at the entire quantum system to see if it was behaving normally. That's like trying to count every grain of sand on a beach to see if the tide is coming in.

This paper shows that you can just watch one tiny part of the system (a single spin or atom). By checking if that one part follows the "Curiosity" and "Wobble" rules, you can instantly tell if the whole system is mixing or stuck.

In short: They found a simple, local "smoke detector" for quantum chaos. If the smoke detector (the local observable) stops working the way it should, you know the fire (ergodicity) has gone out, and the system is broken. This makes it much easier for experimentalists to study these complex quantum phenomena in the real world.

Technical Summary

Here is a detailed technical summary of the paper "Probing the ergodicity breaking transition via violations of random matrix theoretic predictions for local observables" by Pavlov et al.

1. Problem Statement

Quantum many-body systems typically exhibit ergodic behavior, where they thermalize and lose memory of their initial states, adhering to the Eigenstate Thermalization Hypothesis (ETH). However, certain mechanisms can break ergodicity, leading to non-ergodic phases such as Many-Body Localization (MBL), integrable systems, or systems hosting Quantum Many-Body Scars (QMBS).

Current methods to detect these transitions often rely on global quantities that are difficult to measure experimentally, such as:

• Level-spacing statistics (Wigner-Dyson vs. Poisson distributions).
• Von-Neumann entanglement entropy (Volume-law vs. Area-law scaling).
• Out-of-Time-Order Correlators (OTOCs).

The central problem addressed by this work is whether local observables alone can serve as effective witnesses for ergodicity breaking. The authors aim to identify specific, experimentally accessible signatures derived from Random Matrix Theory (RMT) that hold in the ergodic regime but are violated when ergodicity is broken.

2. Methodology

The authors propose a "probe" methodology where a small subsystem (a single spin) is coupled to a larger many-body "bath" (the bulk system). They analyze the dynamics of local observables on this probe spin to infer the global ergodic properties of the total system.

They focus on two specific RMT-based predictions for the ergodic regime:

1. Quantum Fisher Information (QFI) Dynamics:

   • The QFI, $F_Q(\lambda, t)$, measures the distinguishability of quantum states under a parameter perturbation $\lambda$.
   • RMT Prediction: In ergodic systems, the QFI exhibits a distinct intermediate linear growth regime ($F_Q \sim t$) before crossing over to a long-time quadratic growth ($F_Q \sim t^2$). This linear regime is linked to the density of states and the Heisenberg time.
   • Hypothesis: Breaking ergodicity reduces the effective density of states explored, causing the linear regime to vanish, leaving only quadratic growth.

2. Fluctuation-Dissipation Relation (FDR):

   • This relates the long-time fluctuations of an observable, $\delta^2_{\hat{O}}(\infty)$, to the decay rate to equilibrium, $\Gamma$, and the density of states, $D(E)$.
   • RMT Prediction: $\delta^2_{\hat{O}}(\infty) \propto \frac{1}{D(E)\Gamma}$.
   • Hypothesis: In non-ergodic phases (like MBL), the effective Hilbert space dimension is restricted, causing deviations from this scaling relation.

The authors numerically simulate three distinct models representing different ergodicity-breaking mechanisms:

• Model A: A transition from a non-integrable (chaotic) to an integrable spin chain by varying coupling strength.
• Model B: A disordered spin chain exhibiting the MBL transition.
• Model C: The PXP model (Rydberg atom simulator) exhibiting Quantum Many-Body Scars.

3. Key Contributions

• Local Observable Probes: Demonstrated that RMT predictions for local observables (specifically QFI and long-time fluctuations) are robust indicators of global ergodicity, offering a more experimentally feasible alternative to global entanglement or spectral statistics.
• QFI as a Witness: Identified the vanishing of the linear time-growth regime in the QFI as a specific signature of ergodicity breaking.
• FDR Violation: Showed that the scaling of long-time fluctuations deviates from RMT predictions in non-ergodic phases, even when the prefactor depends on the specific shape of eigenstates.
• Universality: Validated these findings across three fundamentally different mechanisms of ergodicity breaking (integrability, disorder-induced localization, and scarred eigenstates).

4. Results

A. Non-Integrable-to-Integrable Transition

• Setup: A probe spin coupled to an integrable spin chain. Weak coupling preserves integrability; strong coupling induces chaos.
• Findings:
  • QFI: For strong coupling (ergodic), the QFI shows the predicted linear-to-quadratic crossover. As coupling weakens (approaching integrability), the linear regime disappears, and the QFI becomes purely quadratic.
  • Fluctuations: The long-time fluctuations of the probe spin follow the RMT fluctuation-dissipation relation in the chaotic regime but deviate significantly in the integrable limit.
  • Transition Point: Both measures detect the transition at the same coupling strength ($J_x^{(SB)} \approx 0.2$) as traditional level-spacing statistics.

B. Many-Body Localization (MBL)

• Setup: A disordered spin chain where disorder strength $W$ drives the system from a thermal phase to an MBL phase.
• Findings:
  • QFI: In the thermal phase, the QFI exhibits the linear regime. As $W$ increases past the critical point ($W_c \approx 2$), the linear regime vanishes completely. The authors used the adjusted coefficient of determination ($R^2$) to statistically confirm that a quadratic fit becomes sufficient in the MBL phase, whereas a linear term is required for the thermal phase.
  • Fluctuations: In the thermal phase, fluctuations decrease as system size increases (scaling with $1/D(E)$). In the MBL phase, fluctuations become independent of system size (saturating) because the effective Hilbert space is no longer extensive.
  • Entropy: While von-Neumann entropy shows logarithmic growth in MBL, the QFI and fluctuation measures provided a sharper, more distinct signature of the phase transition.

C. Quantum Many-Body Scars (QMBS)

• Setup: The PXP model, which hosts a sparse set of non-thermalizing "scarred" eigenstates embedded in a thermal spectrum.
• Findings:
  • Initial State Dependence:
    • Thermal Initial States: For random initial states with low overlap with scars, the system behaves ergodically, and QFI/Fluctuations follow RMT predictions.
    • Scarred Initial States: For initial states like $|Z_2\rangle$ (high overlap with scars), the system fails to thermalize. The QFI shows purely quadratic growth (no linear regime), and the fluctuation-dissipation relation is violated.
  • Prefactor: The authors noted that for the PXP model, the eigenstates are not Lorentzian, leading to a different numerical prefactor ($\chi \approx 5.5$) in the fluctuation relation, yet the scaling behavior remains a valid diagnostic.

5. Significance

This work provides a critical bridge between theoretical Random Matrix Theory and experimental quantum simulation.

• Experimental Feasibility: Measuring global entanglement or full spectral statistics is often impossible in current quantum simulators (e.g., cold atoms, superconducting qubits). However, measuring local observables (like spin magnetization) and their time evolution is standard.
• Diagnostic Tool: The paper establishes that the absence of linear QFI growth and the violation of local fluctuation scaling are reliable, model-agnostic signatures of ergodicity breaking.
• Broad Applicability: By successfully applying these metrics to integrability, MBL, and QMBS, the authors demonstrate that these RMT-based local probes are universal tools for characterizing quantum thermalization and its breakdown.

In conclusion, the paper argues that local observables are sufficient to witness ergodicity breaking, offering a practical pathway for experimentalists to detect phase transitions in complex quantum many-body systems without requiring access to global system properties.