Quantum Chaos in Many-Body Systems Without a Classical Analogue

This dissertation investigates the weak breaking of ergodicity in the constrained PXP spin chain model, demonstrating that while it exhibits signatures of quantum chaos like Wigner-Dyson statistics and ballistic energy spreading, it uniquely hosts a small number of eigenstates that violate the Eigenstate Thermalization Hypothesis and display non-Gaussian eigenvector statistics.

The Gist

The Big Picture: The "Chaotic Dance Floor"

Imagine a massive dance floor filled with people (these are quantum particles). In the world of physics, we want to know: How do these people move and interact?

• The "Chaos" (Ergodicity): In a normal, chaotic party, everyone eventually mixes. If you drop a drop of red dye in one corner, it spreads until the whole room is pink. Everyone visits every part of the room. In physics, this is called thermalization. The system forgets where it started and settles into a comfortable, average state.
• The "Order" (Integrability): In a highly organized military drill, everyone stays in their lane. The red dye stays in a straight line. The system remembers exactly where it started forever.
• The "Weird Middle Ground" (This Paper): This paper studies a specific type of party where the rules are weird. It's not a total free-for-all, and it's not a strict military drill. It's a party where some people get stuck in a loop, while others mix normally. This is called "Weak Ergodicity Breaking."

The specific "party" being studied is called the PXP Model. It's a chain of atoms (like a row of dancers) with a very strange rule: You can only dance (flip your spin) if your immediate neighbors are standing still. If two neighbors are dancing, you are frozen.

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The Three Main Discoveries

The author, Fotis, investigated this "frozen dance floor" and found three surprising things:

1. The "VIPs" Who Don't Mix (Quantum Scars)

The Analogy: Imagine a huge party where 99% of the guests eventually get drunk, mingle, and forget their names (thermalization). But, there is a tiny group of "VIPs" (Special Eigenstates) who stay sober, stand in a perfect circle, and keep dancing the exact same routine forever. They never mix with the crowd.

The Science:

• Most theories say that in a chaotic system, every state should eventually mix and forget its past.
• Fotis found that in the PXP model, there is a small number of special states (called Quantum Many-Body Scars) that refuse to mix.
• If you start the party with the dancers in a specific pattern (like a checkerboard), these VIPs keep the pattern alive for a long time, causing the system to "revive" or oscillate instead of settling down.

2. The "Rhythm" of the Music (Level Statistics)

The Analogy: Imagine the energy levels of the atoms are like musical notes.

• Integrable (Ordered): The notes are like a scale: 1, 2, 3, 4. They are evenly spaced and predictable.
• Chaotic (Random): The notes are like a jazz improvisation. They repel each other; you never hear two notes too close together. This is called Wigner-Dyson statistics.
• The PXP Model: Fotis found the music is in a weird middle zone. It's not a perfect scale, but it's not full jazz either. It sounds like "Semi-Poisson"—a rhythm that has some order but also some chaos. As the party gets bigger (more atoms), the rhythm starts to sound more like the chaotic jazz, but it never quite gets there.

3. The "Wave" of Energy (Ballistic Fronts)

The Analogy: Imagine you shout "Fire!" at one end of the dance floor.

• Diffusive (Normal Chaos): The news spreads slowly, like a rumor passing through a crowd. It takes a long time to reach the other side, and the message gets fuzzy.
• Ballistic (Ordered): The news travels like a bullet in a straight line, hitting the other side instantly.
• The Surprise: Fotis shouted "Fire!" (a "quench") in the middle of the PXP dance floor. He expected the news to spread slowly (diffusively) because the system looked chaotic. Instead, the energy spread in a perfect, straight line at constant speed. It was like a laser beam cutting through the crowd. This was completely unexpected for a system that wasn't perfectly ordered.

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Why Does This Matter?

You might ask, "Why do we care about a weird dance floor of atoms?"

1. It Breaks the Rules: Physics has a "rulebook" (the Eigenstate Thermalization Hypothesis) that says chaotic systems must mix and forget their past. This model proves that rulebook has loopholes.
2. Quantum Memory: Because these "VIP" states (Scars) don't mix, they could be used to store information in a quantum computer. If you can keep a quantum state from "forgetting" itself, you can build better computers.
3. New Physics: It shows that nature has a "middle ground" between total order and total chaos that we didn't fully understand before.

Summary in One Sentence

This paper studies a weird quantum system where atoms are forced to dance under strict rules, discovering that while most of the system behaves chaotically, a few special "ghosts" refuse to mix, and energy travels through the system in perfect, straight lines rather than spreading out like a gas.

Technical Summary

1. Problem Statement

The dissertation investigates the phenomenon of weak ergodicity breaking in quantum many-body systems that lack a classical analogue. While classical chaos is defined by trajectory divergence, quantum chaos is typically diagnosed via spectral statistics (transitioning from Poisson to Wigner-Dyson) and the Eigenstate Thermalization Hypothesis (ETH).

• Standard Behavior: Ergodic systems thermalize (obey ETH), while integrable or Many-Body Localized (MBL) systems do not.
• The Anomaly: Recent experiments with Rydberg atom chains revealed a system that is neither fully ergodic nor integrable. It exhibits "quantum many-body scars" (QMBS)—a small set of non-thermal eigenstates embedded in a thermal spectrum—leading to persistent oscillations (revivals) from specific initial states.
• Objective: The author aims to numerically characterize the PXP model (a kinetically constrained spin chain) to understand the nature of this intermediate behavior, specifically focusing on ETH violations, spectral statistics, eigenvector statistics, and dynamical transport properties.

2. Methodology

The research relies heavily on Exact Diagonalization (ED) of the PXP Hamiltonian for finite system sizes ($L$ up to 32).

• Model: The PXP Hamiltonian is defined as $H_{PXP} = \sum_{i=1}^L P_{i-1} X_i P_{i+1}$, where $X_i$ flips a spin and $P_i$ projects out configurations with adjacent excited spins (Rydberg blockade).
• Symmetry Reduction: To handle the exponential growth of the Hilbert space ($D \sim 2^L$), the author utilizes the model's symmetries:
  • No adjacent excitations constraint (reduces dimension to Fibonacci numbers).
  • Translation symmetry (Zero-momentum sector).
  • Inversion symmetry (Parity sectors).
  • This reduces the effective Hilbert space dimension significantly (e.g., for $L=30$, $D \approx 1.86 \times 10^6$).
• Tools: The simulations were implemented in Julia, utilizing sparse matrix techniques and Krylov subspace methods (via the ExpmV package) for time evolution.
• Diagnostics:
  • ETH Test: Comparison of Eigenstate Expectation Values (EEVs) against the canonical ensemble.
  • Spectral Statistics: Analysis of level-spacing ratios ($r_n$) to distinguish between Poisson (integrable), GOE (chaotic), and intermediate distributions.
  • Eigenvector Statistics: Testing Berry's conjecture regarding the Gaussian distribution of eigenvector components.
  • Quantum Quench: Simulating the time evolution of energy density fronts after a sudden change in initial conditions (splitting the chain into regions of different temperatures/energies).

3. Key Contributions and Results

A. Breakdown of ETH and Quantum Many-Body Scars

• ETH Violation: The study confirms that while the majority of eigenstates in the PXP spectrum obey ETH (thermalizing), a small, discrete set of states (forming "towers") strongly violates it.
• Scars: These special states correspond to the "quantum many-body scars." They exhibit anomalously high overlaps with the period-2 charge density wave state ($|Z_2\rangle = |\uparrow\downarrow\uparrow\downarrow\dots\rangle$).
• Scaling: The standard deviation of the EEVs for the local operator $O_Z$ scales as $\sigma \sim D^{-0.23}$. This is slower than the expected $D^{-0.5}$ for chaotic systems, indicating a failure of full thermalization.

B. Level-Spacing Statistics

• Intermediate Statistics: The level-spacing ratio $\langle r \rangle$ for the PXP model does not match the Poisson limit ($\approx 0.386$) nor the Gaussian Orthogonal Ensemble (GOE) limit ($\approx 0.536$).
• Semi-Poisson to GOE: For smaller systems ($L \le 28$), the statistics resemble the semi-Poisson distribution (characterized by level repulsion but an exponential tail). As system size increases, the statistics trend toward Wigner-Dyson (GOE), suggesting the system is not integrable but possesses unique finite-size features.

C. Eigenvector Component Statistics (Novel Finding)

• Non-Gaussianity: A major finding of this work is that the distribution of eigenvector components in the PXP model is non-Gaussian. This contradicts the standard expectation for chaotic systems (Berry's conjecture), where components should follow a Gaussian distribution.
• Zero-Energy Modes: The non-Gaussian distribution features a prominent spike at zero, caused by a macroscopic degeneracy of zero-energy states (zero modes). Even after excluding these zero modes, the distribution remains non-Gaussian, suggesting a fundamental deviation from random matrix theory expectations for the bulk of the spectrum.

D. Transport and Energy Fronts (Novel Finding)

• Ballistic Fronts: The author performed quantum quenches by preparing the system with a temperature gradient (or energy difference) across the chain.
• Unexpected Linearity: Contrary to the expectation that a generic thermalizing system should exhibit diffusive (non-linear, $\sqrt{t}$) spreading of energy, the PXP model exhibits ballistic (linear, $t$) energy fronts.
• Robustness: This ballistic behavior was observed even when the initial state was prepared in the middle of the spectrum (microcanonical states) and not just near the ground state, ruling out ground-state artifacts. The fronts remain ballistic for significant times before eventually diffusing after multiple reflections.

4. Significance

This dissertation provides a comprehensive numerical characterization of the PXP model, solidifying its status as a unique example of weak ergodicity breaking.

1. Validation of Scars: It numerically reproduces and extends the understanding of Quantum Many-Body Scars, confirming their role in preventing thermalization for specific initial states.
2. Challenge to Random Matrix Theory: The discovery of non-Gaussian eigenvector statistics in a system that is not integrable challenges the universality of Random Matrix Theory (RMT) as the sole descriptor of quantum chaos.
3. Dynamical Anomalies: The observation of ballistic energy fronts in a system that is otherwise diffusive suggests that the "scar" states or the specific kinetic constraints of the PXP model impose a hidden structure on the dynamics that facilitates coherent transport, distinct from standard integrable systems.
4. Methodological Framework: The work provides a robust Julia-based framework for analyzing constrained spin chains, which can be applied to other kinetically constrained models relevant to Rydberg atom experiments and quantum simulation.

In conclusion, the paper argues that the PXP model represents a distinct class of quantum matter that sits between integrability and chaos, characterized by a "weak" breaking of ergodicity that manifests in both static spectral properties and dynamic transport phenomena.