Scar Full Eigenstate Thermalization Hypothesis

This paper proposes a "scar full ETH" framework that extends the standard Eigenstate Thermalization Hypothesis to capture correlations involving non-thermal scar states, establishing their scaling properties and demonstrating the theory's validity through numerical simulations of the PXP model.

The Gist

The Big Picture: A Party That Never Ends

Imagine a chaotic quantum system (like a gas of atoms) as a massive, wild party. Usually, when you leave a room and come back later, the party has settled down. Everyone is mixing randomly, and the room looks "thermalized"—it's just a blur of activity. In physics, this is called thermalization.

For decades, physicists have used a rule called the Eigenstate Thermalization Hypothesis (ETH) to explain this. It says that if you look at any single moment in the party's history, the energy levels look random and mixed up, just like a shuffled deck of cards. This explains why isolated quantum systems eventually act like normal, hot gases.

But there's a glitch.

In some special systems (like the "PXP model" mentioned in the paper), the party doesn't settle down. Instead, a few specific guests (called Quantum Many-Body Scars) keep dancing in a perfect, repeating loop. They refuse to mix with the crowd. They remember their original moves and keep oscillating forever.

The old rules (ETH) fail here because they assume everyone mixes. The authors of this paper realized we need a new rulebook to explain how these "scar" guests interact with the "thermal" crowd. They call this new rulebook the Scar Full ETH (SFETH).

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The Three Types of Interactions

To understand the new rulebook, imagine the party has three types of interactions between guests:

1. Thermal vs. Thermal (The Crowd): Two random guests from the main crowd talking.
   • Old Rule: We already know how this works. They mix randomly.
2. Scar vs. Scar (The VIPs): Two of the special, looping guests talking to each other.
   • New Rule: This is unique to them. It depends entirely on their specific "scar" nature.
3. Scar vs. Thermal (The VIPs talking to the Crowd): This is the tricky part. How does a looping guest interact with a random guest?
   • The Paper's Discovery: The authors found a specific mathematical pattern for this. Even though the VIPs are special, when they talk to the crowd, the conversation follows a predictable structure that combines both the "randomness" of the crowd and the "rhythm" of the VIPs.

The New Rulebook: "Free Cumulants"

The paper introduces a fancy mathematical tool called Free Cumulants. Think of these as "building blocks" for conversations.

• In a normal party (Thermal): You can break down any complex conversation into simple, independent blocks. If you know the blocks, you know the whole conversation.
• In a Scarred party: You need two types of blocks:
  1. Thermal Blocks: For the random crowd parts.
  2. Scar Blocks: For the special looping parts.

The authors proved that any complex interaction involving these special "scar" guests can be built by snapping together these two types of blocks. They showed that you don't need to track every single detail; you just need to know how these blocks fit together.

The "Crossing" Problem (Why Some Things Don't Matter)

In their math, the authors had to deal with "crossing diagrams." Imagine drawing lines connecting guests. Sometimes, lines cross over each other.

• The Analogy: Imagine trying to connect two VIPs to two random guests with strings. If the strings cross, it creates a weird, tangled mess.
• The Finding: The authors proved that in a large system (a huge party), these "crossed" connections are so incredibly weak that they effectively vanish. They are like a whisper in a hurricane. You can ignore them. This simplifies the math immensely, allowing them to focus only on the "non-crossing" (clean) connections.

How They Proved It

The authors didn't just write equations; they ran a computer simulation of the PXP model (a specific type of quantum chain of atoms, often realized in labs with Rydberg atoms).

1. They created a digital version of the party with 22 atoms.
2. They identified the "scar" guests (the ones that don't thermalize).
3. They measured how these guests interacted with each other and the crowd over time.
4. The Result: The messy, real-world data matched their new "block-building" theory perfectly. The complex, oscillating behavior of the scars was exactly what their new formula predicted.

Summary

• The Problem: Old physics rules say everything in a quantum system eventually mixes and forgets its past. But some systems have "scars" that remember and keep oscillating.
• The Solution: The authors created a new framework (SFETH) that treats these scars as special guests who follow their own rules, but still interact with the crowd in a predictable way.
• The Method: They used a mathematical "Lego" approach (free cumulants) to show how to build complex interactions from simple thermal and scar pieces.
• The Proof: They tested this on a computer model of a Rydberg atom chain, and the theory matched the simulation perfectly.

In short, this paper gives us the instruction manual for understanding how "stubborn" quantum particles (scars) behave in a chaotic world, explaining why they don't just blend in with the rest of the crowd.

Technical Summary

Technical Summary: Scar Full Eigenstate Thermalization Hypothesis

Problem Statement
The Eigenstate Thermalization Hypothesis (ETH) provides the standard mechanism for emergent statistical mechanics in isolated chaotic quantum systems, asserting that individual energy eigenstates behave as pseudorandom vectors. Recent developments have refined this into the "full ETH," which characterizes nontrivial correlations among matrix elements in the energy eigenbasis using free cumulants. However, this framework breaks down in systems exhibiting quantum many-body scars. In such systems, while the majority of eigenstates are thermal, a sparse set of non-thermal "scar" states with low entanglement exists within the many-body spectrum. These states violate the conventional ETH description and give rise to persistent temporal oscillations in dynamics. A naive application of the full ETH fails to capture the intrinsic dynamical effects of these scar states, necessitating a generalized framework to characterize the correlations involving them.

Methodology
The authors formulate the "Scar Full Eigenstate Thermalization Hypothesis" (SFETH) to address the correlations among matrix elements involving scar states. The methodology proceeds through the following steps:

1. Classification of Matrix Elements: The matrix elements of a local observable $\hat{O}$ in the energy eigenbasis are classified into three sectors:

   • Thermal sector ($O_{ij}$): Involving only thermal states.
   • Pure scar sector ($O_{ab}$): Involving only scar states.
   • Mixed scar-thermal sector ($O_{ai}, O_{ia}$): Involving one scar and one thermal state.

2. Formulation of SFETH Ansätze:

   • Scaling Ansatz: The authors propose that the average of products of matrix elements involving scar states and distinct thermal indices scales as $e^{-qS(E_{th})}P^{(q+1)}_{ab}(\vec{\omega})$, where $S$ is the thermal entropy and $P$ is a smooth function of energy density and frequency differences.
   • Factorization Ansatz: For products with repeated thermal indices, the average factorizes in the thermodynamic limit. This is justified using typicality arguments, assuming thermal eigenstates behave as orthogonal Haar-random states. The authors utilize Weingarten calculus to explicitly evaluate Haar-random averages, demonstrating that corrections to factorization are exponentially suppressed by the entropy $S$.

3. Decomposition into Cumulants: Applying these ansätze to multipoint correlation functions on scar states, the authors reorganize the correlations into fundamental building blocks. Unlike thermal systems which rely solely on thermal free cumulants, scar correlation functions are decomposed into a hybrid structure of thermal free cumulants and a new class of scar free cumulants.

4. Diagrammatic Analysis: The authors employ a diagrammatic representation to classify contributions to correlation functions. They demonstrate that "crossing diagrams" (where thermal indices cross between scar segments) are parametrically suppressed by the inverse density of states ($e^{-S}$) and vanish in the thermodynamic limit. Only "arc diagrams" and "cactus diagrams" contribute, leading to a decomposition over non-crossing partitions.

5. Numerical Verification: The theory is tested using the paradigmatic PXP model (a spin-1/2 model with Rydberg blockade constraints), which is known to host quantum many-body scars. Using exact diagonalization for system sizes up to $N=22$, the authors verify:

   • The factorization property of three-point correlation functions involving scar states.
   • The negligible contribution of crossing diagrams.
   • The accuracy of the scar free cumulant decomposition in reproducing exact multi-time correlation functions.

Key Contributions

• Theoretical Framework: The paper establishes the SFETH, a generalized ansatz that captures the scaling and factorization properties of matrix elements involving scar states.
• Scar Free Cumulants: The introduction of "scar free cumulants" as fundamental building blocks for multipoint correlation functions on scar states. These cumulants encode the hybrid structure of thermal-scar contributions.
• Reorganization of Correlations: The demonstration that higher-order correlations in scar systems are organized by both thermal and scar free cumulants, providing a systematic reorganization of the correlation functions beyond the traditional ETH.
• Numerical Validation: Rigorous numerical confirmation of the SFETH predictions in the PXP model, specifically validating the factorization relations and the decomposition of multi-time correlation functions.

Results

• The SFETH ansätze successfully predict the scaling behavior of high-order matrix element products involving scar states, showing exponential suppression governed by the thermal entropy.
• Numerical results in the PXP model show excellent agreement between exact diagonalization data and the factorized SFETH predictions. Specifically, the three-point correlation function $F^{(3)}_{aa}(0, t, 0)$ is accurately reconstructed by summing contributions from thermal and scar free cumulants.
• The contribution of crossing diagrams is confirmed to decay rapidly with system size (scaling as $D^{-1}$, where $D$ is the Hilbert space dimension), validating their neglect in the thermodynamic limit.
• The function $P^{(q)}_{ab}(\vec{\omega})$, introduced in the ansatz, is shown to be directly related to the Fourier transform of the scar free cumulants.

Significance
The authors claim that this work provides a unified framework for understanding higher-order correlations and information dynamics in systems with quantum many-body scars. By extending the full ETH to weakly ergodicity-breaking systems, the SFETH allows for a systematic characterization of the "intriguing correlations" that arise from the coexistence of thermal and scar states. The paper posits that this framework paves the way for understanding the persistent temporal oscillations observed in such systems. The authors conclude by suggesting that this framework could potentially be connected to effective descriptions of scar dynamics (e.g., quasiparticle pictures) and generalized to other systems with weak ergodicity breaking, such as those exhibiting Hilbert-space fragmentation, though these remain topics for future work.