Absence of thermalization after a local quench and strong violation of the eigenstate thermalization hypothesis

This paper demonstrates analytically and numerically that a local quantum quench in XX-spin chains with open boundary conditions can prevent thermalization and induce a strong violation of the eigenstate thermalization hypothesis, even when weaker versions of the hypothesis typically hold in integrable systems.

The Gist

Imagine you have a long line of people holding hands, passing a secret message down the chain. This is like a quantum spin chain, a model physicists use to understand how tiny particles interact.

Usually, if you shake one end of this line (a "quench"), the message spreads out, gets jumbled, and eventually, the whole line settles into a calm, predictable state. This is called thermalization. It's like stirring sugar into coffee; eventually, the sugar dissolves evenly, and you can't tell where it started.

However, this paper discovers a surprising exception to this rule. The authors show that under very specific conditions, the system never settles down. The "sugar" stays clumped in one spot forever.

Here is the story of their discovery, broken down with simple analogies:

1. The Setup: The "Impurity"

Imagine the line of people is perfectly uniform. Everyone is identical.

• The Experiment: The researchers suddenly introduce a "weirdo" into the line. Maybe they change the height of the person at the very end of the line, or they change the person right in the middle. This is the impurity.
• The Question: If we shake the system, will the weirdo's influence spread out and disappear (thermalize), or will it stay stuck?

2. The Big Surprise: It Depends on Where You Put the Weirdo

The paper finds that the answer depends entirely on the location of the change:

• Scenario A: The End of the Line (The "Edge" Effect)
  If you change the person at the very end of the line, and you make the change strong enough, the system refuses to thermalize.

  • The Analogy: Imagine the line of people is a long hallway. If you put a giant, heavy boulder at the very end of the hallway, and you try to roll a ball down it, the ball hits the boulder and bounces back. The energy gets trapped near the end. The system stays "excited" and chaotic right next to the boulder forever, while the rest of the hallway might calm down.
  • The Result: The system violates a fundamental rule of physics called the Eigenstate Thermalization Hypothesis (ETH). Usually, this rule says that every state of a system looks like a hot, messy equilibrium. Here, the system breaks that rule so badly that even a "weak" version of the rule doesn't work. It's a "strong violation."

• Scenario B: The Middle of the Line (The "Center" Effect)
  If you change the person right in the middle of the line, the system does thermalize (it calms down), but only if the interactions between people are very simple.

  • The Twist: If the people in the line have slightly more complex interactions (like in the "XXZ model" mentioned in the paper), putting a weirdo in the middle actually breaks the rules of the game entirely. It turns the orderly line into a chaotic mess that does thermalize, but in a way that destroys the system's original "integrable" nature (its ability to be solved mathematically).

3. The "Trapped" Energy (Localized Modes)

Why does the energy stay stuck at the end?

• The Analogy: Think of the quantum particles as waves in a pond. Usually, waves spread out in all directions. But when the "weirdo" (impurity) is strong enough at the edge, it acts like a trap. It creates a "localized mode."
• Imagine a drum. If you hit it in the middle, the sound spreads. But if you clamp the edge of the drum tightly, a vibration can get stuck right at that clamp, vibrating furiously while the rest of the drum stays still. The energy is localized. It cannot escape to the rest of the system to share the heat.

4. Why This Matters

For a long time, physicists thought that if you only messed with a tiny part of a huge system (a "local quench"), the rest of the system would eventually absorb the disturbance and return to normal.

• The Paper's Lesson: This is not always true. If you have a long, one-dimensional chain and you mess with the edge strongly enough, the system can get "stuck" in a non-equilibrium state forever.
• The Implication: This challenges our understanding of how heat and information move in quantum computers and materials. It suggests that by carefully designing the edges of a material, we might be able to trap energy or information in specific spots, preventing it from leaking away.

Summary in One Sentence

The paper shows that if you push hard enough on the edge of a quantum chain, you can create a permanent "traffic jam" of energy that never spreads out, proving that even tiny, local changes can prevent a system from ever reaching a calm, thermal state.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of thermalization in isolated quantum many-body systems. While it is well-established that non-integrable systems thermalize (satisfying the Eigenstate Thermalization Hypothesis, ETH) and integrable systems generally do not (violating ETH) after a global quantum quench, the behavior following a local quench is less clear.

Previous theoretical work and numerical simulations suggested that local perturbations in integrable systems (like the XXZ spin chain) typically lead to thermalization, often satisfying a "weak" version of ETH (wETH), where the vast majority of eigenstates are thermal, with only a negligible fraction of exceptions.

The Core Question: Can a local quench in an integrable system prevent thermalization entirely, and can this lead to a strong violation of the ETH (where even the wETH fails)?

2. Methodology

The authors employ a combination of analytical derivations and numerical simulations focusing on spin-1/2 chains with open boundary conditions.

• Models:
  • XX Model: A non-interacting integrable model ($J_z = 0$).
  • XXZ Model: An interacting integrable model ($J_z \neq 0$).
• Setup:
  • The system starts in a thermal equilibrium state (Gibbs state) defined by a pre-quench Hamiltonian $H_0$ containing a single-spin impurity at site $\nu$.
  • At $t=0$, a local quench occurs by suddenly changing the impurity strength (or switching it on), governed by a post-quench Hamiltonian $H = H_0 + gV$.
  • The authors analyze the long-time behavior of local observables (specifically spin $s^z_\alpha$) to determine if they relax to the thermal expectation value of the post-quench Hamiltonian.
• Analytical Tools:
  • Jordan-Wigner Transformation: Used to map the XX spin chain to a model of free fermions, allowing for exact diagonalization.
  • Onsager's Regression Hypothesis: Utilized to relate the time-dependent expectation values to temporal correlations at thermal equilibrium.
  • Spectral Analysis: Detailed analysis of the eigenvalues and eigenvectors of the transformed Hamiltonian to identify localized modes.
• Numerical Tools:
  • Exact diagonalization for small system sizes ($L \le 24$).
  • Advanced numerical methods exploiting the free-fermion structure to simulate large systems ($L \sim 10^4$).
  • Extrapolation techniques to infer thermodynamic limit behavior.

3. Key Contributions and Results

A. Analytical Findings for the XX Model (End Impurity)

The authors analytically solve the XX model with an impurity at the chain end ($\nu = L$).

• Critical Threshold: They identify a critical impurity strength $p_c = (L+1)/L \approx 1$.
• Localized Modes: When the post-quench impurity strength $|p| > p_c$, the system develops a localized mode (an eigenstate exponentially decaying away from the impurity site).
• Absence of Thermalization: The presence of this localized mode prevents the system from thermalizing. The long-time average of the local observable $s^z_L$ remains distinct from the thermal equilibrium value.
• Strong ETH Violation: This non-thermalization implies a strong violation of the ETH. Unlike the wETH, which allows for a few non-thermal eigenstates, the strong violation here means that the diagonal matrix elements of the observable in the energy eigenbasis differ significantly for states that are close in energy, preventing the system from settling into a thermal state even locally.
• Spatial Localization: The non-thermalization is spatially localized near the impurity. Observables far from the impurity ($\alpha \ll L$) still thermalize, while those near the impurity do not.

B. Analytical Findings for the XX Model (Central Impurity)

When the impurity is placed in the center of the chain ($\nu = L/2$):

• Zero Threshold: The critical threshold for the impurity strength approaches zero as $L \to \infty$.
• Universal Non-thermalization: Even an arbitrarily weak central impurity is sufficient to induce a localized mode and prevent thermalization in the thermodynamic limit.

C. Numerical Findings for the XXZ Model

The authors extend their study to the interacting XXZ model ($J_z \neq 0$) numerically.

• End Impurity: The XXZ model with an end impurity behaves qualitatively similarly to the XX model. For sufficiently strong impurities, thermalization is absent, and a strong ETH violation occurs.
• Central Impurity: A striking difference emerges. For the XXZ model with a central impurity, the system does thermalize.
  • Reason: While the XX model remains integrable with a central impurity, the XXZ model with a central impurity breaks integrability. The system exhibits "quantum chaos" (Wigner-Dyson level statistics) and satisfies the conventional (strong) ETH, leading to thermalization even for weak impurities.

4. Significance and Implications

1. Refutation of Generalized Thermalization: The paper demonstrates that the common belief that "local quenches lead to thermalization in integrable systems" is not universally true. It provides rigorous analytical examples where local quenches prevent thermalization entirely.
2. Strong ETH Violation: The work establishes that non-thermalization in these specific local quench scenarios corresponds to a strong violation of the ETH, not just a weak one. This challenges the notion that wETH is a robust feature of generic local perturbations in integrable systems.
3. Role of Integrability and Geometry: The results highlight the subtle interplay between integrability, boundary conditions, and impurity location.
   • In the XX model, integrability is preserved regardless of impurity position, leading to non-thermalization for both end and central impurities.
   • In the XXZ model, an end impurity preserves integrability (non-thermalization), but a central impurity breaks it (thermalization).
4. Experimental Relevance: The findings suggest that in experimental setups (e.g., cold atoms in optical lattices), the specific location of a defect or impurity can fundamentally alter the long-time dynamics, potentially trapping the system in a non-equilibrium state near the defect.
5. Methodological Advancement: The paper provides a robust framework connecting temporal correlations, localized eigenmodes, and ETH violations, offering a template for analyzing non-thermalization in other integrable models.

Conclusion

Reimann and Eidecker-Dunkel prove that a local quantum quench can permanently prevent a many-body system from thermalizing, provided specific conditions regarding impurity strength and location are met. This phenomenon is rooted in the emergence of localized eigenmodes and constitutes a strong violation of the Eigenstate Thermalization Hypothesis. The study underscores that the fate of thermalization after a local quench is highly sensitive to the model's integrability and the geometric placement of the perturbation.