Basis dependence of eigenstate thermalization

This paper demonstrates that the validity of the eigenstate thermalization hypothesis can depend on the choice of energy eigenbasis in degenerate systems, showing cases where it holds in one basis but fails in another, while also proving that such degeneracies are inevitable in translation- and reflection-symmetric systems and deriving bounds on the resulting basis dependence of thermalization.

The Gist

Imagine you have a massive, complex machine made of billions of tiny, interacting gears. In physics, we call this a "many-body quantum system." For decades, physicists have believed that if you wait long enough, this machine will settle down into a predictable, calm state called thermal equilibrium (like a cup of coffee cooling down to room temperature).

The rulebook for how and why this happens is called the Eigenstate Thermalization Hypothesis (ETH).

Think of the ETH as a promise: "If you look at any single, frozen snapshot of the machine's gears (an 'eigenstate'), the average behavior of the local parts should look exactly like the average behavior of the whole machine when it's calm and hot."

The Big Problem: The "Lens" Issue

The authors of this paper discovered a subtle but massive flaw in how we check this promise. They found that whether the machine actually thermalizes depends entirely on which "lens" or "coordinate system" you use to look at it.

Here is the analogy:

Imagine you are looking at a crowd of people in a stadium.

• View A (The Translator's Lens): You look at the crowd through a lens that groups people by their seat number. From this angle, the crowd looks perfectly mixed and random. Everyone seems to be behaving like a typical fan. The ETH says, "Great, the crowd is thermalized!"
• View B (The Mirror's Lens): Now, you switch to a lens that groups people by their reflection in a giant mirror on the other side of the stadium. Suddenly, you see that for every person standing up, there is a perfect twin sitting down. The crowd is actually highly organized and not random at all. From this angle, the ETH says, "Wait, this crowd is NOT thermalized!"

The paper proves that for certain types of machines (specifically those that look the same if you slide them left/right and flip them like a pancake), both views are mathematically valid, but they tell opposite stories.

The Three Main Discoveries

1. The "Mirror" Trap (Degeneracies)
The authors show that if a system has two specific symmetries (sliding and flipping), it creates "ghost twins" in the math. These are called degeneracies.

• Analogy: Imagine a dance floor where every dancer has a perfect twin doing the exact same move. Because they are identical, you can swap them without changing the dance.
• The Result: In these systems, there are so many "ghost twins" that almost every single state of the machine has a partner. This abundance of twins is the root of the confusion.

2. The "Dangerous" vs. "Safe" Lenses
Because of these twins, you can choose how to describe the machine's state.

• The Safe Lens: You choose a description where the twins are mixed up perfectly. In this view, the ETH holds true. The machine looks thermal.
• The Dangerous Lens: You choose a description where the twins are lined up perfectly side-by-side. In this view, the ETH fails. The machine looks frozen and non-thermal.

The paper provides a concrete example (a spin-1 model) where:

• If you use the "Safe Lens," the system thermalizes.
• If you use the "Dangerous Lens," the system never thermalizes, no matter how long you wait.

3. The Real-World Consequence
This isn't just a math game. The authors show that if you take a real-world system and slightly break the symmetry (like adding a tiny bit of noise so the "ghost twins" aren't perfectly identical anymore), the system still behaves like the "Dangerous Lens" version.

• The Takeaway: If a physicist runs a computer simulation and uses the "Safe Lens" (because it's easier to calculate), they might conclude, "Ah, this system thermalizes!" But in reality, if they built this machine in a lab, it would fail to thermalize. They would be fooled by their own choice of mathematical tools.

Why This Matters

For a long time, physicists thought the ETH was the ultimate explanation for why the universe settles down. This paper says: "Not so fast."

If a system has these specific symmetries, the question "Does it thermalize?" is actually a trick question. The answer depends on how you ask it.

• The Old Way: "Does the ETH hold?" (Answer: It depends on your math.)
• The New Way: We need a new, "lens-proof" version of the ETH that works regardless of how you look at the twins. Until we find that, we can't be sure if a system will actually cool down or stay stuck in a weird, non-thermal state.

Summary in One Sentence

This paper reveals that for certain quantum systems, the laws of thermalization are like a magic trick: they work perfectly if you look at them one way, but completely fail if you look at them another way, potentially leading scientists to draw the wrong conclusions about how real-world materials behave.

Technical Summary

1. Problem Statement

The Eigenstate Thermalization Hypothesis (ETH) posits that individual energy eigenstates of a generic many-body quantum system behave like thermal equilibrium ensembles regarding local observables. While the strong ETH requires all eigenstates to thermalize, the weak ETH only requires that the variance of diagonal matrix elements of local observables vanishes in the thermodynamic limit ($L \to \infty$).

The central problem addressed in this work is the basis dependence of the weak ETH in systems with degenerate energy spectra.

• In non-degenerate systems, the energy eigenbasis is unique (up to phase factors), making the ETH statement unambiguous.
• In degenerate systems (common in translationally invariant systems with spatial reflection symmetry), the choice of basis within a degenerate subspace is arbitrary.
• The authors investigate whether the validity of the weak ETH depends on this arbitrary choice. They demonstrate that for a specific system, the weak ETH can be satisfied in one basis (e.g., translationally invariant) but violated in another (e.g., the basis diagonalizing the observable). This challenges the physical interpretation of ETH as a mechanism for thermalization in degenerate systems.

2. Methodology

A. Theoretical Framework

1. Degeneracy Analysis: The authors prove that for any Hamiltonian $H$ that is both translationally invariant ($[H, T]=0$) and reflection symmetric ($[H, P]=0$), the vast majority of energy eigenvalues are degenerate in the thermodynamic limit. They derive a lower bound showing the fraction of degenerate eigenvalues approaches unity as $L \to \infty$.
2. Optimization of ETH Quantifiers: They analyze the ETH quantifier $\Delta^2_A$ (variance of diagonal elements). Using linear algebra properties (trace invariance and unitary transformations), they identify two extremal bases for any degenerate energy subspace:
   • Maximizing Basis: The basis where the observable $A$ is diagonal within every degenerate subspace. This maximizes the ETH violation (or $\Delta^2_A$).
   • Minimizing Basis: The basis where the diagonal elements of $A$ are constant within every degenerate subspace. This minimizes the ETH violation.
3. Basis-Independent Formulation: They propose that a physically meaningful definition of the weak ETH must rely on the upper bound (the maximizing basis), as physical properties cannot depend on an arbitrary basis choice. They define a basis-independent quantifier $\hat{\Delta}^2_A$.

B. Numerical Model

The authors study a specific spin-1 Hamiltonian with periodic boundary conditions (Eq. 24), known for Hilbert space fragmentation and conservation of magnetization and dipole moment.

• Observable: Single-site magnetization $A = s^z_1$.
• Ensemble: Canonical ensemble (Gibbs state) at various inverse temperatures $\beta$.
• Technique: Dynamical Typicality methods were used to numerically evaluate expectation values and correlation functions in high-dimensional Hilbert spaces, exploiting symmetry sectors (magnetization and dipole moment).

3. Key Contributions and Results

A. Existence of Basis-Dependent Thermalization

The paper provides an explicit counterexample where the weak ETH status depends entirely on the basis:

• Translationally Invariant Basis: When using eigenstates that are also eigenstates of the translation operator $T$, the ETH quantifier $\Delta^2_A$ vanishes as $L \to \infty$. This satisfies the weak ETH.
• Maximally Violating Basis: When using the basis where $A$ is diagonal in degenerate subspaces, the ETH quantifier converges to a non-zero, $\beta$-dependent constant as $L \to \infty$.
• Conclusion: The system satisfies the weak ETH in one basis but violates it in another.

B. Implications for Equilibration and Rethermalization

The authors connect the basis choice to dynamical properties:

• Long-Time Averages: The long-time average of an observable after a quench is given by the diagonal ensemble. This average is only correctly represented by the maximizing basis (where $A$ is diagonal in degenerate subspaces).
• Autocorrelation Functions: The long-time average of the dynamical autocorrelation function equals the ETH quantifier in the maximizing basis.
• Failure of Rethermalization: In the specific model studied, the non-vanishing ETH quantifier in the maximizing basis implies that the system does not rethermalize after a local quench. The system retains memory of the initial state, failing to reach the thermal equilibrium value predicted by the canonical ensemble.

C. Robustness to Perturbations

A critical finding is that this behavior persists even when symmetries are broken.

• The authors introduced a weak perturbation (random local magnetic fields) to the Hamiltonian, breaking translation and reflection symmetries and lifting all degeneracies.
• Result: The ETH quantifier in this non-degenerate system still converged to the same non-zero value observed in the "maximally violating" basis of the unperturbed system.
• Implication: Numerical studies of ETH that exploit symmetries (using the translationally invariant basis) in degenerate systems can lead to physically wrong conclusions. They might falsely conclude that a system thermalizes, whereas the physically relevant (perturbed) system does not.

D. General Bounds

The paper derives general bounds for the ETH quantifier:
$$\frac{1}{|S|} \sum_{E \in W} \left( \frac{[\text{tr}(P_E A)]^2}{g_E} - A_{\text{th}}^2 \right) \leq \Delta^2_A \leq \frac{1}{|S|} \sum_{E \in W} \left( \frac{\text{tr}[(P_E A)^2]}{g_E} - A_{\text{th}}^2 \right)$$
where $P_E$ is the projector onto the energy eigenspace and $g_E$ is the degeneracy. The upper bound (basis-independent) is the physically relevant quantity for determining thermalization.

4. Significance

1. Redefining ETH in Degenerate Systems: The paper argues that the standard formulation of the weak ETH is ill-posed for systems with degeneracies unless the basis-independent upper bound is used. The "weak ETH" proven in previous literature (e.g., for translationally invariant systems) may only hold for specific, non-physical bases.
2. Warning for Numerical Studies: It highlights a major pitfall in numerical ETH studies. Using symmetry-adapted bases (common practice to reduce computational cost) can mask ETH violations. If a system has degeneracies, one must check the basis-independent bound or the behavior in the "diagonal" basis to determine true thermalization properties.
3. Connection to Conservation Laws: The results link the absence of thermalization to the presence of conserved quantities (even non-local ones like quasimomentum) that induce degeneracies. It suggests that systems with such symmetries may fail to thermalize even if they appear to satisfy ETH in a specific basis.
4. Physical Reality: The findings imply that "thermalization" is not just a property of the Hamiltonian's spectrum but depends on the interplay between the Hamiltonian, the observable, and the specific eigenbasis chosen to represent the state. For physical systems (which are never perfectly symmetric), the behavior is dictated by the basis-independent upper bound, which can be non-zero, leading to a lack of thermalization.

In summary, the paper demonstrates that eigenstate thermalization is not an intrinsic property of the Hamiltonian alone in degenerate systems; it is basis-dependent. Consequently, the standard weak ETH is insufficient to guarantee thermalization, and a new, basis-independent formulation is required to correctly predict the long-time behavior of quantum many-body systems.