Numerical evidence for the non-Abelian eigenstate thermalization hypothesis

This paper provides numerical evidence supporting the non-Abelian eigenstate thermalization hypothesis (ETH) through simulations of a 1D Heisenberg chain and offers an analytical proof of its self-consistency, thereby establishing a framework for understanding thermalization in quantum systems with non-commuting conserved quantities.

The Gist

Imagine you have a giant, complex machine made of 18 tiny switches (qubits) that are all connected to each other. In the world of quantum physics, these switches don't just flip on and off; they spin in different directions, and the direction of one switch affects its neighbors.

For a long time, physicists believed that if you let such a machine run for a long time, it would eventually "settle down" into a predictable, average state, much like a cup of hot coffee cooling down to room temperature. This idea is called the Eigenstate Thermalization Hypothesis (ETH). It suggests that no matter how you start the machine, the local parts will eventually act as if they are in a standard "thermal" (hot/cold) balance.

The Problem: The "Non-Commuting" Puzzle
However, there's a catch. In this specific machine, the switches are governed by a special rule called non-Abelian symmetry. Think of this like a game of directions:

• If you turn a compass North, then East, you end up in a different spot than if you turn East, then North.
• In quantum terms, these "directions" (charges) don't get along; they don't commute. They interfere with each other.

Because of this interference, the old rules (the standard ETH) break down. The machine shouldn't settle down the usual way. But, a new theory called the Non-Abelian ETH was proposed to explain how this specific type of machine does eventually settle down, just with a different set of rules.

What This Paper Did
The authors of this paper acted like detectives testing a new theory. They built a computer simulation of this 18-switch machine to see if the new "Non-Abelian ETH" theory was true.

Here is what they found, using simple analogies:

1. The "Smooth" Pattern: They looked at the internal math of the machine. The theory predicted that if you plot the machine's behavior against its energy, the dots should form a smooth, flowing curve (like a gentle hill) rather than a jagged, random mess. Result: The data formed beautiful, smooth bands, just like the theory predicted.
2. The "Random Noise": The theory also said that if you look closely at the tiny differences between the dots, they should look like random static on an old TV screen (Gaussian distribution). Result: When they zoomed in, the "noise" looked exactly like random static.
3. The "Volume" Check: The theory predicted a specific relationship between how "loud" the noise is and how many different states the machine can be in (the density of states). It's like saying, "If the room gets bigger, the echo should get quieter in a very specific way." Result: The echo got quieter at exactly the rate the theory said it should.
4. The "Ratio" Test: They compared the noise inside the machine's main groups versus the noise between groups. The theory said this ratio should be exactly 2. Result: Their measurements came out to 1.99, which is practically 2.

The "Self-Consistency" Proof
Beyond the computer simulation, the authors also did a mathematical proof. They showed that the new theory doesn't contradict itself. They had to tweak a definition of "entropy" (a measure of disorder) slightly—subtracting a small amount related to the machine's spin—to make the math work perfectly. Once they made this small adjustment, the theory held together without any logical holes.

The Bottom Line
This paper provides the first strong, numerical evidence that the Non-Abelian ETH is real. It confirms that even when quantum particles have "clashing" rules (non-commuting charges) that prevent them from behaving normally, they still find a way to thermalize, but they follow a new, slightly more complex set of instructions than we previously thought.

The authors did not claim this leads to new medical treatments or immediate technology. Instead, they successfully proved that this specific theoretical framework for how quantum systems settle down is mathematically sound and matches their computer models.

Technical Summary

Technical Summary: Numerical Evidence for the Non-Abelian Eigenstate Thermalization Hypothesis

Problem Statement
The Eigenstate Thermalization Hypothesis (ETH) explains how generic, isolated quantum many-body systems thermalize internally, positing that local operators' time-averaged expectation values equal their thermal expectation values. However, recent work by Murthy et al. demonstrated that standard ETH fails in the presence of non-Abelian symmetries. In such systems, conserved quantities (charges) fail to commute with one another, leading to degeneracies and the absence of shared eigenbases for all charges. This conflicts with the standard ETH structure and the Wigner–Eckart theorem, which dictates transition frequencies and matrix element structures in systems with continuous symmetries. While Murthy et al. proposed a "non-Abelian ETH" (NA-ETH) to resolve these conflicts, and Noh provided preliminary numerical support in a 2D model, a comprehensive numerical verification in a system with minimal extraneous symmetries was lacking.

Methodology
The authors model a one-dimensional (1D) chain of $N=18$ qubits subject to open boundary conditions. The system evolves under a non-integrable Heisenberg Hamiltonian featuring both nearest-neighbor and next-nearest-neighbor interactions. The Hamiltonian is designed to exhibit $SU(2)$ symmetry (conserving the total spin components $S^x, S^y, S^z$) while explicitly breaking translational and spatial-inversion symmetries to ensure non-integrability.

The study employs exact diagonalization to compute the joint eigenbasis $\{|\alpha, m\rangle\}$ shared by the Hamiltonian $H$, the total spin squared operator $\vec{S}^2$, and the $z$-component of spin $S^z$. Here, $\alpha$ labels the energy $E_\alpha$ and total spin quantum number $s_\alpha$, while $m$ is the magnetic quantum number. Local operators are represented as spherical tensor operators $T^{(k)}_q$. The authors analyze the reduced matrix elements $\langle \alpha || T^{(k)} || \alpha' \rangle$, which are independent of the magnetic quantum numbers due to the Wigner–Eckart theorem.

The analysis focuses on verifying the ansatz proposed by Murthy et al., which posits that reduced matrix elements take the form:
$$\langle \alpha || T^{(k)} || \alpha' \rangle = T^{(k)}(E, S) \delta_{\alpha\alpha'} + e^{-S_{th}(E, S)/2} f^{(T)}_\nu(E, S, \omega) R^{(T)}_{\alpha\alpha'}$$
where $E$ and $S$ are average energy and spin, $\omega$ and $\nu$ are their differences, $R^{(T)}$ represents erratic fluctuations, and $S_{th}$ is a modified thermodynamic entropy defined as the standard entropy minus $\ln(2S+1)$ to account for magnetic degeneracy.

Key Contributions and Results
The paper provides extensive numerical evidence supporting the NA-ETH through seven specific predictions, alongside an analytical proof of self-consistency:

1. Analytical Self-Consistency: The authors prove that the NA-ETH is self-consistent (in the sense of Srednicki) only if the thermodynamic entropy $S_{th}$ is defined to exclude the magnetic degeneracy factor $\ln(2S+1)$. This definition ensures that the scaling of matrix elements derived directly from the ansatz matches the scaling derived indirectly via the composition of operators (using Clebsch–Gordan coefficients).

2. Non-Integrability Verification: Energy-gap statistics within joint eigenspaces of $\vec{S}^2$ and $S^z$ follow the Gaussian Orthogonal Ensemble (GOE) distribution, confirming the system is non-integrable and suitable for ETH testing.

3. Block-Diagonal Smoothness: The block-diagonal reduced matrix elements $\langle \alpha || T^{(1)} || \alpha \rangle$ exhibit smooth dependence on energy density ($E/N$) and spin density ($s_\alpha/N$) in the thermodynamic limit, consistent with the smooth function $T^{(k)}(E, S)$ in the ansatz.

4. Gaussian Fluctuations: After subtracting the smooth mean, the fluctuations of both block-diagonal and off-block-diagonal reduced matrix elements follow Gaussian distributions, supporting the random matrix term $R^{(T)}_{\alpha\alpha'}$.

5. Off-Block-Diagonal Structure: The off-block-diagonal elements also exhibit Gaussian fluctuations and depend on a smooth function $f^{(T)}_\nu(E, S, \omega)$ that decays rapidly as the energy difference $|\omega|$ increases.

6. Variance Ratio: The ratio of the variance of block-diagonal elements to off-block-diagonal elements averages to $2$, a prediction derived from random-matrix theory.

7. Variance Scaling: The variances of both block-diagonal and off-block-diagonal elements decay algebraically with the density of states (DOS). The observed slopes in log-log plots ($\approx -0.83$ and $\approx -0.95$) are close to the predicted slope of $-1$, with deviations attributed to finite-size effects.

8. $q$-Independence: The function $f^{(T)}_\nu$ is shown to be independent of the magnetic quantum number $q$, consistent with the Wigner–Eckart theorem.

Significance
The authors state that this work initiates the observation and application of the non-Abelian ETH. It provides the first thorough numerical support for the hypothesis, extending beyond previous 2D studies to a 1D system with minimal extraneous symmetries. By validating the NA-ETH, the work bridges the gap between quantum thermodynamics (specifically the study of non-commuting charges) and many-body physics. The authors suggest that these results enable future investigations into how non-Abelian symmetries alter thermalization, information retention in quantum memories, and fluctuation-dissipation relations, though they do not propose specific new experiments or applications beyond these theoretical avenues.