Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems

This paper introduces a symmetry-resolved trace distance to probe thermalization in SU(2)-symmetric quantum systems, demonstrating that in the thermal regime, the distance is asymptotically dominated by configurational fluctuations while spin-sector probability fluctuations are exponentially suppressed by non-Abelian eigenstate thermalization.

The Gist

The Big Picture: How Quantum Systems "Cool Down"

Imagine you have a huge, chaotic party (a quantum system) where thousands of guests (particles) are dancing. In physics, we want to know: How does this chaotic party settle down into a calm, predictable state (thermalization)?

Usually, scientists use a rule called the Eigenstate Thermalization Hypothesis (ETH). Think of this as a rule saying: "If you look at any single guest at the party, they look just like the average guest. The specific details of who they are don't matter; they all blend into the crowd."

However, this paper deals with a special kind of party where the guests have strict rules they must follow, like a dance where everyone must move in perfect circles (this is the SU(2) symmetry). In these "non-Abelian" systems, the old rules get a bit messy. The authors wanted to figure out exactly how these rule-bound parties thermalize.

The New Tool: The "Symmetry-Resolved Trace Distance"

To study this, the authors invented a new measuring stick called the Symmetry-Resolved Trace Distance.

The Analogy: The Library and the Bookshelves
Imagine the quantum system is a giant library.

1. The Whole Library: This is the full system.
2. The Sections: The library is divided into sections based on "Spin" (like sorting books by genre: Sci-Fi, History, Mystery). In physics, these are called Spin Sectors.
3. The Books: Inside each section, there are individual books (the specific configurations of particles).

When the system is in a specific state, it's like having a pile of books. The authors ask: "If I look at two slightly different states (two slightly different piles of books), how different are they?"

Because of the symmetry rules, the pile of books is naturally sorted into sections (Sci-Fi, History, etc.). The authors realized they could split the "difference" between two states into two distinct parts:

1. The "Probability" Difference (The Section Counts)

This measures: "Did the number of books in the Sci-Fi section change compared to the History section?"

• The Metaphor: Imagine two librarians organizing the same library. Librarian A has 50 Sci-Fi books and 50 History books. Librarian B has 52 Sci-Fi books and 48 History books. The total number of books is the same, but the distribution (probability) between sections is different.
• The Finding: In these special quantum systems, the rules of the universe (Non-Abelian ETH) force these section counts to become incredibly similar as the library gets bigger. The difference in the "number of books per section" vanishes exponentially fast. It's like saying, "In a huge library, it doesn't matter if you have 50 or 52 Sci-Fi books; the ratio is effectively the same."

2. The "Configurational" Difference (The Book Arrangements)

This measures: "Within the Sci-Fi section, are the specific books arranged differently?"

• The Metaphor: Both librarians have exactly 50 Sci-Fi books. But Librarian A put them in alphabetical order, while Librarian B put them in random order. The count is the same, but the arrangement is different.
• The Finding: This part of the difference does not vanish. Even in a huge, thermalized system, the specific arrangement of particles within a section still has some unique "fingerprint."

The Main Discovery: What Drives the Difference?

The authors proved a mathematical "balance sheet" for these systems:

Total Difference = (Difference in Section Counts) + (Difference in Arrangements)

They found that in large, thermalizing systems:

1. The Difference in Section Counts becomes tiny (exponentially small) because the Non-Abelian ETH forces the system to distribute probabilities perfectly.
2. The Difference in Arrangements remains the dominant factor.

The Takeaway:
In the past, scientists might have looked at the whole library and said, "It's chaotic, everything is different." This paper says, "Wait! If you look closely, the chaos is mostly just about how the books are shuffled inside the sections. The number of books in each section is actually very stable and predictable."

Why Does This Matter?

This is a new way to diagnose if a quantum system is "thermalizing" (settling down) or stuck in a weird, non-thermal state.

• Old Way: Look at the whole mess.
• New Way (This Paper): Separate the "counts" from the "arrangements."
  • If the "counts" are fluctuating wildly, the system is not thermalizing properly.
  • If the "counts" are stable (as predicted by the Non-Abelian ETH) but the "arrangements" are random, the system is thermalizing correctly.

The Experiment

The authors tested this theory on a specific quantum model (the $J_1-J_2$ Heisenberg chain) using powerful computers. They simulated systems of increasing sizes (from 12 to 18 particles).

The Result:
Just as their theory predicted, as the system got bigger, the "Probability Difference" (the section counts) dropped off rapidly, while the "Configurational Difference" (the arrangements) became the main thing left to measure.

Summary in One Sentence

This paper shows that in quantum systems with strict symmetry rules, the "chaos" of thermalization isn't about how many particles are in each group, but rather how those particles are shuffled within their groups; the group sizes become perfectly predictable, leaving only the internal shuffling as the source of difference.

Technical Summary

1. Problem Statement

The paper addresses the challenge of diagnosing thermalization in isolated quantum many-body systems possessing global non-Abelian symmetries (specifically $SU(2)$ spin symmetry).

• Context: While the standard Eigenstate Thermalization Hypothesis (ETH) successfully describes thermalization in systems with commuting conserved charges, systems with non-commuting charges (like total spin) require a Non-Abelian ETH.
• Gap: Standard diagnostics often average over symmetry sectors, potentially obscuring the specific mechanisms of thermalization in non-Abelian systems. There is a need for an eigenstate-level probe that respects the block-diagonal structure of the reduced density matrix induced by $SU(2)$ symmetry.
• Goal: To develop a symmetry-resolved diagnostic tool to distinguish between fluctuations driven by the non-Abelian ETH (which should be suppressed) and intrinsic configurational fluctuations.

2. Methodology

The authors introduce a Symmetry-Resolved Trace Distance derived from the block structure of the reduced density matrix (RDM).

• Theoretical Framework:

  • They utilize the Wigner-Eckart theorem and the Non-Abelian ETH (Eq. 1), which posits that matrix elements of operators in energy eigenstates consist of a smooth function of energy and spin, plus exponentially small random fluctuations.
  • They define a Subsystem-Spin Projector ($\Pi_{S_A}$) that projects the global state onto a subspace where subsystem $A$ has a specific total spin $S_A$.
  • The RDM of subsystem $A$ is decomposed into blocks labeled by $S_A$: $\rho_A = \bigoplus_{S_A} \rho^{(S_A)}_A$.

• Decomposition of Trace Distance:

  • For two neighboring eigenstates $\alpha$ and $\alpha+1$, the trace distance $D_A^\alpha$ is decomposed into two distinct components:
    1. Probability Trace Distance ($D_{\alpha, \text{prob}}$): Arises from the difference in the probabilities ($P_{S_A}$) of finding the subsystem in different spin sectors.
    2. Configurational Trace Distance ($D_{\alpha, \text{conf}}$): Arises from the difference in the normalized density matrices ($\tilde{\rho}^{(S_A)}_A$) within each fixed spin sector.
  • Inequality: The authors prove that $D_{\alpha, \text{prob}} \leq D_A^\alpha \leq D_{\alpha, \text{prob}} + D_{\alpha, \text{conf}}$.

• Analytical Proofs:

  • Proposition 1: Establishes the bounds mentioned above.
  • Proposition 2: Analyzes the microcanonical average of the probability trace distance. They show that $\langle D_{\alpha, \text{prob}} \rangle_W$ is bounded by the variance of the spin-sector probabilities within a microcanonical window. Crucially, under the Non-Abelian ETH, these variances are suppressed by the thermodynamic entropy ($e^{-S_{th}}$), leading to exponential suppression with system size $N$.

• Numerical Simulation:

  • Model: One-dimensional $J_1-J_2$ Heisenberg chain (with $J_1=1$ and varying $J_2$).
  • Regimes: $J_2=0$ (integrable) vs. $J_2 > 0$ (thermalizing).
  • Method: Exact Diagonalization (ED) for system sizes $N=12, 14, 16, 18$ in fixed global symmetry sectors.
  • Analysis: Calculated the finite-size scaling of probability variances and the trace distance components.

3. Key Contributions

1. Novel Diagnostic Tool: Introduced the Symmetry-Resolved Trace Distance, which separates the "probability" (sector occupation) and "configurational" (internal state) contributions to thermalization.
2. Theoretical Bounds: Proved that in non-Abelian thermalizing systems, the probability trace distance is exponentially suppressed with system size due to the Non-Abelian ETH. This provides a rigorous theoretical justification for why spin-sector probabilities become "frozen" or deterministic in the thermodynamic limit.
3. Decoupling Mechanism: Demonstrated that the block structure of the RDM naturally decouples ETH-controlled fluctuations (probabilities) from finer many-body fluctuations (configurations).
4. Numerical Verification: Validated these theoretical predictions using the $J_1-J_2$ Heisenberg chain, showing that while probability fluctuations decay rapidly, the total trace distance is asymptotically dominated by the configurational component.

4. Results

• Probability Fluctuations: Numerical data (Fig. 1a) shows that the sum of variances of spin-sector probabilities, $\sum_{S_A} \text{Var}_W(P_{S_A})$, decays exponentially with system size $N$. This confirms the Non-Abelian ETH prediction.
• Probability Trace Distance: The microcanonical average $\langle D_{\alpha, \text{prob}} \rangle_W$ (Fig. 1b) also decreases with system size. While the numerical data follows a power law for accessible sizes, the authors attribute this to finite-size effects (the inequality bounds are not yet saturated), consistent with the theoretical expectation of exponential decay.
• Dominance of Configurational Distance: The difference between the total trace distance and the configurational trace distance, $\langle D_A^\alpha - D_{\alpha, \text{conf}} \rangle_W$, decays exponentially (Fig. 2).
• Conclusion on Thermalization: In the thermal regime, the total trace distance between neighboring eigenstates is asymptotically dominated by the configurational trace distance. The probability component becomes negligible as the system size increases.

5. Significance

• Refining Thermalization Diagnostics: This work provides a more nuanced understanding of thermalization in systems with non-commuting charges. It suggests that simply averaging over symmetry sectors (as done in standard ETH studies) may hide the specific role of the non-Abelian symmetry.
• Strategy for Analysis: The paper proposes a general strategy: to diagnose thermalization in non-Abelian systems, one should resolve the symmetry structure rather than averaging it out. This allows researchers to identify which parts of the system's fluctuations are controlled by the ETH (and thus vanish in the thermodynamic limit) and which parts retain genuine many-body information.
• Experimental Relevance: Given recent experimental progress in trapped-ion systems probing non-Abelian thermal states, this framework offers a theoretical basis for interpreting how subsystems equilibrate in the presence of non-commuting conserved quantities.

In summary, the paper establishes that for $SU(2)$-symmetric thermalizing systems, the "probability" of finding a subsystem in a specific spin sector becomes rigid (ETH-controlled), while the "shape" of the state within that sector (configurational) carries the remaining fluctuations that define the distance between eigenstates.