Subsystem Thermalization Hypothesis in Quantum Spin Chains with Conserved Charges

This paper extends the universality of quantum thermalization by demonstrating that the subsystem thermalization hypothesis holds generically for small subsystems in quantum spin chains with various symmetries, not only for standard thermal ensembles but also for generalized and partial Generalized Gibbs Ensembles (p-GGEs) that incorporate partial sets of conserved charges.

The Gist

Imagine you have a giant, complex machine made of tiny spinning tops (quantum spins) all connected to each other. In the world of classical physics, if you shake this machine up and let it run for a long time, it eventually settles down into a predictable, "thermal" state—like a cup of coffee cooling to room temperature. This is governed by the laws of thermodynamics.

But in the quantum world, things are weirder. Because the machine is isolated (no outside interference) and follows strict quantum rules, it shouldn't technically "cool down" or forget its starting point. It should just keep evolving forever.

The Big Question:
Despite this, scientists have noticed that if you look at just a small part of this machine (a "subsystem"), that small part often looks like it has cooled down and reached thermal equilibrium, even though the whole machine hasn't. This is the Subsystem Thermalization Hypothesis.

The New Twist in This Paper:
The authors of this paper asked: "What happens if our machine has special 'rules' or 'conserved charges' that it can't break?"

Think of these conserved charges like strict laws of the universe that the machine must obey.

• Z2 Symmetry (Ising Chain): Like a rule that says, "The total number of heads must equal the total number of tails."
• U(1) Symmetry (XXZ Chain): Like a rule that says, "The total spin pointing up minus the total spin pointing down must stay constant."
• SU(2) Symmetry (XXX Chain): A more complex rule where the total spin vector is conserved.

Usually, to predict what a thermal system looks like, scientists use a "Generalized Gibbs Ensemble" (GGE). Think of the GGE as a perfect recipe that includes every single rule (every conserved charge) the system follows. If you bake the cake using this perfect recipe, it should match the behavior of the small part of the machine.

The Innovation: "Partial" Recipes (p-GGE)
The authors realized that maybe we don't need the perfect recipe with all the rules to get a good approximation. They proposed using Partial-GGEs (p-GGEs).

Imagine you are trying to guess the flavor of a soup.

• GGE: You know every single ingredient and spice in the pot.
• p-GGE: You only know some of the ingredients (e.g., you know there's salt and pepper, but you ignore the herbs).

The paper asks: If we use a "partial recipe" that ignores some of the rules, does the small part of the machine still look thermal?

What They Did:
They took three different types of quantum spin chains (Ising, XXZ, and XXX) and ran computer simulations. They created two types of starting points:

1. Energy Eigenstates: The machine in a specific, frozen energy state.
2. Typical States: The machine starting as a random jumble and evolving for a long time (like shaking the machine and letting it settle).

They then compared the "small part" of these machines against the predictions of:

• The full recipe (GGE).
• Partial recipes (p-GGE) that included only some rules, or even excluded the main energy rule (the Hamiltonian) entirely.

The Results (The "Demographics"):
They didn't just look at one case; they looked at thousands of scenarios (the "demographics") to see how often the hypothesis worked.

1. Small Parts Work Best: Just like looking at a single pixel in a high-resolution photo, the hypothesis works very well if the "subsystem" you are looking at is small compared to the whole machine.
2. Partial Recipes Work Surprisingly Well: Even if you use a "partial recipe" (p-GGE) that ignores some of the conserved charges, the small part of the machine still looks thermal.
3. The Hamiltonian Isn't Always Essential: In some cases, they found that even if they ignored the main energy rule (the Hamiltonian) in their recipe, the thermalization still held up. This suggests that for a small part of the system, knowing the total energy isn't always necessary to predict its behavior.
4. Non-Abelian Symmetries: They tested this on systems with complex, non-commuting rules (SU(2) symmetry) and found that the "partial recipe" approach still works.

The Bottom Line:
The paper claims that the idea of quantum thermalization is much more flexible than we thought. We don't need to know every rule of the universe to predict how a small piece of a quantum system will behave. Even "imperfect" descriptions (p-GGEs) that ignore certain conserved quantities can successfully predict that a small part of the system has thermalized.

This expands the "universe" of quantum thermalization, showing that it holds true in a wider variety of scenarios and with less information than previously required.

Technical Summary

Technical Summary: Subsystem Thermalization Hypothesis in Quantum Spin Chains with Conserved Charges

Problem Statement
The paper addresses the thermalization of isolated quantum many-body systems, specifically focusing on non-integrable spin chains with conserved charges. While the Eigenstate Thermalization Hypothesis (ETH) and the concept of Canonical Typicality suggest that local subsystems of energy eigenstates or typical pure states should resemble thermal ensembles, the presence of conserved charges complicates this picture. Standard thermalization frameworks often rely on the Generalized Gibbs Ensemble (GGE), which includes all conserved charges. However, constructing a full GGE is often impractical due to the infinite number of charges in integrable systems or the complexity of non-Abelian symmetries. Furthermore, it remains unclear how the thermalization hypothesis behaves when only a subset of conserved charges is included in the reference thermal ensemble, or when the Hamiltonian itself is excluded from the ensemble definition. The authors aim to quantify the validity of the subsystem thermalization hypothesis across various "partial" ensembles and for different types of states (eigenstates, typical states, and almost superselection states).

Methodology
The authors employ a numerical approach using Exact Diagonalization (ED) on spin-1/2 chains of approximately $L=10$ sites. The study focuses on three non-integrable variants of quantum spin chains:

1. Ising Chain: Possessing a discrete $Z_2$ symmetry (reflection).
2. XXZ Chain: Possessing a continuous $U(1)$ symmetry (total $S^z$).
3. XXX Chain: Possessing a non-Abelian $SU(2)$ symmetry (total spin vector).

The methodology involves:

• State Preparation:
  • Energy Eigenstates: Obtained via ED.
  • Typical States: Generated by unitarily evolving random product states for a sufficiently long time ($T=1000$) to saturate local entanglement entropy.
  • Almost Superselection States: Typical states classified by their expectation values of conserved charges ($\langle Q_a \rangle$), effectively subdividing the Hilbert space into fine-grained sectors.
• Thermal Ensembles: The authors introduce Partial Generalized Gibbs Ensembles (p-GGEs). Unlike the standard GGE which includes all conserved charges, p-GGEs include only a selected subset of charges $\{Q_j\}$ with corresponding chemical potentials $\{\beta_j\}$. Crucially, the framework treats the Hamiltonian ($Q_0$) and other charges on equal footing, allowing for ensembles where the Hamiltonian is excluded.
• Quantification: The validity of the thermalization hypothesis is measured by the relative entropy ($S_A$) between the reduced density matrix of a subsystem $A$ (from the pure state) and the reduced density matrix of the corresponding p-GGE thermal state.
  $$S_A = S \left[ \text{Tr}_{\bar{A}}|\Psi\rangle\langle\Psi| \parallel \text{Tr}_{\bar{A}}\rho_{\{j\}} \right]$$
  A vanishing relative entropy indicates successful thermalization.
• Demographics: Instead of analyzing single states, the authors examine the "demographics" (population distribution) of relative entropy values across ensembles of states with varying charge expectations.

Key Contributions

1. Extension to Partial-GGEs: The paper formalizes and numerically tests the subsystem thermalization hypothesis using p-GGEs, demonstrating that thermalization can hold even when only a subset of conserved charges is included in the thermal ensemble.
2. Hamiltonian Independence: A significant finding is that the Hamiltonian is not strictly necessary for the thermalization hypothesis to hold. p-GGEs constructed without the Hamiltonian (excluding $Q_0$) often yield sharper thermalization demographics (smaller relative entropy) than those including it, particularly in systems with non-Abelian symmetries.
3. Non-Abelian Thermalization: The study extends the thermalization framework to systems with non-Abelian ($SU(2)$) symmetries, comparing typical states and eigenstates against Non-Abelian Thermal States (NATS) and various p-GGEs.
4. Fine-Grained Analysis: The authors distinguish between "coarse-grained" thermalization (averaged over all typical states) and "fine-grained" thermalization (within specific almost superselection sectors). They show that while coarse-grained thermalization may appear successful, fine-grained versions can fail for specific charge sectors.

Results

• Subsystem Size Dependence: Across all models (Ising, XXZ, XXX), the subsystem thermalization hypothesis holds robustly for small subsystems (specifically $A=1$ site). The validity degrades as the subsystem size $A$ increases, eventually breaking down when $A$ approaches half the system size.
• Charge Dependence:
  • Ising ($Z_2$): Thermalization holds for typical states and eigenstates when compared to GGE and p-GGEs. However, fine-grained thermalization fails for sectors with large energy expectations ($\langle Q_0 \rangle$) when compared to p-GGEs defined only by the reflection symmetry.
  • XXZ ($U(1)$): Similar to the Ising case, thermalization is effective for small $A$. A notable swap in behavior is observed: thermalization holds well for all fixed energy sectors ($\langle Q_0 \rangle$) but fails for large total spin sectors ($\langle Q_1 \rangle$) when using specific p-GGEs.
  • XXX ($SU(2)$): The study confirms that Non-Abelian Thermal States (NATS) are valid reference states. Furthermore, p-GGEs involving at least two non-Hamiltonian charges yield effective thermalization. The results suggest that excluding the Hamiltonian from the p-GGE often leads to better thermalization demographics than including it.
• Eigenstates vs. Typical States: In general, the subsystem thermalization hypothesis is found to be more effective for typical states than for energy eigenstates when compared to the same thermal ensemble. Additionally, GGEs generally provide a better match than p-GGEs, though p-GGEs remain effective for small subsystems.

Significance and Claims
The paper claims to extend the universality of quantum thermalization beyond the strict requirements of the Generalized Gibbs Ensemble. By introducing the p-GGE framework, the authors demonstrate that the thermalization hypothesis is more flexible than previously thought:

• It applies to systems where only partial information (a subset of conserved charges) is retained in the thermal description.
• It suggests that the Hamiltonian itself may not be a fundamental requirement for defining a thermal state in the context of subsystem thermalization, as ensembles excluding the energy constraint can still accurately describe local reduced states.
• The framework provides a quantitative tool (relative entropy demographics) to assess the "degree" of thermalization in isolated quantum systems with conserved quantities, bridging the gap between integrable and non-integrable dynamics.

The authors conclude that while the hypothesis holds generically for small subsystems, the choice of the reference ensemble (which charges to include) and the specific charge sector of the state are critical factors in determining the validity of thermalization.