Kubo-Martin-Schwinger relation for energy eigenstates of SU(2)-symmetric quantum many-body systems

By leveraging a recently proposed non-Abelian eigenstate thermalization hypothesis, this paper derives a Kubo-Martin-Schwinger relation for SU(2)-symmetric quantum many-body systems, revealing that finite-size corrections to the fluctuation-dissipation theorem can be polynomially larger than usual in certain cases, a finding supported by numerical simulations of Heisenberg chains.

The Gist

Imagine you have a giant, complex machine made of billions of tiny, dancing gears (these are the particles in a quantum system). In the world of physics, we have a famous rule called the Fluctuation-Dissipation Theorem (FDT).

Think of the FDT as a "cause-and-effect" rule for heat and motion. It says: "If you know how much a machine jiggles around randomly when it's sitting still (fluctuation), you can predict exactly how it will react if you give it a little push (dissipation)." It's like knowing how much a rubber band wobbles when you let go of it, which tells you how hard it will snap back if you pull it.

For this rule to work, the machine needs to follow a specific "symmetry" rule called the KMS relation. Think of the KMS relation as a secret handshake that the machine's gears must perform to ensure the cause-and-effect rule holds true.

The Problem: The "Non-Abelian" Twist

For a long time, physicists thought this handshake was universal. But recently, they discovered a special type of machine where the gears don't just spin; they also have a "spin" direction that interacts in a weird way. In physics, this is called non-Abelian symmetry.

Imagine a group of dancers.

• Normal Symmetry: If you tell them to "spin left," they all spin left. If you tell them to "spin right," they all spin right. The order doesn't matter.
• Non-Abelian Symmetry: If you tell them to "spin left" and then "jump," they end up in a different spot than if you told them to "jump" and then "spin left." The order of operations changes the outcome.

This "order matters" chaos breaks the standard rules of thermodynamics. It's like trying to bake a cake where the order you add the eggs and flour changes the taste of the cake entirely.

The Discovery: A New Handshake

The authors of this paper asked: "If these chaotic, order-sensitive machines exist, do they still follow the FDT rule? Do they still have a KMS handshake?"

They used a new theory called the Non-Abelian ETH (Eigenstate Thermalization Hypothesis). Think of ETH as a theory explaining how a single dancer in a huge crowd eventually starts moving like the rest of the crowd, even if they are alone.

Here is what they found:

1. Yes, they do have a handshake. Even with the chaotic "order matters" rules, these systems still obey a version of the KMS relation. The FDT still works!
2. But, the handshake is slightly different. It's not just about temperature anymore. It also depends on the "spin" of the particles.
3. The "Finite-Size" Surprise. This is the most exciting part.
   • In normal machines, if you make the machine bigger (add more gears), the errors in the rule get smaller very quickly (like $1/N$).
   • In these special non-Abelian machines, the authors found that sometimes the errors get much bigger than expected.
   • The Analogy: Imagine you are trying to balance a tower of blocks.
     • In a normal tower, if you add 100 blocks, the wobble is tiny.
     • In this special "quantum" tower, if you arrange the blocks in a specific way, adding 100 blocks might make the wobble huge (polynomially larger). The system is much more sensitive to its size than we thought.

The Experiment: Simulating the Dance

To prove this, the team didn't build a real quantum machine (which is incredibly hard). Instead, they used a supercomputer to simulate a chain of 16 to 24 "qubits" (quantum bits).

• They watched how these simulated particles danced.
• They measured the "log-ratio" (a fancy way of checking if the handshake is happening).
• The Result: The simulation confirmed that for most cases, the rule holds up well. However, for specific conditions (where the "spin" of the particles is just right), the "wobble" (the error) was indeed larger than the standard rules predicted.

Why Does This Matter?

This paper is like finding a new law of physics for a specific type of chaotic system.

• For Quantum Computers: If we want to build quantum computers that use these "order-sensitive" symmetries, we need to know that the standard rules of heat and noise might be slightly off. We need to account for these larger errors.
• For Thermodynamics: It shows that the universe is more flexible than we thought. Even when things are chaotic and "order matters," there is still an underlying order (the KMS relation), but it comes with a bigger price tag (larger corrections) depending on the size of the system.

In short: The universe has a backup plan for its most chaotic systems. The rules still work, but you have to be careful about the size of your system, or the "wobble" might be bigger than you expect!

Technical Summary

1. Problem Statement

The Fluctuation–Dissipation Theorem (FDT) is a cornerstone of nonequilibrium statistical mechanics, linking a system's response to perturbations with its equilibrium thermal fluctuations. The validity of the FDT relies on the Kubo–Martin–Schwinger (KMS) relation, a symmetry property of thermal states.

While the KMS relation is well-established for thermal mixed states, extending it to energy eigenstates of isolated quantum systems requires the Eigenstate Thermalization Hypothesis (ETH). Recent work proved that systems obeying the standard ETH satisfy a KMS relation with finite-size corrections scaling as $O(N^{-1})$.

However, many physically relevant systems possess non-Abelian symmetries (e.g., SU(2) spin rotation symmetry). These symmetries imply the conservation of non-commuting charges, which conflicts with the standard ETH. A Non-Abelian ETH has been proposed to describe such systems, but it remains unclear whether energy eigenstates of these systems obey a KMS relation, and if so, how the finite-size corrections scale. Specifically, do non-commuting charges alter the standard thermodynamic scaling of the KMS relation?

2. Methodology

The authors employ a combination of analytical derivation based on the Non-Abelian ETH and numerical simulation.

Analytical Framework

• Non-Abelian ETH: They utilize the ansatz proposed by Murthy et al., which posits that matrix elements of local operators in the energy eigenbasis are determined by smooth functions of energy and spin quantum numbers, modulated by random variables and Clebsch–Gordan coefficients arising from SU(2) symmetry.
• Modified Non-Abelian Thermal State (NATS): To handle the non-commuting charges, they introduce a modified thermal state $\rho_{NATS}$ that includes a term for the total spin operator $S$ (in addition to energy $H$ and magnetization $S_z$), controlled by an effective chemical potential $\gamma$.
• Fine-Grained Correlators: Since the standard KMS relation involves continuous energy differences, the authors define a fine-grained correlator that discretizes the changes in magnetic quantum number ($\Delta m$) and total spin quantum number ($\Delta s$). This allows them to isolate the symmetry constraints imposed by SU(2).
• Derivation: They derive the KMS relation for these fine-grained correlators by:
  1. Expressing correlators in the energy eigenbasis.
  2. Applying the Wigner-Eckart theorem and the Non-Abelian ETH.
  3. Performing Taylor expansions of the density of states and smooth functions around the eigenstate parameters ($E_\alpha, s_\alpha$).
  4. Analyzing the scaling of the resulting finite-size corrections based on the magnitude of the total spin $s_\alpha$ relative to system size $N$.

Numerical Simulation

• Model: A 1D Heisenberg XXX chain with nearest- and next-nearest-neighbor interactions (non-integrable for $\lambda=0.25$) subject to periodic boundary conditions.
• System Size: Exact diagonalization was performed for chains of $N = 16$ to $24$ qubits.
• Observables: They calculated fine-grained dynamical correlators for spherical tensor operators $T^{(k)}_q$ (ranks $k=0, 2, 4$).
• Metric: They tested the KMS relation by computing the logarithmic ratio of correlators:
  $$L = \ln \left( \frac{\hat{C}_{AB}(\Omega, \Delta m, \Delta s)}{\hat{C}_{BA}(-\Omega, -\Delta m, -\Delta s)} \right)$$
  According to the theory, this should equal $\beta(\Omega - \mu \Delta m - \gamma \Delta s)$ in the thermodynamic limit.

3. Key Contributions

1. Derivation of Fine-Grained KMS Relation: The paper establishes a rigorous KMS relation for energy eigenstates of SU(2)-symmetric systems. Unlike the standard relation, this "fine-grained" version explicitly depends on the changes in spin quantum numbers ($\Delta s$) and magnetic quantum numbers ($\Delta m$), reflecting the non-Abelian nature of the symmetry.
2. Identification of Anomalous Finite-Size Scaling: The authors prove that the finite-size correction to the KMS relation is not universally $O(N^{-1})$.
   • Standard Scaling: If the total spin $s_\alpha$ scales linearly with system size ($s_\alpha \sim O(N)$), the correction scales as $O(N^{-1})$.
   • Anomalous Scaling: If $s_\alpha$ scales as $O(N^\zeta)$ with $\zeta \in (0, 1)$, the correction scales as $O(N^{-\min\{\zeta, 1-\zeta\}})$. Crucially, for $s_\alpha \sim O(N^{1/2})$, the correction is $O(N^{-1/2})$, which is polynomially larger than the standard case.
3. Numerical Verification: The simulations on 16–24 qubits confirm the $O(N^{-1})$ scaling for specific parameter regimes (e.g., $s_\alpha \sim O(N)$) and provide indirect evidence for the larger corrections in regimes where computational limits prevent direct scaling analysis.

4. Key Results

• The Relation: The fine-grained KMS relation for an eigenstate $|\alpha, m\rangle$ is given by:
  $$\hat{C}_{AB}(\Omega, \Delta m, \Delta s) \approx e^{\beta(\Omega - \mu \Delta m - \gamma \Delta s)} \hat{C}_{BA}(-\Omega, -\Delta m, -\Delta s) \times [1 + \text{correction}]$$
  where the correction term depends on the derivatives of the thermodynamic entropy and the specific scaling of $s_\alpha$.
• Scaling Regimes:
  • When $s_\alpha = O(N)$, the correction is $O(N^{-1})$.
  • When $s_\alpha = O(N^{1/2})$, the correction is $O(N^{-1/2})$. This "anomalous" behavior arises because the density of states and the structure of the Hilbert space (due to non-commuting charges) create a different scaling for the entropy derivatives in this regime.
• Numerical Evidence:
  • For $\Delta s = 0$, the deviation from the predicted slope ($\beta$) decreases as $1/N$.
  • For $\Delta s \neq 0$, the deviations are larger, consistent with the theoretical prediction that non-zero spin changes introduce larger finite-size effects.
  • The data shows that the effective inverse temperature $\beta_{eff}$ converges to the true $\beta$ as $N$ increases, but the rate of convergence depends on the spin sector.

5. Significance

• Extension of Thermodynamics: This work extends the understanding of thermodynamics to systems with non-commuting charges, a rapidly growing field in quantum thermodynamics. It demonstrates that non-Abelian symmetries can fundamentally alter standard results like the FDT.
• Impact on Nonequilibrium Physics: The finding that finite-size corrections can be polynomially larger ($O(N^{-1/2})$) implies that for mesoscopic systems (e.g., quantum simulators, cold atoms), the approach to thermal equilibrium and the validity of linear response theory may be significantly different from Abelian systems.
• Experimental Relevance: The paper suggests that these anomalous corrections could be observable in current experimental platforms (trapped ions, superconducting qubits) where system sizes are limited ($N \sim 20-50$). It provides a specific protocol (measuring fine-grained correlators) to test the Non-Abelian ETH and the modified KMS relation.
• Theoretical Bridge: It bridges the gap between the abstract Non-Abelian ETH and concrete observable quantities (correlators), showing how the "off-diagonal" terms of the ETH (which govern fluctuations) are directly linked to the symmetry properties of the KMS relation.

In summary, the paper proves that while the KMS relation holds for SU(2)-symmetric systems, the non-commutativity of charges can induce anomalously large finite-size corrections, challenging the universality of standard thermodynamic scaling in the presence of non-Abelian symmetries.