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arXiv:2507.07249 · APL Quantum

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

Jae Dong Noh, Aleksander Lasek, Jade LeSchack, Nicole Yunger Halpern

Published Apr 8, 2026• 1 citation
arXivHTMLPDF

Abstract

The fluctuation–dissipation theorem (FDT) is a fundamental result in statistical mechanics. It stipulates that, if perturbed out of equilibrium, a system responds at a rate proportional to a thermal-equilibrium property. Applications range from particle diffusion to electrical-circuit noise. To prove the FDT, one must prove that common thermal states obey a symmetry property, the Kubo–Martin–Schwinger (KMS) relation. Energy eigenstates of certain quantum many-body systems were recently proven to obey a KMS relation. The proof relies on the eigenstate thermalization hypothesis (ETH), which explains how such systems thermalize internally. This KMS relation contains a finite-size correction that scales as the inverse system size. Non-Abelian symmetries conflict with the ETH, so a non-Abelian ETH was proposed recently. Using it, we derive a KMS relation for SU(2)-symmetric quantum many-body systems’ energy eigenstates. The finite-size correction scales as usual under certain circumstances but can be polynomially larger in others, we argue. We support the ordinary-scaling result numerically, simulating a Heisenberg chain of 16–24 qubits. The numerics, limited by computational capacity, indirectly support the larger correction. This work helps extend into nonequilibrium physics the effort, recently of interest across quantum physics, to identify how non-Abelian symmetries may alter conventional thermodynamics.

DOI
10.1063/5.0308181

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The fluctuation–dissipation theorem (FDT) is a fundamental result in statistical mechanics. It stipulates that, if perturbed out of equilibrium, a system responds at a rate proportional to a thermal-equilibrium property. Applications range from particle diffusion to electrical-circuit noise. To prove the FDT, one must prove that common thermal states obey a symmetry property, the Kubo–Martin–Schwinger (KMS) relation. Energy eigenstates of certain quantum many-body systems were recently proven to obey a KMS relation. The proof relies on the eigenstate thermalization hypothesis (ETH), which explains how such systems thermalize internally. This KMS relation contains a finite-size correction that scales as the inverse system size. Non-Abelian symmetries conflict with the ETH, so a non-Abelian ETH was proposed recently. Using it, we derive a KMS relation for SU(2)-symmetric quantum many-body systems’ energy eigenstates. The finite-size correction scales as usual under certain circumstances but can be polynomially larger in others, we argue. We support the ordinary-scaling result numerically, simulating a Heisenberg chain of 16–24 qubits. The numerics, limited by computational capacity, indirectly support the larger correction. This work helps extend into nonequilibrium physics the effort, recently of interest across quantum physics, to identify how non-Abelian symmetries may alter conventional thermodynamics.
The fluctuation-dissipation theorem (FDT) is a hallmark of thermal equilibrium systems in the Gibbs state. We address the question whether the FDT is obeyed by isolated quantum systems in an energy eigenstate. In the framework of the eigenstate thermalization hypothesis, we derive the formal expression for two-time correlation functions in the energy eigenstates or in the diagonal ensemble. They satisfy the Kubo-Martin-Schwinger condition, which is the sufficient and necessary condition for the FDT, in the infinite system size limit. We also obtain the finite size correction to the FDT for finite-sized systems. With extensive numerical works for the XXZ spin chain model, we confirm our theory for the FDT and the finite size correction. Our results can serve as a guide line for an experimental study of the FDT on a finite-sized system.
The eigenstate thermalization hypothesis (ETH) explains how generic quantum many-body systems thermalize internally. It implies that local operators' time-averaged expectation values approximately equal their thermal expectation values, regardless of microscopic details. The ETH's range of applicability therefore impacts theory and experiments. Murthy et al. [Phys. Rev. Lett. 130, 140402 (2023)0031-900710.1103/PhysRevLett.130.140402] recently showed that non-Abelian symmetries conflict with the ETH. Such symmetries have excited interest in quantum thermodynamics lately, as they are equivalent to conserved quantities that fail to commute with each other and noncommutation is a quintessentially quantum phenomenon. Murthy et al. proposed a non-Abelian ETH, which we support numerically. The numerics model a one-dimensional next-nearest-neighbor Heisenberg chain of 18 qubits. We represent local operators with matrices relative to an energy eigenbasis. The matrices bear out seven predictions of the non-Abelian ETH. We also prove analytically that the non-Abelian ETH exhibits a self-consistency property. The proof relies on a thermodynamic-entropy definition different from that of Murthy et al. This work initiates the observation and application of the non-Abelian ETH.
Thermalization of a system when interacting with a thermal bath is an interesting problem. If a system eventually reaches a thermal state in the long time limit, it's expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition, or equivalently, the fluctuation-dissipation relation (FDR). In this paper, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS. In addition, we formulate a nonperturbative approach in the form of a self-consistent approximation that includes a partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition.
We investigate the extent to which the eigenstate thermalization hypothesis (ETH) is valid or violated in the nonintegrable and the integrable spin-1/2 XXZ chains. We perform the energy-resolved analysis of statistical properties of matrix elements of observables in the energy eigenstate basis. The Hilbert space is divided into energy shells of constant width, and a block submatrix is constructed whose columns and rows correspond to the eigenstates in the respective energy shells. In each submatrix, we measure the second moment of off-diagonal elements in a column. The columnar second moments are distributed with a finite variance for finite-sized systems. We show that the relative variance of the columnar second moments decreases as the system size increases in the non-integrable system. The self-averaging behavior indicates that the energy eigenstates are statistically equivalent to each other, which is consistent with the ETH. In contrast, the relative variance does not decrease with the system size in the integrable system. The persisting eigenstate-to-eigenstate fluctuation implies that the matrix elements cannot be characterized with the energy parameters only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.
Despite the unitary evolution of closed quantum systems, long-time expectation of local observables are well described by thermal ensembles, providing the foundation of quantum statistical mechanics. A promising route to understanding this quantum thermalization is the eigenstate thermalization hypothesis (ETH), which posits that individual energy eigenstates already appear locally thermal. Subsequent studies have extended this concept to the full ETH, which captures higher-order correlations among matrix elements through nontrivial relations. In this work, we perform a detailed exact-diagonalization study of finite-size corrections to these relations in the canonical ensemble. We distinguish two distinct sources of corrections: those arising from energy fluctuations, which decay polynomially with system size, and those originating from fluctuations within each energy window, which decay exponentially with system size. In particular, our analysis resolves the puzzle that, for certain observables, finite-size corrections exhibit anomalous growth with increasing system size even in chaotic systems. Our results provide a systematic and practical methodology for validating the full ETH in quantum many-body systems.
The eigenstate thermalization hypothesis (ETH) explains why nonintegrable quantum many-body systems thermalize internally if the Hamiltonian lacks symmetries. If the Hamiltonian conserves one quantity ("charge"), the ETH implies thermalization within a charge sector-in a microcanonical subspace. But quantum systems can have charges that fail to commute with each other and so share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the Hamiltonian will have degeneracies, so the ETH need not imply thermalization. We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and invoking the approximate microcanonical subspace introduced in quantum thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH in calculating local operators' time-averaged and thermal expectation values. In many cases, we prove, the time average thermalizes. However, we find cases in which, under a physically reasonable assumption, the time average converges to the thermal average unusually slowly as a function of the global-system size. This work extends the ETH, a cornerstone of many-body physics, to noncommuting charges, recently a subject of intense activity in quantum thermodynamics.