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arXiv:2607.19861

A Multi-Resolvent Hierarchy for the ETH Smooth Function

Zhiqiang Huang

quant-phSubmitted Jul 22, 2026• Announced Jul 23, 2026• 0 citations• Revised Sep 3, 2026 (v2)
v1 · Jul 22, 2026v2 · Sep 3, 2026
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Abstract

The eigenstate thermalization hypothesis (ETH) parametrizes off-diagonal matrix elements by a smooth function whose microscopic origin remains largely phenomenological. We develop a multi-resolvent hierarchy that derives this smooth structure from the microscopic Hamiltonian. Starting from exact projection identities, we express the ETH variance as $f_{ji}^2=D_{ji}+g_{ji}$, where $D_{ji}$ is the diagonal-overlap baseline and $g_{ji}=\sum_{r\ge2}g_{ji}^{(r)}$ is a systematically improvable hierarchy of multi-channel interference processes. The leading $r=3$ sector generates an odd-parity component in the energy difference, inaccessible to parity-preserving single-resolvent closures. The same resolvent construction yields an exact covariance representation of eigenstate fluctuations. Under amplitude isotropy, decorrelation, and regularity, it reduces to the Gaussian limit, with $q-3=κ_4/\langle c^2\rangle^2$; normalization further fixes the cross-channel covariance $\bar C_i$, including its energy-resolved form. Exact diagonalization verifies these relations in random-matrix and structured systems, while the latter exhibit controlled breakdown of the isotropic Gaussian closure. The framework thus provides a microscopic hierarchy for both the smooth and fluctuation sectors of subsystem ETH.

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36 pages

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The fact that generic isolated many-body quantum systems thermalize is understood using the eigenstate thermalization hypothesis (ETH). In recent years, there has been much interest in the behavior of the ETH spectral functions, which characterize the smooth dependence of the variance of the off-diagonal matrix elements of observables on the associated energy and frequency, and whose low-frequency part contains information about the long-time dynamics. In finite systems described by the ETH, the spectral functions are expected to exhibit plateaus below a characteristic frequency $ω^{}_{\mathrm{ETH}}$. In this regime, the statistics of the matrix elements of observables are expected to be described by random matrix theory. Related frequencies that have been studied in the literature are $ω^{}_{\mathrm{SFF}}$, which controls the onset of random-matrix behavior in the spectral form factor, and the transport frequencies $ω^{}_{\mathrm{tr}}$, which are derived from transport coefficients. However, a direct quantitative comparison of these frequencies is lacking. Using exact diagonalization, we conduct such a comparison for the spectral functions of current operators in clean and disordered quantum spin ladders with diffusive energy and spin transport. We find clear evidence for the expected low-frequency plateaus in the ETH spectral functions. For the accessible system sizes, $ω^{}_{\mathrm{SFF}}$ is consistent with the extent of the plateaus, while the transport frequencies are systematically larger and lie in the nonuniversal regime of the spectral functions. Our findings highlight the need to better understand the origin of these quantitative differences.

cond-mat.stat-mechCross-listedarXiv:2609.24602•Sep 22, 2026•0 citations
The eigenstate thermalization hypothesis (ETH) parametrizes off-diagonal matrix elements by a smooth function whose microscopic origin remains largely phenomenological. We develop a multi-resolvent hierarchy that derives this smooth structure from the microscopic Hamiltonian. Starting from exact projection identities, we express the ETH variance as fji2=Dji+gjif_{ji}^2=D_{ji}+g_{ji}fji2​=Dji​+gji​, where DjiD_{ji}Dji​ is the diagonal-overlap baseline and gji=∑r≥2gji(r)g_{ji}=\sum_{r\ge2}g_{ji}^{(r)}gji​=∑r≥2​gji(r)​ is a systematically improvable hierarchy of multi-channel interference processes. The leading r=3r=3r=3 sector generates an odd-parity component in the energy difference, inaccessible to parity-preserving single-resolvent closures. The same resolvent construction yields an exact covariance representation of eigenstate fluctuations. Under amplitude isotropy, decorrelation, and regularity, it reduces to the Gaussian limit, with q−3=κ4/⟨c2⟩2q-3=κ_4/\langle c^2\rangle^2q−3=κ4​/⟨c2⟩2; normalization further fixes the cross-channel covariance Cˉi\bar C_iCˉi​, including its energy-resolved form. Exact diagonalization verifies these relations in random-matrix and structured systems, while the latter exhibit controlled breakdown of the isotropic Gaussian closure. The framework thus provides a microscopic hierarchy for both the smooth and fluctuation sectors of subsystem ETH.
Eigenstate thermalization fixes the smooth energy-resolved envelope of few-body observables but not the microscopic overlap statistics that realize it. In the multi-resolvent hierarchy, this missing information appears in a two-resolvent fluctuation sector. We solve that sector exactly in a tractable benchmark, random free fermions, obtaining a non-Porter-Thomas intensity hierarchy, a structured channel-distance covariance, and an exact overlap kernel that closes the two-resolvent covariance. We then reconstruct the same sector without using its exact solution: a projected self-consistency scheme fixes the irreducible vertex through independently computable projections, while unconstrained residuals identify the sectors missed by a minimal ansatz and are completed exactly by a finite-size Weingarten evaluation, without fitting. For interacting deformations, we prove a model-independent moment identity showing that the deep-plane coefficients are fixed by Hamiltonian-moment covariances. This yields an exact rigidity law: pair-hopping interactions freeze the sector, dense perturbations add only an isotropic layer, and density-density interactions produce an explicit polynomial deformation in channel geometry. By contrast, the bulk sector requires the interacting eigenstate-overlap kernel and exhibits a logarithmically divergent perturbative response. The fluctuation sector therefore splits into a rigid moment-determined part and an intrinsically nonperturbative eigenstate-resolved part.
The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the rrr-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of rrr and kkk---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at r=3r=3r=3 (k=2,3,4k=2,3,4k=2,3,4), r=4r=4r=4 and r=5r=5r=5 (k=2k=2k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios qrq_rqr​, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.
The exact multi-resolvent hierarchy of dense many-body spectra encodes higher-order spectral-weight correlations, but closing it predictively for interacting Hamiltonians remains a challenge. We introduce the multi-resolvent self-consistency (MRSC) equations, which separate an exact algebraic hierarchy from an approximate finite-parameter closure. The exact Feshbach geometric identity organizes r-resolvent cumulants into a ladder over irreducible vertices Gamma^(r), and the projected hierarchy expresses those vertices as functionals of the cumulants. The closure works in finite parameters: a connectivity basis organizes a general Hamiltonian's seed into class amplitudes, and a causal r-fold Stieltjes representation with a moment-field identity reduce moment matching to a finite-dimensional problem closed at a predictive fixed point. On the free-fermion benchmark the tested exact identities are reproduced and the Faddeeva closure is recovered at r=1. In a blind test on the interacting Ising chain, with the exact two-resolvent data withheld, the Hamiltonian-derived closure predicts the tested probes to 0.35%-0.97%. At the observable level the exact two-resolvent sector supplies a sign-correct, energy-resolved correction to the single-resolvent baseline for the off-diagonal eigenstate-thermalization-hypothesis (ETH) variance, of order one relative to it. Closure errors organize hierarchically: a level-by-level deficit decomposition, anchor measurements of the isolatable terms, and an explicit propagation bound. The same moments further yield a diagnostic chain for the window conditions of local rotational invariance -- an exact drift-kernel identity, a proven single-state identifiability boundary, and a transferability census over couplings and realizations -- a finite-order framework, not a proof of universal emergence.
We analyze thermalization and eigenstate thermalization in the random free fermion model, a quadratic Hamiltonian with a Gaussian Orthogonal Ensemble hopping matrix, by an exact random-matrix calculation. For a single-particle observable AAA we compute the ensemble average and fluctuations of Tr⁡(Aρ(t))\operatorname{Tr}(Aρ(t))Tr(Aρ(t)): we derive the relaxation envelope g2(t/τλ)g^2(t/τ_λ)g2(t/τλ​) with τλ=ℏ/(2ηN)τ_λ= \hbar/(2η\sqrt{N})τλ​=ℏ/(2ηN​) and its oscillatory ∣t∣−3|t|^{-3}∣t∣−3 tail, both properties of the semicircular limiting density, and we show that the variance of the fluctuations is O(1/N)O(1/N)O(1/N), so that the relaxation profile self-averages at fixed scaled time. Incorporating particle-number conservation, we prove a factorization theorem for the energy-shell statistics of the many-body eigenstates: the energy-independent ETH structure follows from the statistical independence of the eigenvector matrix and the spectrum, and at fixed realization the energy constraint enters only through a covariance subtraction of relative order 1/N1/N1/N. A controlled eigenvector--eigenvalue correlation deformation H(ε)=H0+εH1H(\varepsilon)=H_0+\varepsilon H_1H(ε)=H0​+εH1​ then generates, at first order in ε\varepsilonε, the energy dependence of the diagonal energy-resolved slope and of the off-diagonal spectral function, a single-particle diagnostic of the ETH structure in the allowed particle-hole sector. Correlation-function fluctuations and the single-particle spectral form factor finally identify the subleading O(1/N)O(1/N)O(1/N) late-time fingerprints of two-level spectral correlations. The model thus separates eigenbasis chaos from spectral chaos and identifies the statistical origin of energy-independent versus energy-dependent ETH.
We develop a nonperturbative framework for generic nonintegrable many-body systems that reorganizes the expansion of diagonal Green's functions. Starting from exact projection identities and the spectral representation of the resolvent, we derive a recursive hierarchy for the self-energy in which cross-correlated propagation processes are systematically rewritten in terms of diagonal resolvents. Under a diagonal closure approximation, the hierarchy becomes formally closed yet remains systematically improvable. The framework combines two nonperturbative mechanisms. First, a Lanczos continued-fraction representation provides a recursive single-resolvent structure that naturally produces non-Lorentzian spectral features beyond self-consistent Born approximations. Second, an exact projected multi-resolvent hierarchy introduces nonlocal frequency couplings through products of resolvents and their Hilbert transforms. These contributions mix parity sectors under energy reflection and generate spectral skewness, which is absent in single-resolvent closures. To solve the resulting equations, we employ a hierarchy of Lorentzian, Gaussian, and Voigt-type ansätze together with an effective Faddeeva self-energy representation ensuring analyticity and causality. Spectral broadening, distribution tails, and higher-order fluctuations emerge from the interplay between continued-fraction recursion and multi-resolvent correlations. The framework requires no small expansion parameters or diagrammatic truncations, relying instead on ETH-type statistical assumptions appropriate for dense chaotic spectra. It provides a unified route from microscopic interactions to emergent spectral structure, revealing a progression from single-pole self-consistent dynamics to continued-fraction renormalization and finally to multi-resolvent interference effects.
The fact that generic isolated many-body quantum systems thermalize is understood using the eigenstate thermalization hypothesis (ETH). In recent years, there has been much interest in the behavior of the ETH spectral functions, which characterize the smooth dependence of the variance of the off-diagonal matrix elements of observables on the associated energy and frequency, and whose low-frequency part contains information about the long-time dynamics. In finite systems described by the ETH, the spectral functions are expected to exhibit plateaus below a characteristic frequency ωETHω^{}_{\mathrm{ETH}}ωETH​. In this regime, the statistics of the matrix elements of observables are expected to be described by random matrix theory. Related frequencies that have been studied in the literature are ωSFFω^{}_{\mathrm{SFF}}ωSFF​, which controls the onset of random-matrix behavior in the spectral form factor, and the transport frequencies ωtrω^{}_{\mathrm{tr}}ωtr​, which are derived from transport coefficients. However, a direct quantitative comparison of these frequencies is lacking. Using exact diagonalization, we conduct such a comparison for the spectral functions of current operators in clean and disordered quantum spin ladders with diffusive energy and spin transport. We find clear evidence for the expected low-frequency plateaus in the ETH spectral functions. For the accessible system sizes, ωSFFω^{}_{\mathrm{SFF}}ωSFF​ is consistent with the extent of the plateaus, while the transport frequencies are systematically larger and lie in the nonuniversal regime of the spectral functions. Our findings highlight the need to better understand the origin of these quantitative differences.