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arXiv:2306.11829 · Physical review. E

Single-quasiparticle eigenstate thermalization

Piotr Tokarczyk, Lev Vidmar, Patrycja Łydżba

Published Feb 5, 2024• 5 citations
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Abstract

Quadratic Hamiltonians that exhibit single-particle quantum chaos are called quantum-chaotic quadratic Hamiltonians. One of their hallmarks is single-particle eigenstate thermalization introduced in Łydżba et al. [Phys. Rev. B 104, 214203 (2021)2469-995010.1103/PhysRevB.104.214203], which describes statistical properties of matrix elements of observables in single-particle eigenstates. However, the latter has been studied only in quantum-chaotic quadratic Hamiltonians that obey the U(1) symmetry. Here, we focus on quantum-chaotic quadratic Hamiltonians that break the U(1) symmetry and, hence, their "single-particle" eigenstates are actually single-quasiparticle excitations introduced on the top of a many-body state. We study their wave functions and matrix elements of one-body observables, for which we introduce the notion of single-quasiparticle eigenstate thermalization. Focusing on spinless fermion Hamiltonians in three dimensions with local hopping, pairing, and on-site disorder, we also study the properties of disorder-induced near zero modes, which give rise to a sharp peak in the density of states at zero energy. Finally, we numerically show equilibration of observables in many-body eigenstates after a quantum quench. We argue that the latter is a consequence of single-quasiparticle eigenstate thermalization, in analogy to the U(1) symmetric case from Łydżba et al. [Phys. Rev. Lett. 131, 060401 (2023)0031-900710.1103/PhysRevLett.131.060401].

DOI
10.1103/physreve.109.024102

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Quadratic Hamiltonians that exhibit single-particle quantum chaos are called quantum-chaotic quadratic Hamiltonians. One of their hallmarks is single-particle eigenstate thermalization introduced in Łydżba et al. [Phys. Rev. B 104, 214203 (2021)2469-995010.1103/PhysRevB.104.214203], which describes statistical properties of matrix elements of observables in single-particle eigenstates. However, the latter has been studied only in quantum-chaotic quadratic Hamiltonians that obey the U(1) symmetry. Here, we focus on quantum-chaotic quadratic Hamiltonians that break the U(1) symmetry and, hence, their "single-particle" eigenstates are actually single-quasiparticle excitations introduced on the top of a many-body state. We study their wave functions and matrix elements of one-body observables, for which we introduce the notion of single-quasiparticle eigenstate thermalization. Focusing on spinless fermion Hamiltonians in three dimensions with local hopping, pairing, and on-site disorder, we also study the properties of disorder-induced near zero modes, which give rise to a sharp peak in the density of states at zero energy. Finally, we numerically show equilibration of observables in many-body eigenstates after a quantum quench. We argue that the latter is a consequence of single-quasiparticle eigenstate thermalization, in analogy to the U(1) symmetric case from Łydżba et al. [Phys. Rev. Lett. 131, 060401 (2023)0031-900710.1103/PhysRevLett.131.060401].
Eigenstate thermalization has been numerically shown to occur for few-body observables in a wide range of nonintegrable models. For intensive sums of few-body observables, a weaker version of eigenstate thermalization known as weak eigenstate thermalization has been proved to occur in general. Here, we unveil a stricter weak eigenstate thermalization phenomenon that occurs in quadratic models exhibiting quantum chaos in the single-particle sector (quantum-chaotic quadratic models) and in integrable interacting models. In such models, we argue that few-body observables that have a properly defined system-size independent norm are guaranteed to exhibit at least a polynomially vanishing variance (over the entire many-body energy spectrum) of the diagonal matrix elements, a phenomenon we dub normal weak eigenstate thermalization. We prove that normal weak eigenstate thermalization is a consequence of single-particle eigenstate thermalization, i.e., it can be viewed as a manifestation of quantum chaos at the single-particle level. We report numerical evidence of normal weak eigenstate thermalization for quantum-chaotic quadratic models such as the three-dimensional Anderson model in the delocalized regime and the power-law random banded matrix model, as well as for the integrable interacting spin-12\frac{1}{2}21​ XYZ and XXZ models.
We study the matrix elements of local and nonlocal operators in the single-particle eigenstates of two paradigmatic quantum-chaotic quadratic Hamiltonians; the quadratic Sachdev-Ye-Kitaev (SYK2) model and the three-dimensional Anderson model below the localization transition. We show that they display eigenstate thermalization for normalized observables. Specifically, we show that the diagonal matrix elements exhibit vanishing eigenstate-to-eigenstate fluctuations and that their variance is proportional to the inverse Hilbert space dimension. We also demonstrate that the ratio between the variance of the diagonal and the off-diagonal matrix elements is 2, as predicted by the random matrix theory. We study distributions of matrix elements of observables and establish that they need not be Gaussian. We identify the class of observables for which the distributions are Gaussian.
In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians, namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians such as the three-dimensional (3D) Anderson model at weak disorder. Moreover, it describes the average entanglement entropy in Hamiltonians without particle-number conservation, such as the SYK2 model in the Majorana fermion formulation and the 3D Anderson model with additional terms that break particle-number conservation. We extend our analysis to the average bipartite second R\'enyi entanglement entropy of eigenstates of the same quadratic Hamiltonians, which is derived analytically and tested numerically. We conjecture that our results for the entanglement entropies of many-body eigenstates apply to quadratic Hamiltonians whose single-particle eigenstates exhibit quantum chaos, which we refer to as quantum-chaotic quadratic Hamiltonians.
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.
It is an outstanding goal to unveil the key features of quantum dynamics at eigenstate transitions. Focusing on quadratic fermionic Hamiltonians that exhibit localization transitions, we identify physical observables that exhibit scale-invariant critical dynamics at the transition when quenched from the initially localized charge-density-wave states. The identification is based on two ingredients: (a) a relationship between the time evolution of observables in a many-body state and the transition probabilities of single-particle states, and (b) scale invariance of transition probabilities, which generalizes the corresponding recent result for survival probabilities [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 131, 060404 (2023); Phys. Rev. Res. 5, 043301 (2023)]. These properties suggest that the scale-invariant critical dynamics in the quantum-quench dynamics is also exhibited by the observables, which share the common eigenbasis with the Hamiltonian before the quench. Focusing on experimentally relevant observables such as site occupations and the particle imbalance, we numerically demonstrate their critical behavior at the eigenstate transitions in the three-dimensional Anderson model and the one-dimensional Aubry-Andr\'e model.
Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a single itinerant fermion. In our study, we focus on translationally invariant observables including the charge and energy current, thereby also connecting the ETH with transport properties. Considering observables with a Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH in the model taking into account latest developments. A particular emphasis is on the analysis of the structure of the offdiagonal matrix elements $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$ in the limit of small eigenstate energy differences $\ensuremath{\omega}={E}_{\ensuremath{\beta}}\ensuremath{-}{E}_{\ensuremath{\alpha}}$. Removing the dominant exponential suppression of $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$, we find that (1) the current matrix elements exhibit a system-size dependence that is different from other observables under investigation and (2) matrix elements of several other observables exhibit a Drude-like structure with a Lorentzian frequency dependence. We then show how this information can be extracted from the autocorrelation functions as well. Finally, our study is complemented by a numerical analysis of the fluctuation-dissipation relation for eigenstates in the bulk of the spectrum. We identify the regime of $\ensuremath{\omega}$ in which the well-known fluctuation-dissipation relation is valid with high accuracy for finite systems.