Eigenstate Thermalization and Quantum Chaos in the Holstein Polaron Model

Eigenstate thermalization and quantum chaos in the Holstein polaron model

David Jansen,1, 2 Jan Stolpp,3 Lev Vidmar,4 and Fabian Heidrich-Meisner3 1Department of Physics, Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-UniversitätMünchen,D-80333 München,Germany 2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München,Germany 3Institut for Theoretical Physics, Georg-August-UniversitätGöttingen,D-37077 Göttingen,Germany 4Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia The eigenstate thermalization hypothesis (ETH) is a successful theory that provides sufficient cri- teria for ergodicity in quantum many-body systems. Most studies were carried out for Hamiltonians relevant for ultracold quantum gases and single-component systems of spins, fermions, or bosons. The paradigmatic example for thermalization in solid-state physics are phonons serving as a bath for electrons. This situation is often viewed from an open-quantum system perspective. Here, we ask whether a minimal microscopic model for electron-phonon coupling is quantum chaotic and whether it obeys ETH, if viewed as a closed quantum system. Using exact diagonalization, we address this question in the framework of the Holstein polaron model. Even though the model describes only a single itinerant electron, whose coupling to dispersionless phonons is the only integrability-breaking term, we find that the spectral statistics and the structure of Hamiltonian eigenstates exhibit essen- tial properties of the corresponding random-matrix ensemble. Moreover, we verify the ETH ansatz both for diagonal and offdiagonal matrix elements of typical phonon and electron observables, and show that the ratio of their variances equals the value predicted from random-matrix theory.

I. INTRODUCTION ization in solid-state physics is the phonons being a bath for the electrons. There, the distinction between bath Understanding whether and how an isolated quantum and system is not made by a real-space bipartition that many-body system approaches thermal equilibrium after is often considered in ETH studies, but by tracing out one being driven far from equilibrium has been a tremendous type of physical degree of freedom, namely the phonons. theoretical challenge since the birth of quantum mechan- Moreover, in metals, direct electron-electron interactions ics [1,2]. This topic became viral after quantum-gas are often irrelevant while the coupling to phonons leads to experiments [3–5] and other quantum simulators such as eﬀective interactions responsible for thermalization and ion traps became available that are, to a good approxi- transport. mation, closed quantum systems. Using these platforms, A second timely motivation to study nonequilib- a series of experiments on nonequilibrium dynamics was rium problems in electron-phonon coupled systems stems conducted, focusing either on the generic case of thermal- from the experimental advances with time-resolved spec- izing systems [6–9] or the exceptions such as integrable troscopy [47]. Such experiments have stimulated an models [10–12] or many-body localization [13–15]. increased interest in exploring nonequilibrium dynam- An additional impetus to the theoretical community ics of isolated electron-phonon coupled systems theoret- was given by the work of Rigol et al. [16] who demon- ically [48–66]. Among them, systems with a single ex- strated that the eigenstate thermalization hypothesis cited electron coupled to phonons (the so-called polaron (ETH), pioneered by Deutsch [17] and Srednicki [18], pro- case) provide a platform for accurate numerical simula- vides a relevant framework to describe statistical prop- tions [49, 58, 60, 67, 68], allowing us to demonstrate, for erties of eigenstates of lattice Hamiltonians, applicable certain nonequilibrium protocols, equilibration [52, 58] to ongoing experiments with ultracold atoms on optical and indications for thermalization [61]. lattices [7, 13, 14]. As a crucial consequence, if the ETH In this paper, we explore quantum ergodicity of a po- is satisﬁed in a given quantum many-body system, it im- laron system by studying indicators of quantum chaos plies thermalization of local observables after a unitary and the validity of the ETH, which implies thermalization time evolution [19–22].

arXiv:1902.03247v2 [cond-mat.str-el] 26 Apr 2019 So far, the ETH has been verified for a wide number of lattice models such as nonintegrable spin-1/2 chains [23– 33], ladders [26, 34–36] and square lattices [37–39], interacting spinless fermions [40, 41], Bose-Hubbard [26, 42] and Fermi-Hubbard chains [43], dipolar hard-core bosons [44], quantum dimer models [45] and Fibonacci anyons [46]. In these examples, mostly, direct two-body interactions in systems of either spins, fermions or bosons are responsible for rendering the system ergodic. In con- FIG. 1. Quantum chaos and thermalization (right panel) densed matter systems, however, the presence of phonons emerging from coupling of a single electron (upper left) to is ubiquitous. In fact, the textbook example of thermal- independent local harmonic oscillators (lower left). 2 for generic far-from-equilibrium initial states. We focus in the statistics of the neighboring level spacings for dif- on the one-dimensional Holstein polaron model, which ferent parameter regimes of the model. We then focus describes properties of a single electron locally coupled on the analysis of the ETH in Sec.IV and explore both to dispersionless phonons, as sketched in Fig.1. Using diagonal and offdiagonal matrix elements of observables. an exact-diagonalization analysis, we demonstrate that We conclude in Sec.V. despite the apparent simplicity of the model, it exhibits clear manifestations of quantum chaos and the Hamil- tonian eigenstates in the bulk of the spectrum obey the II. THE HOLSTEIN POLARON MODEL ETH. Therefore, the model appears to encompass minimal ingredients for thermalization in condensed matter The one-dimensional Holstein polaron model on a lat- systems including both electrons and phonons. Note that tice with L sites is described by the Hamiltonian dispersionless phonons do not match the standard properties expected for a thermal reservoir [69] since they ex- Hˆ = Hˆkin + Hˆph + Hêph . (1) hibit, e.g., infinite autocorrelation times. The results of the present study, however, suggest that the presence of It consists of the electron kinetic energy operator phonons with a dispersion is not a requirement for thermalization and quantum ergodicity in many-body quan- L ˆ X † † tum systems. Hkin = −t0 cˆjcˆj+1 +c ˆj+1cˆj , (2) We first verify that the statistics of neighboring energy- j=1 level spacings is well described by the Wigner-Dyson dis- wherec ˆ is the electron annihilation operator acting on tribution, and that the averages of restricted gap ratios j site j (in the Holstein model, a spin-polarized gas of elec- agree with predictions of the Gaussian orthogonal ensem- trons is considered). Next, the phonon-energy operator ble (GOE). We then study matrix elements of observables reads in both the electron and phonon sector. For the diagonal matrix elements we use the most restrictive criterion L ˆ X ˆ†ˆ to test the ETH [27, 38], i.e., we show that eigenstate- Hph = ~ω0 bjbj , (3) to-eigenstate fluctuations vanish for all eigenstates in a j=1 finite energy-density window. We then present an exten- ˆ sive analysis of the offdiagonal matrix elements and ex- where bj is the phonon annihilation operator at site j and tract the universal function at different energy densities. ~ ≡ 1, and the local coupling of phonons to the electron Finally, we study variances of both diagonal and offdiag- † densityn ˆj =c ˆ cˆ is onal matrix elements in narrow microcanonical windows j j and find that their ratio approaches a universal number, L which matches predictions of the GOE. ˆ X ˆ† ˆ Heph = γ bj + bj nˆj . (4) An interesting aspect of the Holstein polaron model j=1 is that the electron-phonon coupling is the only mechanism that breaks integrability of the uncoupled elec- We use periodic boundary conditions,c ˆL+j ≡ cˆj and we tronic and bosonic subsystems and gives rise to quan- set the lattice spacing to a = 1. Throughout the paper tum ergodicity. Since there is only a single electron in we use t0 as the energy unit and set t0 = 1 in numerical the system, expectation values of the electron-phonon calculations. coupling energy are not extensive but of order O(1) in We employ full exact diagonalization to numerically every Hamiltonian eigenstate. The question of the mini- obtain all eigenvalues and eigenstates of the Holstein po- mal perturbation strength to render an integrable system laron model, denoted by {Eα} and {|αi}, respectively. quantum chaotic is highly nontrivial and can be traced We truncate the number of local bosonic degrees of back to the pioneering work on ETH by Deutsch [17]. Re- freedom, with M representing the maximal number of cently, a large number of studies addressed the influence phonons per site. We exploit translational invariance of integrability-breaking static impurities on quantum to split the Hamiltonian into L distinct sectors with ergodicity and thermalization (see, e.g., Refs. [70–74]), quasimomenta k. Each sector contains one electron, i.e., ˆ P † and showed that an O(1) integrability-breaking term is hNi = h j cˆjcˆji = 1. The Hilbert-space dimension of enough to induce quantum-chaotic statistics of energy each k sector is D0 = (M + 1)L, and hence the total levels [75–79]. Here, we provide numerical evidence that Hilbert-space dimension is D = L(M + 1)L. an O(1)-integrability-breaking term, introduced by an Since we use exact diagonalization, it is necessary to itinerant electron coupled to noninteracting phonons, is deal with the finite cutoff M in the local phonon number. sufficient to observe perfect ETH properties of all Hamil- This introduces a dependence on M in some quantities. tonian eigenstates in the bulk of the spectrum. The approach taken in this paper is to establish, for a The paper is organized as follows. We introduce the fixed M, quantum-chaotic properties of the model and Holstein polaron model and its basic spectral properties the validity of the ETH for eigenstates that fall within in Sec.II. In Sec.III we study quantum-chaos indicators a finite window of energy densities. We will argue that 3

1.0 random-matrix theory, in particular, to the GOE, which represents the appropriate symmetry class for the Hol- stein polaron model, i.e., models with time-reversal sym-

) metry. Historically, random-matrix theory was shown to E ( 0.5 provide a relevant framework to describe spectral prop- W erties of quantum systems whose classical counterparts are chaotic [80]. As a consequence, all quantum systems with spectral statistics identical to the ones in random 0.0 0 1 2 matrices are called quantum chaotic [81]. Since there is E / Eav no classical counterpart of the Holstein polaron model, we refer to the model as being quantum chaotic if its spectral statistics matches those of the GOE. Note that FIG. 2. The density of eigenstates W (E) of the Holstein the level-spacing statistics (to be studied below) can not polaron model using a bin width ∆E = E /20. The system √ av distinguish between a completely ergodic system and a parameters are ω0/t0 = 1/2, γ/t0 = 1/ 2, L = 8 and M = 3, which corresponds to D = L(M + 1)L ≈ 5 × 105 eigenstates. system with a nonzero number of nonergodic eigenstates The smooth line is a Gaussian function [see Eq. (6)], where Γ whose fraction vanishes in the thermodynamic limit [82– is given by Eq. (8). 89]. In Sec.IV we show that the system under investigation here is indeed completely ergodic, i.e., all eigenstates away from the edges of the spectrum exhibit ETH prop- the qualitative conclusions do not depend on the choice erties [20]. of M. In finite systems, quantum-chaotic properties are usu- We define the distribution of the Hamiltonian eigen- ally observed most unambiguously at intermediate pa- values {Eα}, i.e., the density of eigenstates, as rameter regimes, i.e., when all parameters are of the same order. Moreover, the Holstein polaron model has two in- D X tegrable limits (the noninteracting limit γ → 0 and the W (E) = D−1 δ(E − E ) . (5) α small-polaron limit t0 → 0), where the quantum-chaotic α=1 properties are expected to be absent. Exploring the parameter regimes in which the model exhibits quantum- In Fig.2, we plot W (E) for a system with L = 8, M = 3 √ chaotic properties is one of the main goals of this section. and the model parameters γ/t = 1/ 2, ω /t = 1/2. 0 0 0 In the parameter regime where the model is quan- The density of eigenstates is close to a Gaussian function tum chaotic, this manifests itself for eigenstates at a √ 2 2 nonzero energy density above the ground state and be- g(E) = E /( 2πΓ)e−(E−Eav) /(2Γ ) , (6) av low the highest energy eigenstate [20]. In the Holstein which is shown in the same figure (solid line). The av- polaron model, the ground-state energy density in the 2 limit L → ∞ is Emin/Eav = 0 and the highest-eigenstate erage energy Eav and the energy variance Γ are given by energy density is Emax/Eav = 2. We introduce a control parameter η ∈ [0, 1) to consider eigenstates in a finite ω M energy-density window, such that E = hHˆ i = D−1Tr{Hˆ } = L 0 (7) av 2 Eav − Eα < η if Eα < Eav (9) and Eav − Emin ω2M(M + 2) Γ2 = hHˆ 2i − hHˆ i2 = L 0 + 2t2 + γ2M. (8) and 12 0 Eα − Eav The latter result is consistent with the expectation that < η if Eα > Eav . (10) √ Emax − Eav the relative standard deviation Γ/L vanishes as 1/ L when L → ∞. On the other hand, if L is kept fixed and In our model the corresponding set of eigenstates within M → ∞, the relative standard deviation Γ/M does not the target energy-density window is p vanish but goes to a constant Γ/M → ω0 L/12. We {k} study finite-size dependencies by fixing M and making L Zη = {|αi ; Eα/Eav ∈ [1 − η, 1 + η]} , (11) larger. where {k} denotes the momentum sectors from which eigenstates are selected. It is well known [20, 90] that III. QUANTUM-CHAOS INDICATORS quantum-chaos indicators can only be observed in statistics of Hamiltonian eigenvalues of a single symmetry sec- We first study statistical properties of the spectrum tor. We therefore limit our analysis in this section to the of the Holstein polaron model. A standard approach quasimomentum sector k = 2π/L in the set of eigenstates {k} is to compare these properties with predictions of the Zη in Eq. (11) and fix η = 2/3. 4

0.8 0.56 2.0 ω0 = t0/2 (a) ω0 = t0/2 (b) 1.5 0.54 ) ) v ˜ r a s ( 1.0 ( ˜ 0.4 r P P 0.52 L = 8 0.5 GOE L = 7 Poisson 2 2 L = 6 γ = t0 /2 0.50 0.0 0.0 0 1 2 3 4 0 0.5 1 0 1 2 3 γ/t0 r˜ s 0.56 0.56 L = 8 (c) M = 1, ω0 = t0/2 (d) FIG. 3. Level-spacing statistics P(s) after the unfolding procedure (see the√ text for details) for L = 8, M = 3, ω0/t0 = 0.5 0.54 0.54 and γ/t = 1/ 2. Numerical results are shown as a histogram v v

0 a a ˜ ˜ for the k = 2π/L momentum sector. We consider eigenstates r r within an energy density window given by η = 2/3 in Eq. (11), 0.52 0.52 ω0 = t0/2 L = 16 which for the particular model parameters corresponds to 90% ω0 = t0 L = 14 of all eigenstates. Smooth lines are the Wigner-Dyson distri- ω = 4t L = 12 0.50 0 0 0.50 bution PWD(s), given by Eq. (12). 0 1 2 3 4 0 1 2 3 4 γ/t0 γ/t0

We study quantum-chaos indicators by analyzing the FIG. 4. Statistics of the restricted gap ratior ˜α introduced statistics of neighboring energy-level spacings δα = in Eq. (14). Symbols in (a), (c) and (d) show the average Eα+1 −Eα. We perform an unfolding of the energy levels rãv as a function of γ/t0. We show results at ω0/t0 = 1/2, {k} M = 3 and three different system sizes L in (a), results at in Zη such that the mean level spacing is one [81, 91]. The unfolding procedure is carried out by introducing L = 8, M = 3 and three different values of ω0/t0 in (c), and results at ω /t = 1/2, M = 1 and for three different a cumulative spectral function G(E) = P Θ(E − E ), 0 0 α α system sizes L in (d). Solid lines in (a), (c) and (d) denote where Θ is the unit step function. In the practical anal- r˜GOE ≈ 0.5307, which is obtained numerically for asymptoti- ysis, we find it useful to smoothen it by fitting a polyno- cally large systems [92], while the dashed line√ in (a) denotes mial of degree six g6(E) to G(E). The spectral analysis is R 1 the analytical result 0 r˜PGOE(˜r)d˜r = 4 − 2 3 ≈ 0.5359 ob- then performed on the unfolded neighboring level spac- tained from the GOE distribution PGOE(˜r) for 3 × 3 matri- ings sα = g6(Eα+1) − g6(Eα). We verified that using ces, see Eq. (15). (b) Histogram√ of the distribution P(˜r) at polynomials gn(E) of different degrees has a negligible ω0/t0 = 1/2, γ/t0 = 1/ 2, L = 8 and M = 3. The solid line 2 influence on the results. is PGOE(˜r) and the dashed line is the distribution 2/(1 +r ˜) The distribution P(s) of the unfolded neighboring level obtained using Poisson statistics [92]. spacings sα for the√ Holstein polaron model at ω0/t0 = 1/2 and γ/t0 = 1/ 2 is shown as a histogram in Fig.3. We compare the distribution P(s) to the Wigner-Dyson study a single number (defined below) derived from the distribution (smooth line in Fig.3) distribution. The authors of Ref. [93] considered the ratio of two πs 2 P (s) = e−πs /4 , (12) consecutive level spacings (shortly, the gap ratio) WD 2 δ r = α , (13) which was derived for an ensemble of 2 × 2 random ma- α δ trices and, despite not being exact, is now considered as α−1 a prototypical distribution of the neighboring level spac- and introduced the restricted gap ratio ings in the GOE [20]. We observe a reasonable agreement of P(s) with PWD(s). The agreement may appear sur- min{δα, δα−1} −1 r˜α = = min{rα, rα } . (14) prising since the only integrability-breaking term in the max{δα, δα−1} Holstein polaron model is Hêph (4), which is intensive. Moreover, our observations are qualitatively consistent A convenient property ofr ˜α is that no unfolding is neces- with previous studies of level statistics in spin-1/2 chains sary to eliminate the influence of a finite-size dependence with integrability-breaking terms of the same order [75– through the local density of states. 79]. We studyr ãv = hr˜iη, which represents the average The numerical data shown in Fig.3 are for the largest value of the restricted gap ratior ˜α within the set of k numerically available system at M = 3, and in a regime eigenstates Zη introduced in Eq. (11). A numerically where the model parameters are of the same order. We accurate prediction for the average ofr ˜α in the GOE is now extend our analysis to other parameter regimes and r˜GOE ≈ 0.5307 [92]. This value is different from the aver- system sizes. Instead of the full distribution P(s) we age value of uncorrelated energy levels (relevant for spec- 5 tra of integrable Hamiltonians with a Poisson distribu- While strictly speaking, the thermodynamic limit cor- tion of nearest level spacings), which isr ˜uncorr ≈ 0.3863. responds to sending both L, M → ∞, we observe the The fact thatr ˜GOE > r˜uncorr can be understood from the onset of quantum chaos (Sec.III) and the validity of the level repulsion of the GOE, which suppresses the presence ETH (to be studied in Sec.IV) even when L → ∞ and of small values ofr ˜α in the probability distribution P(˜r) M fixed. A plausible conjecture, which remains to be compared to the spectra with uncorrelated level spacings verified in future work, is that our key observations re- (dashed line), as illustrated in Fig.4(b) [20]. main valid when M → ∞. In the following section, we Figure4(a) shows ˜rav at ω0/t0 = 1/2 as a function of study the ETH properties of the Holstein polaron model γ/t0 for different system sizes L. When increasing L, the using M = 3, which turned out to be the best compro- fluctuations of the averages decrease, and at L = 8, the mise between allowing for relatively large local phonon averages approach the GOE predictionr ˜GOE with high fluctuations and still large enough lattice sizes. accuracy. Deviations fromr ˜GOE are the largest in the limits γ/t0 → 0 and γ/t0 → ∞, which is in agreement with observations in other quantum-chaotic Hamiltoni- IV. EIGENSTATE THERMALIZATION ans [39]. Before exploring the behavior ofr ãv in other parame- The previous section studied quantum-chaos indicators ter regimes, we verify thatr ãv does not only agree with and showed that for a wide range of model parameters predictions of the GOE, but that the entire distribution they agree with predictions of the GOE. We now focus on P(˜r) ofr ˜ does. We plot P(˜r) in Fig.4(b) at ω /t = 1/2 α √ 0 0 a specific√ set of model parameters given by ω0/t0 = 1/2, and γ/t0 = 1/ 2. The data show an excellent agreement γ/t0 = 1/ 2, M = 3, and test the validity of the ETH. with the analytical GOE prediction The ansatz for expectation values of observables in eigenstates of the Hamiltonian, known also as the ETH 27 r˜ +r ˜2 P (˜r) = , (15) ansatz [94], can be written as GOE 4 (1 +r ˜ +r ˜2)5/2 −S(E¯)/2 hα|Oˆ|βi = O(E¯)δα,β + e fO(E,¯ ω)Rα,β , (16) which is exact for 3 × 3 matrices and very accurate for asymptotically large systems [92]. As a side remark, where E¯ = (Eα + Eβ)/2 is the average energy of the pair note thatr ˜GOE = 0.5307 introduced above is a numer- of eigenstates |αi and |βi, ω = Eα − Eβ is the corre- ical result for asymptotically large systems, which on sponding energy difference, and S(E¯) is the thermody- the third digit differs from√ the result for 3 × 3 matri- namic entropy at energy E¯. The requirement for O(E¯) R 1 ¯ ces 0 r˜PGOE(˜r)d˜r = 4 − 2 3 ≈ 0.5359. Our numerical and fO(E, ω) is to be smooth functions of their argu- results in Fig.4(a) are accurate enough to resolve this ments, while Rα,β are random numbers with zero mean difference. and unit variance. When the ETH ansatz is applicable, We next study the influence of the phonon energy O(E¯) from the diagonal part is the microcanonical av- ω0/t0 on quantum-chaotic properties of the model. Fig- erage of an observable Oˆ, while the offdiagonal part can ure4(c) compares results for ˜rav versus γ/t0 at ω0/t0 = be used, e.g., to prove the fluctuation-dissipation relation 1/2, 1 and 4. As a general trend, fluctuations ofr ãv for a single eigenstate [20]. increase with ω0/t0. This is, in particular, evident for We study four different observables, two in the electron ω0/t0 = 4, where the agreement withr ˜GOE is weaker. In sector and two in the phonon sector. In the electron the antiadiabatic regime of the Holstein polaron model, sector, we study the electron kinetic energy Hˆkin, defined ω0 > 4t0, the phonon energy becomes larger than the in Eq. (2), and the quasimomentum occupation operator electron bandwidth and gaps in the spectrum increase. As a consequence, some values ofr ˜ in Eq. (14) become L α 1 X very small and the overall value ofr ˜ decreases. Since mˆ = e−i(l−j)q cˆ†cˆ (17) av q L j l finite-size effects are expected to increase with increas- j,l=1 ing gaps, we do not study quantum-chaotic properties in the antiadiabatic regime using the existing numerical at q = 0. Note that the expectation values of both elec- method. tron observables are intensive quantities, which is a con- Finally, we study the influence of the local phonon cut- sequence of having only a single electron in the Holstein ˆ off M on the restricted gap ratio. Figure4(d) shows re- polaron model. The kinetic energy Hkin is the sum over sults for M = 1 (hard-core bosons) at L = 12, 14 and 16, allm ˆ q. Therefore, if ETH behavior is observed form ˆ q=0 which leads to identical Hilbert-space dimensions as for it will not immediately imply ETH behavior for Hˆkin (and the results for M = 3 and L = 6, 7, 8, respectively, shown vice versa). Nevertheless, both quantities are related and in Fig.4(a). We observe increased fluctuations when M it may not be surprising to see ETH behavior in both decreases. Still, fluctuations decrease in both cases when cases. fixing M and increasing L, suggesting that in the limit In the phonon sector, we study the average phonon L → ∞, the model exhibits quantum-chaotic properties number per site Nˆph = Hˆph/(ω0L), where Hˆph was in- even in the case of hard-core bosons. troduced in Eq. (3), and the nearest-neighbor offdiagonal 6

√ FIG. 5. Diagonal matrix elements of observables hα|Oˆ|αi for ω0/t0 = 1/2, γ/t0 = 1/ 2, and M = 3 in the quasimomentum sector k = 2π/L. Results are plotted as a function of Eα/Eav, where the average energy Eav is deﬁned in Eq. (7). Symbols from the back to the front represent results for L = 6 (red), L = 7 (dark blue) and L = 8 (light blue), respectively. Arrows in (a) indicate the trend of the width of the matrix-element distribution as L increases.

(a) (b) (c) (d) 100 100 100 100 ) ) ) 0 ) n h i = 1 p k

-1 q -1 -1 -1 T

10 10 N 10 10 ( H m ( ( z ( z z

η = 1/3, zmax z

η = 1/3, z η -2 η = 2/3, z -2 -2 -2 10 m®ax 10 10 10 η = 2/3, z η 4 6® 8 10 4 6 8 10 4 6 8 10 4 6 8 10 L L L L

FIG. 6. Statistics of eigenstate-to-eigenstate fluctuations zα(O), see Eq. (20), for the same observables and model parameters as in Fig.5. We show the mean hziη(O), see Eq. (21), and the maximum zmax(O), see Eq. (22). We include eigenstates from the quasimomentum sectors with 0 ≤ k ≤ π that belong to the subspace Zη, defined in Eq. (19). At η = 1/3, the fraction of eigenstates included in Zη (out of the total number of eigenstates from the quasimomentum sectors 0 ≤ k ≤ π) are 37%, 44%, 50% and 55% for L = 5, 6, 7, 8, respectively, while at η = 2/3 the fractions are 70%, 79%, 85% and 90%, respectively. Solid lines are guides to the eyes, signaling an exponential decay with L. matrix element of the phonon one-body correlation ma- expectation values become smooth and sharp functions trix, with the underlying operator of the energy density in the limit L → ∞. The key step in validating the ETH for the diagonal L 1 X matrix elements of observables is to quantify finite-size Tˆ = ˆb†ˆb + H.c. . (18) 1 L j j+1 fluctuations of the results in Fig.5. We study whether j=1 (and how) eigenstate-to-eigenstate fluctuations in the The observables in both sectors are chosen such that one bulk of the spectrum vanish when L → ∞. To this end, observable is part of the Hamiltonian (1)[Hˆ and Nˆ ] we use the most restrictive criterion [27] defined below. kin ph The ETH is expected to be satisfied for the same set while the other one is not [m ˆ and Tˆ ]. q 1 of eigenstates in the bulk of the spectrum for which quantum-chaos indicators were found to exhibit GOE behavior in Sec.III. In Eq. (11), we defined these sets of A. Diagonal matrix elements of observables {k} eigenstates Zη , where η defines the interval of energy densities and {k} denotes the set of quasimomentum sec- We first focus on the eigenstate-expectation values tors, from which eigenstates are selected. For the analysis hα|Oˆ|αi, i.e., on the diagonal part of the ETH ansatz in of eigenstate-expectation values, we include eigenstates Eq. (16). Figures5(a)-5(d) show results for observables from quasimomentum sectors 0 ≤ k ≤ π and denote this in the k = 2π/L quasimomentum sector for three differ- set of eigenstates as ent system sizes L = 6, 7, 8, plotted versus the eigenstate [ energy density Eα/Eav. In all cases we observe clear Z = Z{k} . (19) manifestations of ETH, namely, eigenstate-expectation η η 0≤k≤π values are only functions of the energy density. In particular, results for different L suggest that the fluctua- Eigenstates from quasimomentum sectors k < 0 are ex- tions decrease with L and as a consequence, eigenstate- cluded from Zη since the eigenstate-expectation values 7 at k and −k are identical due to time-reversal symmetry. We introduce a measure of eigenstate-to-eigenstate- fluctuations of diagonal expectation values

zα(O) = hα + 1|Oˆ|α + 1i − hα|Oˆ|αi (20) and study two statistical properties of zα(O), the mean and the maximum. The mean is deﬁned as

−1 X hziη(O) = ||Zη|| |zα(O)| , (21)

|αi∈Zη FIG. 7. Sketch of a matrix of size D0 with matrix elements where ||Zη|| is the number of eigenstates in Zη, and the |hα|Oˆ|βi|. The shaded region in (a) represents the matrix maximum is elements included in the sum in Eq. (23). In (b), each square contains µ eigenstates, for which we calculate the variance of zmax(O) = max |zα(O)| . (22) 2 |αi∈Z diagonal matrix elements σdiag, Eq. (25), and the variance of η 2 offdiagonal matrix elements σoffdiag, Eq. (26). The statistics of eigenstate-to-eigenstate fluctuations, Eq. (20), was first studied in Ref. [27]. The number of eigenstates analyzed in that work was a finite frac- ETH ansatz (16). The moving average shown in the same tion of the total number of eigenstates, implying that panel indicates that the averaged function is almost flat those eigenstates correspond to infinite temperature in at small ω/Eav and then rapidly decays at larger ω/Eav, the thermodynamic limit. Here, in contrast, we study consistent with results observed in the two-dimensional the statistics of fluctuations in a window of finite energy transverse field Ising model [39] and in the hard-core bo- densities (similar to Ref. [38]), defined by the parameter son model with dipolar interactions [44]. η in Eq. (11). By fixing η, the fraction f of eigenstates The overall prefactor of the matrix elements |hα|Oˆ|βi| involved in the statistics relative to the total number of depends on system size. To extract that prefactor we eigenstates increases with L and eventually approaches define the average offdiagonal matrix element 1 when L → ∞. For example, at η = 2/3 and L = 8 1 X the fraction f is around 0.9 (see the caption of Fig.6 for |O | = |hα|Oˆ|βi| , (23) α,β N details). α,β; α6=β Figures6(a)-6(d) show the finite-size scaling of zmax |(Eα+Eβ )/(2Eav)−ε¯|<∆/2 and hzi for the same observables as in Figs.5(a)-5(d). η where the normalization N equals the number of ele- We show results for the set of eigenstates Z using η = η ments included in the sum andε ¯ = E/E¯ is the target 1/3 and 2/3. Remarkably, in all cases, we observe an av average energy density of the pair of eigenstates. The exponential decay of fluctuations with the system size L, ¯ ETH ansatz predicts the prefactor to be ∼ e−S(E)/2, for while in fact the electronic density decreases from 1/5 √ which the leading term atε ¯ = 1 scales as ∼ 1/ D0. Fig- to 1/8. These results support the validity of the ETH ure8(a) shows our numerical results for |Oα,β| atε ¯ = 1 as in the Holstein polaron model, i.e., they suggest that all 0 −3 −1 eigenstates in the bulk of the spectrum obey the ETH. a function of D . We use ∆ = 10 and 10 in Eq. (23) and show that results for different ∆ are almost identical. The data points can, for all observables, accurately 0 −b B. Offdiagonal matrix elements of observables be fitted using the function a(D ) . We find that b is very close to the predicted value b = 1/2 (see the caption of Fig.8 for the actual numerical values of b). However, We now turn our focus to offdiagonal matrix elements if one also aims at resolving eventual subleading correc- hα|Oˆ|βi of observables and test the second part of the tions to the scaling controlled by Eq. (16), as discussed ETH ansatz in Eq. (16). We consider the symmetry sec- in Ref. [29], more data points would be needed. tor k = 2π/L, which includes D0 = (M +1)L eigenstates. The universal properties of the offdiagonal matrix ele- We focus on matrix elements of pairs of eigenstates with ments |hα|Oˆ|βi| encoded in f (E,¯ ω) can be studied after a similar average energy E¯. An example of the set of O randomness and system-size dependent prefactors are re- eigenstates for which E¯ ≈ E is sketched as a shaded av moved. We define region in Fig.7(a). ! Figure8(b) shows results for the matrix elements |hα|Oˆ|βi| FO(E,¯ ω) = MA , (24) |hα|Hˆkin|βi| of the kinetic energy as a function of ω/Eav |Oα,β| (we only plot results for ω = Eα − Eβ > 0). We re- strict the eigenstates to a narrow energy window |(Eα + which is a moving average of the renormalized offdiagonal −3 Eβ)/(2Eav) − 1| < ∆/2 using ∆ = 10 . The results re- matrix elements of observables. The function FO(E,¯ ω) veal strong fluctuations of |hα|Hˆkin|βi|, which is in agree- is, up to a constant prefactor, identical to the univer- ment with the presence of the random term Rα,β in the sal function fO(E,¯ ω) used in the ETH ansatz (16). We 8

10-1 (a) (b) ˆ 100 Hˆ Hkin kin -2 10 ) | ω β ˆ , ˆ , T 1

T 1 ¯ E α

( -4 O O | -3 Filled ¯ε = 1 10 Solid ¯ε = 1

10 F ¯ε = 0.8 ¯ε = 0.8 ¯ε = 0.6 ¯ε = 0.6 ¯ε = 0.4 ¯ε = 0.4 10-4 10-8 102 103 104 0 1 2 N¯ε ω / Eav

FIG. 9. Oﬀdiagonal matrix elements of observables |hα|Oˆ|βi|

for the electron kinetic energy Hˆkin and the phonon correlator Tˆ1, using identical system parameters as in Fig.8. Here, we vary target average energies of pairs of eigenstates ε¯ = E/E¯ av = 1, 0.8, 0.6, 0.4. (a) Averages |Oα,β |, defined in Eq. (23), using ∆ = 10−1. Results are plotted versus the number of eigenstates Nε¯ within a given microcanonical win- −b dow. Lines are fits to a(Nε¯) for L ≥ 6 andε ¯ = 1, where the exponent is b = 0.45 (Hˆkin) and 0.53 (Tˆ1). (b) The function FO(E,¯ ω), defined in Eq. (24), versus ω/Eav = (Eα−Eβ )/Eav. We use ∆ = 10−3, L = 8 and calculate the moving average in ˆ a window δω/Eav = 0.1. FIG. 8. Offdiagonal matrix√ elements of observables |hα|O|βi| for ω0/t0 = 1/2, γ/t0 = 1/ 2, and M = 3 in the quasimomentum sector k = 2π/L with the Hilbert-space dimension ¯ D0 = (M + 1)L. The target average energy of pairs of eigen- On the other hand, the dependence of fO(E, ω) on the average energy E¯ is not described by the ETH ansatz states isε ¯ = E/E¯ av = 1. (a) Averages |Oα,β |, defined in Eq. (23), for system sizes L = 5, 6, 7, 8. Filled symbols are ob- and rarely studied in the literature (see Ref. [20] for a −3 ¯ tained using ∆ = 10 in Eq. (23) while open circles behind discussion on how fO(E, ω) is related to the fluctuation- the filled symbols are obtained using ∆ = 10−1. Lines are fits dissipation relation). Our results in Fig.9(b) suggest 0 −b −3 to a(D ) for L ≥ 6 and ∆ = 10 , where the exponent is that FO(E,¯ ω) in the Holstein polaron model is roughly ¯ b = 0.49 (Hˆkin), 0.53 (m ˆ q=0), 0.58 (Nˆph) and 0.58 (Tˆ1). (b) independent of the pair energy averageε ¯ = E/Eav. We Matrix elements |hα|Hˆkin|βi| versus ω/Eav = (Eα − Eβ )/Eav. note that a recent study ad hoc assumed a slowly varying The dashed line in (b) is a moving average in a window fO(E,¯ ω) with E¯ in a nonintegrable spin chain [31]. Our ¯ δω/Eav = 0.1. Lines in (c) are functions FO(E, ω) defined results lend numerical support to this assumption. The in Eq. (24), which are moving averages of renormalized offdi- study of fO(E,¯ ω) and its E¯-dependence in other models agonal matrix elements, averaged in the same window as in remains an open problem. (b). Results in (b) and (c) are for L = 8 and ∆ = 10−3.

C. Variances of matrix elements of observables therefore plot FO(E,¯ ω) in Fig.8(c) for different observables as a function of ω/E . The results reveal that the av Finally, we study fluctuations of both diagonal and off- offdiagonal matrix elements of both electron and phonon diagonal matrix elements of observables, focusing on the observables exhibit a similar scaling with ω. quasimomentum sector k = 2π/L. We define the vari- We complement our previous results by calculating the ance of a set of consecutive diagonal matrix elements offdiagonal matrix elements of the pairs of eigenstates at average energy densitiesε ¯ < 1. In Fig.9, we show results (α,µ) 2 2 2 [σdiag (O)] = h(Oα,α) iµ − hOα,αiµ , (25) for two observables (the electron kinetic energy and the phonon correlator), while results for the other two ob- where α determines the first matrix element and µ de- servables studied in Fig.8(c) exhibit a similar behavior termines the number of matrix elements in the set. The −1 Pα+µ−1 ˆ (not shown here). According to the ETH ansatz (16), average hOα,αiµ = µ ρ=α hρ|O|ρi is defined in a the scaling of |hα|Oˆ|βi| with the system size is governed microcanonical window, i.e., it is an equal-weight aver- ¯ age over µ consecutive eigenstates, starting at the index by e−S(E)/2. In the context of Eq. (23), S(E¯) is the log- α. Similarly, the variance of the submatrix of offdiagonal arithm of the number of microstates N in the energy ε¯ matrix elements is defined as density window [¯ε − ∆/2, ε¯+ ∆/2]. In Fig.9(a), we plot ˆ (α,µ) 2 2 2 |hα|O|βi| versus Nε¯ and observe a reasonably good col- [σoffdiag(O)] = h|Oα,β| iµ − |hOα,βiµ| , (26) lapse of the data for differentε ¯. We fit the results to −b 2 −1 Pα+µ−1 0 ˆ 0 a(Nε¯) and obtain b close to 0.5, as predicted by the where hOα,βiµ = (µ − µ) ρ,ρ0=α hρ |O|ρi and ρ 6= ρ . ETH ansatz. Note that the offdiagonal matrix elements can be com- 9

√ FIG. 10. Variances of matrix elements of observables for ω0/t0 = 1/2, γ/t0 = 1/ 2 and M = 3 in the quasimomentum 2 sector k = 2π/L. Main panels: Ratio of variances Σα,µ, defined in Eq. (27), for four different observables at L = 8. We fix 2 0 µ = 100 and plot Σα,µ=100 for every α ∈ [1, D −µ]. Horizontal dashed lines are averages over those ratios of variances for which eα,µ=100/Eav ∈ [1 − η, 1 + η]. We use η = 2/3 such that the set of eigenstates included in the average roughly corresponds to the set of eigenstates analyzed in Fig.6 (i.e., 90% of all eigenstates). We get 2 .05, 2.04, 2.07, 2.03 in panels (a)-(d), respectively. 2 Insets: Histograms of Σα,µ=100 for lattice sizes L = 7 and 8, using the same set of α as the one that was used to obtain the horizontal lines in the main panel. plex as a consequence of diagonalizing the Hamiltonian physical systems, one may ask whether the same re- in the translationally invariant basis. Both variances are sult can be obtained for matrix elements of Hamiltonian sketched in Fig.7(b) for a given α and µ. We associate eigenstates [95] and if the answer is affirmative, at which the mean energy eα,µ to every microcanonical window energy densities this behavior is found. defined by (α, µ), where eα,µ = hHα,αiµ. 2 The numerical extraction of Σα,µ obeying predictions In our calculations, we fix µ and calculate the variances from the random-matrix theory is a nontrivial task, and 0 for every α in the interval [1, D − µ]. Our main focus is the results depend on the parameters α and µ as well as on the ratio of variances defined as on the Hilbert-space dimension D0. In the main panels of

(α,µ) Figs. 10(a)-10(d), we fix µ = 100 and set L = 8, M = 3, [σ (O)]2 0 0 2 diag which yields D = 65536 and hence µ D . Our choice Σα,µ(O) = . (27) [σ(α,µ) (O)]2 of µ is consistent with the one used in a recent study offdiag of a two-dimensional spin-1/2 system [39]. We compute 2 Within random-matrix theory one can show [20] that the Σα,µ=100(O) for four different observables as a function 2 of the eigenstate index α and observe Σ2 (O) ≈ 2 ratio of variances takes a universal value ΣGOE(O) = 2 α,µ=100 for generic local observables in the GOE. Since the ETH for the majority of eigenstates for all observables. generalizes random-matrix theory to Hamiltonians of real We quantify the excellent agreement with predictions of random-matrix theory, observed in Figs. 10(a)-10(d), by computing the averages over those ratios of variances for which eα,µ=100/Eav ∈ [1 − η, 1 + η] and η = 2/3. The resulting averages are shown as horizontal dashed lines in Fig. 10 and yield, for all four observables, almost perfect 2 agreement with ΣGOE(O). The insets of Figs. 10(a)-10(d) 2 show the distributions of Σα,µ=100(O) for the same set of α as the one that was used to obtain the horizontal lines in the main panel, and two system sizes L = 7 and 8. They reveal that, by increasing the system size, the distributions become narrower and their means approach the random-matrix theory result. It is remarkable that the agreement of the ratio of vari- 2 FIG. 11. (a) Ratio of variances Σα,µ=100, defined in Eq. (27), ances for Hamiltonian eigenstates with the GOE predic- for the phonon one-body correlation operator Tˆ1 at L = 8 and tions carries over to eigenstates away from the middle M = 3. The data are identical to the ones in Fig. 10(d), but of the spectrum. Namely, in contrast to eigenstates in plotted versus the mean energy density eα,µ=100/Eav. (b) Av- the middle of the spectrum, which can be well approx- 2 erage ratio of variances Σµ, defined in Eq. (28), as a function imated by pure states that are random superpositions of µ for the same model parameters as in Fig. 10. The aver- of base kets in some simple basis [96], eigenstates away age is taken over the shaded region in panel (a). The num- from the middle of the spectrum are no longer entirely bers of ratios of variances included in the average are roughly random. Figure 11(a) shows the same data for Σ2 2 3 4 α,µ=100 7.4 × 10 , 3.0 × 10 and 1.2 × 10 for L = 6, 7, 8, respectively. as in Fig. 10(d), but plotted versus the mean energy den- 10 sity eα,µ=100/Eav. It reveals a broad region away from that quantum ergodicity and thermalization in a many- 2 eα,µ=100/Eav = 1 where Σα,µ=100 ≈ 2. body system of electrons and phonons does not require 2 phonons to exhibit properties of a bath in isolation, as Finally, we study the dependence of Σα,µ on µ and the system size, and we only consider eigenstates α away from usually assumed in studies of open quantum systems [69]. the middle of the spectrum. To this end, we consider a The second is to demonstrate that an integrability break- 2 ing term of the order O(1) is sufficient to induce perfect subset of ratios of variances Sµ = {Σα,µ ; eα,µ/Eav ∈ [1/3, 2/3]}, which corresponds to a shaded region in ETH behavior. [In our case, the eigenstate expectation 2 value of the electron-phonon coupling operator is O(1) in Fig. 11(a). The average Σα,µ in this region is defined as every eigenstate.] This extends previous studies [75–79] that showed that an O(1) integrability breaking term ren- X ders the spectral level statistics to be quantum chaotic. Σ2 (O) = ||S ||−1 Σ2 (O) (28) µ µ α,µ There are several interesting extensions to our paper. Σ2 ∈S α,µ µ First, one may study generic conditions for an integrability breaking term of the order O(1) to induce ETH be- and plotted in Fig. 11(b). The results reveal that when havior. Of particular interest may be systems where the L increases, a larger number of eigenstates µ can be in- uncoupled electron-phonon system is not as highly de- 2 2 cluded in the variances to observe Σα,µ → ΣGOE. Re- generate as in the Holstein model. This can be achieved, 2 2 markably, this suggests that Σα,µ = ΣGOE in the limit e.g., by considering dispersive or disordered phonons. L → ∞, for eigenstates both in the middle and away Second, extending the analysis to finite electronic den- from the center of the spectrum. sities would be interesting, but also numerically much more challenging. Third, exact diagonalization studies are obviously limited to small systems. Therefore, complementary analytical insights into the problem of V. CONCLUSIONS ETH in electron-phonon models would be desirable. Fi- nally, concrete studies of quench problems in the polaron In this paper, we addressed the question whether the case with an extensive quench energy are lacking. This hallmark features of the ETH can be observed in a could be carried out by using, e.g., matrix-product-state paradigmatic condensed matter model that includes both (MPS) methods [97–99] (see Refs. [100–103] for exam- electron and phonon degrees of freedom. We studied ples of recent developments). References [60, 104–106] the Holstein polaron model, which is a minimal model discuss MPS methods that are specifically designed for describing a single electron that locally couples to dis- electron-phonon coupled systems. These questions are persionless phonons. In spite of the apparent simplicity left for future work. of the model, we showed that it represents an example where ETH is very well fulfilled, to a quantitative degree that is observed in single-component models as well [39]. ACKNOWLEDGMENTS In particular, we showed that: (i) the statistics of neighboring energy-level spacings agree with predictions of the We acknowledge useful discussions with J. Bonˇca,I. GOE; (ii) the eigenstate-to-eigenstate fluctuations of the de Vega, M. Goldstein, D. J. Luitz, S. Kehrein, M. diagonal matrix elements of observables decay exponen- Mierzejewski, P. Prelovˇsek,M. Rigol and R. Steinigeweg. tially for all eigenstates in the bulk of the spectrum; (iii) This work was supported by the Deutsche Forschungsge- the ’universal’ part of the offdiagonal matrix elements of meinschaft (DFG, German Research Foundation) under all four observables under investigation exhibits a simi- Project No. 207383564 via Research Unit FOR 1807, and lar decay as a function of the energy difference of pairs via CRC 1073 under Project No. 217133147. D.J. ac- of eigenstates, and exhibits an almost negligible depen- knowledges support by the Deutsche Forschungsgemein- dence on the average energy of the pair of eigenstates; schaft (DFG, German Research Foundation) under Ger- and (iv) the ratio of variances of fluctuations of diago- many’s Excellence Strategy No. EXC-2111-390814868. nal to offdiagonal matrix elements of observables fulfills L.V. acknowledges support from the Slovenian Research predictions of the random-matrix theory. The latter is a Agency (ARRS), Research Core Funding No. P1-0044. robust feature of Hamiltonian eigenstates also away from Part of this research was conducted at KITP at the Uni- the middle of the spectrum. versity of California at Santa Barbara. This research was Our results establish two important aspects of quan- supported in part by the National Science Foundation tum many-body systems: The first is to demonstrate under Grant No. NSF PHY-1748958.

[1] J. von Neumann, Beweis des Ergodensatzes und des H- [2] S. Goldstein, J. L. Lebowitz, R. Tumulka, and Theorems, Z. Phys. 57, 30(1929). N. Zangh`ı,Long-time behavior of macroscopic quantum 11

systems, Eur. Phys. J. H 35, 173(2010). [20] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, [3] I. Bloch, J. Dalibard, and W. Zwerger, Many-body From quantum chaos and eigenstate thermalization to physics with ultracold gases, Rev. Mod. Phys. 80, 885 statistical mechanics and thermodynamics, Adv. Phys. (2008). 65, 239(2016). [4] T. Langen, R. Geiger, and J. Schmiedmayer, Ultracold [21] T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Ther- atoms out of equilibrium, Annual Rev. of Condensed malization and prethermalization in isolated quantum Matt. Phys. 6, 201(2015). systems: a theoretical overview, J. Phys. B 51, 112001 [5] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- (2018). body systems out of equilibrium, Nat. Phys. 11, 124 [22] J. M. Deutsch, Eigenstate thermalization hypothesis, (2015). Rep. Prog. Phys. 81, 082001(2018). [6] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, [23] M. Rigol, Breakdown of thermalization in finite one- U. Schollwöck, J. Eisert, and I. Bloch, Probing the re- dimensional systems, Phys. Rev. Lett. 103, 100403 laxation towards equilibrium in an isolated strongly cor- (2009). related 1D Bose gas, Nat. Phys. 8, 325(2012). [24] M. Rigol and L. F. Santos, Quantum chaos and ther- [7] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, malization in gapped systems, Phys. Rev. A 82, 011604 R. Schittko, P. M. Preiss, and M. Greiner, Quan- (2010). tum thermalization through entanglement in an isolated [25] R. Steinigeweg, J. Herbrych, and P. Prelovˇsek,Eigen- many-body system, Science 353, 794(2016). state thermalization within isolated spin-chain systems, [8] C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodru- Phys. Rev. E 87, 012118(2013). betz, Z. Chen, A. Megrant, R. Barends, B. Campbell, [26] W. Beugeling, R. Moessner, and M. Haque, Finite-size B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, scaling of eigenstate thermalization, Phys. Rev. E 89, P. J. J. OMalley, C. Quintana, D. Sank, A. Vainsencher, 042112(2014). J. Wenner, T. C. White, A. Polkovnikov, and J. M. [27] H. Kim, T. N. Ikeda, and D. A. Huse, Testing whether Martinis, Ergodic dynamics and thermalization in an all eigenstates obey the eigenstate thermalization hy- isolated quantum system, Nat. Phys. 12, 1037(2016). pothesis, Phys. Rev. E 90, 052105(2014). [9] Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, [28] D. J. Luitz, Long tail distributions near the many-body M. Rigol, S. Gopalakrishnan, and B. L. Lev, Thermal- localization transition, Phys. Rev. B 93, 134201(2016). ization near integrability in a dipolar quantum Newton’s [29] D. J. Luitz and Y. Bar Lev, Anomalous thermalization cradle, Phys. Rev. X 8, 021030(2018). in ergodic systems, Phys. Rev. Lett. 117, 170404(2016). [10] T. Kinoshita, T. Wenger, and S. D. Weiss, A quantum [30] J. R. Garrison and T. Grover, Does a single eigenstate newton’s cradle, Nature (London) 440, 900(2006). encode the full Hamiltonian?, Phys. Rev. X 8, 021026 [11] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, (2018). B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Dem- [31] J. Richter, J. Gemmer, and R. Steinigeweg, Impact of ler, and J. Schmiedmayer, Relaxation and prethermal- eigenstate thermalization on the route to equilibrium, ization in an isolated quantum system, Science 337, arXiv:1805.11625 (2018). 1318(2012). [32] I. M. Khaymovich, M. Haque, and P. A. McClarty, [12] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, Eigenstate thermalization, random matrix theory and M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, behemoths, arXiv:1806.09631 (2018). and J. Schmiedmayer, Experimental observation of a [33] R. Hamazaki and M. Ueda, Random-matrix behav- generalized Gibbs ensemble, Science 348, 207(2015). ior of quantum nonintegrable many-body systems with [13] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, Dyson’s three symmetries, arXiv:1901.02119 (2019). M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and [34] R. Steinigeweg, A. Khodja, H. Niemeyer, C. Gogolin, I. Bloch, Observation of many-body localization of in- and J. Gemmer, Pushing the limits of the eigenstate teracting fermions in a quasi-random optical lattice, Sci- thermalization hypothesis towards mesoscopic quantum ence 349, 842(2015). systems, Phys. Rev. Lett. 112, 130403(2014). [14] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- [35] A. Khodja, R. Steinigeweg, and J. Gemmer, Relevance Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, of the eigenstate thermalization hypothesis for thermal and C. Gross, Exploring the many-body localization relaxation, Phys. Rev. E 91, 012120(2015). transition in two dimensions, Science 352, 1547(2016). [36] W. Beugeling, R. Moessner, and M. Haque, Off-diagonal [15] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. matrix elements of local operators in many-body quan- Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Mon- tum systems, Phys. Rev. E 91, 012144(2015). roe, Many-body localization in a quantum simulator [37] K. R. Fratus and M. Srednicki, Eigenstate thermaliza- with programmable random disorder, Nat. Phys. 12, tion in systems with spontaneously broken symmetry, 907(2016). Phys. Rev. E 92, 040103(2015). [16] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization [38] R. Mondaini, K. R. Fratus, M. Srednicki, and M. Rigol, and its mechanism for generic isolated quantum sys- Eigenstate thermalization in the two-dimensional trans- tems, Nature (London) 452, 854(2008). verse field Ising model, Phys. Rev. E 93, 032104(2016). [17] J. M. Deutsch, Quantum statistical mechanics in a [39] R. Mondaini and M. Rigol, Eigenstate thermalization closed system, Phys. Rev. A 43, 2046(1991). in the two-dimensional transverse field Ising model. II. [18] M. Srednicki, Chaos and quantum thermalization, Phys. Off-diagonal matrix elements of observables, Phys. Rev. Rev. E 50, 888(1994). E 96, 012157(2017). [19] M. Rigol and M. Srednicki, Alternatives to eigenstate [40] M. Rigol, Quantum quenches and thermalization in one- thermalization, Phys. Rev. Lett. 108, 110601(2012). dimensional fermionic systems, Phys. Rev. A 80, 053607 (2009). 12

[41] C. Neuenhahn and F. Marquardt, Thermalization of Phys. Rev. B 91, 104302(2015). interacting fermions and delocalization in Fock space, [59] P. Werner and M. Eckstein, Field-induced polaron for- Phys. Rev. E 85, 060101(2012). mation in the Holstein-Hubbard model, EPL (Euro- [42] S. Sorg, L. Vidmar, L. Pollet, and F. Heidrich-Meisner, physics Letters) 109, 37002(2015). Relaxation and thermalization in the one-dimensional [60] C. Brockt, F. Dorfner, L. Vidmar, F. Heidrich-Meisner, Bose-Hubbard model: A case study for the interaction and E. Jeckelmann, Matrix-product-state method with quantum quench from the atomic limit, Phys. Rev. A a dynamical local basis optimization for bosonic systems 90, 033606(2014). out of equilibrium, Phys. Rev. B 92, 241106(2015). [43] R. Mondaini and M. Rigol, Many-body localization [61] J. Kogoj, L. Vidmar, M. Mierzejewski, S. A. Trug- and thermalization in disordered Hubbard chains, Phys. man, and J. Bonˇca,Thermalization after photoexcita- Rev. A 92, 041601(2015). tion from the perspective of optical spectroscopy, Phys. [44] E. Khatami, G. Pupillo, M. Srednicki, and M. Rigol, Rev. B 94, 014304(2016). Fluctuation-dissipation theorem in an isolated system [62] J. Kogoj, M. Mierzejewski, and J. Bonˇca, Nature of of quantum dipolar bosons after a quench, Phys. Rev. bosonic excitations revealed by high-energy charge car- Lett. 111, 050403(2013). riers, Phys. Rev. Lett. 117, 227002(2016). [45] Z. Lan and S. Powell, Eigenstate thermalization hypoth- [63] Z. Huang, L. Chen, N. Zhou, and Y. Zhao, Transient esis in quantum dimer models, Phys. Rev. B 96, 115140 dynamics of a one-dimensional Holstein polaron under (2017). the influence of an external electric field, Ann. Phys. [46] A. Chandran, M. D. Schulz, and F. J. Burnell, The (Berlin) 529, 1600367(2017). eigenstate thermalization hypothesis in constrained [64] H. Hashimoto and S. Ishihara, Photoinduced charge- Hilbert spaces: A case study in non-Abelian anyon order melting dynamics in a one-dimensional interacting chains, Phys. Rev. B 94, 235122(2016). Holstein model, Phys. Rev. B 96, 035154(2017). [47] C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, [65] L. Chen, R. Borrelli, and Y. Zhao, Dynamics of coupled F. Parmigiani, and D. Mihailovic, Ultrafast optical electronboson systems with the multiple Davydov D1 spectroscopy of strongly correlated materials and high- ansatz and the generalized coherent state, J. Phy. Chem. temperature superconductors: a non-equilibrium ap- A 121, 8757(2017). proach, Adv. Phys. 65, 58(2016). [66] Z. Huang, M. Hoshina, H. Ishihara, and Y. Zhao, Tran- [48] L.-C. Ku and S. A. Trugman, Quantum dynamics of sient dynamics of super Bloch oscillations of a 1D Hol- polaron formation, Phys. Rev. B 75, 014307(2007). stein polaron under the influence of an external AC elec- [49] L. Vidmar, J. Bonˇca,M. Mierzejewski, P. Prelovˇsek, tric field, Ann. Phys. (Berlin) 531, 1800303(2019). and S. A. Trugman, Nonequilibrium dynamics of the [67] J. Bonˇca,S. A. Trugman, and I. Batistić, Holstein po- Holstein polaron driven by an external electric field, laron, Phys. Rev. B 60, 1633(1999). Phys. Rev. B 83, 134301(2011). [68] H. Fehske and S. A. Trugman, Numerical solution of the [50] H. Fehske, G. Wellein, and A. R. Bishop, Spatiotempo- holstein polaron problem, Polarons in Advanced Materi- ral evolution of polaronic states in finite quantum sys- als, volume 103 of Springer Series in Materials Science, tems, Phys. Rev. B 83, 075104(2011). 393–461 (Springer Netherlands, 2007). [51] L. Vidmar, J. Bonˇca,T. Tohyama, and S. Maekawa, [69] I. de Vega and D. Alonso, Dynamics of non-Markovian Quantum dynamics of a driven correlated system cou- open quantum systems, Rev. Mod. Phys. 89, 015001 pled to phonons, Phys. Rev. Lett. 107, 246404(2011). (2017). [52] D. Goleˇz,J. Bonˇca,L. Vidmar, and S. A. Trugman, Re- [70] B. Bertini and M. Fagotti, Determination of the laxation dynamics of the Holstein polaron, Phys. Rev. nonequilibrium steady state emerging from a defect, Lett. 109, 236402(2012). Phys. Rev. Lett. 117, 130402(2016). [53] G. Li, B. Movaghar, A. Nitzan, and M. A. Ratner, Po- [71] M. Fagotti, Charges and currents in quantum spin laron formation: Ehrenfest dynamics vs. exact results, chains: late-time dynamics and spontaneous currents, J. Chem. Phys. 138, 044112(2013). J. Phys. A. 50, 034005(2017). [54] M. Hohenadler, Charge and spin correlations of a Peierls [72] A. Bastianello and A. De Luca, Nonequilibrium steady insulator after a quench, Phys. Rev. B 88, 064303 state generated by a moving defect: The supersonic (2013). threshold, Phys. Rev. Lett. 120, 060602(2018). [55] A. F. Kemper, M. Sentef, B. Moritz, C. C. Kao, Z. X. [73] A. Bastianello and A. De Luca, Superluminal moving Shen, J. K. Freericks, and T. P. Devereaux, Mapping of defects in the Ising spin chain, Phys. Rev. B 98, 064304 unoccupied states and relevant bosonic modes via the (2018). time-dependent momentum distribution, Phys. Rev. B [74] A. Bastianello, Lack of thermalization for integrability- 87, 235139(2013). breaking impurities, arXiv:1807.00625 (2018). [56] M. Sentef, A. F. Kemper, B. Moritz, J. K. Freericks, [75] L. F. Santos, Integrability of a disordered Heisenberg Z.-X. Shen, and T. P. Devereaux, Examining electron- spin-1/2 chain, J. Phys. A. 37, 4723(2004). boson coupling using time-resolved spectroscopy, Phys. [76] O. S. Bariˇsić,P. Preloˇsek,A. Metavitsiadis, and X. Zo- Rev. X 3, 041033(2013). tos, Incoherent transport induced by a single static im- [57] Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, In- purity in a Heisenberg chain, Phys. Rev. B 80, 125118 teraction quench in the Holstein model: Thermaliza- (2009). tion crossover from electron- to phonon-dominated re- [77] L. F. Santos and A. Mitra, Domain wall dynamics in laxation, Phys. Rev. B 91, 045128(2015). integrable and chaotic spin-1/2 chains, Phys. Rev. E [58] F. Dorfner, L. Vidmar, C. Brockt, E. Jeckelmann, and 84, 016206(2011). F. Heidrich-Meisner, Real-time decay of a highly excited [78] E. J. Torres-Herrera and L. F. Santos, Local quenches charge carrier in the one-dimensional Holstein model, with global effects in interacting quantum systems, 13

Phys. Rev. E 89, 062110(2014). random matrix ensembles, Phys. Rev. Lett. 110, 084101 [79] M. Brenes, E. Mascarenhas, M. Rigol, and J. Goold, (2013). High-temperature coherent transport in the XXZ chain [93] V. Oganesyan and D. A. Huse, Localization of inter- in the presence of an impurity, Phys. Rev. B 98, 235128 acting fermions at high temperature, Phys. Rev. B 75, (2018). 155111(2007). [80] O. Bohigas, M. J. Giannoni, and C. Schmit, Character- [94] M. Srednicki, The approach to thermal equilibrium in ization of chaotic quantum spectra and universality of quantized chaotic systems, J. Phys. A. 32, 1163(1999). level fluctuation laws, Phys. Rev. Lett. 52, 1(1984). [95] T. Prosen, Ergodic properties of a generic noninte- [81] L. F. Santos and M. Rigol, Onset of quantum chaos grable quantum many-body system in the thermody- in one-dimensional bosonic and fermionic systems and namic limit, Phys. Rev. E 60, 3949(1999). its relation to thermalization, Phys. Rev. E 81, 036206 [96] L. Vidmar and M. Rigol, Entanglement entropy of (2010). eigenstates of quantum chaotic Hamiltonians, Phys. [82] M. Kormos, M. Collura, G. Takacs, and P. Calabrese, Rev. Lett. 119, 220603(2017). Real time confinement following a quantum quench to [97] U. Schollwöck, The density-matrix renormalization a non-integrable model, Nat. Phys. 13, 246(2017). group, Rev. Mod. Phys. 77, 259(2005). [83] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Ser- [98] U. Schollwöck, The density-matrix renormalization byn, and Z. Papić,Weak ergodicity breaking from quan- group in the age of matrix product states, Ann. Phys. tum many-body scars, Nat. Phys. 14, 745(2018). 326, 96(2011). [84] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Ser- [99] S. Paeckel, T. Köhler,A. Swoboda, S. R. Manmana, byn, and Z. Papić,Quantum scarred eigenstates in a U. Schollwöck, and C. Hubig, Time-evolution methods Rydberg atom chain: Entanglement, breakdown of ther- for matrix-product states, arXiv:1901.05824. malization, and stability to perturbations, Phys. Rev. B [100]M. Znidariˇc,A.ˇ Scardicchio, and V. K. Varma, Diffusive 98, 155134(2018). and subdiffusive spin transport in the ergodic phase of [85] W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Pe- a many-body localizable system, Phys. Rev. Lett. 117, riodic orbits, entanglement, and quantum many-body 040601(2016). scars in constrained models: Matrix product state ap- [101] B. Kloss, Y. B. Lev, and D. Reichman, Time-dependent proach, Phys. Rev. Lett. 122, 040603(2019). variational principle in matrix-product state manifolds: [86] V. Khemani, C. R. Laumann, and A. Chandran, Sig- Pitfalls and potential, Phys. Rev. B 97, 024307(2018). natures of integrability in the dynamics of Rydberg- [102] E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D. blockaded chains, arXiv:1807.02108. Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi, [87] N. J. Robinson, A. J. A. James, and R. M. Konik, Sig- Many-body localization and delocalization in large natures of rare states and thermalization in a theory quantum chains, Phys. Rev. B 98, 174202(2018). with confinement, arXiv:1808.10782. [103] A. Hallam, J. Morley, and A. G. Green, The Lyapunov [88] C.-J. Lin and O. I. Motrunich, Exact quantum many- spectrum of quantum thermalisation, arXiv:1806.05204. body scar states in the Rydberg-blockaded atom chain, [104] F. A. Y. N. Schröder and A. W. Chin, Simulating arXiv:1810.00888. open quantum dynamics with time-dependent varia- [89] T. Iadecola and M. Znidariˇc,Exactˇ localized and bal- tional matrix product states: Towards microscopic cor- listic eigenstates in disordered chaotic spin ladders and relation of environment dynamics and reduced system the Fermi-Hubbard model, arXiv:1811.07903. evolution, Phys. Rev. B 93, 075105(2016). [90] L. F. Santos and M. Rigol, Localization and the effects [105] M. L. Wall, A. Safavi-Naini, and A. M. Rey, Simulating of symmetries in the thermalization properties of one- generic spin-boson models with matrix product states, dimensional quantum systems, Phys. Rev. E 82, 031130 Phys. Rev. A 94, 053637(2016). (2010). [106] B. Kloss, D. Reichman, and R. Tempelaar, Multi-set [91] T. A. Brody, J. Flores, J. B. French, P. A. Mello, matrix product state calculations reveal mobile Franck- A. Pandey, and S. S. M. Wong, Random-matrix physics: Condon excitations under strong Holstein-type cou- spectrum and strength fluctuations, Rev. Mod. Phys. pling, arXiv:1812.00011. 53, 385(1981). [92] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in