Strong Eigenstate Thermalization within a Generalized Shell in Noninteracting Integrable Systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Strong Eigenstate Thermalization within a Generalized Shell in Noninteracting IntegrableSystems
Takashi Ishii ∗ and Takashi Mori † Department of Physics, Graduate School of Science, University of Tokyo, Kashiwa 277-8574, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
Integrable systems do not obey the strong eigenstate thermalization hypothesis (ETH), which has been proposedas a mechanism of thermalization in isolated quantum systems. It has been suggested that an integrable systemreaches a steady state described by a generalized Gibbs ensemble (GGE) instead of thermal equilibrium. Weprove that a generalized version of the strong ETH holds for noninteracting integrable systems with translationinvariance. Our generalized ETH states that any pair of energy eigenstates with similar values of local conservedquantities looks similar with respect to local observables, such as local correlations. This result tells us that anintegrable system relaxes to a GGE for any initial state that has subextensive fluctuations of macroscopic localconserved quantities. Contrary to the previous derivations of the GGE, it is not necessary to assume the clusterdecomposition property for an initial state.
Introduction.—
Out-of-equilibrium dynamics of isolatedquantum systems and their steady states have been exploredrecently [1–3]. Various experiments [4–7] as well as numer-ical calculations [8–10] have revealed that nonintegrable sys-tems thermalize under unitary time evolution. As a possiblemechanism of thermalization, the eigenstate thermalizationhypothesis (ETH) has been studied [9, 11–18]. The ETH inthe strong (weak) sense, or namely “the strong (weak) ETH”,states that all (almost all) energy eigenstates have thermalproperties when we look at local observables. It has beenrecognized that the strong ETH ensures thermalization, whilethe weak ETH does not because the initial state may have animportant weight on nonthermal energy eigenstates [13]. In-deed, the weak ETH can be proved for generic translationallyinvariant systems including integrable systems [13, 19, 20],although it is known that integrable systems generally do notthermalize [13, 21–23] [24]. Meanwhile, the strong ETH hasbeen numerically verified in nonintegrable models [15, 16].For integrable systems, it is suggested that the steadystate is given by the so-called generalized Gibbs ensemble(GGE) [21], which is constructed by using a set of localand quasi-local conserved quantities of the system. Numeri-cal studies on specific integrable models support the validityof the GGE [21, 22]. By analytically calculating the timeevolution of local observables, the validity of GGE is evenproved for noninteracting integrable systems with translationinvariance when the initial state satisfies some certain prop-erties. More precisely, proofs were given for Gaussian initialstates [25–29], followed by extensions to initial states thatsatisfy the cluster decomposition property [30–32]; see alsoRef. [33] for continuous models. Though the present workfocuses on noninteracting integrable systems, it should benoted that the validity of the GGE has also been investigatedfor interacting integrable systems [34–43], which cannot bemapped to free particles but exactly solvable via the Betheansatz method.Here, a set of questions naturally arises. Can one con-struct a generalized version of the ETH as a mechanism thatexplains the relaxation to a GGE, just as the standard ETHexplained thermalization in nonintegrable systems? If so,can we remove the assumption of the cluster decompositionproperty for the initial state in deriving the relaxation to aGGE? The removal of this assumption is important in con- sidering a spin system that can be mapped to a quadraticfermion Hamiltonian (e.g., the transverse-field Ising model)because it is not obvious whether a physically realistic ini-tial state, which satisfies the cluster decomposition property with respect to spin operators , also satisfies it with respect tofermion operators [44]; indeed, there are cases where a non-local transformation reveals nontrivial correlations [45–47].In Ref. [22], a generalization of the ETH has been pro-posed. Their generalized ETH has been numerically veri-fied [22] and also proved for various local operators in thetranslationally invariant transverse-field Ising model [48], but only in the weak sense . It has not been clarified yet whetherit is valid in the strong sense. Although the concept of thegeneralized ETH helps us to understand the validity of theGGE [22, 48], the weak generalized ETH does not ensure initself the relaxation to a GGE in an integrable system. It istherefore desirable to formulate the generalized ETH that isvalid in the strong sense.In the present paper, by constructing a generalized shellthat is specified by a set of macroscopic conserved quantities,we reformulate the generalized ETH and analytically provethat our generalized ETH proposed is valid in the strong sense in integrable models of the quadratic form with translationinvariance. It is shown that our strong generalized ETH en-sures the relaxation to a GGE for initial states that have subex-tensive fluctuations of macroscopic local conserved quanti-ties [49]. We manage to remove the assumption of the clusterdecomposition property here, and thus our result is beyondthe previous rigorous results [30–32]. In Table I we show forhelp of understanding a comparison between the strong ETHand our strong generalized ETH.
Model and Setup.—
We consider a bilinear fermion systemdescribed by the translationally invariant Hamiltonian H = L Õ x , y = (cid:16) c † x A x − y c y + c † x B x − y c † y + c x B ∗ y − x c y (cid:17) (1)under the periodic boundary condition; the analysis is almostunchanged for the anti-periodic boundary condition. Thecoefficients A l satisfies A l = A ∗− l because H = H † . Weassume the locality of the Hamiltonian, i.e., A l = B l = | l | P > r H with a finite range r H >
0, where | l | P : = min {| l | , L − | l |} denotes the distance l under the periodic Hilbert subspace steady state validitystrong ETH [9, 13–18] energy shell Gibbs ensemble nonintegrable: valid but with counterexamples.integrable: invalid.strong generalizedETH (present study) shell defined by manymacroscopic conserved quantities generalizedGibbs ensemble translationally invariantnoninteracting integrable: valid.TABLE I. A comparison between the strong ETH [9, 13–18] and our strong generalized ETH. While the usual strong ETH is discussed forstates in the energy shell, in the formulation of our strong generalized ETH we consider a generalized shell, which is defined as a Hilbertsubspace specified by a set of macroscopic conserved quantities. The strong ETH and our strong generalized ETH are sufficient conditionsfor relaxation to the steady state described by the Gibbs ensemble and the generalized Gibbs ensemble, respectively. In the column indicated“validity,” we explain the current understanding on the validity of the two concepts. As for the strong ETH, in nonintegrable systems, itsvalidity has been numerically confirmed [15, 16], although there exists some counterexamples [17, 18]. In integrable systems, numericaldemonstrations and analytical calculation show that the strong ETH does not hold [9, 13, 14, 16]. As for our strong generalized ETH, weanalytically prove in this paper its validity in translationally invariant noninteracting integrable systems. boundary conditions. This form of Hamiltonian includes,for example, a fermionic system with on-site potential andnearest-neighbor hopping terms. The XY model, a hard-coreboson system, and the transverse-field Ising model can also bemapped to this form using the Jordan-Wigner transformation.We first consider the case of B l =
0, for which the total par-ticle number is conserved. This system can be diagonalizedby the Fourier transform as H = Õ p ε p f † p f p , (2)where f † p = ( /√ L ) Í Lx = c † x e − ipx and ε p = Í Lx = A x e ipx .The summation over p = π m / L is taken over integers m with −( L − )/ ≤ m ≤ ( L − )/
2, where we consider thecase of odd L throughout the paper, although this restrictionis not essential.The occupation-number operator of each of the L eigen-modes { f † p f p } is a conserved quantity. Although f † p f p arenot spatially local, we can construct macroscopic local con-served quantities out of them as Q ( + ) n = Õ p cos ( np ) f † p f p , n = , , . . . , L − , Q (−) n = Õ p sin ( np ) f † p f p , n = , . . . , L −
12 ; (3)see Ref. [23]. We then define Q ( + )− n = Q ( + ) n , Q (−)− n = −Q (−) n ,and Q (−) =
0. Note that Q ( + ) coincides with the total particlenumber: Q ( + ) = Õ p f † p f p = ˆ N . (4)We denote an eigenvalue of Q (±) n for the Fock eigenstates by Q (±) n .When B l ,
0, the Bogoliubov transformation followingthe Fourier transformation diagonalizes the Hamiltonian as H = Õ p ˜ ε p η † p η p + const ., (5)where ˜ ε p and η † p are given by a p : = Í Lx = A x e ipx and b p : = i Í Lx = B x sin ( px ) as˜ ε p = a p − a − p + q ( a p + a − p ) + | b p | , (6) η † p = s ( p ) f † p + t ( p ) f − p , (7) with the functions s ( p ) and t ( p ) defined as s ( p ) = | b p | q | b p | + ( ˜ ε p − a p ) , (8) t ( p ) = | b p | b p ˜ ε p − a p q | b p | + ( ˜ ε p − a p ) . (9)Macroscopic local conserved quantities in this case are givenby Q ( + ) n = Õ p cos ( np )( ˜ ε p + ˜ ε − p ) η † p η p , Q (−) n = Õ p sin ( np ) η † p η p , (10)where we use the same notations as in Eq. (3), but there willbe no confusion.The locality of Q ( + ) n in Eq. (10) is proved as follows. First,we divide it into two parts as follows: Q ( + ) n = Õ p ˜ ε p cos ( np ) η † p η p + Õ p ˜ ε p − ˜ ε − p ( np ) η † p η p . (11)It is known and explicitly confirmed that the first term ofEq. (11) is local [36]. As for the second term, we no-tice that ˜ ε p − ˜ ε − p = a p − a − p is written as a finite sum Í r H x = − r H A x ( e ipx − e − ipx ) because of the fact that ˆ H is a localoperator with the maximum range r H . Therefore, the secondterm of Eq. (11) is written as a linear combination of {Q (−) m } with m ≤ n + r H , which is a local operator. Thus, for anyfixed n , both the first and the second terms of Eq. (10) arelocal in the thermodynamic limit.In terms of these local conserved quantities, the GGE isgiven as the density matrix ρ GGE = e − Í ( L − )/ n = (cid:16) Λ ( + ) n Q ( + ) n + Λ (−) n Q (−) n (cid:17) Z GGE , (12)where Z GGE is the normalization constant. The parameters Λ (±) p are determined from the initial state | ψ ( )i by the con-dition that h ψ ( )|Q (±) n | ψ ( )i = Tr [Q (±) n ρ GGE ] . Strong generalized ETH.—
In order to formulate our gen-eralized ETH, we first define a Hilbert subspace called an n c -shell with the notation S n c . Let us denote the set of thesimultaneous eigenstates of {Q (±) n } by E . An n c -shell is thendefined as the Hilbert subspace spanned by all the eigen-states in E with the eigenvalues located around the center { ¯ Q (±) n } n c n = : S n c : = Span n | α i ∈ E : for all 0 ≤ n ≤ n c , Q (±) n ∈ [ ¯ Q (±) n − ∆ (±) n , ¯ Q (±) n + ∆ (±) n ] o . (13)Here, the half width of the shell ∆ (±) n is arbitrary as longas it is microscopically large but macroscopically small; forexample, we can choose ∆ n ∝ L / . Note that n runs upto n c ≤ ( L − )/ . The n c -shell can be regarded as a gen-eralization of the usual energy shell in the microcanonicalensemble.Now we formulate the strong generalized ETH. It statesthat all the energy eigenstates in S n c are locally indistin-guishable from each other in the limit of n c → ∞ takenafter the thermodynamic limit L → ∞ . For convenience,we also say that a local observable ˆ o satisfies the n c -ETH when h α | ˆ o | α i = h α ′ | ˆ o | α ′ i for any pair of eigenstates | α i , | α ′ i ∈ S n c in the thermodynamic limit.It should be noted that another generalization of ETH hasbeen proposed in the previous work [22], stating that energyeigenstates with similar distributions of the mode occupationnumber look similar with respect to local observables. Belowwe explain the relation between our generalized ETH basedon the n c -shell and the generalized ETH based on the modeoccupation number distibutions, which is a simplified versionof the one originally proposed in Ref. [22].For simplicity, we consider the case in which the to-tal particle number is conserved with B l =
0. Then,each energy eigenstate | α i consists of N occupied levels { p α , p α , . . . , p α N } , where p α i = π n α i / L with integers { n α i } Ni = satisfying − π ≤ p α < p α < · · · < p α N < π . In short, h α | f † p f p | α i = p ∈ { p α , p α , . . . , p α N } . Let ussay that two eigenstates | α i and | α ′ i have ‘similar’ distribu-tions of the mode occupation number if and only if δ ( α, α ′ ) = " N N Õ i = (cid:16) p α i − p α ′ i (cid:17) / (14)is smaller than a threshold ǫ , which can be set to zero in thethermodynamic limit. The generalized ETH formulated inRef. [22] essentially states that two eigenstates with similardistributions of the mode occupation number are locally in-distinguishable. Now we begin the explanation of its relationwith our generalized ETH. Let us consider the difference ofa macroscopic conserved quantity in the states | α i and | α ′ i : δ q (±) n : = L (cid:12)(cid:12)(cid:12) h α |Q (±) n | α i − h α ′ |Q (±) n | α ′ i (cid:12)(cid:12)(cid:12) . (15)If | δ q (±) n | ≤ ∆ (±) n / L for all n ≤ n c , the two eigenstates | α i and | α ′ i belong to the same n c -shell under a suitable choiceof the center of the shell { ¯ Q (±) n } n c n = . By using p α i , we can rewrite δ q ( + ) n as δ q ( + ) n = L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Õ i = [ cos ( np α i ) − cos ( np α ′ i )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L N Õ i = (cid:12)(cid:12)(cid:12) cos ( np α i ) − cos ( np α ′ i ) (cid:12)(cid:12)(cid:12) (16)By using | cos θ − cos φ | ≤ | θ − φ | , we obtain δ q ( + ) n ≤ nL N Õ i = | p α i − p α ′ i |≤ n ρδ ( α, α ′ ) , (17)where ρ = N / L and we have used δ ( α, α ′ ) ≥( / N ) Í Ni = | p α i − p α ′ i | . Similarly, δ q (−) n ≤ n ρδ ( α, α ′ ) holds.From these inequalities, we can immediately conclude thattwo eigenstates | α i and | α ′ i belong to the same n c -shellunder a suitable choice of { ¯ Q (±) n } n c n = as long as δ ( α, α ′ ) ≤ ∆ (±) n /( n c N ) . Since ∆ (±) n is chosen so that ∆ (±) n / N → | α i and | α ′ i with similar distributions of the mode occupationnumber belong to the same n c -shell. This implies that if thegeneralized ETH based on the n c -shell holds in the strongsense, then the generalized ETH based on the similarity ofthe distributions of the mode occupation number also holds inthe strong sense [50]. Thus the proof of the strong generalizedETH based on the n c -shell complements the numerical resultin Ref. [22], in which the generalized ETH based on the modeoccupation number distribution has been confirmed only inthe weak sense. Proof.—
We consider local observables ˆ o which consistsof fermionic operators { c † , c } with the maximum range r .For example,ˆ o = L L Õ j = (cid:16) c † j + c j + c † j c j + c † j + c j + c † j c j + (cid:17) (18)is the case of r =
2. As a shorthand notation, we write h ˆ o i : = h α | ˆ o | α i for a fixed eigenstate | α i .We first consider the case B l =
0, in which the total par-ticle number is conserved. In this case, the diagonalizedHamiltonian is given by Eq. (2) and macroscopic conservedquantities are given by Eq. (3). We shall prove that the eigen-state expectation value of a local observable can be writtenas a smooth function of the eigenvalues { Q (±) m / L } of the con-structed conserved quantities with m ≤ r . In other words,any local observable with the maximum range r satisfies the r -ETH.By virtue of Wick’s theorem (see the note in [51]), theeigenstate expectation value h ˆ o i of any local observable ˆ o with the maximum range r can be decomposed into productsof two-point functions of the form h c † x c y i with | x − y | P ≤ r .More precisely, if we denote by X i a linear superposition of { c x , c † x } , h X X . . . X n i = Õ (− ) P h X i X j ih X i X j i . . . h X i n X j n i , (19)where the sum is over all partitions of 1 , , . . . n into pairs {( i , j ) , ( i , j ) , . . . , ( i n , j n )} with i < j , i < j , . . . i n < j n , and P is the parity of the permutation ( , , . . . , n ) →( i , j , i , j , . . . , i n , j n ) [52]. It should be noted that Eq. (19)also holds even when B l , η p and η † p can be written as a linear superpositionof { c x , c † x } .We can express the two-point function in terms of theconserved quantities in Eq. (3) as in h c † x c y i = L Õ p e ip ( x − y ) h f † p f p i = L ( Q ( + ) x − y + iQ (−) x − y ) . (20)Therefore, h ˆ o i is generally a smooth function of { Q (±) m / L } with m ≤ r .This immediately leads to the validity of the strong gener-alized ETH. Moreover, any local operator with the maximumrange r ≤ n c satisfies the n c -ETH. Therefore, as far as weconsider local operators with a fixed maximum range r , thesteady state is described by the microcanonical ensemblewithin the n c -shell, which is in the thermodynamic limitequivalent to the truncated GGE, ρ ( n c ) GGE : = exp h − Í n c n = (cid:16) Λ ( + ) n Q ( + ) n + Λ (−) n Q (−) n (cid:17)i Z ( n c ) GGE , (21)for an arbitrary n c ≥ r , where Z ( n c ) GGE is the normalizationfactor. In the limit of n c → ∞ after the thermodynamiclimit, the GGE reproduces expectation values of arbitrarylocal operators in the steady state.Next, we consider free fermion models in which the totalparticle number is not conserved ( B l , h c † x c y i and h c † x c † y i with | x − y | P ≤ r , the latter ofwhich appears because B x − y ,
0. By expressing these two-point functions using the mode occupation numbers η † p η p ,we have h c † x c y i = L Õ p cos [ p ( x − y )] (cid:16) s ( p ) − | t ( p )| (cid:17) h η † p η p i + iL Õ p sin [ p ( x − y )]h η † p η p i + const ., (22)and h c † x c † y i = iL Õ p sin [ p ( x − y )] s ( p ) t ( p )h η † p η p i + const . (23)By performing the Fourier series expansion, we can express h c † x c y i and h c † x c † y i as h c † x c y i = v L Q ( + ) x − y + L ( L − )/ Õ n = v n (cid:16) Q ( + ) x − y + n + Q ( + ) x − y − n (cid:17) + iL Q (−) x − y + const ., (24) and h c † x c † y i = − iL ( L − )/ Õ n = w n (cid:16) Q ( + ) x − y + n − Q ( + ) x − y − n (cid:17) + const . (25)Here, v n in Eq. (24) and w n in Eq. (25) are the Fouriercoefficients of 2˜ ε ( p ) + ˜ ε (− p ) (cid:16) s ( p ) − | t ( p )| (cid:17) (26)and 4 i ˜ ε ( p ) + ˜ ε (− p ) s ( p ) t ( p ) , (27)respectively, where the Fourier coefficient of φ ( p ) is definedby φ n = ( / L ) Í p φ ( p ) e − ipn . It is noted that the relations v n = v − n and w n = − w − n , which follows from the parity ofthe functions (26) and (27), are used in deriving Eqs. (24)and (25). According to the Riemann-Lebesgue lemma, v n and w n tend to zero in the limit of | n | → ∞ taken after thethermodynamic limit (see the note in [53]). Therefore, we canapproximately truncate the summations over n in Eqs. (24)and (25) at a sufficiently large n ∗ , e.g., h c † x c y i ≈ v L Q ( + ) x − y + L n ∗ Õ n = v n (cid:16) Q ( + ) x − y + n + Q ( + ) x − y − n (cid:17) + iL Q (−) x − y + const . (28)This approximation becomes exact in the limit of n ∗ → ∞ taken after the thermodynamic limit.In this way, the eigenstate expectation value of a localoperator with a maximum range r is approximately writtenas a linear combination of Q (±) n / L with n ≤ r + n ∗ , and thisapproximation becomes exact in the limit of n ∗ → ∞ . Itimplies that any local operator satisfies n c -ETH in the limitof n c → ∞ after the thermodynamic limit. Thus, the stronggeneralized ETH has been proved. Conclusion.—
The strong generalized ETH proved in thiswork ensures that if the initial state is in a generalized shellconstructed by local conserved quantities, the system relaxesto a steady state that is described by the GGE, either trun-cated or not. Since a physically relevant initial state, e.g., astate prepared by a quench, has subextensive fluctuations ofmacroscopic quantities, such an initial state is necessarily ina generalized shell. Therefore, a steady state after relaxationis described by a GGE in a translationally invariant noninter-acting integrable system. Our results can be generalized to d -dimensional systems and noninteracting bosons.In the previous studies, the validity of the GGE has beenproved for noninteracting integrable models with translationinvariance by requiring the cluster decomposition propertyfor the initial state [30–32]. In contrast, our result appliesto dynamics with the initial state which can be any state ina single generalized shell. Since the cluster decompositionproperty does not hold for all of such states, our result showsthat the GGE is valid for a wider class of initial states thanexpected previously. It should be noted that the removal ofthe assumption of the cluster decomposition property is par-ticularly important when we consider a spin model that ismapped to quadratic fermions, e.g., the transverse-field Isingchain and the XY chain. In these models, a physically realis-tic initial state should obey the cluster decomposition prop-erty with respect to the spin operators , but it is not obviouswhether the same initial state obeys the cluster decompositionproperty with respect to the fermion operators [44].In this work, we have assumed the translation invarianceand the locality of the quadratic Hamiltonian. It is a futureproblem to clarify whether these assumptions are essential forthe relaxation towards a GGE. Considering the case of non-local Hamiltonians is important in validating the relaxationto the Floquet GGE [54, 55] in the low-frequency regimeof time-periodic systems, where the effective Hamiltoniangenerally becomes nonlocal [56].It is also open to extend the strong generalized ETH tointeracting integrable systems, which cannot be mapped tofree particles but exactly solvable via the method of the Betheansatz. Although the idea of the generalized ETH has beenapplied to interacting integrable systems [57, 58], it has notbeen proven in the strong sense. A recent finding of quasi-local charges in the XXZ chain has advanced our under-standing on the validity of the GGE in interacting integrablesystems [43]. The quasi-local charges should be taken intoaccount properly; otherwise, the generalized ETH cannot betrue [37, 38] and the GGE fails [39–42].We are grateful to Professor N. Hatano for fruitful discus-sions and careful reading of the manuscript. We also thankDr. T. N. Ikeda and Dr. Y. Watanabe for useful discussions.We thank Professor M. Rigol for valuable comments on themanuscript. T.I. was supported by the Program for LeadingGraduate Schools, MEXT, Japan. ∗ [email protected] † [email protected][1] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Ad-vances in Physics , 239 (2016).[2] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. , 124(2015).[3] T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, J. Phys. B , 112001 (2018).[4] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. Mcculloch, U. Scholl-wöck, J. Eisert, and I. Bloch, Nat. Phys. , 325 (2012).[5] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko,P. M. Preiss, and M. Greiner, Science , 794 (2016).[6] C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodrubetz,Z. Chen, A. Megrant, R. Barends, B. Campbell, B. Chiaro,A. Dunsworth, E. Jeerey, J. Kelly, J. Mutus, P. J. J. O ’malley,C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White,A. Polkovnikov, and J. M. Martinis, Nat. Phys. , 1037(2016).[7] G. Clos, D. Porras, U. Warring, and T. Schaetz, Phys. Rev.Lett. , 170401 (2016).[8] K. Saito, S. Takesue, and S. Miyashita, J. Phys. Soc. Jpn. ,1243 (1996).[9] M. Rigol, V. Dunjko, and M. Olshanii, Nature , 854 (2008).[10] F. Jin, H. De Raedt, S. Yuan, M. I. Katsnelson, S. Miyashita,and K. Michielsen, J. Phys. Soc. Jpn. , 124005 (2010). [11] J. M. Deutsch, Phys. Rev. A , 2046 (1991).[12] M. Srednicki, Phys. Rev. E , 888 (1994).[13] G. Biroli, C. Kollath, and A. M. Läuchli, Phys. Rev. Lett. ,250401 (2010).[14] R. Steinigeweg, J. Herbrych, and P. Prelovšek, Phys. Rev. E , 012118 (2013).[15] H. Kim, T. N. Ikeda, and D. A. Huse, Phys. Rev. E , 052105(2014).[16] W. Beugeling, R. Moessner, and M. Haque, Phys. Rev. E ,042112 (2014).[17] N. Shiraishi and T. Mori, Phys. Rev. Lett. , 030601 (2017).[18] T. Mori and N. Shiraishi, Phys. Rev. E , 022153 (2017).[19] T. Mori, arXiv:1609.09776 (2016).[20] E. Iyoda, K. Kaneko, and T. Sagawa, Phys. Rev. Lett. ,100601 (2017).[21] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys.Rev. Lett. , 050405 (2007).[22] A. C. Cassidy, C. W. Clark, and M. Rigol, Phys. Rev. Lett. , 140405 (2011).[23] F. H. L. Essler and M. Fagotti, J. Stat. Mech. , 064002(2016).[24] We note that some studies have shown [59, 60] that a kind ofrelaxation that is defined in terms of certain statistical indexesmay not be qualitatively affected by integrability.[25] T. Barthel and U. Schollwöck, Phys. Rev. Lett. , 100601(2008).[26] S. Sotiriadis, P. Calabrese, and J. Cardy, EPL (EurophysicsLetters) , 20002 (2009).[27] P. Calabrese, F. H. L. Essler, and M. Fagotti, Phys. Rev. Lett. , 227203 (2011).[28] P. Calabrese, F. H. L. Essler, and M. Fagotti, Journal ofStatistical Mechanics: Theory and Experiment , P07016(2012).[29] P. Calabrese, F. H. L. Essler, and M. Fagotti, Journal ofStatistical Mechanics: Theory and Experiment , P07022(2012).[30] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Phys.Rev. Lett. , 030602 (2008).[31] S. Sotiriadis and P. Calabrese, J. Stat. Mech. , P07024(2014).[32] M. Gluza, C. Krumnow, M. Friesdorf, C. Gogolin, and J. Eis-ert, Phys. Rev. Lett. , 190602 (2016).[33] A. Bastianello and S. Sotiriadis, J. Stat. Mech. , 023105(2017).[34] B. Pozsgay, J. Stat. Mech. , P07003 (2013).[35] M. Fagotti and F. H. Essler, J. Stat. Mech. , P07012 (2013).[36] M. Fagotti and F. H. L. Essler, Phys. Rev. B , 245107 (2013).[37] G. Goldstein and N. Andrei, Phys. Rev. A , 043625 (2014).[38] B. Pozsgay, J. Stat. Mech. , P09026 (2014).[39] B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto,M. Rigol, and J.-S. Caux, Phys. Rev. Lett. , 117202 (2014).[40] B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos,G. Zaránd, and G. Takács, Phys. Rev. Lett. , 117203 (2014).[41] M. Brockmann, B. Wouters, D. Fioretto, J. De Nardis,R. Vlijm, and J.-S. Caux, J. Stat. Mech. , P12009 (2014).[42] M. Mestyán, B. Pozsgay, G. Takács, and M. Werner, J. Stat.Mech. , P04001 (2015).[43] E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L.Essler, and T. Prosen, Phys. Rev. Lett. , 157201 (2015).[44] C. Murthy and M. Srednicki, arXiv:1809.03681 (2018).[45] M. den Nijs and K. Rommelse, Phys. Rev. B , 4709 (1989).[46] K. Hida and S. Takada, J. Phys. Soc. Jpn. , 1879 (1992).[47] Y. Hatsugai, J. Phys. Soc. Jpn. , 3856 (1992).[48] L. Vidmar and M. Rigol, J. Stat. Mech. , 064007 (2016).[49] The condition of subextensive fluctuations of macroscopic lo-cal conserved quantities is much weaker than the conditionof the cluster decomposition property. The latter implies the former, but the former does not imply the latter.[50] It should be noted that the converse is not true in general.[51] Although Wick’s theorem is usually applied to the vacuumstate, it can be applied to individual eigenstates too in thepresent systems because the eigenstates can be expressed asa vacuum state by redefining the particles and holes for eachmode. See Ref. [52].[52] L. G. Molinari, arXiv:1710.09248 (2017).[53] The value Λ ε ( p ) + Λ ε (− p ) is non-negative for all p . In the casein which Λ ε ( p ) + Λ ε (− p ) touches the p -axis, the functions (26)and (27) may not be L -integrable. In this case, the followingarguments remain valid by replacing Λ ε ( p ) + Λ ε (− p ) by a constant δ c > ε ( p ) + Λ ε (− p ) becomes smaller than δ c , and taking the limit δ c → L → ∞ and n c → ∞ .[54] A. Lazarides, A. Das, and R. Moessner, Phys. Rev. Lett. ,150401 (2014).[55] T. Ishii, T. Kuwahara, T. Mori, and N. Hatano, Phys. Rev. Lett. , 220602 (2018).[56] L. D’Alessio and M. Rigol, Phys. Rev. X , 041048 (2014).[57] J.-S. Caux and F. H. L. Essler, Phys. Rev. Lett. , 257203(2013).[58] E. Ilievski, E. Quinn, J. De Nardis, and M. Brockmann, J.Stat. Mech. , 063101 (2016).[59] G. P. Berman, F. Borgonovi, F. M. Izrailev, and A. Smerzi,Phys. Rev. Lett. , 030404 (2004).[60] L. F. Santos, F. Borgonovi, and F. M. Izrailev, Phys. Rev. Lett.108