Quantum quenches and thermalization in one-dimensional fermionic systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Quantum quenches and thermalization in one-dimensional fermionic systems
Marcos Rigol
Department of Physics, Georgetown University, Washington, DC 20057, USA
We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional latticesafter a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- andtwo-body observables such as the momentum distribution function [ n ( k ) ] and the density-density structurefactor [ N ( k ) ], respectively, and study the effects of approaching an integrable point. We show that while therelaxation dynamics and thermalization of N ( k ) for fermions is very similar to the one of hardcore bosons,the behavior of n ( k ) is distinctively different. The latter observable exhibits a slower relaxation dynamics infermionic systems. We identify the origin of this behavior, which is related to the off-diagonal matrix elementsof n ( k ) in the basis of the eigenstates of the Hamiltonian. More generally, we find that thermalization occursfar away from integrability and that it breaks down as one approaches the integrable point. PACS numbers: 03.75.Ss, 05.30.Fk, 02.30.Ik, 67.85.Lm
I. INTRODUCTION
In recent years, the study of the nonequilibrium dynamicsof isolated quantum systems has attracted a great deal of at-tention. The main motivation for these studies lies behindthe spectacular success that experimentalists have achievedin trapping and manipulating ultracold quantum gases. Thishas allowed them to, for example, load ultracold bosons inoptical lattices and study the collapse and revival of the mat-ter wave interference after quenching the interaction strengthfrom deep in the superfluid regime into deep in the Mott in-sulating regime [1], and to observe the damping of the dipoleoscillations when the center of mass of the gas was displacedaway from the center of the trap [2–4].Since these atomic gases are trapped, cooled, and manip-ulated in a very high vacuum, i.e., in no contact with anykind of thermal or particle reservoir, a fundamental questionthat arises is whether after a sudden perturbation the systemwill relax to thermal equilibrium (an equilibrium in which ob-servables are described by standard statistical mechanical en-sembles). The usual assumption is that thermalization occursin general. However, recent experiments in one-dimensional(1D) geometries (created by a deep two-dimensional opticallattice) have failed to observe relaxation to a thermal distribu-tion after a quench [5]. The absence of thermalization can beunderstood in the very special regime in which the system isat an integrable point [6]. Interestingly, in Ref. [5] the gas wasperturbed away from integrability and thermalization did notoccur. Other experiments in 1D geometries (generated by anatom chip) have reported the indirect observation of thermal-ization [7]. In the latter case, the transverse confinement wasnot as strong as the one induced by the optical lattice in Ref.[5]. The question of whether strict one dimensionality couldaffect the outcome of the relaxation dynamics has not beenfully addressed experimentally.The particular case in which the system is at an integrablepoint lends itself to combined analytical and numerical stud-ies, which have allowed theorists to show that in that case therelaxation dynamics after a quench lead to nonthermal equi-librium distributions of few-body observables [6]. In addi-tion, a generalization of the Gibbs ensemble, which takes into account the conserved quantities that make the system inte-grable, has been shown to successfully describe experimentalobservables after relaxation [6, 8]. Further analytical and nu-merical works have analyzed the relevance of the generalizedGibbs ensemble to different models and observables and ad-dressed its limits of applicability [9–18].When the systems are far away from integrability, for exam-ple in higher dimensions, the expectation is that they shouldthermalize. Recent numerical work has shown this to be thecase for bosons in two dimensions, studied with exact diago-nalization [19], and for fermions studied within the dynamicalmean-field theory (DMFT) approximation [20]. In addition,thermalization in the isolated two-dimensional bosonic casecould be understood within the eigenstate thermalization hy-pothesis (ETH) proposed by Deutsch [21] and Srednicki [22],in which the individual eigenstates of the many-body Hamil-tonian exhibit thermal behavior [19].In one-dimensional nonintegrable systems, the situation isa bit more subtle as many models can be tuned (by chang-ing some Hamiltonian parameters) to be arbitrarily close tointegrable points. An early study of the relaxation dynamicsin fermionic systems when breaking integrability, using time-dependent density renormalization group (tDMRG) [23], con-cluded that thermalization does not occur even if the system isperturbed away from the integrable point. Another early workin 1D, also using tDMRG, studied the relaxation dynamics ofthe (nonintegrable) Bose-Hubbard model when quenching thesystem across the superfluid-to-Mott-insulator transition [24].In that case the authors concluded that thermalization occursin some regimes but not in others.More recently, we have performed a systematic study ofthermalization in finite 1D systems of hardcore bosons. Ourresults indicate that far away from integrability the systemdoes thermalize [25]. However, thermalization breaks downas one approaches the integrable point. An important ques-tion that still needs to be answered is what happens with thepoint at which thermalization breaks down as one approachesthe thermodynamic limit. It may either move toward the in-tegrable point or toward some point away from integrability.More powerful numerical techniques or analytical approachesmay be required to answer that question, which will not beaddressed here.In this work, we extend the analysis in Ref. [25] to thefermionic case, which allows us to discuss some of the is-sues that remained open in Ref. [23]. In particular, the timescales required for different observables to relax to a station-ary “equilibrium” distribution, and the description of the ob-servables after relaxation. We also study how the closeness toan integrable point affects the dynamics and thermalization af-ter a quantum quench. As in earlier works [19, 25], we showthat the breakdown of thermalization for our observables isdirectly linked to the breakdown of ETH.The presentation is organized as follows. In Sec. II, weintroduce the lattice model to describe 1D fermions withnearest- and next-nearest-neighbor hopping and repulsive in-teractions. We define our observables of interest and brieflydiscuss our numerical approach. The nonequilibrium dynam-ics of this model is presented in Sec. III. Section IV is devotedto the analysis of statistical mechanical approaches to describeobservables after relaxation. In Sec. V, we justify these statis-tical mechanical approaches, or their failure, within ETH. Anexplanation of why the momentum distribution function of thefermions has a distinctively slow relaxation dynamics is pre-sented in Sec. VI in terms of the off-diagonal elements of thisobservable in the basis of the eigenstates of the Hamiltonian.Finally, the summary and outlook are presented in Sec. VII.
II. HAMILTONIAN AND OBSERVABLES
The Hamiltonian of spin-polarized fermions in a 1D latticewith periodic boundary conditions can be written as ˆ H = L X i =1 (cid:26) − t (cid:16) ˆ f † i ˆ f i +1 + H.c. (cid:17) + V (cid:18) ˆ n i − (cid:19) (cid:18) ˆ n i +1 − (cid:19) − t ′ (cid:16) ˆ f † i ˆ f i +2 + H.c. (cid:17) + V ′ (cid:18) ˆ n i − (cid:19) (cid:18) ˆ n i +2 − (cid:19)(cid:27) . (1)where the fermionic creation and annihilation operators at site i are denoted by ˆ f † i and ˆ f i , respectively, and the local-densityoperator by ˆ n i = ˆ f † i ˆ f i . In Eq. (1), the nearest- and next-nearest-neighbor hopping parameters are denoted by t and t ′ ,respectively, and the nearest- and next-nearest-neighbor inter-actions are denoted by V and V ′ , respectively. In our study,we only consider repulsive interactions ( V, V ′ > ), and thenumber of lattice sites is denoted by L .In Ref. [25], we have already studied a very similar Hamil-tonian for hardcore bosons (bosons with an infinite on-site re-pulsion), in which case ˆ H b = L X i =1 (cid:26) − t (cid:16) ˆ b † i ˆ b i +1 + H.c. (cid:17) + V (cid:18) ˆ n bi − (cid:19) (cid:18) ˆ n bi +1 − (cid:19) − t ′ (cid:16) ˆ b † i ˆ b i +2 + H.c. (cid:17) + V ′ (cid:18) ˆ n bi − (cid:19) (cid:18) ˆ n bi +2 − (cid:19)(cid:27) . (2)where the hardcore boson creation and annihilation operatorsat site i are denoted by ˆ b † i and ˆ b i , respectively, and the local-density operator by ˆ n bi = ˆ b † i ˆ b i . The parameters t, V, t ′ , and V ′ in Eq. (2) have exactly the same meaning for hardcorebosons as for the fermions in Eq. (1). For hardcore bosons,the creation and annihilation operators commute as usual forbosons [ˆ b i , ˆ b † j ] = [ˆ b i , ˆ b j ] = [ˆ b † i , ˆ b † j ] = 0 , for i = j, (3)but on the same site, the hardcore bosons operators satisfyanticommutation relations typical of fermions, n ˆ b i , ˆ b † i o = 1 , ˆ b † i = ˆ b i = 0 . (4)These constraints avoid double or higher occupancy of the lat-tice sites.Similar to the hardcore boson case studied in Ref. [25], theHamiltonian (1) is integrable for t ′ = V ′ = 0 . In this section,we restrict our analysis to systems with 1/3 filling, and studythe effects of finite, but small, values of t ′ = V ′ = 0 whenone departs from the integrable point. In this case, the groundstate of our Hamiltonian is always a Luttinger liquid and nometal-insulator transition occurs in the system [26].For t ′ = 0 , the fermionic Hamiltonian (1) can be mappedonto the identical Hamiltonian for hardcore bosons (2) (up toa boundary effect), i.e., the fermionic operators in Eq. (1) justneed to be replaced by operators describing hardcore bosons.This can be done using the Jordan-Wigner transformation[27], and implies that in the thermodynamic limit both sys-tems have the same spectrum and identical diagonal correla-tions. However, off-diagonal correlations and related observ-ables, such as the momentum distribution function, are verydifferent for fermions and bosons. A finite value of t ′ still al-lows for a mapping of the fermionic Hamiltonian (1) onto aHamiltonian of hardcore bosons. However, the new hardcoreboson Hamiltonian for t ′ = 0 will be different to the one inEq. (2). The mapping introduces additional operators in theterm proportional to t ′ . In addition, in the fermionic case, afinite value of t ′ breaks the particle-hole symmetry present for t ′ = 0 and present for any value of t ′ in the hardcore bosoncase.The study of the nonequilibrium dynamics and thermody-namics of these systems is performed using full exact diago-nalization of the Hamiltonian (1). For our largest system sizes,eight fermions in 24 sites, the total Hilbert space has dimen-sion D = 735 471 . We take advantage of the translationalsymmetry of the lattice, which allows us to block diagonalizethe full Hamiltonian while the dimension of the largest block(in k space) is D k = 30 667 .In this work, we focus our analysis on two observables. Thefirst one in the momentum distribution function ˆ n ( k ) = 1 L X i,j e − k ( i − j ) ˆ f † i ˆ f j , (5)which is the Fourier transform of the one-body density matrix( ˆ ρ ij = ˆ f † i ˆ f j ). As mentioned before, this observable behavesvery differently for fermions as compared with the hardcorebosons to which they can be mapped. We should add that n ( k ) is usually measured in experiments with ultracold quantumgases via time-of-flight expansion.The second observable of interest is the density-densitystructure factor ˆ N ( k ) = 1 L X i,j e − k ( i − j ) ˆ n i ˆ n j , (6)which is the Fourier transform of the density-density correla-tion matrix. Since we work at fixed number of fermions N f ,the expectation value of ˆ N ( k = 0) is always h ˆ N ( k = 0) i = N f /L and, as usual, we set it to zero by subtracting that trivialconstant. The structure factor can be measured in cold gasesexperiments by means of noise correlations [28]. III. NONEQUILIBRIUM DYNAMICS
We will restrict our study of the nonequilibrium dynamicsto the case in which the system is taken out of equilibrium bymeans of a quench. Quenches are a special way to induce dy-namics by starting with an eigenstate of some initial Hamilto-nian ˆ H ini , and then instantaneously changing the Hamiltonian(at time τ = 0 ) to some final time-independent Hamiltonian ˆ H fin .As mentioned previously, our model is integrable for t ′ = V ′ = 0 . Since we are interested in studying the effect that ap-proaching an integrable point has on the dynamics and ther-malization of the system, we take the initial state to be aneigenstate of a system with t = t ini , V = V ini , t ′ , V ′ ,and then quench the nearest-neighbor parameters t and V to t = t fin , V = V fin without changing t ′ , V ′ , i.e., we onlychange t ini , V ini → t fin , V fin . This quench is performedfor different values of t ′ , V ′ as one approaches the integrablepoint t ′ = V ′ = 0 .Notice that our quenches do not break translational symme-try. For that reason, we always select the initial state to be oneof the eigenstates of the initial Hamiltonian in the total k = 0 sector. Since translational symmetry is preserved, only stateswith zero total momentum are required to calculate the exacttime evolution of the system | ψ ( τ ) i = e − i b H fin τ | ψ ini i = X α C α e − iE α τ | Ψ α i , (7)where | ψ ( τ ) i is the time-evolving state, | ψ ini i is the initialstate, | Ψ α i are the eigenstates of the final Hamiltonian withzero total momentum, energy E α , and C α = h Ψ α | ψ ini i . Thesum runs over all the eigenstates of the total k = 0 sector.In this section, we perform the exact time evolution of up toeight fermions in lattices with up to 24 sites. This means thatthe largest total k = 0 sector diagonalized had 30 666 states.In contrast to classical systems, where one has to performthe time evolution in order to compute the long-time aver-age of any observable, in isolated quantum systems (wherethe time evolution is unitary) one can predict such time av-erages without the need of calculating the dynamics. Sincethe wave function of any initial state can be written in termsof the eigenstates of the final Hamiltonian | Ψ α i as | ψ ini i = P α C α | Ψ α i , one finds that the time evolution of the quantum expectation value of an observable ˆ O can be written as h b O ( τ ) i ≡ h ψ ( τ ) | b O | ψ ( τ ) i = X α, β C ⋆α C β e i ( E α − E β ) τ O αβ , (8)where O αβ are the matrix elements of ˆ O in the basis of thefinal Hamiltonian. This in turn implies that the infinite timeaverage of the observable can be written as h b O ( τ ) i ≡ O diag = X α | C α | O αα . (9)We have assumed that, as in the case of generic (noninte-grable) systems, the spectrum is nondegenerate and incom-mensurate. This means that if the expectation value of ˆ O re-laxes to some kind of equilibrium value (up to the recurrencesthat must occur if the system is isolated), that value should bethe one predicted by Eq. (9). Following previous work, weprefer to think of this exact result as the prediction of a “diag-onal ensemble,” where | C α | is the weight of each state withinthis ensemble [19, 25].One objection that may arise at this point is that even ifthe system is nondegenate and incommensurate, the spectrummay have arbitrarily close levels and it may take an unrealisti-cally long time for the prediction in Eq. (9) to apply, i.e., that itmay not be relevant to describe experiments. In order to studyhow the dynamics drives our observables of interest [ n ( k ) and N ( k ) ] toward the prediction of Eq. (9), we follow the schemein Ref. [25]. We study the normalized area between n ( k, τ ) and N ( k, τ ) , during the time evolution, and their infinite timeaverage, i.e., at different times we compute δn k ( τ ) = P k | n ( k, τ ) − n diag ( k ) | P k n diag ( k ) (10)and δN k ( τ ) = P k | N ( k, τ ) − N diag ( k ) | P k N diag ( k ) (11)In Fig. 1, we show results for δn k ( τ ) and δN k ( τ ) as a func-tion of time τ for four different quenches as one approachesthe integrable point. Besides undergoing the same change t ini = 0 . , V ini = 2 . → t fin = 1 . , V fin = 1 . , thesequenches have another very important property in common.The initial state for each was selected to be one eigenstate ofthe initial Hamiltonian in such a way that the effective tem-perature T of the system [29] is always the same ( T = 2 . in Fig. 1). Given the energy of the time-evolving state in thefinal Hamiltonian, which is conserved, E = h ψ ini | b H fin | ψ ini i , (12)the effective temperature T [29] is defined by the expression E = 1 Z Tr n ˆ H fin e − ˆ H fin /T o , (13)where Z = Tr n e − ˆ H fin /T o (14)is the partition function, and we have set the Boltzmann con-stant k B to unity, and t fin = 1 sets the energy scale in oursystem.In Figs. 1(a) and 1(f), we compare the initial momentumdistribution and structure factors with the predictions of Eq.(9) away from integrability ( t ′ = V ′ = 0 . ). (The re-sults for other values of t ′ , V ′ are similar and not shownhere.) The dynamics of those observables for four differentvalues of t ′ , V ′ is presented in Figs. 1(b)–1(e) for δn k ( τ ) andin Figs. 1(g)–1(j) for δN k ( τ ) . They show that the dynam-ics of n ( k ) and N ( k ) are different. Away from integrabil-ity ( t ′ = V ′ = 0 ), δN k ( τ ) quickly relaxes [in a time scale τ ∼ ( t fin ) − = 1 . ] to a value ∼ δn k ( τ ) , on the other hand, slowly drifts to-ward δn k ( τ ) ∼ . – . in a much longer time scale, ordersof magnitude longer than the one seen for the relaxation of δN k ( τ ) . - π - π/2 π/2 π n ( k ) - π - π/2 π/2 π N ( k ) InitialDiagonal δ n k δ N k δ n k δ N k δ n k δ N k τ δ n k τ δ N k (b)(c)(e) (g)(d) (h)(i)(j)(a) (f) t’=V’=0t’=V’=0.04t’=V’=0.12t’=V’=0.32t’=V’=0.32T=2.0, L=24 FIG. 1: (Color online) Quantum quench t ini = 0 . , V ini = 2 . → t fin = 1 . , V fin = 1 . , with t ′ ini = t ′ fin = t ′ and V ′ ini = V ′ fin = V ′ in a system with N f = 8 and L = 24 . The initial statewas chosen in such a way that after the quench the system has aneffective temperature T = 2 . [29] in all cases. Given the energyof the initial state in the final Hamiltonian E = h ψ ini | b H fin | ψ ini i ,the effective temperature is computed following Eq. (13). (a) Initialand diagonal ensemble results for n ( k ) when t ′ = V ′ = 0 . . (b)–(e) Time evolution of δn k for t ′ = V ′ = 0 . , 0.12, 0.04, and 0.0,respectively. (f) Initial and diagonal ensemble results for N ( k ) when t ′ = V ′ = 0 . . (g)–(j) Time evolution of δN k for t ′ = V ′ = 0 . ,0.12, 0.04, and 0.0, respectively. The behavior of δN k ( τ ) in these spin-polarized systems isqualitatively (and quantitatively) very similar to the one ob-served for hardcore bosons in Ref. [25]. The time evolutionof δn k for the fermions, on the other hand, is different fromthe one seen for the hardcore bosons in Ref. [25]. For thelatter systems, n ( k ) quickly relaxed toward the predictions ofthe diagonal ensemble, even at integrability. In Fig. 1(j), onecan see that at integrability δN k ( τ ) for the fermions also re-laxes quickly toward the predictions of the diagonal ensembleand then exhibits fluctuations between δN k ( τ ) = 0 . and δN k ( τ ) = 0 . , while δn k ( τ ) in Fig. 1(e) exhibits very largeoscillations between δn k ( τ ) = 0 . and δn k ( τ ) = 0 . . Ad-ditionally, we studied the dynamics of δn k for a time scale ten times longer than the one depicted in Fig. 1(e) and observedexactly the same behavior.For hardcore bosons [25], we have shown that increasingthe effective temperature [29] (or the final energy E ) de-creases the mean value of δn k ( τ ) and δN k ( τ ) after relaxationand also decreases the temporal fluctuations of those quanti-ties. In Fig. 2, we show results for quenches similar to the onesin Fig. 1, but with a higher effective temperature ( T = 3 . ).The comparison between Figs. 2 and 1 also reveals a reductionin the mean values of δn k ( τ ) and δN k ( τ ) and their fluctua-tions, which is particularly obvious at integrability. Still, dif-ferences remain between the relaxation dynamics of δn k ( τ ) and δN k ( τ ) . Also, one can notice that at integrability themean values of δn k ( τ ) and δN k ( τ ) and their fluctuations arein general larger than away from integrability, and they all de-crease the further away one moves from the integrable point. - π - π/2 π/2 π n ( k ) - π - π/2 π/2 π N ( k ) InitialDiagonal δ n k δ N k δ n k δ N k δ n k δ N k τ δ n k τ δ N k (b)(c)(e) (g)(d) (h)(i)(j)(a) (f) t’=V’=0t’=V’=0.04t’=V’=0.12t’=V’=0.32t’=V’=0.32T=3.0, L=24 FIG. 2: (Color online) Quantum quench t ini = 0 . , V ini = 2 . → t fin = 1 . , V fin = 1 . , with t ′ ini = t ′ fin = t ′ and V ′ ini = V ′ fin = V ′ in a system with N f = 8 and L = 24 . The initialstate was chosen in such a way that after the quench the system hasan effective temperature T = 3 . [29] in all cases (see the captionof Fig. 1). (a) Initial and diagonal ensemble results for n ( k ) and t ′ = V ′ = 0 . . (b)–(e) Time evolution of δn k for t ′ = V ′ = 0 . ,0.12, 0.04, and 0.0, respectively. (f) Initial and diagonal ensembleresults for N ( k ) and t ′ = V ′ = 0 . . (g)–(j) Time evolution of δN k for t ′ = V ′ = 0 . , 0.12, 0.04, and 0.0, respectively. The improvement of the relaxation dynamics with increas-ing the effective temperature (energy) of the isolated state[29], discussed above, can be related in part to the increasein the density of states in the final system with increasing en-ergy. This increases the number of eigenstates of the Hamil-tonian that participate in the dynamics making dephasing inEq. (8) more effective, and hence reducing temporal fluctua-tions after relaxation. This can be better seen in Fig. 3, whichclearly shows that the number of states with the largest val-ues of | C α | increases as the temperature decreases. Since P α | C α | = 1 , this means that the lower the temperature thesmaller the total number of states that participate in the dy-namics. This can be also seen in Fig. 3, where the numberof states with | C α | > − and | C α | > − is larger for T = 3 . [Fig. 3(b)] than for T = 2 . [Fig. 3(a)] (notice thelogarithmic scale in both axes). Overall, one can conclude thatdephasing becomes less effective with decreasing temperatureand this leads to an enhancement of temporal fluctuations.A similar argument, based on the number of states partic-ipating in the dynamics, can help us understand why the av-erage values and temporal fluctuations of δn k ( τ ) and δN k ( τ ) are in general larger after relaxation when one is at integra-bility, or close to it. Figure 3 also shows that for any giventemperature, the total number of states that participate in thedynamics decreases as one approaches the integrable point. Atintegrability, this may be related to the presence of additionalconserved quantities, which restrict the number of eigenstatesof the final Hamiltonian that can have a significant overlapwith the initial state. -5 -4 -3 -2 | C α | N u m b e r o f S t a t e s t’=V’=0 t’=V’=0.03 t’=V’=0.06 t’=V’=0.12 t’=V’=0.24 -4 -3 -2 -1 | C α | (a) (b) T=2.0 T=3.0
FIG. 3: (Color online) Number of states with | C α | greater than thevalue presented in the x axis, for an effective temperature (a) T =2 . and (b) T = 3 . , and for the same quenches studied in Figs. 1and 2, respectively. Here, N f = 8 and L = 24 . It would be desirable to study how the mean values andtemporal fluctuations of δn k ( τ ) and δN k ( τ ) behave after re-laxation as one increases the system size. Unfortunately, thecomputational requirements of our approach increase expo-nentially with system size and a rigorous finite-size scaling isnot possible. As a step in understanding how finite-size effectsaffect our results here, we show in Fig. 4 results for a smallersystem with seven fermions in 21 lattice sites (with T = 3 . ).By comparing those results with the ones in Fig. 2 one cansee that, as expected, decreasing the system size increases themean values of δn k ( τ ) and δN k ( τ ) after relaxation and thetemporal fluctuations of both quantities. This supports the ex-pectation that with increasing the system size, after relaxation,the mean values of δn k ( τ ) and δN k ( τ ) should become arbi-trarily small and so should the temporal fluctuations.A different question is what happens to the time scale for n ( k ) to relax to the diagonal ensemble prediction as the sys-tem size increases. Previous work on that direction [30, 31]has suggested a possible intermediate quasisteady regime thatoccurs before full relaxation. This has been seen in recent nu-merical work [20], and our results for the behavior of n ( k ) insmaller systems may be an indication in that direction. Some-thing that is important to keep in mind from our results in thissection is that different observables may exhibit different re-laxation times. In particular, and in contrast to the hardcoreboson case, the momentum distribution function of fermionsis an observable that takes a long time to relax to an equi-librium distribution. This may have influenced the failure to - π - π/2 π/2 π n ( k ) - π - π/2 π/2 π N ( k ) InitialDiagonal δ n k δ N k δ n k δ N k δ n k δ N k τ δ n k τ δ N k (b)(c)(e) (g)(d) (h)(i)(j)(a) (f) t’=V’=0t’=V’=0.03t’=V’=0.12t’=V’=0.32t’=V’=0.32T=3.0, L=21 FIG. 4: (Color online) Quantum quench t ini = 0 . , V ini = 2 . → t fin = 1 . , V fin = 1 . , with t ′ ini = t ′ fin = t ′ and V ′ ini = V ′ fin = V ′ in a system with N f = 7 and L = 21 . The initial states wereselected in the same way as the ones in Fig. 2 ( T = 3 . ). (a) Initialand diagonal ensemble results for n ( k ) and t ′ = V ′ = 0 . . (b)–(e) Time evolution of δn k for t ′ = V ′ = 0 . , 0.12, 0.04, and 0.0,respectively. (f) Initial and diagonal ensemble results for N ( k ) and t ′ = V ′ = 0 . . (g)–(j) Time evolution of δN k for t ′ = V ′ = 0 . ,0.12, 0.04, and 0.0, respectively. observe thermalization in the momentum distribution functionof a nonintegrable fermionic system in Ref. [23].It is interesting to note that if V = t ′ = V ′ = 0 (noninter-acting case) the momentum distribution function of fermionsin a periodic system is a conserved quantity, i.e., for any ini-tial state it will not change in time. However, the n ( k ) ofhardcore bosons, to which the fermions can be mapped, canhave a nontrivial dynamics and relaxes to the predictions ofa generalized Gibbs ensemble, as shown in Refs. [6] and [8].Having V, V ′ = 0 (in our case V = t = 1 . and V ′ variesbetween 0 and 0.32) allows the n ( k ) of fermions to changewith time (since now the fermions are interacting), but thetime scale for relaxation to the equilibrium distribution stillseems to be affected by the fact that in k space the scatteringbetween fermions is very special. We find the relaxation timefor n ( k ) to be much longer than the corresponding time scalefor other observables such as N ( k ) . The differences betweenthe time scales for the relaxation of different observables (andtheir temporal fluctuations) can be understood in terms of theoff-diagonal matrix elements of the observables, which willbe discussed in Sec. VI. IV. THERMODYNAMICS
In Sec. III, we have shown that, for our systems of inter-est, the diagonal ensemble provides an accurate descriptionof observables after relaxation. Some observables may takelonger to relax, and under some circumstances they may noteven relax at all [see, e.g. , n ( k ) in Fig. 1(e)], but wheneverwe find relaxation to an equilibrium value, it is well describedby the predictions of Eq. (9). This was expected and we re-fer the reader to Refs. [14, 19, 25] for similar results in otherquantum systems. In what follows, we study how the predic-tions of the diagonal ensemble compare to standard statisticalmechanical ensembles. A great advantage of the infinite timeaverage [Eq. (9)] is that all time dependence has been removedfrom the time evolution of the quantum-mechanical problemand one has a unique result to test statistical mechanics.Since the systems on which we have performed the dy-namics are isolated, the most appropriate statistical ensembleto compare observables after relaxation is the microcanon-ical ensemble. As usual, the computations in the micro-canonical ensemble are performed averaging over all eigen-states (from all momentum sectors) that lie within a window [ E − ∆ E, E + ∆ E ] , where E = h ψ ini | ˆ H fin | ψ ini i , and wehave taken ∆ E = 0 . in all cases. Similarly to what was donein Refs. [19] and [25], we have checked that our results are in-dependent of the exact value of ∆ E in the neighborhood of ∆ E = 0 . .The main panel in Fig. 5(a) depicts how the difference be-tween the diagonal and the microcanonical ensembles behavesas one moves away from integrability. Once again, we find adifferent behavior for n ( k ) and N ( k ) . For N ( k ) , we find thatfor both effective temperatures [29] considered, the differencebetween the diagonal and microcanonical ensemble is alwayssmaller than 1% for t ′ = V ′ > . , and one can say that thesystem exhibits thermal behavior. This is very similar to theresults for the same quantity obtained in the hardcore bosonsystems analyzed in Ref. [25]. On the other hand, n ( k ) ex-hibits a larger difference between the diagonal and the micro-canonical ensemble for the same values of t ′ , V ′ , in particularat lower temperatures. For both observables, one can still seethat as one approaches the integrable point the difference be-tween both ensembles increases, signaling the breakdown ofthermalization in all cases.In the inset in Fig. 5(a), we also compare the predictions oftwo diagonal ensembles generated by different initial states,which are chosen in such a way that the effective temperatureof both system after the quench [29] are the same. The behav-ior is qualitatively similar to the one seen in the main panel ofthe same figure and shows that the breakdown of thermaliza-tion for both observables is accompanied by a dependence onthe initial state.Results for a smaller system size, with the same densityand effective temperatures, are shown in the main panel andinset in Fig. 5(b). They show that, as expected, the differencesbetween the two ensembles for any given value of t ′ , V ′ in-crease with decreasing system size. Notice that the y axis inFigs. 5(a) and 5(b) has a different scale, so that the contrastbetween the results in both panels is stronger than may appearat first sight.For t ′ = V ′ > . , the comparison between the results fordifferent systems sizes led us to expect that the predictions ofthe microcanonical ensemble, for both observables, will co-incide with the ones of the diagonal ensemble as the systemsize is increased, i.e., that thermalization takes place if one issufficiently far from integrability. We do note that n ( k ) forthese fermionic systems is more sensitive to finite-size effectsthan N ( k ) , something that is in contrast to the behavior seenfor hardcore bosons [25]. R e l . D i ff . D i a g . v s M i c r o ca n . T=2.0, n(k) T=2.0, N(k) T=3.0, n(k) T=3.0, N(k) R e l . D i ff . t w o i n iti a l s t a t e s t’,V’ R e l . D i ff . D i a g . v s M i c r o ca n . T=2.0, n(k) T=2.0, N(k) T=3.0, n(k) T=3.0, N(k) R e l . D i ff . t w o i n iti a l s t a t e s (a)(b) L=24L=21
FIG. 5: (Color online) (Main panels) Comparison between the pre-diction of the microcanonical and diagonal ensembles, for n ( k ) and N ( k ) , as a function of increasing t ′ , V ′ for T = 2 . and T = 3 . .Results are shown for (a) L = 24 , N f = 8 , and (b) L = 21 , N f = 7 .The diagonal ensembles correspond to the quenches in Figs. 1, 2,and 4. (Insets) Comparison between the prediction of diagonal en-sembles generated by two different initial states, for T = 2 . and T = 3 . . The results shown are for (a) L = 24 , N f = 8 , and(b) L = 21 , N f = 7 . As in the main panels, one of the diagonalensembles is generated by initial states selected from the eigenstatesof a Hamiltonian with t ini = 0 . , V ini = 2 . , the other diagonalensemble is generated by initial states selected from the eigenstatesof a Hamiltonian with t ini = 2 . , V ini = 0 . . The final Hamil-tonian (with t fin = 1 . , V fin = 1 . ) and the effective temperature[29] are identical for both initial states. By relative differences in thisfigure we mean the normalized area between the different ensemblepredictions for n ( k ) and N ( k ) . Relative differences are computed inexactly the same way as δn k ( τ ) and δN k ( τ ) are computed in Eqs.(10) and (11), respectively. For values of t ′ = V ′ closer to the t ′ = V ′ = 0 integrablepoint, the outcome of increasing system size is less clear, atleast for the system sizes that we are able to analyze here.Two possibilities emerge. (i) As the system size increases thepoint at which the results for the diagonal an microcanoni-cal ensembles start to differ will move toward t ′ = V ′ = 0 .(ii) The very same point will move toward a nonzero value of t ′ , V ′ . Here, we are not able to discriminate between thosetwo scenarios, and further studies will be needed to addressthat question.It is relevant to notice that, for the small systems sizes stud-ied in this work, it is important to select the microcanonicalensemble when comparing with the outcome of the dynamics.This is because finite-size effects can make the predictionsof different standard statistical mechanical ensembles differ-ent from each other. In Fig. 6, we compare the results of themicrocanonical ensemble, which we have been using up tothis point, with the ones obtained with the canonical ensem-ble. One can see there that depending on the observable underconsideration the results of both ensembles can differ up to ∼ n ( k ) . Interestingly, they are not as large as theones observed for the same quantity in the case of hardcorebosons [25]. t’,V’ R e l . D i ff . C a n . v s M i c r o ca n . T=2.0, L=21 T=2.0, L=24 T=3.0, L=21 T=3.0, L=24
N(k)n(k)
FIG. 6: (Color online) Relative differences (normalized area) be-tween the predictions of the microcanonical and canonical ensem-bles for n ( k ) [upper four curves] and N ( k ) [lower four curves] insystems with L = 21 , N f = 7 , and L = 24 , N f = 8 . The rela-tive differences are computed in exactly the same way as δn k ( τ ) and δN k ( τ ) are computed in Eqs. (10) and (11), respectively. V. EIGENSTATE THERMALIZATION HYPOTHESIS
In what follows, we will analyze the reason underlying ther-malization in these isolated quantum systems, and the causefor the differences seen between n ( k ) and N ( k ) when com-paring the diagonal and microcanonical ensembles.In Sec. IV, we have shown that away from integrabilitythere are regimes in which the system thermalizes, i.e., inwhich the predictions of the diagonal ensemble and the mi-crocanonical ensemble coincide. This means that O diag = O microc , X α | C α | O αα = 1 N E, ∆ E X α | E − E α | < ∆ E O αα , (15)where N E, ∆ E is the number of energy eigenstates with ener-gies in the window [ E − ∆ E, E + ∆ E ] . This result is cer-tainly surprising because the left-hand side of Eq. (15) de-pends on the initial conditions through the overlaps of the ini-tial state with the eigenstates of the final Hamiltonian ( C α ),while the right-hand side of Eq. (15) only depends on the en-ergy E (as mentioned before, our results do not depend on thespecific value of ∆ E ).A possible solution to this paradox was advanced byDeutsch [21] and Srednicki [22] in terms of the eigenstate thermalization hypothesis (ETH). Within ETH, the differencebetween the eigenstate expectation values of generic few-bodyobservables [ O αα ] are presumed to be small between eigen-states that are close in energy. This implies that if one takes asmall window of energy ∆ E , as it is done in the microcanoni-cal ensemble, all the eigenstate expectation values within thatwindow will be very similar to each other. Hence, the mi-crocanonical average and a single eigenstate will provide es-sentially the same answer, i.e., the eigenstates already exhibitthermal behavior. The same will happen with the predictionof the diagonal ensemble as long as the | C α | ’s are stronglypeaked around the energy E . The latter has been shown tobe the case for quenches in hardcore boson systems in twodimensions [19], hardcore bosons in one dimension [25, 32],softcore bosons in one dimension [33], and generic quenchesin the thermodynamic limit [19]. FIG. 7: (Color online) (a),(d) Distribution of | C α | for two of thequenches depicted in Fig. 2, for (a) t ′ fin = V ′ fin = 0 . and (d) t ′ fin = V ′ fin = 0 . . In both cases, the final effective temperatureof the system is T = 3 . [29]. The vertical dashed lines signal theenergy E of the time-evolving state. (b),(e) Eigenstate expectationvalues of ˆ n ( k = 0) in the full spectrum (including all momentumsectors) of the Hamiltonian (1) with t = V = 1 . and, (b) t ′ = V ′ =0 . and (e) t ′ = V ′ = 0 . . (c),(f) Eigenstate expectation values of ˆ N ( k = π ) in the full spectrum (including all momentum sectors) ofthe Hamiltonian (1) with t = V = 1 . and, (c) t ′ = V ′ = 0 . and (f) t ′ = V ′ = 0 . . These results were obtained for systems with L = 24 and N f = 8 , i.e., the total Hilbert space consists of 735 471states. In Figs. 7(a) and 7(d), we show the distributions of | C α | for two of the quenches studied in Fig. 2. One quench isfar away from integrability and the other one at integrability.There are some features of the distributions of | C α | that areimportant to mention here. (i) They are neither similar to themicrocanonical nor to the canonical distributions. (ii) Theyare strongly peaked around the energy E of the time-evolvingstate. Actually, one can see in the figures that the values of | C α | decay exponentially as one moves away from E (sig-naled by the vertical line in the figures). (iii) They are notrelated to the effective temperature of the system, which is ex-actly the same in both cases ( T = 3 . ) while the distributionsof | C α | are clearly different. The specific values of | C α | ,and the exponent of their decay, depend on the initial state[32]. Because of these properties of the | C α | distributions,they alone cannot explain why thermalization takes place, forexample, for N ( k ) when t ′ = V ′ = 0 . in Fig. 5.In the bottom four panels of Fig. 7, we show the eigen-state expectation values of ˆ n ( k = 0) [(b),(e)] and ˆ N ( k = π ) [(c),(f)] for a system far from integrability, with t = V = 1 . and t ′ = V ′ = 0 . [(b),(c)], and for a system at integrability,with t = V = 1 . and t ′ = V ′ = 0 . [(e),(f)]. Those results,together with the | C α | distributions in Figs. 7(a) and 7(d),can help us understand the differences between the diagonaland the microcanonical ensembles in Fig. 5(a).The most striking feature in the bottom panels in Fig. 7 isthe contrast between the eigenstate expectation values at in-tegrability and far away from it. The former ones exhibit avery wide distribution of eigenstate expectation values. Thisis true even if one selects a narrow window at the center ofthe spectrum (equivalent to very high temperatures). As onemoves away from integrability, one can clearly see that thefluctuations between eigenstate expectation values of nearbyeigenstates reduces dramatically and ETH starts to be valid.For t ′ = V ′ = 0 . , a comparison between the expectationvalues of ˆ n ( k = 0) and ˆ N ( k = π ) shows that for the ener-gies of the final states created in our quenches, the distributionof the eigenstate expectation values of ˆ n ( k = 0) is relativelybroader than the one seen for ˆ N ( k = π ) , which explains whythe differences between the diagonal and microcanonical en-sembles in Fig. 5(a) are larger for the former one. It also helpsin understanding why ˆ n ( k ) is more sensitive to the selectionof the initial state.In order to gain a more quantitative understanding of howETH breaks down as one approaches integrability, we havecomputed the average relative deviation of the eigenstate ex-pectation values with respect to the microcanonical predic-tion, ∆ mic n ( k = 0) and ∆ mic N ( k = π ) . For any givenenergy of the microcanonical ensemble, these quantities arecomputed as ∆ mic n ( k = 0) = P α | n αα ( k = 0) − n mic ( k = 0) | P α n αα ( k = 0) (16)and ∆ mic N ( k = π ) = P α | N αα ( k = π ) − N mic ( k = π ) | P α N αα ( k = π ) (17)where n αα ( k = 0) and N αα ( k = π ) are the eigenstate expec-tation values of ˆ n ( k = 0) and ˆ N ( k = π ) , respectively, and n mic ( k = 0) and N mic ( k = π ) are the microcanonical expec-tation values at any given energy E . The sum over α contains all states with energies in the window [ E − ∆ E, E + ∆ E ] . Asdiscussed before, we have taken ∆ E = 0 . .Clearly, for different values of the Hamiltonian parameters,the lowest and highest-energy states as well as the level spac-ing are different. Hence, in order to make a meaningful com-parison between different systems as one moves away fromthe integrable point, we study the behaviors of ∆ mic n ( k = 0) and ∆ mic N ( k = π ) as a function of the effective tempera-ture T of a canonical ensemble that has the same energy E as the microcanonical ensemble. Given an energy E withinthe microcanonical ensemble, the effective temperature canbe computed by means of Eq. (13). We should stress that theeffective temperature is only used here as an auxiliary toolfor comparing different systems, i.e., it just provides a uniqueenergy scale for comparing observables independently of theHamiltonian parameters, which change the ground-state en-ergy and the level spacing. FIG. 8: (Color online) Average relative deviation of eigenstate ex-pectation values with respect to the microcanonical prediction vs T (see text). Results are presented for: (a),(c) ∆ mic n ( k = 0) and(b),(d) ∆ mic N ( k = π ) in systems with: (a),(b) L = 24 , N f = 8 and (c),(d) L = 21 , N f = 7 . The inset in (d) shows a direct compar-ison of ∆ mic N ( k = π ) between the systems with L = 24 , N f = 8 (thin lines) and the systems with L = 21 , N f = 7 (thick lines). Thevalues of t ′ , V ′ in the three cases depicted in the inset are, from topto bottom, t ′ = V ′ = 0 , t ′ = V ′ = 0 . , and t ′ = V ′ = 0 . , andfollow the same legends of the main panel shown in (b). In all cases t = V = 1 . . In Fig. 8, we present results for ∆ mic n ( k = 0) [(a),(c)]and ∆ mic N ( k = π ) [(b),(d)] vs T for six different values of t ′ = V ′ in systems with 24 lattice sites and eight fermions[(a),(b)] and 21 lattice sites and seven fermions [(c),(d)]. Theaverage relative deviations for both observables are presentedin the same scale, which immediately allows one to see whatwas already evident in Fig. 7, namely, that the average relativedeviations for ∆ mic n ( k = 0) are larger (more than two timeslarger) than the corresponding ones for ∆ mic N ( k = π ) forany given value of t ′ , V ′ . For temperatures T & . , one cansee that those deviations for both observables saturate with in-creasing t ′ , V ′ for t ′ = V ′ > . . Comparing the resultsfor the same observables but for different systems sizes, onecan see that with increasing system size the average relativedeviations are decreasing. This can be better seen in the insetin Fig. 8(d), where we have presented a direct comparison of ∆ mic N ( k = π ) for three different values of t ′ , V ′ in the sys-tems with 21 (thick lines) and 24 (thin lines) sites. At integra-bility, one does not see much of a change with changing sys-tem size for ∆ mic N ( k = π ) , but some reduction can be seenfor ∆ mic n ( k = 0) [Figs. 8(a) and 8(c)]. For t ′ = V ′ > ,we do see a clear reduction in all cases. At low temperatures T < . , the density of states is low and in many cases theuse of the microcanonical ensemble is not justified. From Fig.7, one can see that in that regime (of low energies) the fluc-tuations of the observables are very large and thermalizationis not expected to occur. An important question that needsto be addressed in the future is what will happen with thatwindow of temperatures (energies) with increasing the sys-tem size. For our small systems, we do not see a noticeablechange with increasing system size. t’,V’ ∆ m i c N ( k = π ) ∆ m i c n ( k = ) T=2.0, L=21 T=2.0, L=24 T=3.0, L=21 T=3.0, L=24 (a)(b)
FIG. 9: (Color online) Average relative deviation of eigenstate ex-pectation values with respect to the microcanonical prediction at twofixed temperatures ( T = 2 . and T = 3 . ) and for two system sizes( L = 21 , N f = 7 , and L = 24 , N f = 8 ) vs t ′ , V ′ . Results arepresented for: (a) ∆ mic n ( k = 0) and (b) ∆ mic N ( k = π ) . In allcases t = V = 1 . . An alternative way to present some of the results depictedin Fig. 8, which can help us make direct contact with the re-sults discussed in Fig. 5, is to take two values of the effec-tive temperature within the microcanonical ensemble and plotthe average relative deviation of eigenstate expectation valueswith respect to the microcanonical prediction for those tem- peratures as a function of the integrability breaking parame-ters t ′ , V ′ . This is done in Fig. 9 for T = 2 . and T = 3 . ,and L = 21 and L = 24 . Figure 9 clearly shows that thebreakdown of thermalization seen in Fig. 5 as one approachesintegrability is directly related to the increase in the relativedeviation of eigenstate expectation values with respect to themicrocanonical prediction, i.e., to the increase in the fluctua-tions of the eigenstate to eigenstate expectation values of ˆ n ( k ) and ˆ N ( k ) , or what is the same to the failure of ETH. A com-parison between Figs. 9(a) and 9(b) also helps in understand-ing why in Fig. 5 the difference between the diagonal and mi-crocanonical ensembles is larger for n ( k ) than for N ( k ) , evenwhen one is far from integrability. Figure 9(a) shows that forthe effective temperatures selected in Fig. 5, ∆ mic n ( k = 0) ismore than two times larger than ∆ mic N ( k = π ) . The reduc-tion in ∆ mic n ( k = 0) and ∆ mic N ( k = π ) , with increasingsystem size, can also be seen in Fig. 9 when comparing theresults for L = 21 and L = 24 . VI. OFF-DIAGONAL MATRIX ELEMENTS
While the diagonal elements of the observables, in the ba-sis of the eigenstates of the Hamiltonian, helped us understandwhether after relaxation the expectation values of generic few-body observables can be described by standard statistical me-chanical approaches, it follows from Eqs. (8) and (9) that theoff-diagonal elements of the observables can help us under-stand the relaxation dynamics and temporal fluctuations of ob-servables after relaxation, h b O ( τ ) i − h b O ( τ ) i = X αβα = β C ⋆α C β e i ( E α − E β ) τ O αβ . (18)As we pointed out in Sec. III, one of our findings in thiswork is that the momentum distribution function of fermionsexhibits a slower relaxation dynamics than the one seen forthe same observable in an identical system of hardcore bosons[25]. These differences were seen far from the integrable pointas well as at the integrable point. In Figs. 10(a) and 10(c),we compare the off-diagonal elements of ˆ n ( k = 0) betweena system of spinless fermions [Fig. 10(a)] and an identicalsystem with hardcore bosons [Fig. 10(c)], both systems arefar from integrability [with t ′ = V ′ = 0 . ] and the centraleigenstate has an energy for which the temperature of a canon-ical ensemble with the same energy would be T = 3 . . It isapparent in these two figures that the off-diagonal elementsof ˆ n ( k = 0) are larger for the fermions than for the hardcorebosons, in particular close to the diagonal.The relaxation dynamics of our other observable of inter-est N ( k ) , on the other hand, was seen to be very similarbetween the fermions studied in this work and the hardcorebosons studied in Ref. [25]. In both systems, this observ-able relaxed very quickly (in a time scale τ ∼ /t ). Figures10(b) and 10(d) depict results for the off-diagonal elements of ˆ N ( k = π ) . The results for fermions and hardcore bosons arealmost indistinguishable in this case, with all off-diagonal el-ements being much smaller than the diagonal ones. The con-0 α β (a) |n αβ (0)| −50 −25 0 25 50−50−25 0 25 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α (c) −50 −25 0 25 50 α β (b) |N αβ ( π )| −50 −25 0 25 50−50−25 0 25 50 0 0.05 0.1 0.15 0.2 0.25 0.3 α (d) −50 −25 0 25 50 FIG. 10: (Color online) Off-diagonal elements of ˆ n ( k = 0) [(a),(c)]and ˆ N ( k = π ) [(b),(d)] for fermions [(a),(b)] and hardcore bosons[(c),(d)] far from integrability. Results are shown for the 100 eigen-states closest to the one with energy E = − . ( T = 3 . , see text)for the fermions [(a),(b)] and E = − . ( T = 3 . , see text) for thehardcore bosons [(c),(d)]. In all cases t = V = 1 . , t ′ = V ′ = 0 . , L = 24 , and N f = N b = 8 . trast between the magnitude of the off-diagonal elements of ˆ n ( k = 0) in Fig. 10(a) and ˆ N ( k = π ) in Fig. 10(b) also ex-plains the difference between the relaxation dynamics of bothobservables in fermionic systems depicted in Fig. 1.Results for the same quantities for fermions and hardcorebosons at the integrable point are shown in Fig. 11. We findthe results at integrability in stark contrast with those far awayfrom the integrable point. At integrability, the off-diagonalelements of both observables are in most cases very small orzero, but then there are some states between which the off-diagonal elements can be very large (much larger than anyoff-diagonal element seen in the nonintegrable case). In ad-dition, very large off-diagonal elements can be seen betweenstates that have quite different energies. In Fig. 10, the largestoff-diagonal elements in the nonintegrable case always occurbetween states that are close in energies, and they are seen toreduce as the energy of the states depart from each other. Thisis not the case at integrability.A more quantitative understanding of these issues can begained by taking one state from Figs. 10 and 11 and plottingthe off-diagonal elements between that state (which we take tobe the one at the center) and all the closest energy eigenstatesfor our two observables of interest. These are depicted in Fig.12 for the nonintegrable case with t ′ = V ′ = 0 . [(a)–(c)]and at integrability t ′ = V ′ = 0 [(d)–(f)]. α β (a) |n αβ (0)| −50 −25 0 25 50−50−25 0 25 50 0 0.2 0.4 0.6 0.8 1 α (c) −50 −25 0 25 50 α β (b) |N αβ ( π )| −50 −25 0 25 50−50−25 0 25 50 0 0.1 0.2 0.3 0.4 0.5 0.6 α (d) −50 −25 0 25 50 FIG. 11: (Color online) Off-diagonal elements of ˆ n ( k = 0) [(a),(c)]and ˆ N ( k = π ) [(b),(d)] for fermions [(a),(b)] and hardcore bosons[(c),(d)] at integrability. Results are shown for the 100 eigenstatesclosest to the one with energy E = − . ( T = 3 . , see text) forthe fermions [(a),(b)] and E = − . ( T = 3 . , see text) for thehardcore bosons [(c),(d)]. In all cases t = V = 1 . , t ′ = V ′ = 0 . , L = 24 , and N f = N b = 8 . First, let us focus on the behavior of n αβ ( k = 0) . Resultsfor that quantity in the nonintegrable fermionic and hardcoreboson cases are shown in panels (a) and (b) for two latticeswith L = 21 and L = 24 , respectively. For each lattice size,one can clearly see that the off-diagonal elements of ˆ n ( k = 0) of the fermions are always larger than the ones of the bosons,and the differences are largest close to the diagonal. As the en-ergies between the states depart from each other n αβ ( k = 0) for bosons and fermions become similar. The comparison be-tween the two lattices also shows that with increasing systemsize the off-diagonal elements of the observables decrease, butthe fact that n αβ ( k = 0) for the fermions is larger close to thediagonal remains. Similarly, we find that with decreasing theenergy of the state the off-diagonal elements of n αβ ( k = 0) also become larger, although for temperatures T > . thiseffect is smaller than the finite-size effect difference betweenpanels (a) and (b) in Fig. 12.Results for n αβ ( k = 0) at integrability are presented inpanels (d) and (e) of Fig. 12. There one can also see that thevalues of n αβ ( k = 0) are in general larger for fermions thanfor hardcore bosons and that they decrease with increasingthe system size. However, here fermions and bosons share acommon feature, namely, large off-diagonal elements can beseen between states that have energies that are increasinglydifferent from each other. This, in conjunction with the fewernumber of states that have a significant overlap with the ini-1 FIG. 12: (Color online) Off-diagonal elements of ˆ n ( k = 0) [(a),(b),(d),(e)] and ˆ N ( k = π ) [(c),(f)] for fermions and hardcorebosons away from integrability [(a)–(c)] and at integrability [(d)–(f)].Panels (b),(c),(e),(f) correspond to cuts across Figs. 10 and 11, witheither α = 0 or β = 0 . Panels (a) and (d) depict results for theoff-diagonal elements of ˆ n ( k = 0) in systems with L = 21 . In allcases, the eigenstate defining the center of the window has the energyclosest to that of a canonical ensemble with temperature T = 3 . . tial state, can lead to larger temporal fluctuations during thetime evolution after a quench at integrability. This effect isenhanced at lower temperatures where we see some increasein the magnitude of the off-diagonal elements and a decreasein the number of states that overlap with the initial state (seeFig. 3).Finally, the off-diagonal elements for ˆ N ( k = π ) far fromintegrability and at integrability are presented in Fig. 12, pan-els (c) and (f), respectively. Those panels show that, for thedensity-density structure factor, off-diagonal elements are al-ways much smaller than the diagonal ones, and they are rel-atively smaller than the ones of ˆ n ( k ) . At integrability [Fig.12(f)], one can see a few values of N αβ ( k = π ) that areclearly larger than the rest, but they are still less and relativelysmaller than the ones seen for n αβ ( k = 0) in Fig. 12(e). Theseresults explain why ˆ N ( k ) takes less time to relax to the diag-onal ensemble prediction and why the temporal fluctuationsafter relaxation are also smaller for this observable. VII. SUMMARY
We have presented a detailed study of the relaxation dy-namics after a quantum quench and the description after re- laxation of the momentum distribution function n ( k ) (a one-particle observable) and the density-density structure factor N ( k ) (a two-particle observable) of spinless fermions withnearest and next-nearest hoppings and interactions in one-dimensional periodic lattices. We have also compared someresults for fermions with those of a similar hardcore bosonsystem. We should stress that those models are typical fordescribing the physics of one-dimensional systems. For ex-ample, they can be mapped onto the spin-1/2 XXZ linearchain with next-nearest-neighbor interactions. (The observ-ables analyzed here are related to the different spin-spin cor-relation functions.) Hence, we expect the results reported inthis manuscript to be generic and apply to other gapless mod-els and observables in one-dimensional systems.We have shown that, in general, n ( k ) for fermions exhibitsa slower relaxation dynamics, i.e., it takes longer to relax to anequilibrium distribution than other observables for fermions[such as N ( k ) ] and than n ( k ) for a similar one-dimensionalsystem of hardcore bosons (studied in Ref. [25]). Close to andat integrability, we have also found that n ( k ) may even fail torelax to an equilibrium distribution while N ( k ) , and n ( k ) fora similar bosonic system, do relax to equilibrium. We haveshown that this behavior of the dynamics of n ( k ) is related tothe particularly large off-diagonal elements of ˆ n ( k ) in the ba-sis of the eigenstates of the Hamiltonian. Those off-diagonalelements are larger than the ones for the same observable inhardcore boson systems, and much larger than the ones for ˆ N ( k ) in the same fermionic system. From the contrast be-tween the dynamics of n ( k ) and N ( k ) emerges a general pic-ture in which some few-body observables in isolated quan-tum systems may quickly relax to an equilibrium distributionwhile other observables may take longer to relax, or even failto relax, to an equilibrium distribution.We have shown that far from integrability observables af-ter relaxation are well described by standard ensembles fromstatistical mechanics, such as the microcanonical ensemble,which is particularly relevant to our small and isolated sys-tems. The ability of the microcanonical ensemble to predictthe expectation value of few-body observables after relaxationwas shown to be related to the validity of the eigenstate ther-malization hypothesis, within which the expectation valuesof observables in the eigenstates of the Hamiltonian alreadyexhibit thermal behavior. We have also shown that as oneapproaches the integrable point thermalization breaks down,with the differences between the microcanonical predictionsand the results of the relaxation dynamics increasing continu-ously as one converges toward the integrable point. We haveestablished that there is a clear correlation between the failureof the system to thermalize and the failure of the eigenstatethermalization hypothesis, i.e., as one approaches integrabil-ity the differences between the eigenstate expectation valuesof few-body observables increases between eigenstates thatare close in energy, and this happens over the full spectrum ofthe Hamiltonian.Our results here have been obtained for small one-dimensional lattices, which are relevant to current experi-ments in one-dimensional geometries [34]. However, severalimportant questions remain to be addressed in future works.2For example, what happens to the point at which thermaliza-tion breaks down (at all temperatures) as one increases thesystem size. Far from integrability, we find that for our smallsystems thermalization occurs whenever the effective temper-ature is T & . [29]. Another question that needs to be ad-dressed is what happens with the window of energies wherethermalization fails to occur far from integrability as one in-creases the system size. Finally, there is the question of thetime scale that n ( k ) of the fermions takes to relax to the ther-mal distribution, and the emergence of intermediate quasista-tionary distributions [30, 31], as the size of the system is in-creased. Experiments with ultracold gases will generate many new questions and help address the current ones. Acknowledgments
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