Emergent eigenstate solution to quantum dynamics far from equilibrium

Emergent eigenstate solution to quantum dynamics far from equilibrium

Lev Vidmar Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA    Deepak Iyer Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837, USA    Marcos Rigol Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA

The quantum dynamics of interacting many-body systems has become a unique venue for the realization of novel states of matter. Here we unveil a new class of nonequilibrium states that are eigenstates of an emergent local Hamiltonian. The latter is explicitly time dependent and, even though it does not commute with the physical Hamiltonian, it behaves as a conserved quantity of the time-evolving system. We discuss two examples of integrable systems in which the emergent eigenstate solution can be applied for an extensive (in system size) time: transport in one-dimensional lattices with initial particle (or spin) imbalance, and sudden expansion of quantum gases in optical lattices. We focus on noninteracting spinless fermions, hard-core bosons, and the Heisenberg model. We show that current-carrying states can be ground states of emergent local Hamiltonians, and that they can exhibit a quasimomentum distribution function that is peaked at nonzero (and tunable) quasimomentum. We also show that time-evolving states can be highly-excited eigenstates of emergent local Hamiltonians, with an entanglement entropy that does not exhibit volume-law scaling.

I Introduction

Experiments with ultracold gases greiner_mandel_02b (); kinoshita_wenger_06 (); will_best_10 (); trotzky_chen_12 (); gring_kuhnert_12 (); will_iyer_15_97 (); langen_erne_15 (), photonic Christodoulides03 (); rechtsman_zeuner_13 () and solid state systems wang_steinberg_13 (); fausti_tobey_11 (); stojchevska14 (), and foundational theoretical developments polkovnikov_sengupta_review_11 (); dalessio_kafri_16 (); eisert_friesdorf_review_15 (); vidmar16 () are driving the study of the dynamics of quantum many-body systems at a rapid pace. When taken far from equilibrium, generic isolated quantum systems typically thermalize rigol_dunjko_08 (), whereas integrable systems (characterized by an extensive number of local conserved quantities) do not. They are instead described by generalized Gibbs ensembles rigol_dunjko_07 (); ilievski15 (). For nonintegrable systems close to integrable points, one expects relaxation to long-lived states (prethermal states berges_borsanyi_04 ()), that can also be described using generalized Gibbs ensembles kollar_wolf_11 (); essler_14 (); nessi_iucci_14 (). Drawing from notions of the equilibrium renormalization group, prethermal states can be understood as being nonthermal fixed points berges_rothkopt_08 (). More recently, the discovery of dynamical phase transitions heyl_polkovnikov_13 (); canovi_14 (); heyl_15 (), which are the result of nonanalytic behavior in time, has added another dimension to the connection between quantum dynamics and traditional statistical mechanics.

Here, we add yet another paradigm to this already rich phenomenology. It is motivated by recent theoretical studies that have revealed an intriguing emergence of power-law correlations (like those in ground states) in transport far from equilibrium. Such a phenomenon has been observed in various one-dimensional lattice systems of hard-core rigol04 (); rigol05a () and soft-core rodriguez_manmana_06 () bosons, spinful fermions hm08 (), and spins antal97 (); lancaster10 (); sabetta_misguich_13 (). Some of the observed behavior can be reproduced using ground states of effective Hamiltonians (different from the ones considered here) antal97 (); antal98 (); hm08 (); eisler09 (); sabetta_misguich_13 (). Another motivation for our work are recent experiments exploring the sudden expansion of ultracold fermionic and bosonic gases in optical lattices schneider12 (); ronzheimer13 (); xia_zundel_15 (); vidmar15 (). One of those experiments, which is of particular relevance to this work, studied the sudden expansion of a Mott insulator of strongly interacting bosons vidmar15 (). It was observed that peaks develop in the momentum distribution at nonzero momenta, signaling unconventional quasicondensation rigol04 (). Since the expansion in the experiments occurs at energies far above the ground-state energy, it has remained a mystery why quasicondensation (revealing the emergence of power-law correlations) occurs in such systems.

In this work, we provide an explanation for this phenomenon. We unveil the existence of a class of nonequilibrium states that are eigenstates of an emergent local Hamiltonian. We use the term emergent to highlight that it is not trivially related to the physical Hamiltonian dictating the dynamics. The emergent Hamiltonian is explicitly time dependent and behaves as a local conserved quantity, even though it does not commute with the physical Hamiltonian. The novelty of this class of states is that they exhibit nontrivial time evolution despite being eigenstates of a local conserved operator.

The concept of the emergent eigenstate solution provides new insights into several physical phenomena. It explains why power-law correlations with ground-state character emerge in current-carrying states of integrable (or nearly integrable) one-dimensional models. It also elucidates the dynamics of quasimomentum occupations in current-carrying states, and suggests a way to dynamically tune the position of the peak of the quasimomentum distribution function. In the context of entanglement entropy, the emergent eigenstate solution shifts the focus from the entanglement entropy of time-evolving states to the entanglement entropy of eigenstates of local Hamiltonians. It also highlights the physical relevance of highly-excited eigenstates of integrable systems in which the entanglement entropy does not exhibit a volume-law scaling.

The paper is organized as follows. In Sec. II, we introduce the framework of the emergent eigenstate description, and discuss properties of the emergent local Hamiltonian. We also consider conditions for the emergent eigenstate description to be valid. We then discuss two physical applications in Secs. III and IV. In Sec. III, we study transport and current-carrying states of noninteracting fermions, hard-core bosons, and in the Heisenberg model. We devote special attention to cases in which the time-evolving state is the ground state of the emergent local Hamiltonian. In Sec. IV, we study the sudden expansion of quantum gases in a setup close to the one realized in recent experiments in optical lattices. Finally, in Sec. V, we focus on the entanglement entropy of the current-carrying states studied in Sec. III. We show that highly-excited eigenstates that do not exhibit volume-law scaling of the entanglement entropy are the ones of relevance to the problems studied here. We summarize our results, and discuss other possible applications of the emergent eigenstate description, in Sec. VI.

Ii Emergent eigenstate description

ii.1 Construction of the emergent Hamiltonian

We consider systems initially described by a Hamiltonian


where and are extensive sums of local operators, namely, of operators with support on lattice sites. The parameter may take any value, including (for which ). We therefore do not require any of the two operators or to act as a perturbation. We consider initial states that are eigenstates of ,


where is the corresponding energy eigenvalue.

In the quantum quenches of interest here, at time , the time-independent Hamiltonian is changed instantaneously into the time-independent Hamiltonian ( after the quench), and the initial state evolves as (we set ). We relate the operators involved in such quenches by means of the equation


where is a local operator and is some constant.

We now manipulate Eq. (2), by inserting an identity and multiplying by on the left, leading to


where is a time-dependent operator in the Schrödinger picture (it is time independent in the Heisenberg picture, ). In general, is highly nonlocal and, hence, of no particular interest. Its nonlocal character is apparent in the series expansion of , which is an infinite sum of nested commutators of and . Each nonvanishing higher-order commutator usually extends the spatial support of the products of operators involved in . Using Eq. (3), we can write


where represents the -th order commutator of with


Even though the expansion (5) has, in general, zero radius of convergence, there are physically relevant problems for which becomes a local operator. This can occur either if vanishes at some finite or if the nested commutators close the sum (5). If is a local operator, we define . Physically, we interpret as being an emergent local Hamiltonian. is a local conserved operator of the time-evolving system (it is time independent in the Heisenberg picture, ). Its novelty comes from the fact that, despite being conserved, it does not commute with the Hamiltonian that governs the dynamics. This is possible only because is time dependent in the Schrödinger picture.

Whenever the emergent Hamiltonian description can be invoked, instead of explicitly time evolving the wavefunction, one only needs to find a single eigenstate of satisfying


Equations (1)-(4) imply that, in the absence of degeneracies, this eigenstate is . We call this scenario the emergent eigenstate solution to quantum dynamics. It gives rise to a new class of nonequilibrium states: states that simultaneously exhibit nontrivial time evolution and are eigenstates of a time-dependent local operator that is conserved during the time evolution of the system. By conserved we mean that the expectation value of is time independent under dynamics generated by .

The simplest family of quantum quenches in which the emergent eigenstate solution to quantum dynamics can be used is the one in which


It results in an emergent local Hamiltonian of the form


There are families of quantum quenches for which is not exactly conserved ( is nonzero for all ), but for which the time-dependent state is exponentially close to an eigenstate of the emergent local Hamiltonian for times that are proportional to the system size. In the context of physical applications, we are interested in families of quantum quenches that fall into this category. In such quenches is conserved up to boundary terms. Next, we discuss a criterion for the applicability of the emergent eigenstate description for those quenches, and clarify the role of the initial state.

ii.2 Emergent eigenstate description for approximately conserved operators

If is not exactly conserved, i.e., for all , but it is approximately conserved (e.g., results in terms with support only at the boundaries of the system), one may still invoke the emergent eigenstate description by truncating the series (5) at some . For the discussion in this section, we assume that one can truncate the series at . As a result, the emergent local Hamiltonian has the form (9). In principle, such is not conserved since in the Heisenberg picture


Nevertheless, there are families of initial states for which behaves as being approximately conserved for an extensive (in system size) time. In these cases, one can show that




and denotes the (linear) system size.

To see how Eqs. (11) and (12) imply that eigenstate of the emergent local Hamiltonian becomes indistinguishable from the time-evolving state with increasing system size, let us write in the basis defined by the eigenstates of :


where, from the normalization of , we have that , , and is the corresponding Hilbert-space dimension.

Using Eqs. (7) and (13), Eq. (11) can be rewritten as


where is the eigenenergy corresponding to eigenstate , . We then see that for Eq. (12) to be satisfied one generally needs


yielding .

Equation (11) can also be written as the expectation value of in the initial state as


Hence, even though is explicitly time dependent, the emergent local Hamiltonian can behave as a conserved quantity in some nonequilibrium states for extensively (in the system size) long times.

For most practical applications, the emergent eigenstate description provides a useful framework if the lowest-order terms in Eq. (16) are exactly zero. Moreover, an extensive time of validity will be obtained if an extensive number of expectation values vanish independently. This can be realized for systems in which is conserved up to boundary terms (see the next two sections).

The concept of the emergent eigenstate description formulated in this section opens a new window for studies of dynamics far from equilibrium. For example, if were the ground state of an emergent local Hamiltonian, then all the tools developed to study ground states of quantum systems could immediately be applied to understand a system far from equilibrium. As we discuss next, there are experimentally relevant time-evolving states that are ground states of such emergent local Hamiltonians.

In the following, we report two applications of the emergent eigenstate solution: we study transport and current-carrying states in Sec. III, and the sudden expansion of quantum gases in optical lattices in Sec. IV. In both applications, we make use of the property that even though is not exactly conserved (because of boundary terms), the emergent Hamiltonian satisfies Eq. (11) with an exponentially small over extensively (in the system size) long times. In Sec. V, we study entanglement properties of the emergent local Hamiltonian.

Iii Transport in integrable lattice systems

Transport of particles (or spin, energy, etc) far from equilibrium is a topic that has been attracting increasing attention in the context of isolated quantum systems. A convenient way to simulate current-carrying states in such systems is to prepare initial states with particle (spin, energy) imbalance that mimics large reservoirs, and let them evolve under homogeneous Hamiltonians. One of the most studied setups is the “melting” dynamics of a sharp domain wall antal99 (); karevski02 (); ogata02 (); rigol04 (); hunyadi04 (); hm08 (); eisler13 (); mossel10 (); santos11 (); vasseur15 (); hauschild15 (). This setup is a particular case of the more general one considered below.

iii.1 Noninteracting spinless fermions and hard-core bosons

We first construct an emergent eigenstate solution for systems of noninteracting spinless fermions and hard-core bosons. We focus on one-dimensional chains with sites ( is the particle number) and open boundary conditions. We prepare the initial state with particle imbalance by applying a linear gradient along the chain eisler09 (); lancaster10 (); lancaster16 ().

The initial state for noninteracting spinless fermions (SF) is the ground state of


where the kinetic energy and the potential energy terms, respectively, are


Here, creates a spinless fermion at site , and the site occupation operator is . The prefactor in Eq. (19) ensures that the expectation value of is extensive in system size. The strength of the linear gradient is measured in units of the hopping amplitude , and we set . In the limit , we simplify (17) and consider , see Appendix A for details.

We require to be large enough such that, in the ground state of , there exist regions with site occupations one and zero at the chain boundaries. This is achieved for , where is the critical value needed for the th single-particle Wannier-Stark state to be a Bessel function with support on sites. The time evolution of such initial state under produces a current-carrying state . We are interested in this current-carrying state before the propagating front of particles reaches the chain boundary.

We also study the same setup for hard-core bosons, which can be mapped onto spin-1/2 systems, and noninteracting spinless fermions cazalilla_citro_review_11 (). The interest in this model, which is the infinite on-site repulsion limit of the Bose-Hubbard model, is two fold: (i) it has been studied in experiments with ultracold atoms ronzheimer13 (); vidmar15 (), and (ii) in and out of equilibrium, the correlation functions can be very different from those of noninteracting fermions cazalilla_citro_review_11 (); vidmar16 ().

For hard-core bosons, we replace  (18) by


while, as a consequence of the mapping, the potential term is identical to  (19). Infinite repulsion is enforced by the constraints , where is the boson creation operator at site . We calculate expectation values of observables by expressing operators in terms of spinless fermions and following Refs. rigol04 (); rigol05a ().

Within the setup in this section, Eq. (3) results in and


which is the particle current operator in an open lattice. In finite systems with open boundaries, is not exactly conserved since


As a result, all higher-order commutators  (6) are nonzero. However, we show in the following that an extensive (in system size) number of expectation values of vanish in the initial state. Equation (5) can then be truncated at to get the emergent local Hamiltonian


Note that includes the difference between site-occupation operators at the lattice boundaries (times ). As we argue below, the emergent eigenstate solution is accurate as long as the propagating front of particles (holes) does not reach the chain boundary, i.e., as long as and . Therefore, in what follows, we replace and in Eq. (III.1), which ensures that the target eigenstate of the ensuing emergent local Hamiltonian ,


is the ground state. is the ground state of because the latter state is nondegenerate at all times. Hence, since is the ground state at , it must be the ground state at all times.

In Eq. (24), we merged the kinetic and the current operators from Eq. (III.1) into a single operator, characterized by the hopping amplitude


and the phase


In the ground state of , the phase determines the position of the maximum of the quasimomentum distribution function.

To prove that and are indeed relevant to the dynamics of interest here, we computed the expectation values of higher-order commutators in the initial state (16). The results are reported in Appendix B for the case in which , i.e., for the initial sharp domain wall. The analysis reveals that the emergent eigenstate description is exponentially accurate for . Since the maximal group velocity in the lattice is (in units of , where is the lattice spacing), the physical picture consistent with this time restriction is that the emergent Hamiltonian description (III.1) is valid so long as the expanding particles (holes) do not reach the edge of the lattice, . In that case, the dynamics is expected to be identical to that of a semi-infinite domain wall, for which the boundaries are irrelevant.

We complement the analytical results in Appendix B, and the physical picture that has emerged from them, by numerically calculating the subtracted overlap and the expectation value . In Fig. 1, we plot the numerical results for those quantities versus the rescaled time for various system sizes, where is given by the expression eisler09 (); lancaster10 ()


Note that for .

In Fig. 1, and are zero within machine precision at short times and start departing from zero when approaches , as advanced by our physical picture for . The same argument applies for nonzero . However, that case is more involved since: (i) the initial state exhibits a metallic interface between the left/right regions with maximal/vanishing site occupations, and (ii) the velocity of the propagating front is not constant [see Fig. 7(a) in Appendix D]. In Appendix D, we show that the time at which the site occupations at the boundaries of the lattice depart from their initial values is .

Figure 1: Times of validity of the emergent eigenstate description. (a) Subtracted overlap , where , of the time-evolving state with the ground state of the emergent local Hamiltonian  (24). (b) Expectation value of the emergent local Hamiltonian in the time-evolving state . Results in both panels are shown for different values of and , and three different system sizes , as indicated in the legend. For , is taken as explained in Appendix A. We rescale time dividing by  (27), which is proportional to the system size, to demonstrate the validity of the emergent eigenstate solution for when .

The results in Fig. 1 demonstrate that, as long as the emergent local Hamiltonian behaves as a conserved operator, which is the case whenever particles (or holes) have not reached the chain boundary, the ground state of is indistinguishable from the time-evolving state , as advanced in Sec. II.2.

We now turn our focus to physical properties of current-carrying states that can be described using the emergent eigenstate solution. We study one-particle properties such as site occupations, particle currents, decay of one-body correlations, and the quasimomentum distribution function.

The mapping of hard-core bosons onto noninteracting spinless fermions cazalilla_citro_review_11 () implies identical site occupations for bosons and fermions. Their dynamics, , from initial states with particle imbalance has been studied in the past antal99 (); karevski02 (); ogata02 (); rigol04 (); hunyadi04 (); eisler09 (); lancaster10 (). In many instances, plotting site occupations versus site positions divided by a function of the evolution time results in data collapse. In the setup under consideration here, this is achieved (for any ) by plotting site occupations versus , where . This results in a time- and -independent site occupation profile for , as shown in Fig. 2(a). One can use to derive the velocity of the propagating front of particles (holes) as a function of time [see Appendix D and Fig. 7(a)].

Another observable that yields identical expectation values for noninteracting fermions and hard-core bosons is the particle current  (21). Since does not commute with on a lattice with open boundaries, its expectation value is time dependent antal99 (). By invoking the Heisenberg representation and properties of higher-order commutators of with , as discussed in Appendix B, one gets that


i.e., the particle current increases linearly in time and is proportional to the difference in site occupations at boundary. This result was generalized in Ref. vasseur15 () to different families of current-carrying states, including interacting spinless-fermion systems. The relation (28) can already be inferred from Eq. (III.1) by requiring the expectation value of the emergent local Hamiltonian on the time-evolved state to be time independent. This suggests that Eq. (28) is accurate for the same times for which the emergent eigenstate description is, and it breaks down when the front of propagating particles (holes) reaches the lattice boundary. Hence the particle current in this setup belongs to the class of observables whose expectation values out of equilibrium are controlled, for an extensive (in system size) time, by expectation values of observables in the initial state.

Figure 2: Site occupations and nonlocal correlations in current-carrying states. (a) Site occupations versus . Results are shown for particles, different values of , and different times. The black solid line is the scaling solution for . The main panels in (b) and (c) display , where for noninteracting spinless fermions (b) and for hard-core bosons (c), as a function of for particles, , at time . The solid lines overlapping with the numerical results are in (b) and in (c) lancaster10 (). The insets in (b) and (c) display the absolute values of all elements of the one-body density matrix for the same model parameters and time as the main panels, but calculated for particles.

Next we study one-body correlations in the current-carrying states. This is of particular interest since an intimate relation between current-carrying states and power-law correlations has been observed for over the last thirty years rigol04 (); rodriguez_manmana_06 (); hm08 (); antal97 (); lancaster10 (); sabetta_misguich_13 (); schmittmann95 (); spohn83 (); prosen_znidaric_10 (). For noninteracting fermions antal99 () and hard-core bosons rigol04 (), previous studies revealed that the emergent (when starting from the sharp domain wall) power-law correlations in the current-carrying states exhibit ground-state exponents (ground state with respect to the physical Hamiltonian). This is an intriguing result because: (i) in thermal equilibrium in one dimension, power-law correlations can only be found in the ground state, and (ii) the current-carrying states considered here have energy densities well above that of the ground state of the physical Hamiltonian. The emergent eigenstate solution introduced in this work explains why ground-state-like correlations emerge during the far-from-equilibrium dynamics: the time-evolving states are ground states of emergent local Hamiltonians.

Figures 2(b) and 2(c) show the spatial decay of one-body correlations at time for a sharp domain wall (a product state with no correlations at all) as initial state. We define , where for noninteracting fermions and for hard-core bosons. For the former [Fig. 2(b)], one-body correlations decay as  antal99 (), while for the latter [Fig. 2(c)], they decay as  lancaster10 (). The spatial decay of (for ) is identical for all and . The insets in Figs. 2(b) and 2(c) display density plots of the absolute values of all elements of the one-body density matrix.

When computing the quasimomentum distribution function


in which for noninteracting fermions and for hard-core bosons, not only do the absolute values of [see Fig. 2] matter, but also their phases. Results for are shown in Fig. 3 at different times and for different values of . The maximum of exhibits two generic features during the dynamics: (i) it increases with time (coherence is dynamically enhanced), and (ii) its position shifts towards higher quasimomenta.

In the ground state of the emergent local Hamiltonian (24), the peak position is determined by the phase , which is identical for noninteracting fermions and hard-core bosons. We discuss its properties in Appendix D and Figs. 7(c)–7(d). Our analysis reveals that the peak emerges and stays at if the initial state is a sharp domain wall (a perfect product state, i.e., ). For all nonzero values of , the position of the peak changes with time and is limited to quasimomenta below .

Figure 3: Dynamics of the quasimomentum distribution function . (a) and (b): for noninteracting spinless fermions. (c) and (d): for hard-core bosons. Results are shown for in (a) and (c), and for in (b) and (d). Solid lines depict the numerical results, while the dashed lines in (a) and (b) were obtained using Eq. (31). Dotted vertical lines in all panels represent the shift of the peak in at time , and is given by Eq. (59).

For noninteracting fermions, increasing increases the spatial extent of the metallic interface in the initial state, which results in an increase of , see Figs. 3(a) and 3(b). In Appendix C we analytically show that


The dynamics of the quasimomentum distribution is obtained by a straightforward generalization of Eq. (30): (i) we change due to the time-dependent phase (26), and (ii) replace to account for the time-dependent hopping amplitude (25). This results in


The predictions of Eq. (31), plotted as dashed lines in Figs. 3(a) and 3(b), are indistinguishable from the numerical results (solid lines) obtained by time-evolving the initial state.

The quasimomentum distribution of hard-core bosons is markedly different from the fermionic one. This is a consequence of quasicondensation in the ground state of the emergent local Hamiltonian (24). Such a dynamical quasicondensation was studied theoretically in Ref. rigol04 (), in which the maximum value (in time) of the largest eigenvalue of the one-body density matrix was shown to scale as , and observed experimentally in Ref. vidmar15 (). The specific setup of Refs. rigol04 (); vidmar15 () is discussed in Sec. IV.

Summarizing our presentation so far, we have introduced a physically relevant example in which the emergent eigenstate solution is applicable and for which the target eigenstate is the ground state of the emergent local Hamiltonian. This allowed us to understand why power-law correlations (and quasicondensation) can occur in transport problems far from equilibrium. It also helped us gain an analytic understanding of the behavior of quasimomentum distribution functions, for which phase factors in one-body correlations lead to peaks at nonzero quasimomenta.

iii.2 Heisenberg model

The generality of the framework in Sec. II suggests that the emergent local Hamiltonian description is not restricted to noninteracting models or models mappable onto them. Here we demonstrate that this is indeed the case. We focus on one of the most widely studied models of quantum magnetism, the spin-1/2 XXZ chain. In contrast to the setup studied in Sec. III.1, the (approximately) conserved operator in this case is not the particle current, but the energy current.

To make an explicit connection with the results obtained for noninteracting fermions, we map the spin-1/2 XXZ model onto interacting spinless fermions cazalilla_citro_review_11 (), and follow the notation from Sec. III.1


where denotes the amplitude of nearest-neighbor interaction. We consider systems with open boundaries and lattice sites.

The operator , which is the boost operator zotos97 () for the energy current , plays the role of for noninteracting fermions. is defined as


and satisfies the relation . The energy current operator


is exactly conserved () for periodic boundary conditions, and it is sometimes denoted as because each of its terms has support on three lattice sites zotos97 (); mierzejewski15 ().

Figure 4: Emergent Hamiltonian description of the domain-wall melting for interacting spinless fermions. (a) Overlap as a function of rescaled time for (from full exact diagonalization) and (noninteracting system), and different system sizes. The inset depicts . For and (see inset), the results for interacting and noninteracting fermions are virtually indistinguishable at short times. Results for larger system sizes are only available for the noninteracting case. (b) and (c) Site and quasi-momentum occupations for obtained from (symbols) and from (lines; data shown for at time ). (b) Site occupations for and different times as a function of the rescaled coordinate . The data collapses onto a line gobert05 () as for noninteracting particles vidmar15 (). (c) Quasi-momentum distribution function for different system sizes and the same rescaled time .

disconnects the left from the right part of the chain at the bond denoted by . As a consequence, the sharp domain wall of spinless fermions, , is an eigenstate of with eigenvalue . We set the initial Hamiltonian and focus on the sharp domain wall as initial state so that Eq. (2) is satisfied with .

We construct the emergent local Hamiltonian by following the derivation in Sec. II, starting from . In analogy to Eq. (9), we define


We refer to the eigenstate of that describes the dynamics as .

The first question to be answered is the accuracy of the eigenstate to describe the time-evolving state . We follow the analysis presented in Sec. II.2. In the setup considered here (i.e., a finite system with open boundaries), the operator is not exactly conserved since


with being the leftmost site. The expectation value of this commutator in the initial state (16) vanishes. To demonstrate the validity of the emergent eigenstate description for an extensive (in system size) time (12), one needs to show that an extensive number of expectation values of higher-order commutators vanishes. This procedure is analogous to the one carried out for noninteracting fermions in Appendix B. Even though obtaining a general expression for high-order commutators of the Heisenberg model is a daunting task, two consecutive commutators beyond Eq. (III.2) confirm that the largest support of the operators in Eq. (16) grows linearly with the power of in Eq. (16), and that site occupation operators emerge in pairs (see Appendix E). We therefore conjecture that the time regime in which the emergent Hamiltonian description is exponentially accurate increases with the size of the initial domain wall, and hence with the total particle number , as for noninteracting fermions.

We complement the analysis above with numerical calculations. We use full exact diagonalization to calculate the time-evolved wavefunction and the eigenstate of the emergent local Hamiltonian . We add a small onsite potential to in the leftmost site to break the degeneracy of with other eigenstates with zero eigenenergy. This does not change . It only changes its eigenenergy as (maximal site occupancy) for the times for which the emergent Hamiltonian description is valid.

In Fig. 4(a), we plot the overlap between and for and different system sizes. All the overlaps are nearly one for (see also the inset), independently of whether the system is interacting () or not (). The results in the inset of Fig. 4(a) reveal that, as increases, the ratio for which and are identical within computer precision increases. These results are qualitatively similar to those in Fig. 1(a) and support the expectation that the emergent eigenstate description is also valid for interacting systems.

The emergent local Hamiltonian  (35) includes the noninteracting point . The sharp domain wall melting of noninteracting fermions can therefore be described either by an eigenstate of  (24) with , or by an eigenstate of . The two Hamiltonians describe different physics, nevertheless, they share at least one identical eigenstate . Note, however, that used in Sec. III.1 is the ground state of , while used here is a highly excited eigenstate of .

We now turn to physical properties of the current-carrying state. The shape of the propagating front has been already studied in the literature gobert05 (); sabetta_misguich_13 (); alba14 (). In Fig. 4(b), we plot the time evolution of the site occupations for . The results collapse onto the same curve when the lattice positions are divided by time. For  gobert05 (), as in Fig. 4(b), the site occupations do not differ significantly from the noninteracting case displayed in Fig. 2(a). In contrast, the quasimomentum distribution [see Fig. 4(c)] exhibits a pronounced difference with respect to the noninteracting case. This is because develops a peak at . For noninteracting fermions the peak emerges and remains at at all times. Further studies are needed to understand in a systematic way the effects of interactions and the role of initial states in long-lived current-carrying states of interacting spinless fermions.

Iv Sudden expansion of quantum gases in optical lattices

In this section, we discuss the application of the emergent eigenstate description to the symmetric expansion of trapped ultracold spinless fermions, or hard-core bosons, after the harmonic confining potential is suddenly turned off but the optical lattice remains on. This is different from traditional time-of-flight measurements in which all potentials, including the optical lattice, are turned off and the particles expand in free space.

In recent experiments studying such setups, strongly-interacting bosons were initially prepared in one-dimensional Mott insulating states with one boson per site in the center of the trap ronzheimer13 (); vidmar15 (). For very strong on-site repulsive interactions, one can think of such bosons as hard-core bosons and their expansion dynamics is that of a domain wall “melting” symmetrically. So long as there remain sites with occupation one, the melting on each side can be described independently using the emergent eigenstate solution from Sec. III.1. The question that remains is whether an emergent eigenstate solution exists after the occupation of the sites in the center of the system drops below one (but before the expanding fronts reach the edges of the lattice).

Figure 5: Sudden expansion of noninteracting spinless fermions, or hard-core bosons, in optical lattices. (a) Site occupations at different times. (b) Overlap between the time-evolving state and the eigenstate of the emergent local Hamiltonian (39), and in the inset. The results are plotted versus the rescaled time for four system sizes. When increasing the number of particles , within machine precision (see inset) until .

To answer that question, we study the dynamics of noninteracting spinless fermions (or, equivalently, hard-core bosons) on an open lattice with sites, running from to . We consider the initial state , with the central sites occupied, see Fig. 5(a), and the remaining ones empty. In analogy to Sec. III.1, the calculations can be straightforwardly extended to initial eigenstates that contain metallic (superfluid) domains surrounding the region with one fermion per site. Similar setups have been studied for hard-core bosons rigol04 (); rigol05a (); vidmar13 (), as well as interacting soft-core bosons rodriguez_manmana_06 (); vidmar13 (); brandino15 () and spinful fermions hm08 (); langer12 (); mei16 ().

The initial product state is an eigenstate of any Hamiltonian that is a sum of site occupation operators with arbitrary coefficients, which includes the linear and harmonic potentials. The emergent eigenstate solution can be constructed for both cases vidmar_xu_17 (). Here, we consider the linear potential, for which the analysis is simpler, , where the operator


is essentially the one defined in Eq. (19). An important difference w.r.t. Sec. III.1 is that, here, the initial state is not the ground state of , but a highly excited state with eigenenergy . The dynamics is studied under


Figure 5(a) shows the site occupations of the fermions, or, equivalently, hard-core bosons, at different times.

The emergent local Hamiltonian for this setup is


Even though this Hamiltonian is essentially the one in Eq. (24) in the limit , the target eigenstate lies in the center of the spectrum, and it is highly degenerate. Nevertheless, since the single-particle spectrum is nondegenerate, can be uniquely obtained as the Slater determinant comprising consecutive single-particle states and giving . This is the way is constructed.

As for the domain wall melting, the emergent local Hamiltonian (39) is in general not a conserved operator. The particle current operator does not commute with due to the open boundary conditions (22), which makes each higher-order commutator in Eq. (10) nonzero. However, the support of operators in grows linearly from the lattice boundaries, as shown by the analysis in Appendix B. As a result, the number of expectation values of that vanish in our initial state can be made arbitrarily large by just increasing the number of empty sites in the initial state. This results in the emergent eigenstate description being correct for arbitrarily long times independently of whether the site occupations in the center of the trap are one or below one during the expansion dynamics.

We verify the conclusions above by numerically calculating the overlap [see Fig. 5(b)]. The results in Sec. III.1 for may lead one to conclude that the emergent eigenstate description will break down at [see Fig. 1(a)]. In contrast, in Fig. 5(b), no change is observed in the dynamics of at time , i.e., when the region with one atom per site in the center of the lattice has melted. Instead, starts deviating from 1 around . This is a consequence of particles reaching the lattice boundaries (the initial state has empty sites on each side).

The emergent eigenstate description of the sudden symmetric expansion is therefore a nontrivial extension of the transport problems studied in Sec. III.1. The fact that the dynamics for within each half of the lattice in the symmetric expansion is identical to the domain-wall melting studied in Sec. III.1 allows us make an interesting observation. Namely, expectation values of observables in the ground state of Eq. (24) at are identical to the ones in a highly-excited eigenstate of Eq. (39) for a system that is two times larger. As for the XXZ chain, this implies that power-law correlations, which in thermal equilibrium can only be observed in the ground state, also occur in highly-excited eigenstates of integrable Hamiltonians.

V Entanglement entropy of the target eigenstate

Figure 6: Entanglement entropy of eigenstates of the emergent local Hamiltonian  (35) for . (a) Entanglement entropy of the eigenstate that describes the domain-wall melting, after subtracting , for different system sizes and . The thick solid line displays the ansatz (41) with as an average of for . [For a given , we obtain by fitting the numerical result to Eq. (41).] (b) Entanglement entropy per site at time as a function of the inverse system size. Diamonds depict the average entanglement entropy per site of eigenstates of the many-body spectrum (see text for details; for we average over eigenstates). The solid line is obtained from Eq. (41) for the same value of as in (a). The dashed line is a linear fit to for , with and .

The evolution of the entanglement entropy in transport problems has been studied both for noninteracting gobert05 (); eisler09 (); alba14 () and interacting fermions sabetta_misguich_13 (); alba14 (). In the following, we focus on the domain-wall melting in the Heisenberg model and the underlying emergent local Hamiltonian  (35). The entanglement entropy is defined as


where the reduced density matrix of the left/right subsystem is , i.e., a trace over the complement subsystem of the total density matrix . The length of both subsystems is set to . We focus on the entanglement entropy of excited eigenstates of the emergent local Hamiltonian , where the density matrix of eigenstate is . We are interested in the scaling of the entanglement entropy with the system size. In particular, we would like to distinguish between states that exhibit a volume-law scaling ( when ), and states with vanishing entropy density ( when ), which include area-law states and critical states .

We first focus on the entanglement entropy of the eigenstate of that describes the domain-wall melting. Previous studies of the domain-wall melting in quantum spin chains suggested a logarithmic growth with time sabetta_misguich_13 (); alba14 (). Here, we are interested in the entanglement entropy at a fixed rescaled time to extract the dependence on the system size. Following Ref. alba14 (), we take the following ansatz


where is a constant. This heuristic ansatz is motivated by, but not equal to, the result for the local quench eisler07 (); calabrese_cardy_07 (). It was shown to provide accurate results, up to small temporal oscillations, for noninteracting fermions and the Heisenberg chain alba14 ().

Equation (41) predicts that, for a fixed , the quantity should be independent of the lattice size. The numerical results in Fig. 6(a) are, up to small temporal fluctuations, consistent with this expectation. This confirms that the entanglement entropy per site vanishes in the thermodynamic limit [see circles in Fig. 6(b)]. The novel aspect about this result is that it demonstrates that there are highly excited energy eigenstates of the interacting Hamiltonian that exhibit a non-volume law scaling of the entanglement entropy. They are of relevance to transport problems like the ones studied here. Their contribution is expected to be negligible in the context of statistical mechanics.

On the other hand, the majority of eigenstates are expected to exhibit a volume-law scaling with system size. In Fig. 6(b), we calculate the average entanglement entropy of eigenstates of the many-body spectrum. Here, is the Hilbert space dimension, and we choose the eigenstates , where is the eigenstate that describes the domain-wall melting. The dashed line in Fig. 6(b) is a linear extrapolation of the entanglement entropy density vs to . It makes apparent that the entanglement entropy of the majority of the eigenstates of the emergent local Hamiltonian exhibits a volume-law scaling with the system size.

Our results thereby highlight the importance of gaining a better understanding of the entanglement entropy of highly-excited eigenstates of interacting integrable models alba09 (); santos_polkovnikov_12 (); deutsch13 (); storms14 (); ares14 (); beugeling15 (); lai15 (); garrison15 (). Eigenstates with non-volume-law scaling of the entanglement entropy exist throughout the spectrum, and they may play a special role in nonequilibrium states such as the ones studied here.

Vi Discussion and Outlook

The emergent eigenstate solution to quantum dynamics introduced in this work uncovers a new class of nonequilibrium states. The key property of this class of states is that they are eigenstates of a local operator we call the emergent Hamiltonian, which behaves as a conserved quantity. We have introduced a general framework that can be used to construct emergent local Hamiltonians. They are explicitly time dependent and do not commute with the physical Hamiltonian.

We constructed emergent local Hamiltonians for simple experimentally relevant setups in which the Hamiltonian before the quench is either the boost operator or the boost operator plus the final Hamiltonian. Those setups are relevant to transport in systems with initial particle (or spin) imbalance, and to the sudden expansion of quantum gases in optical lattices. We should stress, however, that emergent local Hamiltonians can also be constructed to describe the dynamics of initial states that are stationary states of Hamiltonians that do not contain the boost operator vidmar_xu_17 ().

We used the emergent eigenstate solution in the context of noninteracting spinless fermions, hard-core bosons, and the Heisenberg model. We have shown that: (i) time-evolving current-carrying states can be ground states of emergent local Hamiltonians. The description of this family of states does not require one to invoke the concept of local equilibrium, which is often applied to describe nonequilibrium steady states. The quasimomentum distribution of those states exhibits a maximum at nonzero quasimomentum, which can be tuned by modifying the initial state. Our results explain the main features observed in a recent experiment with ultracold bosons in optical lattices vidmar15 (). (ii) Time-evolving states can also be highly-excited eigenstates of emergent local Hamiltonians, with entanglement entropies that do not exhibit volume law scaling. This highlights the physical relevance of some non-volume-law states in the bulk of the spectrum of integrable Hamiltonians to quantum dynamics.

There are problems of current interest in various fields to which the emergent eigenstate description introduced in this work can be applied: (i) The quench dynamics of thermal equilibrium states (relevant to theory and experiments dealing with ultracold quantum gases). Two recent studies of that problem were undertaken in Refs. xu17 (); vidmar_xu_17 (), in which (mixed) time-evolving states were shown to be Gibbs states of the emergent local Hamiltonian in Eq. (24), and an effective cooling observed numerically was explained analytically using  xu17 (). (ii) In the context of periodically driven systems, related ideas have been recently used to engineer integrable Floquet dynamics. Namely, to engineer driven systems that do not exhibit chaotic behavior under a periodic drive gritsev17 (). (iii) As pointed out by a referee, the construction introduced in this work can also be used to design fast forward Hamiltonians, which aim at bringing a system from the ground state of one Hamiltonian to the ground state of another Hamiltonian. This has potential applications in atomic, molecular and optical physics to design protocols that are much faster than the ones relying on adiabatic processes Torrontegui13 (). (iv) Even though the systems considered here are integrable, the emergent eigenstate construction only requires finding one conserved (or almost conserved) operator for a given initial state. This hints possible applications to (weakly) nonintegrable systems. The emergence of power-law correlations in the (nonintegrable) one-dimensional Bose-Hubbard model rodriguez_manmana_06 (), lends support to this expectation.

This work was supported by the Office of Naval Research. We acknowledge insightful discussions with M. Mierzejewski, C. D. Batista, F. Heidrich-Meisner, A. Polkovnikov, T. Prosen, and U. Schneider.

Appendix A Initial sharp domain wall

A possible initial state to study current-carrying states is a sharp domain wall (a product state), in which the first sites () of the lattice with open boundaries are occupied, and there are empty sites (). This state is an eigenstate of  (17) in the limit in (17) and results in . To avoid confusion with the limit , we consider in this case the initial Hamiltonian and the emergent local Hamiltonian