Emergent Eigenstate Solution to Quantum Dynamics Far from Equilibrium

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.

Popular Summary

Statistical mechanics describes systems in equilibrium, which means that they can be characterized by only a few parameters such as temperature. This is not true for systems that are far from equilibrium. These systems can exhibit surprising behavior, especially when they are quantum in nature. One known example of such behavior—first predicted theoretically and recently observed experimentally—is the emergence of correlations among atoms in an expanding Bose gas. Naively, such a far-from-equilibrium system is “very hot.” However, it exhibits quantum coherence like a Bose-Einstein condensate, a phase of matter seen when a dilute gas is cooled to near absolute zero temperature. We have developed a theoretical understanding of this phenomenon.

Our analysis shows that, in fact, the “high-temperature” far-from-equilibrium quantum state in this expanding Bose gas is also a “low-temperature” state but of a somehow different emergent system (or, more technically, Hamiltonian). This theoretical framework can be applied to a wide range of phenomena that have recently been studied both experimentally and theoretically. Specifically, we apply it to understanding transport and expansion dynamics of ultracold gases in optical lattices.

Moving forward, we also see our theoretical insights as providing a tool to engineer and manipulate complex dynamical quantum states with tailored properties. These insights have already been applied to periodically driven systems—ones where some microscopic parameter changes periodically in time—in order to avoid heating to a featureless infinite-temperature state.

Figure 1

Times of validity of the emergent eigenstate description. (a) Subtracted overlap $|1-O(t)|$, where $O(t)=|⟨{\mathrm{\Psi }}_t|\psi (t)⟩|$, of the time-evolving state $|\psi (t)⟩$ with the ground state $|{\mathrm{\Psi }}_t⟩$ of the emergent local Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\prime }(t)$ [Eq. (24)]. (b) Expectation value of the emergent local Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\prime }(t)$ in the time-evolving state $|\psi (t)⟩$. Results in both panels are shown for different values of ${\gamma }^{-1}=0$, 0.1, and 0.2, and three different system sizes $L=2N+1$, as indicated in the legend. For ${\gamma }^{-1}=0$, ${\widehat{\mathcal{H}}}_{\mathrm{SF}}^{\prime }(t)$ is taken as explained in Appendix pp1. We rescale time dividing by $\tau$ [Eq. (27)], which is proportional to the system size, to demonstrate the validity of the emergent eigenstate solution for $t/\tau \lesssim 0.5$ when $L\to \infty$.

Figure 2

Site occupations and nonlocal correlations in current-carrying states. (a) Site occupations $n_l(t)=⟨\psi (t)|{\widehat{n}}_l|\psi (t)⟩$ versus $\tilde{l}=l\gamma (t)/(2L)$. Results are shown for $N=200$ particles, different values of $\gamma$, and different times. The black solid line is the scaling solution $n_{\tilde{l}}(t)=\arccos (\tilde{l})/\pi$ for $-1<\tilde{l}<1$. The main panels in (b) and (c) display $\mathcal{C}(x)\equiv {\mathcal{C}}_{0,x}$, where ${\mathcal{C}}_{j,l}=|⟨{\widehat{f}}_j^{†}{\widehat{f}}_l⟩|$ for noninteracting spinless fermions (b) and ${\mathcal{C}}_{j,l}=|⟨{\widehat{b}}_j^{†}{\widehat{b}}_l⟩|$ for hard-core bosons (c), as a function of $x$ for $N=2000$ particles, ${\gamma }^{-1}=0$, at time $t/\tau =0.35$. The solid lines overlapping with the numerical results are $\mathcal{C}(x)=1/(\pi x)$ in (b) and $\mathcal{C}(x)=0.29/\sqrt{x}$ in (c) [33]. 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 $N=200$ particles.

Figure 3

Dynamics of the quasimomentum distribution function $m_q(t)$. (a),(b) $m_q(t)$ for $N=1000$ noninteracting spinless fermions. (c),(d) $m_q(t)$ for $N=1000$ hard-core bosons. Results are shown for ${\gamma }^{-1}=0.10$ in (a) and (c) and for ${\gamma }^{-1}=0.20$ in (b) and (d). Solid lines depict the numerical results, while the dashed lines in (a) and (b) are obtained using Eq. (31). Dotted vertical lines in all panels represent the shift of the peak in $m_q(t)$ at time $t/\tau =0.5$, and is given by Eq. (d4).

Figure 4

Emergent Hamiltonian description of the domain-wall melting for interacting spinless fermions. (a) Overlap $O(t)=|⟨{\mathrm{\Psi }}_t^V|{\psi }^V(t)⟩|$ as a function of rescaled time $t/N$ for $V=1.2$ (from full exact diagonalization) and $V=0$ (noninteracting system), and different system sizes. The inset depicts $|1-O(t)|$. For $N=6$, 7, and 8 (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),(c) Site and quasimomentum occupations for $V=1.2$ obtained from $|{\psi }^V(t)⟩$ (symbols) and from $|{\mathrm{\Psi }}_t^V⟩$ (lines) (data shown for $N=8$ at time $t/N=0.35$). (b) Site occupations $n_l(t)$ for $N=8$ and different times as a function of the rescaled coordinate $(l-1/2)/t$. The data collapse onto a line [57] as for noninteracting particles [40]. (c) Quasimomentum distribution function $m_q(t)$ for different system sizes and the same rescaled time $t/N=0.35$.

Figure 5

Sudden expansion of noninteracting spinless fermions, or hard-core bosons, in optical lattices. (a) Site occupations $n_l(t)=⟨\psi (t)|{\widehat{n}}_l|\psi (t)⟩$ at different times. (b) Overlap $O(t)=|⟨{\mathrm{\Psi }}_t|\psi (t)⟩|$ between the time-evolving state $|\psi (t)⟩$ and the eigenstate $|{\mathrm{\Psi }}_t⟩$ of the emergent local Hamiltonian in Eq. (39), and $|1-O(t)|$ in the inset. The results are plotted versus the rescaled time $t/N$ for four system sizes. When increasing the number of particles $N$, $O(t)=1$ within machine precision (see inset) until $t/N\approx 1$.

Figure 6

Entanglement entropy of eigenstates of the emergent local Hamiltonian ${\widehat{\mathcal{H}}}_V(t)$ [Eq. (35)] for $V=1.2$. (a) Entanglement entropy $S_0$ of the eigenstate $|{\mathrm{\Psi }}_t⟩$ that describes the domain-wall melting, after subtracting $(1/4)\log (L/2)$, for different system sizes $L$ and $N=L/2$. The thick solid line displays the ansatz in Eq. (41) with $k^{\prime }=0.529$ as an average of $k^{\prime }(L)$ for $L=14$, 16, 18. [For a given $L$, we obtain $k^{\prime }(L)$ by fitting the numerical result to Eq. (41).] (b) Entanglement entropy per site at time $t/N=0.35$ as a function of the inverse system size. Diamonds depict the average entanglement entropy $S_{\mathrm{av}}$ per site of $\mathcal{D}/5$ eigenstates of the many-body spectrum (see text for details) (for $L=18$, we average over $10^3$ eigenstates). The solid line is obtained from Eq. (41) for the same value of $k^{\prime }$ as in (a). The dashed line is a linear fit to $S_{\mathrm{av}}/(L/2)=a_1/L+a_2$ for $L\ge 14$, with $a_1=0.076$ and $a_2=0.308$.

Figure 7

Site and quasimomentum occupations of noninteracting spinless fermions for $N/L=1/2$. (a) Velocity of the propagating front $v^{*}(t)$ [Eq. (d1)]. (b) Amplitude of the quasimomentum distribution function for noninteracting fermions $\mathrm{\Delta }m(t)$ [Eq. (d2)]. (c),(d) Position of the peak of the quasimomentum distribution function $\phi (t)$ [Eq. (d4)]. Results are shown versus $t/\tau$ in (a)–(c), and versus ${\gamma }^{-1}$ in (d).