Periodic Orbits, Entanglement, and Quantum Many-Body Scars in Constrained Models: Matrix Product State Approach

We analyze quantum dynamics of strongly interacting, kinetically constrained many-body systems. Motivated by recent experiments demonstrating surprising long-lived, periodic revivals after quantum quenches in Rydberg atom arrays, we introduce a manifold of locally entangled spin states, representable by low-bond dimension matrix product states, and derive equations of motion for them using the time-dependent variational principle. We find that they feature isolated, unstable periodic orbits, which capture the recurrences and represent nonergodic dynamical trajectories. Our results provide a theoretical framework for understanding quantum dynamics in a class of constrained spin models, which allow us to examine the recently suggested explanation of “quantum many-body scarring” [Nat. Phys. 14, 745 (2018)], and establish a possible connection to the corresponding phenomenon in chaotic single-particle systems.

Figure 1

(a) Flow diagrams of ${\dot{\theta }}_e(t)$, ${\dot{\theta }}_o(t)$ for the model Eq. (1) with $s=1/2$. The color map gives the error $\gamma$, Eq. (5). There is an isolated, unstable periodic orbit (red curve) describing oscillatory motion between $|{\mathbb{Z}}_2⟩$ (green dot) and $|{\mathbb{Z}}_2^{\prime }⟩$ (blue dot), with numerically extracted period $T\approx 2\pi \times 1.51{\mathrm{\Omega }}^{-1}$. Conversely, motion from $|\mathbf{0}⟩$ (red dot) proceeds towards a saddle point where the error is large. (b) Dynamics of local observable $S_i^z(t)$. There are persistent, coherent oscillations in the local observable for $|{\mathbb{Z}}_2⟩$ with similar period, while $|\mathbf{0}⟩$ instead shows quick relaxation and equilibration towards a thermal value predicted by ETH [39].

Figure 2

(a) Level spacing statistics in the momentum-zero, inversion-symmetric sector. Plotted is the $r$ statistics defined by the average of $r_n=[\min (s_n,s_{n-1})/\max (s_n,s_{n-1})]$ where $s_n=E_{n+1}-E_n$. There is a clear albeit slow trend with Hilbert space dimension $d$ towards Wigner-Dyson statistics in the Gaussian Orthogonal Ensemble (GOE) class, indicated by $r\approx 0.53$, away from the integrable Poissonian (POI) limit of $r\approx 0.39$ (for discussion of the slow convergence, see Refs. [49, 50]). (b),(c) Growth of entanglement entropy $S_A$ following quenches from the $|\mathbf{0}⟩$ and $|{\mathbb{Z}}_2⟩$ states, of subregions $A$ being (b) six contiguous sites, (c) a single-site, for the $s=1/2$ model. Total system size is $L=30$.

Figure 3

(a) Geometrical depiction of the TDVP over a manifold of states $|\psi (\mathit{z})⟩$ parametrized by $\mathit{z}$. The instantaneous motion $-iH|\psi (\mathit{z})⟩$ is projected onto the tangent space at the point, leading to motion on the manifold (green trajectory). The norm of the vector orthogonal to the manifold, $\mathrm{\Gamma }=\gamma \sqrt{L}$ [cf. Eq. (5)], is a measure of its accuracy. (b) MPS representation of states $|\psi (\mathit{\theta },\mathit{\varphi })⟩$ [cf. Eq. (3)] used.

Figure 4

(a) and (c) Flow diagrams [Eq. (4)] and error $\gamma$ for (a) $s=1$, (c) $s=2$. The indicated periodic orbits (red curves) have periods (a) $T\approx 2\pi \times 1.64{\mathrm{\Omega }}^{-1}$, and (c) $T\approx 2\pi \times 1.73{\mathrm{\Omega }}^{-1}$. Note that points ${\theta }_{o/e}={\theta }_{o/e}\pm 2\pi$ are identified. (b),(d) Relaxation of local observable $S_i^z(t)$ for (b) $s=1$, (d) $s=2$. One sees, similarly to Fig. 1, quick relaxation of the $|\mathbf{0}⟩$ state toward a thermal value predicted by ETH [39], while persistent oscillations for $|{\mathbb{Z}}_2⟩$, with similar periods in (a),(c).