Ground state and persistent oscillations in the quantum East model

Abstract

For the 1D quantum East model with open boundaries, we show that in the limit $s \to -\infty$, the ground state is accurately captured by a simple spin-coherent product state. We further identify a low-entanglement excited eigenstate that differs from the ground state only by a $π$-rotation of the boundary spin, remaining well approximated by a spin-coherent state. For a range of $-\infty<s<0$, the edge-coherent product state overlaps with two eigenstates separated by a size-independent energy gap, leading to persistent coherent oscillations of both global and local observables in the thermodynamic limit. These oscillations originate from boundary physics and are distinct from quantum many-body scars or hypercube-like Fock-space mechanisms.

Figures (8)

• Figure 1: (a) Bipartite entanglement entropy of energy eigenstates for $L = 16$ and $s\to -\infty$ (Eq. \ref{['eq:H_East_L']}). The dashed line denotes the Page value. Circles mark the ground state ($\ket{\mathrm{GS}}$), anti-ground state ($\ket{\mathrm{AS}}$), excited states, $\ket{\mathrm{ES_1}}$ and $\ket{\mathrm{ES_2}}$ with unusually low entanglement compared to its neighboring states. (b) Plot shows the overlap of $\ket{\Theta}$ (Eq. \ref{['eq:spin-cohere_1']}) with $\ket{\mathrm{GS}}$ as a function of $s$ for $L = 18$. Inset shows the optimum angle of the individual spins of $\ket{\Theta}$. Above it is a cartoon of $\ket{\Theta}$ approximating $\ket{\mathrm{GS}}$ for $s \leq 0$. (c) Plot shows the overlap of $\ket{\bullet_L}$ with eigenstates ordered in energy ($n$ is the index) for $L = 14$. Colorbar denotes $|\left\langle \Psi_n|\bullet_L \right\rangle|^2$. Note how $\ket{\bullet_L}$ for negative-enough $s$ mainly overlaps a single eigenstate, consistent with it being itself an eigenstate at $s\to-\infty$, while it overlaps strongly with two eigenstates for $s<0$, leading to the oscillatory dynamics in panel (d). Above is a cartoon of the spin-coherent state, $\ket{\bullet_L}$ which approximates the eigenstate $\ket{\mathrm{ES_1}}$ and differs from $\ket{\mathrm{GS}}$ (see panel (b)) at a single edge spin. (d) Time evolution of survival probability (solid lines) and magnetization along X-direction (dashed lines) for different values of $s$ and $L = 14$ where the initial state is $\ket{\bullet_L}$.
• Figure 2: All panels in this figure for $s\to-\infty$. (a) Number density and magnetization along X-directions for $\ket{\mathrm{ES_1}}$ (see Fig. \ref{['fig1']}(a)) versus lattice site for $L = 16$. Horizontal dashed lines denote $\left\langle \hat{n} \right\rangle = \frac{3}{4}$ and $\left\langle \hat{\sigma^x} \right\rangle = \frac{\sqrt{3}}{2}$, as expected for the spin-polarized state $\ket{\frac{\pi}{3}}^{\otimes L}$. Note that the last lattice site deviates significantly from the dashed lines for both observables. Inset shows the entanglement entropy of $\ket{\mathrm{ES_1}}$ vs. system size where the dashed line denotes the Page values. (b) Optimum angles of a spin-coherent state (Eq. \ref{['eq:def_spin-cohere']}) maximizing the overlap with $\ket{\mathrm{ES_1}}$. Inset [i] shows the energy of the spin-coherent state $\ket{\bullet_L}$ (solid line) and $\ket{\mathrm{ES_1}}$ (markers). Inset [ii] shows the overlap of $\ket{\mathrm{ES_1}}$ with $\ket{\bullet_L}$ as a function of inverse system size along with linear fit (solid line). (c) Fractal dimensions of $\ket{\mathrm{ES_1}}$ in both Z- and X-bases. Horizontal dashed lines denote $D_\infty$. Error-bars denote 95% confidence interval. (d) Survival probability of the spin-coherent state. Vertical dashed line denotes the Heisenberg time. Inset shows the overlap of the spin-coherent state with the energy eigenstates where the vertical line denotes the energy of $\ket{\mathrm{ES_1}}$.
• Figure 3: (a) Spectral function for $\pi$-pulse perturbation of the (real) ground state at $s\to-\infty$. Inset shows the energy ($\omega_0$) and inverse lifetime ($\Gamma$) of the edge mode vs. system size obtained from Eq. (\ref{['eq:A']}). (b) Fractal dimensions of the $\ket{\mathrm{GS}}$ in Z-basis for different values of $s$. Inset shows $D_\infty$ vs. $s$ capturing QPT at $s = 0$.
• Figure 4: DOS for $\hat{H}_\mathrm{East}(s\to-\infty)$: DOS for different system sizes. The energy axis is centered and scaled to ensure zero mean and unit variance. Grey solid line denotes Gaussian fit. Inset shows the measures of skewness ($\mu_3$) and excess kurtosis ($\mu_4$) as a function of $L^{-1}$.
• Figure 5: Ground state for $\hat{H}_\mathrm{East}(s\to-\infty)$: (a) Site density (circle) and magnetization in X-direction (cross) w.r.t. ground state for different system sizes shown via different colors. Dashed lines denote the average over the sites and different system sizes. (b) Fractal dimensions of the ground state w.r.t. Z and X bases. Horizontal dotted lines denote $D_\infty$ extracted from the system size scaling of the maximum intensity. Dashed lines denote the fractal dimensions of the spin-coherent state for $\theta = \frac{\pi}{3}$ (Eqs. \ref{['eq:Dq_Z_spin_cohere']} and \ref{['eq:Dq_X_spin_cohere']}). (c) Ground state energy as a function of system size via markers while the solid gray line shows the energy of the polarized spin-coherent state, $\ket{\Theta}$. Inset shows the overlap between the ground state and $\ket{\Theta}$. (d) Fractal dimensions of $\ket{\overline{\mathrm{GS}}}$, which is obtained by projecting out $\ket{\Theta}$ from the ground state. (e) von Neumann entanglement entropy of $\ket{\overline{\mathrm{GS}}}$ w.r.t. different subsystem size $L_A$ for $L = 20$. Dashed line denotes the Page curve. Inset shows the entanglement entropy of the ground state (red) and $\ket{\overline{\mathrm{GS}}}$ (blue) for equal bipartition. Solid lines denote linear fit ($1.261 - 1.464 \ln L$ for $\ket{\overline{\mathrm{GS}}}$). (e) Survival probability of $\ket{\Theta}$. Vertical dashed line denotes the Heisenberg time. Inset shows the overlap of $\ket{\Theta}$ with the energy eigenstates where the vertical dashed line denotes the ground state energy. We only show the energy levels for which the overlap is more than $N^{-1}$.