Towers of quantum many-body scars under stochastic resetting

Abstract

Towers of quantum many-body scars are sets of highly-excited eigenstates of nonintegrable Hamiltonians whose dynamics shows athermal behavior and persistent oscillations in time. The preparation of such states is, however, challenging due to their entanglement content. In this work, we show that local properties of such states can be prepared by interspersing the scarred dynamics with stochastic resets to much simpler unentangled product states. Stochastic resetting amounts to reinitializing the many-body wavefunction of the system at random times to a predefined state, which we choose to be in the scarred subspace. We derive several analytical results for the ensuing dynamics, e.g., for the time evolution of the fidelity and of local observables. Resetting damps the scarred oscillations and generates spatial off-diagonal long-range order in the ensuing stationary state. The latter shows mixedness that scales logarithmically as a function of the system size, which follows from the structure of the scarred eigenstates. We prove that such stationary states are locally equivalent, in the sparse-resetting limit, to a single pure scarred eigenstate, which is determined by the reset state. This protocol thereby might represent a route to the experimental preparation of the local properties of correlated and entangled states through resetting.

Figures (5)

• Figure 1: Fidelity between the time-evolved state and the reset state as a function of time. Plot of the fidelity \ref{['eq:fidelity_def']} as function of time $t$ for different values of the resetting rate: $r=0.0$ (purely unitary evolution), $r=0.02$, $r=0.2$ and $r=2.0$. In the unitary case, the fidelity shows perfects revivals of the reset state as a consequence of the equispaced scarred energy levels. For any nonzero resetting rate, the frequency of the oscillations is still visible, but the associated amplitude gets damped in time and the system eventually reaches a NESS. The other parameters are fixed as $\alpha=1.0$ for the reset state, $\omega=1.0$ and $L=50$ for the system size.
• Figure 2: Dynamics of the scar quasi-particle operator as function of time. We plot the expectation value of the observable $\hat{O}_j= \hat{V}_j + \hat{V}_j^{\dag}=e^{-i\pi j}\hat{c}_{j,\downarrow} \hat{c}_{j,\uparrow} + h.c.$ (Fermi-Hubbard) or $\hat{O}_j=e^{-i \pi j} (\hat{S}^{-}_j)^2/2 + h.c.$ (spin $1$ XY chain) as a function of time $t$ for the resetting dynamics interspersed with the Hamiltonian in Eqs. \ref{['Eq:F_H_model']} and \ref{['Eq:spin_1_XY_model']}. The initial condition is $\ket{\alpha}$ in Eq.\ref{['Eq:coh_state']} and one takes various values of the resetting rate $r=0.0$ (unitary evolution), $r=0.02$, $r=0.2$ and $r=2.0$. The other parameters are fixed as $\alpha=1.0$, $\omega=1.0$ and $L=50$.
• Figure 3: (a) Logarithmic scaling of Rényi-$2$ entropy as a function of size $L$. The parameters are $\omega = 2.0$ and $\alpha = 1.0$, while various values of the resetting rate $r$ are reported. Data are obtained from the numerical evaluation of the exact formula of Eq. \ref{['Eq:2_Renyi']}. The dashed curves are the results of fitting to a function form $f(x)=a+b\log(x)$, where $a$ and $b$ are fitting parameters. The fitting has been performed for $250<L\leq 500$. (b) Fitting parameters $b$ as a function of the resetting rate $r$ for different ranges of system size $L$ where the fitting is performed. We see that as the resetting grows, one needs to consider larger sizes for the fitting region in order for the fitting parameter $b$ to converge to the theoretical value $1/2$ of Eq. \ref{['eq:entropy_asymptotic_scaling']}.
• Figure 4: (a): Time evolution of the Rényi-$2$ entropy. The parameters are $\omega = 1.0$, $\alpha = 2.0$ and $L=30$ for all the panels. (a) Density plot of the Rényi-$2$ entropy as a function of time $t$ and resetting rate $r$. (b): Rényi-$2$ entropy as a function of time $t$ for various values of the resetting rate $r$. Mixedness of the stationary state is maximized for small resetting rate. (c): Rényi-$2$ entropy as a function of $r$ for various values of the time $t$. The maximum of $S_2(\hat{\rho}_{r}(t))$ is generated by the fact that both for $r\to 0^{+}$ (approximately unitary dynamics) and $r\to \infty$ (quantum Zeno regime) the state is pure.
• Figure 5: Local equivalence between the diagonal ensemble and a single scar for weak resetting. Parameters are $\alpha=1.0$ and arbitrary $\omega\in \mathbb{R}$. Solid lines represent the expectation value of $\hat{O}_j^{(1)} = \hat{n}_j \hat{n}_{j+1}$ and $\hat{O}_j^{(2)}=\hat{n}_j \hat{n}_{j+1} \hat{n}_{j+2}$ over the stationary mixed state $\hat{\rho}_r (+\infty)$ with $r=0^+$, in Eqs. \ref{['eq:O_1_left']} and \ref{['eq:O_l_2']}, respectively. Dotted lines represent the expectation values of the same operators over the scar eigenstate with quasiparticle number $n^{\ast}=L/2$, Eqs. \ref{['eq:O1_right']} and \ref{['eq:O2_r']}, respectively.