Floquet scars and prethermal fragmentation in a driven spin-one chain

Abstract

We study the periodic dynamics of a spin-one chain driven using a square-pulse protocol with amplitude $Q_0$ and frequency $ω_D$. The Hamiltonian of the spin chain hosts a thermodynamically large number of $Z_2$-valued conserved quantities $W_{\ell}$ on the links $\ell$. This allows us to study the Floquet dynamics of this chain within a given sector with fixed values of $W_{\ell}$. For the sector with all $W_{\ell}=1$, we find signatures of quantum many-body scar states for $\hbar ω_D \gg Q_0$; they lead to oscillatory dynamics and fidelity revival for specific initial states. Upon lowering $ω_D$, we find an ergodic regime exhibiting fast thermalization consistent with the prediction of the (Floquet) eigenstate thermalization hypothesis. In addition, we identify special drive frequencies $ω_n^{\ast}= Q_0/(2n \hbar)$ (where $n = 1, 2, 3, \cdots$) at which the Floquet Hamiltonian exhibits prethermal strong Hilbert space fragmentation (HSF) with the largest fragment being ergodic; in contrast, a weak HSF is found at $ω'_n= Q_0/[\hbar(2n+1)]$ (where $n = 0, 1, 2, \cdots$). We also study the sector with $W_{\ell} =\{\cdots 1,1,-1,1,1,-1 \cdots \}$ which shows strong HSF at $ω_n^{\ast}$ but no fragmentation at $ω'_n$. Our analysis indicates that the strong HSF in this sector harbors an integrable largest fragment. We provide numerical support for our analytical and perturbative results using exact-diagonalization (ED) studies on finite chains of length $L\le 24$. Our numerical results for entanglement entropy, fidelity, and correlation functions of the driven chain provide definitive signatures of prethermal strong HSF for both sectors.

Figures (11)

• Figure 1: Schematic representation of the spin-one Kitaev chain. The top panel shows the representation in terms of $S^x_{2j} S^x_{2j+1}$ and $S^y_{2j} S^y_{2j-1}$ for even site $2j$ (shown as $XX$ and $YY$ respectively). For each of the links $\ell$ between any two sites, $W_{\ell}$, defined in Eq. \ref{['cons1']} of the main text, is conserved. The bottom panel shows an equivalent representation in terms of $XY \equiv S_{j}^x S_{j+1}^y$ on all sites of the chain; these two Hamiltonians are unitarily related as discussed in the text. An interaction term of the form $(XY)^2$ is also added; together they form the Hamiltonian $H$ studied in the text.
• Figure 2: (a) Plot of the half-chain entanglement $S_{L/2}(mT)/S_p \equiv S/S_p$ as a function of $m$ at the special frequency $\omega_D= \omega_1^{\ast}= Q_0/(2\hbar)$ and for several different representative initial Fock states. The plot indicates that the saturation value of $S/S_p <1$ at large $m$; moreover, it depends on the chosen initial state. (b) In contrast, for $\omega_D= 1.2 \omega_1^{\ast}$, an analogous plot for the same initial states leads to $S/S_p \to 1$. For both plots, $L=24$, the initial states are chosen from the $W_{\ell}=1$ sector, $Q_0/J=100$, and all energies are in units of $J$.
• Figure 3: (a) Plot of $S_{L/2}(mT)/S_p \equiv S/S_p$ as a function of $n$ at the special frequency $\omega_D= \omega_1^{\ast}= Q_0/(2\hbar)$ and for several representative values of $Q_0/J$. The initial state is chosen to be a Fock state lying in the largest fragment of $H_a$; the corresponding Page value of the fragmented sector, $S_p^f$, is indicated by the red-dashed line. (b) Plot of $S_{\rm av}/S_p$ as a function of $Q_0/J$ showing a gradual crossover from ergodic to fragmented dynamics. For both plots, $L=24$, the initial states are chosen from the $W_{\ell}=1$ sector, $\omega_D=\omega_1^{\ast}$, and all energies are scaled in units of $J$.
• Figure 4: (a) Plot of $S_{L/2}(mT)/S_p \equiv S/S_p$ as a function of $m$ at the special frequency $\omega_D= \omega_1^{\ast}= Q_0/(2\hbar)$ and $Q_0/J=35$ (blue), $40$ (magenta), $45$ (yellow), and $50$ (green) for an initial frozen state $|\psi(0)\rangle= |\cdots zzz\cdots\rangle$ lying in the $W_{\ell}=1$ sector. The dynamics of $S/S_p$ is a consequence of higher order terms in the Floquet Hamiltonian. The prethermal timescale $T_p= m_p T$ is estimated from the number of drive cycles $m=m_p$ for which $S(mT)/S_p \simeq 0.05$. (b) Plot of $JT_p/\hbar$ as a function of $Q_0$ keeping $\omega_D$ fixed at $\omega_1^{\ast}$; the data corresponding to panel (a) is shown by circles with same color code as in (a). For both plots, $L=24$ and all energies are scaled in units of $J$. See text for details.
• Figure 5: (a) Plot of $S_{L/2}(mT)/S_p \equiv S/S_p$ as a function of $m$ at $\hbar \omega_D=Q_0$, $L=24$ and $Q_0/J=40$ for several initial Fock states. $S/S_p$ saturates to three distinct values showing signatures of weak fragmentation. (b) Plot of the half-chain entanglement $S_{L/2}/S_p$ for Floquet eigenstates as a function of their quasienergy $E_F$ for $\hbar \omega_D=Q_0=40J$ for $L=18$, showing two distinct clusters separated by a gap in $S$. For both plots, the initial states are chosen from the $W_{\ell}=1$ sector, and all energies are scaled in units of $J$. See text for details.