Local-to-Global Entanglement Dynamics by Periodically Driving Impurities

Abstract

We study the entanglement dynamics of a one-dimensional spin chain subject to a local Floquet drive of a two-site impurity. We uncover a sharp transition in the entanglement dynamics as a function of the driving frequency. For large drive periods $T$, we observe a linear growth in entanglement entropy (EE), indicating a heating phase with volume law entanglement. Surprisingly, for driving periods below a critical value $T_\ast$, the EE grows subextensively with time, characteristic of a local quantum quench. In the non-interacting limit, we analytically trace the origin of this phenomenon to a transition in the single-particle Floquet quasi-energy spectrum. We also find that for $T>T_*$, the so-called ``average energy" operator develops non-local, rainbow-like couplings that are responsible for the rapid entanglement growth in the heating phase, but remains local for $T<T_*$. Using extensive matrix-product-state simulations, we show that the non-heating phase and the subextensive entanglement growth persist in the presence of weak interactions for numerically accessible timescales. Our results establish that local Floquet engineering can generate emergent bulk phenomena, shedding new light on energy localization and thermalization in driven many-body systems.

Figures (10)

• Figure 1: (a) Setup: We consider a periodically driven two-site impurity (in yellow) immersed in a uniform chain of length $2L$. (b) Main result: Below a critical period $T_*$, the drive only acts locally and leads to local quench dynamics characterized by a subextensive growth of EE ('local quench' scenario). Above this critical period, the drive acts as a non-local perturbation which leads to linear growth of EE ('global quench' scenario). In the non-interacting setting, the entanglement transition corresponds to a gap closure of the Floquet Hamiltonian.
• Figure 2: The entanglement phases of the 2-step driven impurity model Eq. \ref{['eq:floquetdrive_free']} with system size $2L=400$ and $\lambda=0.5$. In the nonheating phase ($T<\pi$), (a) the EE exhibits logarithmic scaling over time (plotted every six Floquet cycles at $T=2.5$, from orange to purple) and (c) the half-partition EE shows partial revivals that are independent of the driving frequency. In contrast, in the heating phase $(T > \pi)$, (b) the entanglement follows a linear scaling of entanglement after an initial transient spreading from the defect (plotted every 6 Floquet cycles at $T = 4.2$) and (d) the half partition entanglement exhibits a steady linear growth at early times.
• Figure 3: Average Energy spectrum $\theta_n(T)$ as a function of Floquet period $T$ for a 2-step drive between $\lambda=1$ and 0.5, in (a) the free fermionic setting for the lowest $10^5$ states with system size $2L=50$ with (b) representing the zoom-in around the transition point $T=\pi$, and (c-d) with additional nearest-neighbor interactions $\Delta=0.1,0.15$, for system size $2L=14$. The dashed vertical line on each plot marks the exact transition point in the noninteracting case, and the dashed horizontal lines in (c-d) mark the bandwidth for $T>\pi$. The color indicates the overlap between the ground state of the infinite frequency effective Hamiltonian $\bar{H} = (1/T)\int_0^T H(t) \mathrm{d}t$ with the ground state of $\Theta$ at a given frequency $T$. Grey dots indicate that the overlap is smaller than $0.002$.
• Figure 4: Entanglement evolution of the driven defect in an interacting chain within the nonheating regime, with system size $2L=100$ and $\lambda=0.8$. (a) In the nonheating regime ($T=2.8$), the half-partition entanglement remains bounded after a few hundred Floquet cycles across different $\Delta$ values, akin to the noninteracting case. Notably, the revival period now develops a dependence on the anisotropy $\Delta$. (b) For $\Delta=0.1$, the EE scales logarithmically over space for intermediate times (plotted every cycle, from yellow to purple). The MPS parameters used are: an initial DMRG bond dimension of $\chi_D = 100$, which determines the ground state and a TEBD bond dimension of $\chi_T=400$ for $\Delta=0.05, 0.1$ and $\chi_T=500$ for $\Delta = 0.15$.
• Figure S1: The entanglement phases for the harmonically driven impurity model with system size $2L=400$. (a)(b) The entanglement scaling (plotted every six Floquet cycles, from orange to purple) in the nonheating phase at $T=2.5$ and the heating phase at $T=4.2$, respectively. (c)(d) The half-partition entanglement evolution across different driving periods $T$ in the nonheating and heating regimes.