Cooperative Shielding in Many-Body Systems with Long-Range Interaction

In recent experiments with ion traps, long-range interactions were associated with the exceptionally fast propagation of perturbation, while in some theoretical works they have also been related with the suppression of propagation. Here, we show that such apparently contradictory behavior is caused by a general property of long-range interacting systems, which we name cooperative shielding. It refers to shielded subspaces that emerge as the system size increases and inside of which the evolution is unaffected by long-range interactions for a long time. As a result, the dynamics strongly depends on the initial state: if it belongs to a shielded subspace, the spreading of perturbation satisfies the Lieb-Robinson bound and may even be suppressed, while for initial states with components in various subspaces, the propagation may be quasi-instantaneous. We establish an analogy between the shielding effect and the onset of quantum Zeno subspaces. The derived effective Zeno Hamiltonian successfully describes the short-ranged dynamics inside the subspaces up to a time scale that increases with system size. Cooperative shielding can be tested in current experiments with trapped ions.

Figure 1

Density plots for the evolution of $⟨{\sigma }_n^x(t)⟩$; $L=13$; $\mathcal{B}=1/2$; $W=0$. Initial state: $⟨{\sigma }_7^x(0)⟩=-1$ and $⟨{\sigma }_{n\ne 7}^x(0)⟩=+1$. A light cone typical of short-range interaction is seen in (a), as expected, but also in (d)–(f) where the evolution is shielded from the present long-range interaction. Freezing occurs for very long times in (b); it also happens in (c) where the bands of $V$ are not degenerate.

Figure 2

Probability for the initial state to remain in [(a) and (b)] or leave [(c) and (d)] the band $b=1$. The initial state is a random superposition of states $|V_k^b⟩$. In (a) $J_z=0$, $W=2$ and in (b) $J_z=1$, $W=0$; $L=10,12,14$ from bottom to top. In (c) $P_{\text{leak}}$ vs $W$ for $J_z=0$ and in (d) $P_{\text{leak}}$ vs $J_z$ for $W=0$. Symbols represent numerical results and full lines are analytical estimates [39] with an overall fitting multiplicative factor. In all panels: averages over 50 realizations, $\mathcal{B}=0$, $\alpha =0$.

Figure 3

Fidelity decay and time for it to reach the value $1/2$; initial states are random superpositions of $|V_k^b⟩$. Upper panels: $F(t)$ for $b=3$ for $J_z=0$, $W=2$ (a) and for $W=0$, $J_z=1$ (d). From bottom to top: $L=10$, 12, 14. Numerical results: full lines. Gaussian decay: dashed lines. Lower panels have $J_z=0$ and give $T_{1/2}$ vs $W$ for $L=12$ (b), and vs $L$ for $W=2$ (c), for $|\mathrm{\Psi }(0)⟩$ from different bands. Numerical data: symbols. Analytical estimate $T_{1/2}=c_1/\delta E$ with $c_1$ being a fitting parameter: dashed lines. All panels: averages over 50 realizations, $\alpha =0$, $\mathcal{B}=0$.