Protected quasilocality in quantum systems with long-range interactions

We study the out-of-equilibrium dynamics of quantum systems with long-range interactions. Two different models describing, respectively, interacting lattice bosons and spins are considered. Our study relies on a combined approach based on accurate many-body numerical calculations as well as on a quasiparticle microscopic theory. For sufficiently fast decaying long-range potentials, we find that the quantum speed limit set by the long-range Lieb-Robinson bounds is never attained and a purely ballistic behavior is found. For slowly decaying potentials, a radically different scenario is observed. In the bosonic case, a remarkable local spreading of correlations is still observed, despite the existence of infinitely fast traveling excitations in the system. This is in marked contrast to the spin case, where locality is broken. We finally provide a microscopic justification of the different regimes observed and of the origin of the protected locality in the bosonic model.

Figure 1

Correlation spreading in long-range spin and boson models for various values of $\alpha$. (a) Connected spin-spin correlation function in the LRTI model for the quench $V_i=h/2\to V_f=h/10$. (b) Connected density-density correlation function in the LRBH model for the quench $U_i=V_i=J\to U_f=V_f=J/4$. Results were obtained using the t-VMC approach for systems of $L=400$ sites (for visibility, only a part is shown). The length unit is the lattice spacing and the time units are $\hslash /h$ for (a) and $\hslash /J$ for (b).

Figure 2

Panels (a1,b1): Time-integrated spin-spin correlation functions ${\overline{G}}_c^{\sigma \sigma }(R,t)$ for the same data as in Fig. 1. Superimposed lines show the activation time $t^{☆}\left(\varepsilon \right)$ (see text), for $\varepsilon$ ranging from $2\times {10}^{-2}$ (lighter lines) down to $5\times {10}^{-3}$ (darker lines). Panels (a2,b2): Behavior of the fitted exponent $\beta$, computed within linear spin-wave theory, as a function of the cutoff parameter $\varepsilon$ and for the system size $L=2^{14}$. The length unit is the lattice spacing and the time unit is $\hslash /h$.

Figure 3

QP analysis of the group velocity for the LRBH model with exponent $\alpha =1/2$, density $n=1/2$, and $U=V=J/4$. In the inset, the stationary phase weights corresponding to the three solution branches of the equation $2v_g\left(k\right)=v$ are shown. Wave vectors are in units of the inverse lattice spacing and velocities are in units of the hopping amplitude $J$.