Spread of correlations in strongly disordered lattice systems with long-range coupling

We investigate the spread of correlations in a one-dimensional lattice system with high on-site energy disorder and long-range couplings with a power-law dependence on the distance ($\propto r^{-\mu }$). The increase in correlation between the initially quenched node and a given node exhibits three phases: quadratic in time, linear in time, and saturation. No further evolution is observed in the long time regime. We find an approximate solution of the model valid in the limit of strong disorder and reproduce the results of numerical simulations with analytical formulas. We also find the time needed to reach a given correlation value as a measure of the propagation speed. Because of the triple-phase evolution of the correlation function, the propagation changes its time dependence. In the particular case of $\mu =1$, the propagation starts as a ballistic motion and then, at a certain crossover time, turns into standard diffusion.

Figure 1

Schematically illustrated physical system: a chain of uniformly distributed spins with periodic boundary conditions. (a) When one of the spins is rotated (single local quench), interactions occur in the system. (b) The rotated central spin pulls on the other spins, causing excitation to move through the system. In a system with long-range interactions, information about the quench is expected to reach distant spins immediately.

Figure 2

(a)–(c) Map of time evolution of the absolute value of the correlation function $|C_{\alpha }\left(t\right)|$ [Eq. (4)] for every single node in the system of $N=1001$ spins with disorder strength $W=200$. Color maps indicate the values of the correlation function as a function of the spin number and time for three different values of the interaction exponent ($\mu =1.0,2.0,3.0$). The dashed lines indicate the propagation fronts defined by $C_{\alpha }\left(t\right)=\tilde{C}$ (with $\tilde{C}$ given in the upper-right corner of each subfigure). The same correlation fronts are shown again in log-log scale in subfigures [(d)–(f)], where dashed lines represent analytical results obtained by employing central spin model (see Sec. 3b), and the axes have been swapped to better represent the propagation of the correlation. Additionally, in panels [(a)–(c)] we represent by dotted line the light cone region given by the time $t_1$ [see Fig. 3 and Eq. (29)], outside which correlations diminish much faster (but still power law) than inside the cone [see Figs. 3–3, and Eqs. (24), (26), and (28)]. (g)–(i) Correlation function $|C_{\alpha }\left(t\right)|$ as a function of time for spins $\alpha =0,1,4,16,64,256,500$. Dashed lines here indicate the numerical solution of the central spin model, where simulations were performed for $500000$ disorder realizations.

Figure 3

Parameters of the evolution of correlations as a function of the distance from the central spin for several values of the exponent $\mu =1.0,1.5,2.0,2.5,3.0$. In each subfigure, black dashed lines indicate the results obtained from the approximate analytical solution (see Sec. 4), described by formulas given in the panels. (a) Proportionality factor $A_{\alpha }$ for the quadratic regime, see Eq. (8); (b) proportionality factor $B_{\alpha }$ for the linear regime, see Eq. (9); (c) saturation level $C_{\alpha }^{\infty }$, see Eq. (10); (d) crossover time $t_0^{\left(\alpha \right)}$ between quadratic and linear regime; and (e) crossover time $t_1^{\left(\alpha \right)}$ between linear and saturating regime.

Figure 4

Schematic plot of the probability distribution of eigenenergies separation: (a) for the first phase of motion $[C_{\alpha }\left(t\right)\propto t^2]$; (b) for the second phase of motion $[C_{\alpha }\left(t\right)\propto t]$; and (c) for the saturation phase $[C_{\alpha }\left(t\right)=C_{\alpha }^{\infty }]$.

Figure 5

(a)–(c) Map of time evolution of the entanglement entropy $S_{\alpha }\left(t\right)$ [Eq. (A4)] for every single node in the system of $N=1001$ spins with disorder strength $W=200$. Color maps indicate the values of the entropy as a function of the site number and time for three different values of the interaction exponent ($\mu =1.0,2.0,3.0$). The dashed lines indicate the propagation fronts defined by $S_{\alpha }\left(t\right)=\tilde{S}$ (with $\tilde{S}$ given in the upper-right corner). The same correlation fronts are shown again in log-log scale in subfigures (d)–(f), and the axes have been swapped to better represent the propagation of the correlation. (g)–(i) Entanglement entropy $S_{\alpha }\left(t\right)$ as a function of time for spins $\alpha =1,4,16,64,256,500$.

Figure 6

(a)–(c) Map of time evolution of the absolute value of the correlation function $|C_{\alpha }\left(t\right)|$ [Eq. (4), cf. Fig. 2] for every single node in the system of $N=1001$ spins, coupling exponent $\mu =1$, and for three different values of the disorder strength, $W=2,0.2,0$. The dashed lines indicate the propagation fronts defined by $C_{\alpha }\left(t\right)=\tilde{C}$ (with $\tilde{C}$ given in the upper-right corner of each subfigure). The same correlation fronts are shown again in log-log scale in subfigures (d)–(f), where dashed lines represent analytical result obtained in Sec. 4 [see. Eq. (30)], and the axes have been swapped to better represent the propagation of the correlation. (g)–(i) Correlation function $|C_{\alpha }\left(t\right)|$ as a function of time for spins $\alpha =0,1,4,16,64,256,500$. The dashed lines here indicate the numerical solution of the central spin model where simulations were performed for $500000$ disorder realizations. The dotted lines in (h) and (i) indicate limit values of correlation functions $C_0\left(\infty \right)$ and $|C_{\alpha }\left(\infty \right)|$ (the same for all $\alpha$) in the disorder-free system [cf. Eq. (B1)].