Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent $S_i=\pm 1$ positioned at a lattice vertex $i$, copies the state of another one located at a distance $r$, selected randomly with a probability $P(r)\propto r^{-\alpha }$. Employing both analytical and numerical methods, we compute the two-time correlation function $G(r;t,s)$ ($t\ge s$) between the state of a variable $S_i$ at time $s$ and that of another one, at distance $r$, at time $t$. At time $t$, the memory of an agent of its former state at time $s$, expressed by the {\it autocorrelation function} $A(t,s)=G(r=0;t,s)$, decays algebraically for $\alpha >1$ as $[L(t)\mathrm{/}L(s){]}^{-\lambda }$, where $L$ is a time-increasing coherence length and $\lambda$ is the Fisher-Huse exponent. We find $\lambda =1$ for $\alpha >2$, and $\lambda =1\mathrm{/}(\alpha -1)$ for $1<\alpha \le 2$. For $\alpha \le 1$, instead, there is an exponential decay, as in mean-field. Then, at variance with what is known for the related Ising model, here we find that $\lambda$ increases upon decreasing $\alpha$. The space-dependent correlation $G(r;t,s)$ obeys a scaling symmetry $G(r;t,s)=g[r\mathrm{/}L(s);L(t)\mathrm{/}L(s)]$ for $\alpha >2$. Similarly, for $1<\alpha \le 2$ one has $G(r;t,s)=g[r\mathrm{/}\mathcal{L}(t);\mathcal{L}(t)\mathrm{/}\mathcal{L}(s)]$, where now the length $\mathcal{L}$ regulating two-time correlations differs from the coherence length as $\mathcal{L}\propto L^{\delta }$, with $\delta =1+2(2-\alpha )$.