Bipartite particle number fluctuations in dephased long-range lattice systems

We investigate the dynamics of subsystem particle number fluctuations in a long-range system with power-law decaying hopping strength characterized by exponent $\ensuremath{\mu}$ and subjected to a local dephasing at every site. We introduce an efficient bond length representation for the four-point correlator, enabling the large-scale simulation of the dynamics of particle number fluctuations for both fermionic and bosonic systems and for translationally invariant initial states. Our results show that the particle number fluctuation dynamics exhibit one-parameter Family-Vicsek scaling, with superdiffusive scaling exponents for $\textmu{}<1.5$ and diffusive scaling exponents for $\textmu{}\ensuremath{\ge}1.5$. Finally, exploiting the bond-length representation, we provide an exact analytical expression for the particle number fluctuations and their scaling exponents in the short-range limit ($\ensuremath{\mu}\ensuremath{\rightarrow}\ensuremath{\infty})$.