Hyperuniformity in the Manna Model, Conserved Directed Percolation and Depinning

Hyperuniformity is an emergent property, whereby the structure factor of the density $n$ scales as $S(q) \sim q^\alpha$, with $\alpha>0$. We show that for the conserved directed percolation (CDP) class, to which the Manna model belongs, there is an exact mapping between the density $n$ in CDP, and the interface position $u$ at depinning, $n(x)=n_0+\nabla^2 u(x)$, where $n_0$ is the conserved particle density. As a consequence, the hyperuniformity exponent equals $\alpha={4-d-2\zeta}$, with $\zeta$ the roughness exponent at depinning, and $d$ the dimension. In $d=1$, $\alpha=1/2$, while $0.6> \alpha\ge0$ for other $d$. Our results fit well simulations in the literature, except in $d=1$, where we perform our own to confirm this result. Such an exact relation between two seemingly different fields is surprising, and paves new paths to think about hyperuniformity and depinning. As corollaries, we get results of unprecedented precision in all dimensions, exact in $d=1$. This corrects earlier work on hyperuniformity in CDP.