Periodic elastic medium in which periodicity is relevant

We analyze, in both $\mathrm{}(1+1)\mathrm{}$ and $\mathrm{}(2+1)\mathrm{}$ dimensions, a periodic elastic medium in which the periodicity is such that at long distances the behavior is always in the random-substrate universality class. This contrasts with the models with an additive periodic potential in which, according to the field-theoretic analysis of Bouchaud and Georges and more recently of Emig and Nattermann, the random manifold class dominates at long distances in $\mathrm{}(1+1)\mathrm{}$ and $\mathrm{}(2+1)\mathrm{}$ dimensions. The models we use are random-bond Ising interfaces in hypercubic lattices. The exchange constants are random in a slab of size $L^{d-1}\times \lambda$ and these coupling constants are periodically repeated, with a period λ, along either $\{10\}$ or $\{11\}$ [in $\mathrm{}(1+1)\mathrm{}$ dimensions] and $\{100\}$ or $\{111\}$ [in $\mathrm{}(2+1)\mathrm{}$ dimensions]. Exact ground-state calculations confirm scaling arguments which predict that the surface roughness w behaves as $w\sim L^{2/3},L\ll L_c$ and $w\sim L^{1/2},L\gg L_c$ with $L_c\sim {\lambda }^{3/2}$ in $\mathrm{}(1+1)\mathrm{}$ dimensions, and $w\sim L^{0.42},L\ll L_c$ and $w\sim \ln \mathrm{}\left(L\right),L\gg L_c$ with $L_c\sim {\lambda }^{2.38}$ in $\mathrm{}(2+1)\mathrm{}$ dimensions.