Elastic properties of a two-dimensional lattice in a weak random pinning potential: origin of the pinning force

The problem of an elastic system in the presence of a random pinning potential is investigated using weak-disorder perturbation theory. The occurrence of a restoring pinning force is derived from first principles. A quantitative estimate of the pinning force is obtained and agrees with the collective pinning theory of Larkin and Ovchinnikov. The renormalization of the elastic properties is calculated. Contribution from the anharmonicity of the lattice can have both possible signs whereas contribution from the pinning force always reduces the effective shear modulus.