Application of Polynomial Algorithms to a Random Elastic Medium

A randomly pinned elastic medium in two dimensions is modeled by a disordered fully-packed loop model. The energetics of disorderinduced dislocations is studied using exact and polynomial algorithms from combinatorial optimization. Dislocations are found to become unbound at large scale, and the elastic phase is thus unstable giving evidence for the absence of a Bragg glass in two dimensions. Randomly pinned elastic media are used to model various condensedmatter systems with quenched disorder including the vortex phase of dirty type-II superconductors[1]. Much analytical progress on these systems has been made within the elastic approximation where dislocations are excluded by fiat. The intriguing possibility of spontaneous formation of disorderinduced dislocations (pairs and loops respectively in two and three dimensions) at large scale, however, remains a challenging question[2]. To address this issue at zero temperature requires a detailed understanding of the energetics of dislocations in terms of their elastic-energy cost and disorder-energy gain in the ground stateBG. In recent years, we have witnessed a fruitful exploration of novel algorithms in for complex disordered systems whose ground state itself is dominated by random disorder. One class of these efficient algorithms is based on network flow optimization. It includes the min-cost-flow, max-flow and matching algorithms which compute the exact ground state in time that grows only polynomialy in the system size, an attibution of great practical importance. Some recent applications include studies of the roughness and topography of random manifolds and 2d random elastic media by max-flow[3], matching[4] and min-cost-flow algorithms[5], the sensitivity exponents of the random-field-Ising model by max-flow algorithms[6], the domain-wall energy in the gauge glass by min-cost-flow algorithms[7],