The two-dimensional Ising spin glass at zero temperature

This thesis reports on a study of the Ising spin glass in two dimensions. Since the critical temperature for this system in known to be zero, only ground-state calculations are considered. Ground states for the Ising spin glass in two dimensions can be determined in polynomial time by a recently proposed mapping to an auxiliary graph decorated with Kasteleyn cities, as long as periodic boundary conditions are applied at most in one direction. Using this method, ground states for systems with open-periodic boundary conditions for lattices of linear sizes up to L = 10000 have been determined, and defect energies as well as domain-wall lengths have been calculated. A new algorithm based on a combination of the matching approach and a windowing technique is proposed, and quasi-exact ground-states for lattices with periodic-periodic boundary conditions up to L = 3000 are determined. The run-time of this windowing algorithm is also polynomial. By using these techniques, high-precision estimates of the spin-stiffness exponent and the domain-wall fractal dimension for Gaussian couplings have been achieved. The 2D Ising spin glass with bimodal couplings has a multitude of degenerate ground states, with the number of degenerate states growing exponentially with increasing system size. It is hence necessary to develop techniques for sampling the ground-state manifold uniformly. A new efficient algorithm serving this purpose is presented. The algorithm is based on an exact analysis of clusters of free spins in a disorder configuration and a subsequent sampling step based on parallel tempering Monte Carlo. Using this algorithm together with the mapping approach, high-precision estimates of the spin-stiffness exponent and the domain-wall fractal dimension for bimodal couplings are obtained. The estimates of the spin-stiffness exponent and the domain-wall fractal dimension for both Gaussian and bimodal couplings are the most accurate estimates which have been reported to date. The geometry of the domain walls of both Gaussian and bimodal couplings is compared to the detailed predictions given for random curves in the plane in the framework of Schramm-Loewner Evolution (SLE). Different boundary conditions are considered, and for each case the fractal dimension and the SLE diffusion constant of the corresponding Brownian motion are calculated. Correlations between different domain-wall segments are explicitly checked by testing for independence of the increments of the Loewner driving function