Exact results on spin glass models

Exact and/or rigorous results are reviewed for the Ising and XY/Villain spin glasses in finite dimensions, such as the exact energy, correlation identities and a functional relation between the distribution functions of ferromagnetic and spin glass order parameters. This last relation is useful to prove that the phase space is not complex on a line in the phase diagram. The spin wave theory neglecting periodicity is shown to give exact results for the Villain model on a line in the phase diagram. Implications of this fact on the renormalization-group results for the random-phase XY/Villain model in two dimensions are discussed.Comment: 9 pages, Proceedings of STATPHYS21, to be published in Physica

Error counting in a quantum error-correcting code and the ground-state energy of a spin glass

Upper and lower bounds are given for the number of equivalence classes of error patterns in the toric code for quantum memory. The results are used to derive a lower bound on the ground-state energy of the +/-J Ising spin glass model on the square lattice with symmetric and asymmetric bond distributions. This is a highly non-trivial example in which insights from quantum information lead directly to an explicit result on a physical quantity in the statistical mechanics of disordered systems.Comment: 15 pages, 7 figures, JPSJ style, latex style file include

Nonequilibrium relations in spin glasses

The applications of nonequilbrium relations such as the Jarzynski equality and the fluctuation theorem to spin glasses are considered. The spin glass is a basic platform where we consider an application of an approximate solver of combinatorial optimization problems, simulated annealing. We find a novel relationship between an average through a nonequilibrium process where the temperature changes as in simulated annealing and a thermal average in equilibrium with different amounts of quenched randomness. The results shown in the present study may serve as an alternative way to overcome critical slowing down in spin glasses. It means that this way may mitigate difficulties in several hard optimization problems.Comment: 4 pages, Proceeding of International Symposium on Nanoscience and Quantum Physics (nanoPHYS'09) Version 3 is the final on