Ground States, Energy Landscape, and Low-Temperature Dynamics of ±J Spin Glasses

Abstract

The previous three chapters have focused on the analysis of computational problems using methods from statistical physics. This chapter largely takes the reverse approach. We turn to a problem from the physics literature, the spin glass, and use the branch-and-bound method from combinatorial optimization to analyze its energy landscape. The spin glass model is a prototype that combines questions of computational complexity from the mathematical point of view and of glassy behavior from the physical one. In general, the problem of finding the ground state, or minimal energy configuration, of such model systems belongs to the class of NP-hard tasks. The spin glass is defined using the language of the Ising model, the fundamental description of magnetism at the level of statistical mechanics. The Ising model contains a set of n spins, or binary variables Si, each of which can take on the value “up” (Si = 1) or “down” (Si = −1). Finding the ground state means finding the spin variable values minimizing the Ising Hamiltonian energy (cost) function, written in general as