Ordering and broken symmetry in short-ranged spin glasses

In this topical review we discuss the nature of the low-temperature phase in both infinite-ranged and short-ranged spin glasses. We analyse the meaning of pure states in spin glasses, and distinguish between physical, or 'observable', states and (probably) unphysical, 'invisible' states. We review replica symmetry breaking (RSB), and describe what it would mean in short-ranged spin glasses. We introduce the notion of thermodynamic chaos, which leads to the metastate construct. We apply these tools to short-ranged spin glasses, and conclude that RSB, in any form, cannot describe the low-temperature spin glass phase in any finite dimension. We then discuss the remaining possible scenarios that could describe the low-temperature phase, and the differences they exhibit in some of their physical properties-in particular, the interfaces that separate them. We also present rigorous results on metastable states and discuss their connection to thermodynamic states. Finally, we discuss some of the differences between the statistical mechanics of homogeneous systems and those with quenched disorder and frustration.