  It is generally believed (but not yet proved) that Ising spin glasses with
nearest-neighbor interactions have a phase transition in three and higher
dimensions to a low-temperature spin glass phase, but the nature of this phase
remains controversial, especially whether it is characterized by multiple
incongruent Gibbs states. Of particular relevance to this question is the
behavior of the typical free energy difference restricted to a finite volume
between two such putative Gibbs states, as well as the nature of the
fluctuations of their free energy difference as the couplings within the volume
vary. In this paper we investigate these free energy difference fluctuations by
introducing a new kind of metastate which classifies Gibbs states through their
edge overlap values with a reference Gibbs state randomly chosen from the
support of the periodic boundary condition (PBC) metastate. We find that the
free energy difference between any two incongruent pure states, regardless of
the details of how they're organized into mixed states within the PBC
metastate, converges to a Gaussian (or Gaussian-like) distribution whose
variance scales with the volume, proving a decades-old conjecture of Fisher and
Huse. The same conclusion applies, though with some additional restrictions, to
both mixed Gibbs states and ground states. We discuss some implications of
these results.
