Low energy excitations in a long prism geometry: testing the lower critical dimension of the Ising spin glass

We propose a general method for studying systems that display excitations with arbitrarily low energy in their low-temperature phase.

We argue that in a rectangular right prism geometry, with longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal dimension, but the scaling of the correlation length with the transverse size carries crucial information from which the lower critical dimension can be inferred.

The method is applied in the particularly demanding context of Ising spin glasses at zero magnetic field.

The lower critical dimension and the multifractal spectrum for the correlation function are computed from large-scale numerical simulations.

Several technical novelties (such as the unexpectedly crucial performance of Houdayer's cluster method or the convenience of using open - rather than periodic - boundary conditions) allow us to study three-dimensional prisms with transverse dimensions up to $L=24$ and effectively infinite longitudinal dimensions down to low temperatures.

The value that we find for the lower critical dimension turns out to be in agreement with expectations from both the Replica Symmetry Breaking theory and the Droplet model for spin glasses.

We argue that our novel setting holds promise in clarifying which of the two competing theories more accurately describes three-dimensional spin glasses.