Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal $(\ifmmode\pm\else\textpm\fi{}J)$ distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to
N=4802 spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension
dc. For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly
dc=2.We propose a general method for studying systems that display low-energy excitations in their low-temperature phase. We argue that in a rectangular right-prism geometry (the 3-dimensional generalization of a strip), with one longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal direction. Still, 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 functions 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 exponent controlling the behavior of the correlation length agrees with predictions from the Replica Symmetry Breaking (RSB) theory. We argue that our novel setting holds promise in clarifying which of the two competing theories, RSB or the Droplet Model, more accurately describes three-dimensional spin glasses.
The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.