Overlap locking and nonperturbative effects in spin glasses

We study the phenomenon of the locking of the order parameter (or synchronization) in spin glasses at low temperatures. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Nonperturbative phenomena are present when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≪</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>H</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , being <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> the coupling Hamiltonian and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>N</mml:mi> </mml:math> the system size. In this intermediate-coupling region, the effect is related to finite-size free-energy corrections in mean-field spin-glass models and to correlations in the Dyson hierarchical spin glass, a model that mimics the physics of finite-dimensional systems. We study this phenomenon in the mean-field approach, both analytically and numerically, and we finally compute the critical exponents for finite-volume corrections in mean-field theory and for the decay of correlations in the Dyson hierarchical model.