Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature

Abstract

The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the M -layer construction whose first diagrams are evaluated numerically and analytically. The Ginzburg criterion, from both the paramagnetic and spin-glass phase, reveals that the upper critical dimension below which mean-field theory fails is DU = 8, at variance with the classic results DU = 6 yielded by finite-temperature replica field theory. The different outcome follows from two crucial properties: finite-connectivity of the lattice and zero temperature. They both also lead to specific features of the replica-symmetry-breaking phase.

Field theory for zero temperature soft anharmonic spin glasses in a field

“…Another possibility is that the transition changes nature from being continuous to discontinuous. As way to circumvent this problem, it has been suggested [10,11] to consider directly the theory at zero temperature and applied field and test the stability of this zero temperature critical point when finite dimensional fluctuations are switched on. This route has been applied to the Random Field Ising model [12], where it is known that the zero temperature critical point exists and its attractive from a renormalization point of view, but one needs to clarify its critical nature (when, for example, dimensional reduction applies [13][14][15]).…”