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The spin-glass transition in a field in finite dimension is analyzed directlyat zero temperature using a perturbative loop expansion around the Bethelattice solution. The loop expansion is generated by the $M$-layer constructionwhose first diagrams are evaluated numerically and analytically. Thegeneralized Ginzburg criterion reveals that the upper critical dimension belowwhich mean-field theory fails is $D_U \le 8$, at variance with the classicalresult $D_U = 6$ yielded by finite-temperature replica field theory. Ourexpansion around the Bethe lattice has two crucial differences with respect tothe classical one. The finite connectivity $z$ of the lattice is directlyincluded from the beginning in the Bethe lattice, while in the classicalcomputation the finite connectivity is obtained through an expansion in $1/z$.Moreover, if one is interested in the zero temperature ($T = 0$) transition,one can directly expand around the $T = 0$ Bethe transition. The expansiondirectly at $T = 0$ is not possible in the classical framework because thefully connected spin glass does not have a transition at $T = 0$, being in thebroken phase for any value of the external field.

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