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

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 generalized Ginzburg criterion reveals that the upper critical dimension below which mean-field theory fails is $D_U \le 8$, at variance with the classical result $D_U = 6$ yielded by finite-temperature replica field theory. Our expansion around the Bethe lattice has two crucial differences with respect to the classical one. The finite connectivity $z$ of the lattice is directly included from the beginning in the Bethe lattice, while in the classical computation 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 expansion directly at $T = 0$ is not possible in the classical framework because the fully connected spin glass does not have a transition at $T = 0$, being in the broken phase for any value of the external field.