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 >= 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.

Automated summary

This study analyzes the spin-glass transition in finite dimension at zero temperature, revealing crucial differences with classical results such as the upper critical dimension D-U >= 8 for mean-field theory failure.