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 -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 , at variance with the classical result 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 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 . Moreover, if one is interested in the zero temperature () transition, one can directly expand around the Bethe transition. The expansion directly at is not possible in the classical framework because the fully connected spin glass does not have a transition at , being in the broken phase for any value of the external field.
- Received 16 April 2021
- Revised 18 January 2022
- Accepted 26 January 2022
DOI:https://doi.org/10.1103/PhysRevLett.128.075702
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