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 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.

14 pages, 4 figures

References in corpus (9)

• Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses
• Diluted one-dimensional spin glasses with power law decaying interactions
• Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions
• Thermodynamic glass transition in a spin glass without time-reversal symmetry
• Spin glasses in a field: Three and four dimensions as seen from one space dimension
• The three dimensional Ising spin glass in an external magnetic field: the role of the silent majority
• The correspondence between long-range and short-range spin glasses
• Spin Glass in a Field: a New Zero-Temperature Fixed Point in Finite Dimensions
• Geometry of large-scale low-energy excitations in the one-dimensional Ising spin glass with power-law interactions

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