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.

arXiv comment14 pages, 4 figures

References in corpus (31)

• Relations between Short Range and Long Range Ising models
• 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
• Probing the Almeida-Thouless line away from the mean-field model
• Disappearance of the de Almeida-Thouless line in six dimensions
• Ising spin glass transition in magnetic field out of mean-field
• 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
• Replica symmetry breaking in and around six dimensions
• The three dimensional Ising spin glass in an external magnetic field: the role of the silent majority
• Nontrivial critical fixed point for replica-symmetry-breaking transitions
• Spin glass transition in a magnetic field: a renormalization group study
• The correspondence between long-range and short-range spin glasses
• Spin Glass in a Field: a New Zero-Temperature Fixed Point in Finite Dimensions
• Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function
• Ensemble renormalization group for disordered systems
• Generic replica symmetric field-theory for short range Ising spin glasses
• Dynamical transition in the D = 3 Edwards-Anderson spin glass in an external magnetic field
• Morphology of renormalization-group flow for the de Almeida-Thouless-Gardner universality class
• Geometry of large-scale low-energy excitations in the one-dimensional Ising spin glass with power-law interactions
• Diluted Mean-Field Spin-Glass Models at Criticality
• Loop expansion around the Bethe approximation through the $M$-layer construction
• Solvable Models of Supercooled Liquids in Three Dimensions
• One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses
• The fate of the bootstrap percolation hybrid critical point in finite dimension
• Finite-size critical scaling in Ising spin glasses in the mean-field regime
• Real Space Migdal-Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment
• New loop expansion for the Random Magnetic Field Ising Ferromagnets at zero temperature
• Physical observables of the Ising spin glass in 6-epsilon dimensions: asymptotical behavior around the critical fixed point
• One-loop topological expansion for spin glasses in the large connectivity limit

Cited by in corpus (14)

• Universality in Anderson localization on random graphs with varying connectivity
• Numerical test of the replica-symmetric Hamiltonian for the correlations of the critical state of spin glasses in a field
• Spin-glass dynamics: experiment, theory and simulation
• Field theory for zero temperature soft anharmonic spin glasses in a field
• Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field
• The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase
• Bethe $M$-layer construction for the percolation problem
• Thermodynamics of the Fredrickson-Andersen Model on the Bethe Lattice
• Bethe $M$-layer construction on the Ising model
• Critical exponents of the spin glass transition in a field at zero temperature
• Logarithmic critical slowing down in complex systems: from statics to dynamics
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Evidence of de Almeida-Thouless line below six dimensions
• Long-range spin glass in a field at zero temperature