Bethe $M$-layer construction on the Ising model

Abstract

In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the $M$-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension $D_U$ of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around $D_U$ to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the $M$-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant $g$ and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.

References in corpus (10)

• Disappearance of the de Almeida-Thouless line in six dimensions
• Spin Glass in a Field: a New Zero-Temperature Fixed Point in Finite Dimensions
• Non-backtracking random walk
• Morphology of renormalization-group flow for the de Almeida-Thouless-Gardner universality class
• Loop expansion around the Bethe approximation through the $M$-layer construction
• Solvable Models of Supercooled Liquids in Three Dimensions
• Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature
• The fate of the bootstrap percolation hybrid critical point in finite dimension
• New loop expansion for the Random Magnetic Field Ising Ferromagnets at zero temperature
• One-loop topological expansion for spin glasses in the large connectivity limit

Cited by in corpus (4)

• Bethe $M$-layer construction for the percolation problem
• Analytical solution to Heisenberg spin glass models on sparse random graphs and their de Almeida-Thouless line
• Critical exponents of the spin glass transition in a field at zero temperature
• Long-range spin glass in a field at zero temperature