Phase Diagrams and Crossover in Spatially Anisotropic d=3 Ising, XY Magnetic and Percolation Systems: Exact Renormalization-Group Solutions of Hierarchical Models

TL;DR

This paper introduces hierarchical lattices to exactly solve and approximate phase diagrams of anisotropic 3D magnetic and percolation models, revealing crossovers between different orderings.

Contribution

It provides exact solutions for hierarchical models and approximates phase diagrams for anisotropic 3D systems, including crossovers from algebraic to long-range order.

Findings

Global phase diagrams with crossovers from d=2 to d=1-3

Exact solutions for hierarchical models

Approximate phase diagrams for physical models

Abstract

Hierarchical lattices that constitute spatially anisotropic systems are introduced. These lattices provide exact solutions for hierarchical models and, simultaneously, approximate solutions for uniaxially or fully anisotropic d=3 physical models. The global phase diagrams, with d=2 and d=1 to d=3 crossovers, are obtained for Ising, XY magnetic models and percolation systems, including crossovers from algebraic order to true long-range order.