Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures

1992/06/15 by Davide Cassi · 2 citations
Physics and Astronomy · Mathematics · #Theoretical and Computational Physics #Physics of Superconductivity and Magnetism #Quantum many-body systems #Corollary #Generalization #Random walk #Fractal #Quantum walk #Physics #Symmetry (geometry) #Spontaneous magnetization #Dimension (graph theory) #Heisenberg model #Ferromagnetism #Quantum #Statistical physics #Mathematical physics #Mathematics #Quantum mechanics #Magnetization #Pure mathematics #Mathematical analysis #Magnetic field

paper · doi:10.1103/physrevlett.68.3631

openalex publication_date 1992/06/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/22

Abstract

It is rigorously proven that classical O(n) and quantum spin-s Heisenberg ferromagnetic models on generic networks cannot have spontaneous magnetization at any finite temperature if random walks on the same structure are recursive. It follows as a corollary that on fractals a continuous symmetry can be spontaneously broken only if the spectral dimension is greater than 2.

Citations

• Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models

Cited by

• Physical limitations of the Hohenberg–Mermin–Wagner theorem
• Mermin–Wagner theorem [wikipedia]