Frustrated magnets in the limit of infinite dimensions: dynamics and disorder-free glass transition

• Published in: arXiv.org
• Main Authors: Mauri, Achille, Katsnelson, Mikhail I
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.04.2024
• Subjects: Anisotropy; Antiferromagnetism; Exchanging; Fourier transforms; Glass transition temperature; High temperature; Liquid phases; Low temperature; Magnets; Order parameters; Phase transitions; Physics - Disordered Systems and Neural Networks; Random noise; Statistical mechanics; Temperature
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2311.09124

We study the statistical mechanics and the equilibrium dynamics of a system of classical Heisenberg spins with frustrated interactions on a \(d\)-dimensional simple hypercubic lattice, in the limit of infinite dimensionality \(d \to \infty\). In the analysis we consider a class of models in which the matrix of exchange constants is a linear combination of powers of the adjacency matrix. This choice leads to a special property: the Fourier transform of the exchange coupling \(J(\mathbf{k})\) presents a \((d-1)\)-dimensional surface of degenerate maxima in momentum space. Using the cavity method, we find that the statistical mechanics of the system presents for \(d \to \infty\) a paramagnetic solution which remains locally stable at all temperatures down to \(T = 0\). To investigate whether the system undergoes a glass transition we study its dynamical properties assuming a purely dissipative Langevin equation, and mapping the system to an effective single-spin problem subject to a colored Gaussian noise. The conditions under which a glass transition occurs are discussed including the possibility of a local anisotropy and a simple type of anisotropic exchange. The general results are applied explicitly to a simple model, equivalent to the isotropic Heisenberg antiferromagnet on the \(d\)-dimensional fcc lattice with first and second nearest-neighbour interactions tuned to the point \(J_{1} = 2J_{2}\). In this model, we find a dynamical glass transition at a temperature \(T_{\rm g}\) separating a high-temperature liquid phase and a low temperature vitrified phase. At the dynamical transition, the Edwards-Anderson order parameter presents a jump demonstrating a first-order phase transition.