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

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. 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 (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 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 face-centered-cubic lattice with first- and second-nearest-neighbor interactions tuned to the point J₁=2J₂. In this model, we find a dynamical glass transition at a temperature T₆ 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.