Spin systems on hypercubic Bethe lattices: A Bethe-Peierls approach

We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary o...

Bethe approximation is shown to violate Bravais lattices translational invariance. A new scheme is then presented which goes over the one-site Weiss model yet preserving initial lattice symmetry. A mapping to a one-dimensional finite closed chain in an external field is obtained. Lattice topology determines the chain size. Using recent results in percolation, lattice connectivity between chains is argued to be $(q(d-1)-2)/(d)$ where $q$ is the coordination number and $d$ is t...

We present a thorough study of the static properties of 2D models of spin-ice type on the square lattice or, in other words, the sixteen-vertex model. We use extensive Monte Carlo simulations to determine the phase diagram and critical properties of the finite dimensional system. We put forward a suitable mean-field approximation, by defining the model on carefully chosen trees. We employ the cavity (Bethe-Peierls) method to derive self-consistent equations, the fixed points ...

We study the influence of an external magnetic field h on the phase diagram of a system of Fermi particles living on the sites of a Bethe lattice with coordination number z and interacting through on-site U and nearest-neighbor V interactions. This is a physical realization of the extended Hubbard model in the narrow-band limit. Our results establish that the magnetic field may dramatically affect the critical temperature below which a long-range charge ordered phase is obser...

The magnetic properties of the mixed spin-$\frac{1}{2}$ and spin-$\frac{7}{2}$ Ising model with a crystal-field in a longitudinal magnetic field are investigated on the Bethe lattice using exact recursion relations. The ground-state phase diagram is constructed. The temperature-dependent one is displayed in the case of uniform crystal-field on the $(k_{\text{B}}T/|J|, D/|J|)$ plane in the absence of the external constraint for lattice coordination numbers $z = 3, 4, 6$. The o...

Glauber dynamics of a bond-diluted Ising model on a Bethe lattice (a random graph with fixed connectivity) is investigated by an approximate theory which provides exact results for equilibrium properties. The time-dependent solutions of the dynamical system derived by this method are in good agreement with the results obtained by Monte Carlo simulations in almost all situations. Furthermore, the derived dynamical system exhibits a remarkable phenomenon that the magnetization ...

The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for a general d-dimensional (hyper)-simple-cubical lattice. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general $d$ expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion ...

We propose a generic framework to describe classical Ising-like models defined on arbitrary graphs. The energy spectrum is shown to be the Hadamard transform of a suitably defined sparse "coding" vector associated with the graph. We expect that the existence of a fast Hadamard transform algorithm (used for instance in image ccompression processes), together with the sparseness of the coding vector, may provide ways to fasten the spectrum computation.. Applying this formalism ...

We develop a transfer matrix method to compute exactly the spin-spin correlation functions of Bethe lattice spin models in the external magnetic field h and for any temperature T. We first compute the correlation function for the most general spin - S Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin - S Ising model includes the spin-1/2 simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. F...

We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from minus infinity to plus infinity by setting up the self-consistent field equations, which we show are exact in this case. We find that for a 3-coordinated Bethe lattice, there is no jump discontinuity in magnetization for arbitrarily small gaussia...