Exact correlation functions of Bethe lattice spin models in external magnetic fields

Abstract

We develop a transfer matrix method to compute exactly the spin-spin correlation functions < s 0 s n > of Bethe lattice spin models in the external magnetic field h and for any temperature T . We first compute < s 0 s n > 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. From the spin-spin correlation functions, we obtain functions of correlation length, ξ(T, h), for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as h → 0 and T approaches the critical temperature T c . Our method to compute exact spinspin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.

Semiflexible polymer on an anisotropic Bethe lattice

“…It should be mentioned that our proposal of defining the Euclidean distance between two points of the Cayley tree is similar to earlier results in the literature relating this distance to the chemical distance, measured along the chain [9]. However, the distinction between the chemical and the Euclidean distances is not always properly considered in the literature, and this may lead to contradicting results [10], as we will discuss in more detail in the conclusion. In section II we define the model and calculate the mean square end-to-end distance recursively on the anisotropic Bethe lattice.…”

“…We would thus have ν = 1, the one-dimensional value, and the identity between the results for the Bethe lattice and for walks without immediate return on hypercubic lattices would break down. This definition of distance was used recently in the exact calculation of correlation functions for a general spin-S magnetic model [10], leading to ν = 1, in opposition to the generally accepted mean-field value ν = 1/2 [12].…”

Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory

“…Nevertheless, it plays an important role in statistical and condensed-matter physics because some problems involving disorder and/or interactions can be solved exactly when defined on a Bethe lattice, e.g., Ising models, 1,2,6 percolation, 7,8,9 or Anderson localization. 10,11,12,13 Such exact solutions on the Bethe lattice for Z < ∞ sometimes, 1,2 but not always, 10,11,12,13 have mean-field character.…”

Residual entropy and waterlike anomalies in the repulsive one dimensional lattice gas

“…First of all, the two transitions observed in our model can be seen as remanent from phase transitions on higher dimensional systems collapsing to T → 0 as dimensionality is reduced to d → 1. This collapse is also evident on the analytical solution of a lattice gas inside the Bethe lattice: in this case, while continuously reducing the coordination number the liquid-gas phase transition continuously shrinks to T → 0 [18,25]. This is a relevant issue for one dimensional systems because the observed ground state phase transitions incorporate elements from both continuous and discontinuous phase transitions [26].…”