2014 A Bethe lattice with a given coordination number can be described in terms of a regular tiling of the hyperbolic plane. An interest of this description is the possibility of intermediate lattices between regular lattices in Euclidean plane and Bethe lattices with the same coordination number. Tome 43 N° 8 15 AVRIL 1982 LE JOURNAL DE PHYSIQUE LETTRES J. Physique LETTRES 43 (1982) L-249 L-25...

Recently in a series of papers [DFM], [FM1], [FM2] the Bethe ansatz equations and Bethe vectors for the six-vertex model with the anisotropy commensurable with π or, as usually said, at roots of unity, were studied. In that case the spectrum of the transfer-matrix, which is a generating function of the standard commuting conservation laws in the model, becomes highly degenerate. In [FM2] a cons...

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9×9, rational, gl(1|2)-invariant R-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for in...

There are (at least) two main approaches to solvable lattice models and their associated quantum spin chains: the Bethe ansatz [6] and the corner transfer matrix method [5]. In his paper [6], Bethe solved the Heisenberg spin chain based on the string hypothesis which asserts that the eigenvalues of the Hamiltonian form certain strings in the complex plane as the size of the system tends to infi...

We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likel...

Bayesian belief propagation in graphical models has been recently shown to have very close ties to inference methods based in statistical physics. After Yedidia et al. demonstrated that belief propagation fixed points correspond to extrema of the so-called Bethe free energy, Yuille derived a double loop algorithm that is guaranteed to converge to a local minimum of the Bethe free energy. Yuille...

We investigate different ways of generating approximate solutions to the inverse Ising problem (IIP). Our approach consists in to take as a starting point for further perturbation procedures, a Bethe mean-field solution obtained with a maximum spanning tree (MST) of pairwise mutual information which we refer to as the Bethe reference point. We consider three different ways of following this ide...

We solve the A (2) 2n vertex model with all kinds of diagonal reflecting matrices by using the algebraic Behe ansatz, which includes constructing the multi-particle states and achieving the eigenvalue of the transfer matrix and corresponding Bethe ansatz equations. When the model is Uq(Bn) quantum invariant, our conclusion agrees with that obtained by analytic Bethe ansatz method.

An integrable quantum group deformation of the supersymmetric t-J model is introduced. Open boundary conditions lead to an spl q (2; 1) invariant hamiltonian. A general procedure to obtain such invariant models is proposed. To solve the model a generalized nested algebraic Bethe ansatz is constructed and the Bethe ansatz equations are obtained. The quantum supergroup structure of the model is i...

The XXX Heisenberg model is studied at finite temperature. The free energy is derived without recourse to Thermal Bethe Ansatz method and Quantum Transfer Matrix method. The result perfectly agrees with the free energy derived by Thermal Bethe Ansatz method. An explicit expression of the cluster expansion coefficient in arbitrary order is presented for the first time.