Integrable boundary conditions for the B_2 model

New integrable B_2 model with off-diagonal boundary reflections is proposed. The general solutions of the reflection matrix for the B_2 model are obtained by using the fusion technique. Wefind that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. The new quantization conditions will induce the novel structure of the energy spectrum and the boundary states. The corresponding boundary effects can be studied based on the results in this paper. Meanwhile, the reflection matrix of high rank models associated with Bnalgebra can also be obtained by using the method suggested in this paper.     

Bethe ansatz results for Hubbard chains with toroidal boundary conditions

We solve exactly the problem of a one-dimensional repulsive-U Hubbard chain with toroidal boundary conditions (HTB) using the Bethe ansatz approach. We calculate analytically the finite-size corrections to the ground-state energy in the half-filled case and use this expression to derive charge and spin stiffnesses with no assumptions. We then use a particle-hole transformation to calculate the finite-size corrections for the half-filledattractive- U case, and again derive the resulting expressions for the charge and spin stiffnesses. Lastly, we discuss how the repulsive-U corrections relate to those of a Heisenberg model with toroidal boundary conditions.On leave from Departamento de Fisica, Universidade Federal de S. Carlos, S. Carlos, 13560, Brazil.     

Six-vertex model with rotated boundary conditions

We investigate the six-vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes, a model recently introduced by Litvin and Priezzhev. Auxiliary vertices are used to define an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. A product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system. Using an equation for commuting transfer matrices, we calculate their eigenvalues. Finite-size properties of the model are discussed from the viewpoint of the conformal field theory.     

The Bethe ansatz for the six-vertex model with rotated boundary conditions

A method is suggested for derivation of the Bethe ansatz equations for the six-vertex model on a square lattice rotated at an arbitrary angle with respect to the coordinate axes. The method is based on the random walk representation for configurations of the model. The equations for the ice model on the rotated lattice are derived and some numerical results are obtained.     

Reduction of the XXZ model with twisted boundary conditions

The XXZ model with twisted boundary conditions is considered. The method of energy spectrum calculation based on the functional equation for the transfer matrix is analyzed. The Hamiltonian eigenvalues are obtained in an explicit form.     

Two-dimensional Ising model: Duality and boundary conditions

Duality relations are derived between the Ising model on a square lattice wrapped on a cylinder with a defect and with free boundary conditions at the ends and the Ising model with magnetic fields applied at the ends of the cylinder. We derive as a consequence of these relations the duality relations for Ising models with free, mixed, and fixed boundary conditions at the ends of the cylinder. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 369–374 (10 March 1996)     

Chiral Potts model with skewed boundary conditions

We obtain the transfer matrix functional relations for the chiral Potts model with skewed boundary conditions and find that they are the same as for periodic boundary conditions, but with modified selection rules. As a start toward calculating the interfacial tension in general, we here evaluate it in a low-temperature limit, performing a Bethe-ansatz-type calculation. Finally, we specialize the relations to the superintegrable case, verifying the ansatz proposed by Albertiniet al.     

Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions

By using the intertwiner and face-vertex correspondence relation, we obtain the Bethe ansatz equation of the eight-vertex model with open boundary conditions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for the eight-vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries.     

Finite-size corrections in the Ising model with special boundary conditions

The Ising model in two dimensions with the special boundary conditions of Brascamp and Kunz (BK) is analyzed. We derive exact finite-size corrections for the free energy F   of the critical ferromagnetic Ising model on the M×N$M\times N$ square lattice with Brascamp–Kunz boundary conditions [H.J. Brascamp, H. Kunz, J. Math. Phys. 15 (1974) 66]. We show that finite-size corrections strongly depend not only on the boundary conditions but also on the shape and pattern of the lattice. In the limit N→∞$N\to \infty$ we obtain the expansion of the free energy and the inverse correlation lengths for infinitely long strip with BK boundary conditions. Our results are consistent with the conformal field theory prediction for the mixed boundary conditions.     

Integrable open boundary conditions for the Bariev model of three coupled XY spin chains

The integrable open-boundary conditions for the Bariev model of three coupled one-dimensional XY spin chains are studied in the framework of the boundary quantum inverse scattering method. Three kinds of diagonal boundary K-matrices leading to nine classes of possible choices of boundary fields are found and the corresponding integrable boundary terms are presented explicitly. The boundary Hamiltonian is solved by using the coordinate Bethe ansatz technique and the Bethe ansatz equations are derived.     

A long-range ferromagnetic spin model with periodic boundary conditions

A possible way to study long-range interacting particles in finite–infinite periodic systems is applied to a modified Ising model with ferromagnetic interaction that decays as a 1/r^α law. We verify, by Monte Carlo heat-bath simulations for cases in D dimensions, that the thermodynamic quantities scale in a way proposed by Tsallis, and that the mean field theory is exact in the last model for all 0⩽α⩽D/2 suggested previously by other authors.     

Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as fe=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.