Fixed-spin Boundary Conditions Research Articles

Algebraic Reduction of the Ising Model

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.

Eigenvectors of the Baxter-Bazhanov-Stroganov τ(2)(t q ) model with fixed-spin boundary conditions

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤN quantum chain model.

Transfer Matrix Functional Relations for the Generalized 2(tq) Model

The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.

Anisotropic colloidal particles in critical fluids.

We consider anisotropic colloidal particles with dumbbell or lens shapes that are immersed in a critical binary fluid mixture. The orientation-dependent long-ranged universal interactions mediated by the critical solvent between a particle and a wall or between two particles are investigated for mesoscopic particle sizes small compared to the correlation length and interparticle distances. Exact results are obtained using a "small particle operator expansion." The amplitudes of the isotropic and anisotropic operators in the expansion depend on the size and aspect ratio of the dumbbell or lens and are determined by density profiles in the Ising model at the critical point in a wedge geometry with symmetry-breaking fixed-spin boundary conditions. Dumbbells and ellipsoids with a symmetry preserving surface are also considered.

CYCLIC MONODROMY MATRICES FOR THE R-MATRIX OF THE SIX-VERTEX MODEL AND THE CHIRAL POTTS MODEL WITH FIXED SPIN BOUNDARY CONDITIONS

Irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model are described. As a consequence, the direct computation of spectra for transfer-matrices of the chiral Potts model with special fixed-spin boundary conditions is done. The generalization of simple Baxter's Hamiltonian is proposed.

Conformal theory of spin correlations in the semi-infinite 3-state Potts and self-dual ZNmodels

In the conformal theory the magnetization operator of the critical 3-state Potts or self-dual Z3 model is degenerate at level 6. Solving a sixth-order differential equation, the authors calculate the bulk 4-spin correlation function and the spin-spin correlation function in the half space. The spin-spin correlation function of the semi-infinite self-dual ZN model is also obtained for arbitrary N. Both free- and fixed-spin boundary conditions are considered.

A simple solvable ZN Hamiltonian

The “superintegrable” case of the N-state chiral Potts model has various intriguing properties. In particular, for fixed-spin boundary conditions the column-to-column transfer matrix has a very simple eigenvalue spectrum: a direct product of N-by- N diagonal matrices. Here we present an associated non-Hermitian Hamiltonian: all its coefficients are arbitrary, it is a generalization of the Ising Hamiltonian to N-state systems, and its eigenvalue spectrum is a corresponding direct sum.

Duality of ordinary and extraordinary surface critical behaviour in the two-dimensional Potts model

The dual relationship between semi-infinite two-dimensional q-state Potts models with free and fixed spin boundary conditions is used to study the surface critical behaviour at the bulk critical temperature with fixed boundary spins, i.e. the extraordinary transition. Both the magnetisation and energy densities have singularities of the same form mod T-Tc mod 2- alpha as the bulk free energy. At T=Tc both the spin-spin and energy-energy correlations decay as r-4 parallel to the surface.