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arXiv:math-ph/9904006 · International Journal of Modern Physics A

QUANTUM MODEL OF INTERACTING "STRINGS" ON THE SQUARE LATTICE

Herman E. Boos

Published Jan 10, 2000• 1 citation
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Abstract

The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the "string" states is studied. The solution to the eigenvalue problem is obtained for the single "string" in cases of the "string" with fixed ends and "string" of types (1, 1) and (1, 2) living on the torus. The latter case has the features of a self-interacting system and does not seem to be integrable while the previous two cases are equivalent to the free-fermion model.

DOI
10.1142/s0217751x00000069

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The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the "string" states is studied. The solution to the eigenvalue problem is obtained for the single "string" in cases of the "string" with fixed ends and "string" of types (1, 1) and (1, 2) living on the torus. The latter case has the features of a self-interacting system and does not seem to be integrable while the previous two cases are equivalent to the free-fermion model.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.
We consider the general -symmetric free-fermion model on the finite periodic lattice, which includes as special cases the Ising model on the square and triangular lattices and the -symmetric BBS τ (2) -model with n = 2. Translating Kaufman's fermionic approach to diagonalization of Ising-like transfer matrices into the language of Grassmann integrals, we determine the transfer matrix eigenvectors and observe that they coincide with the eigenvectors of a square lattice Ising transfer matrix. This allows us to find exact finite-lattice form factors of spin operators for the statistical model and the associated finite-length quantum chains, of which the most general is equivalent to the XY chain in a transverse field.
We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.