Linear-Algebraic Results Associated with Antiferromagnetic Heisenberg Chains
Man Duen Choi, Jeffrey S. Rosenthal, Peter Rosenthal
This paper analyzes the $3^N \times 3^N $ selfadjoint matrix\[ H_N = \sum\limits_{j = 1}^{N - 1} S_x^{( j)} S_x^{( j + 1 )} + S_y^{( j )} S_y^{( j + 1 )} + S_z^{( j )} S_z^{( j + 1 )} , \] the Hamiltonian of a spin-1, one-dimensional, Heisenberg antiferromagnet. Various results are obtained, including alternative representations of $H_N $, families of operators commuting with $H_N $, a complete description of the spin-0 subspace (which includes all one- and two-dimensional eigenspaces of $H_N $), and a proof that $H_N $ must have an eigenvalue smaller than $ - \frac{4}{3}( N - 1 )$.
SIAM Journal on Matrix Analysis and Applications•Jul 1, 1993•0 citations