Orthogonality constraints and entropy in the SO(5)-Theory of HighTc-Superconductivity

S.C. Zhang has put forward the idea that high-temperature-superconductors can be described in the framework of an SO(5)-symmetric theory in which the three components of the antiferromagnetic order-parameter and the two components of the two-particle condensate form a fivecomponent order-parameter with SO(5) symmetry. Interactions small in comparison to this strong interaction introduce anisotropies into the SO(5)-space and determine whether it is favorable for the system to be superconducting or antiferromagnetic. Here the view is expressed that Zhang’s derivation of the effective interaction Veff based on his Hamiltonian Ha is not correct. However, the orthogonality constraints introduced several pages after this ’derivation’ give the key to an effective interaction very similar to that given by Zhang. It is shown that the orthogonality constraints are not rigorous constraints, but they maximize the entropy at finite temperature. If the interaction drives the ground-state to the largest possible eigenvalues of the operators under consideration (antiferromagnetic ordering, superconducting condensate, etc.), then the orthogonality constraints are obeyed by the ground-state, too.