Duality relation for frustrated spin models.

We consider discrete spin models on arbitrary planar graphs and lattices with frustrated interactions. We first analyze the Ising model with frustrated plaquettes. We use an algebraic approach to derive the result that an Ising model with some of its plaquettes frustrated has a dual model which is an Ising model with an external field ipi/2 applied to the dual sites centered at frustrated plaquettes. In the case where all plaquettes are frustrated, this leads to the known result that the dual model has a uniform field ipi/2, whose partition function can be evaluated in the thermodynamic limit for regular lattices. The analysis is extended to a Potts spin glass with analogous results obtained.