Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Abstract

Here we study the partition function zeros of the one-dimensional Blume-Emery-Griffiths model close to their edge singularities. The model contains four couplings (H, J, Delta, K) including the magnetic field H and the Ising coupling J. We assume that only one of the three couplings (J, Delta, K) is complex and the magnetic field is real. The generalized zeros z(i) tend to form continuous curves on the complex z-plane in the thermodynamic limit. The linear density at the edges z(E) diverges usually with rho(z) similar to vertical bar z - z(E)vertical bar(sigma) and sigma = -1/2. However, as in the case of complex magnetic fields (Yang-Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with sigma = -2/3 can appear as we prove here explicitly for the cases where either Delta or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J) are more involved; in practice, we have not been able to find an explicit example with sigma = -2/3 as far as the other couplings (H, Delta, K) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of sigma = -2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.