Quasi-classical determination of the in-plane magnetic field phase   diagram of superconducting Sr_2RuO_4
  We have carried out a determination of the magnetic-field-temperature (H-T) phase diagram for realistic models of the high field superconducting state of tetragonal Sr_2RuO_4 with fields oriented in the basal plane. This is done by a variational solution of the Eilenberger equations.This has been carried for spin-triplet gap functions with a {\bf d}-vector along the c-axis (the chiral p-wave state) and with a {\bf d}-vector that can rotate easily in the basal plane. We find that, using gap functions that arise from a combination of nearest and next nearest neighbor interactions, the upper critical field can be approximately isotropic as the field is rotated in the basal plane. For the chiral {\bf d}-vector, we find that this theory generically predicts an additional phase transition in the vortex state. For a narrow range of parameters, the chiral {\bf d}-vector gives rise to a tetracritical point in the H-T phase diagram. When this tetracritical point exists, the resulting phase diagram closely resembles the experimentally measured phase diagram for which two transitions are only observed in the high field regime. For the freely rotating in-plane {\bf d}-vector, we also find that additional phase transition exists in the vortex phase. However, this phase transition disappears as the in-plane {\bf d}-vector becomes weakly pinned along certain directions in the basal plane. 
R.P. Kaur, D.F. Agterberg, and H. Kusunose