Title: Anisotropy of magnetic interactions and symmetry of the order parameter in unconventional superconductor Sr$_{2}$RuO$_{4}$,
Abstract: Sr$_2$RuO$_4$ is the best candidate for spin-triplet superconductivity, an
unusual and elusive superconducting state of fundamental importance. In the
last three decades Sr$_2$RuO$_4$ has been very carefully studied and despite
its apparent simplicity when compared with strongly correlated high-$T_{c}$
cuprates, for which the pairing symmetry is understood, there is no scenario
that can explain all the major experimental observations, a conundrum that has
generated tremendous interest. Here we present a density-functional based
analysis of magnetic interactions in Sr$_{2}$RuO$_{4}$ and discuss the role of
magnetic anisotropy in its unconventional superconductivity. Our goal is
twofold. First, we access the possibility of the superconducting order
parameter rotation in an external magnetic field of 200 Oe, and conclude that
the spin-orbit interaction in this material is several orders of magnitude too
strong to be consistent with this hypothesis. Thus, the observed invariance of
the Knight shift across $T_{c}$ has no plausible explanation, and casts doubt
on using the Knight shift as an ultimate litmus paper for the pairing symmetry.
Second, we propose a quantitative double-exchange-like model for combining
itinerant fermions with an anisotropic Heisenberg magnetic Hamiltonian. This
model is complementary to the Hubbard-model-based calculations published so
far, and forms an alternative framework for exploring superconducting symmetry
in Sr$_{2}$RuO$_{4}.$ As an example, we use this model to analyze the
degeneracy between various $p-$triplet states in the simplest mean-field
approximation, and show that it splits into a single and two doublets with the
ground state defined by the competition between the "Ising" and "compass"
anisotropic terms.