# Anisotropy of magnetic interactions and symmetry of the order parameter in unconventional superconductor Sr$_{2}$RuO$_{4}$

## TL;DR

This paper investigates magnetic interactions and anisotropy in Sr$_{2}$RuO$_{4}$, challenging existing assumptions about its superconducting pairing symmetry and proposing a new model to explore its unconventional superconductivity.

## Contribution

It introduces a density-functional analysis of magnetic anisotropy and proposes a double-exchange-like model to better understand the superconducting order parameter in Sr$_{2}$RuO$_{4}$.

## Key findings

- Magnetic anisotropy is too strong for the order parameter rotation hypothesis.
- Knight shift invariance across Tc cannot be explained by spin-orbit interaction.
- The proposed model reveals degeneracy splitting among triplet states due to anisotropic terms.

## 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.