Effect of a staggered spin-orbit coupling on the occurrence of a nematic phase in Sr${}_3$Ru${}_2$O${}_7$

Ultra-clean crystals of Sr${}_3$Ru${}_2$O${}_7$ undergo a metamagnetic transition at low temperatures. This transition shows a strong anisotropy in the applied field direction with the critical field $H_c$ ranging from $\sim 5.1$T for $H\perp c$ to $\sim 8$T for $H\parallel c$. In addition, studies on ultra-pure samples revealed a bifurcation of the metamagnetic line for fields in $c$-direction and it is argued that a nematic phase emerges between the magnetization jumps. The aim of this study is to explain the field-direction anisotropy of these phenomena. Based on a microscopic tight-binding model, we introduce the metamagnetic transition by means of a van Hove singularity scenario. We show that the rotation of the O-octahedra around the c-axis expected for this material introduces a staggered spin-orbit coupling within the planes and naturally leads to an anisotropy of the magnetic response. We describe the low-temperature phase as a nematic state favored by forward scattering processes. The spin-orbit coupling shows an influence on both, the critical field $H_c$ and the occurrence of the nematic phase.