Influence of spin-orbit coupling on the metamagnetic transition in Sr$_3$Ru$_2$O$_7$

Clean single 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 a critical field $H_c$ ranging from $5.1$T for the case of $H\perp c$ to almost $8$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 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 Rashba-like 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 occurence of the nematic phase.