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 $Hperp c$ to $sim 8$T for $Hparallel c$. 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.
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