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Mean-field theory on a coupled system of ferromagnetism and electronic nematic order

We analyze an effective model on a square lattice with two types of forward scattering interactions, which, respectively, drive ferromagnetism (FM) and electronic nematic order via a d-wave Pomeranchuk instability (dPI). The FM and dPI in general compete with each other and they are typically separated by a first order phase boundary in the plane of the chemical potential and temperature. Nevertheless there is a parameter region where the dPI occurs inside the FM phase, leading to their coexistence. We also study the effect of a magnetic field by choosing a chemical potential where the ground state is paramagnetic without a field. In this case, instead of FM, the dPI competes with a metamagnetic instability. The latter occurs above a threshold strength of the FM interaction and otherwise the dPI is stabilized with a dome-shaped phase diagram in the plane of a magnetic field and temperature. The FM interaction shifts the center of the dome to a lower field, accompanied by a substantial reduction of the field range where the dPI is stabilized and by an extension of the first order part of the transition line, although the maximal critical temperature does not change. The experimental phase diagram of the bilayer ruthenate Sr3Ru2O7 can be well captured by the present theory.