Theories and models of superconducting state PACS 74.20.Rp -Pairing symmetries (other than s-wave) PACS 74.70.Pq -RuthenatesAbstract -We investigate the combined effect of Hund's and spin-orbit (SO) coupling on superconductivity in multi-orbital systems. Hund's interaction leads to orbital-singlet spin-triplet superconductivity, where the Cooper pair wave function is antisymmetric under the exchange of two orbitals. We identify three d-vectors describing even-parity orbital-singlet spin-triplet pairings among t2g-orbitals, and find that the three d-vectors are mutually orthogonal to each other. SO coupling further assists pair formation, pins the orientation of the d-vector triad, and induces spinsinglet pairings with a relative phase difference of π/2. In the band basis the pseudospin d-vectors are aligned along the z-axis and correspond to momentum-dependent inter-and intra-band pairings. We discuss quasiparticle dispersion, magnetic response, collective modes, and experimental consequences in light of the superconductor Sr2RuO4.

An anisotropic metallic phase dubbed electronic nematic phase bounded by two consecutive metamagnetic transitions has been reported in the bilayer ruthenate Sr3Ru2O7. It has also been shown that the nematic and the accompanying metamagnetic transitions are driven by an effective momentum-dependent quadrupole-type interaction. Here, we study the microscopic origin of such an effective interaction. To elucidate the mechanism behind the spontaneous Fermi surface distortion associated with the nematic, we identify a simple tight binding model based on t2g orbitals, spin-orbit coupling and the rotation of RuO6 octahedra as starting point, consistent with the Fermi surface obtained from recent angle-resolved photoemission data. Within an extended Hubbard model the nematic state, characterized by an anisotropy between the bands near (±π, 0) and (0, ±π), then strongly competes with ferromagnetic order but pre-empts it via a finite nearest neighbor interaction. We discuss experimental means to confirm our proposal.

It was suggested that the two consecutive metamagnetic transitions and the large residual resistivity discovered in Sr 3 Ru 2 O 7 can be understood via the nematic order and its domains in a single layer system. However, a recently reported anisotropy between two longitudinal resistivities induced by tilting the magnetic field away from the c axis cannot be explained within the single layer nematic picture. To fill the gap in our understanding within the nematic order scenario, we investigate the effects of bilayer coupling and in-plane magnetic field on the electronic nematic phases in a bilayer system. We propose that the in-plane magnetic field in the bilayer system modifies the energetics of domain formation, since it breaks the degeneracy of two different nematic orientations. Thus, the system reveals a pure nematic phase with a resistivity anisotropy in the presence of an in-plane magnetic field. In addition to the nematic phase, the bilayer coupling opens a different route to a hidden nematic phase that preserves the x-y symmetry of the Fermi surfaces.

Electronic nematicity, proposed to exist in a number of transition metal materials, can have different microscopic origins. In particular, the anisotropic resistivity and meta-magnetic jumps observed in Sr 3 Ru 2 O 7 are consistent with an earlier proposal that the isotropic-nematic transition is generically first order and accompanied by meta-magnetism when tuned by a magnetic field. However, additional striking experimental features such as a non-Fermi liquid resistivity and critical thermodynamic behaviour imply the presence of an unidentified quantum critical point (QCP). Here we show that orbital degrees of freedom play an essential role in revealing a nematic QCP, even though it is overshadowed by a nearby meta-nematic transition at low temperature. We further present a finite temperature phase diagram including the entropy landscape and discuss our findings in light of the phenomena observed in Sr 3 Ru 2 O 7 .A variety of transition metal materials such as cuprates [1], Ru-oxides [2] and Fe-pnictides [3] have been proposed to harbour an electronic nematic phase [4]. Electronic nematic phases are broadly characterized by the presence of spontaneously broken rotational symmetry and viewed as the quantum counterpart of nematic classical liquid crystal phases. The theoretical proposal of nematic quantum liquid crystals became more concrete when experiments on ultra-pure bilayer ruthenate (Sr 3 Ru 2 O 7 ) samples subjected to a magnetic field along the c-axis revealed an unusual phase characterized by a pronounced residual resistivity in place of a putative meta-magnetic quantum critical point (QCP) [5]. Interestingly, Sr 3 Ru 2 O 7 was initially viewed as a prototype for the study of quantum phase transitions, exhibiting a striking non-Fermi liquid resistivity thought to originate from the putative magnetic field tuned QCP [6]. The unusual phase found in ultra-pure samples is delimited by two consecutive first order meta-magnetic transitions at low temperature and, remarkably, exhibits a significant inplane magnetoresistive anisotropy when the external field is slightly tilted towards one of the in-plane crystal axes [2]. These observations strongly imply the formation of an anisotropic metallic, i.e. electronic nematic, phase in the bilayer ruthenate compound.Based at first on the two consecutive meta-magnetic transitions, an electronic nematic phase was proposed and generic features of nematic phase formation were theoretically explored early on [5,7,8]. It was found that the transition between the isotropic and nematic phase is generally first order, and that nematic order typically develops near a van Hove singularity (vHS) to avoid a Lifshitz transition. Varying the chemical potential, the nematic phase is bounded at low and high values by two isotropic phases, while the concomitant first order transitions lead to jumps in the electron density. When a magnetic field is applied (and the chemical potential is held fixed at, say, some low value), Zeeman coupling acts as a spin-dependent chemical potential te...

Electronic nematic phases are broadly characterized by spontaneously broken rotational symmetry. Although they have been widely recognized in the context of high temperature cuprates, bilayer ruthenates, and iron-based superconductors, the focus so far has been exclusively on the uniform nematic phase. Recently, however, it was proposed that on a square lattice a nematic instability in the d-wave charge channel could lead to a spatially modulated nematic state, where the modulation vector q is determined by the relative location of the Fermi level to the van Hove singularity.[1] Interestingly, this finite-q nematic phase has also been identified as an additional leading instability that is as strong as the superconducting instability near the onset of spin density wave order.[2] Here we study the electrical conductivity tensor in the modulated nematic phase for a general modulation vector. Our results can be used to identify modulated nematic phases in correlated materials.