Possible 3D nematic odd-parity pairing in Sr$_2$RuO$_4$: experimental evidences and predictions

Abstract

Due to the presence of a nontrivial three-dimensional spin-orbital entanglement, Sr2RuO4 may be a timereversal invariant nematic p-wave superconductor with coexisting in-plane and out-of-plane pairings. Here we discuss various signatures of such a state if the out-of-plane pairing component is dominant. First, the enhancement of the superconducting Tc under in-plane uniaxial strains is nearly a quadratic function of the strain, because the out-of-plane pairing lacks a linear-order coupling to the strain. Second, when the strain applies along a certain in-plane direction, the nematic p-wave pairing exhibits only a single phase transition as the temperature is lowered. These are consistent with several recent uniaxial strain measurements, which are otherwise hard to reconcile with chiral p-wave order. We further show that the nematic p-wave state can be distinguished from the chiral p-wave state through the velocity jumps of certain sound waves at the onset of superconductivity. Possible implications for µSR experiment under strain and NMR Knight shift measurement are also discussed.

Thermodynamic Evidence for a Two-Component Superconducting Order Parameter in Sr$_2$RuO$_4$

“…In this section, we calculate expressions for the elastic moduli discontinuities and attenuation, starting from a Ginzburg-Landau theory. Similar expressions for the chiral state have been calculated by Sigrist [21] and for nematic superconducting states by Huang et al [22].…”

“…Our expressions match those of Sigrist for the chiral state, and we correct the expressions derived by Huang et al by including all possible order parameter fluctuations. In particular, we find that both in-plane shear moduli ((c 11 − c 12 )/2 and c 66 ) should show a discontinuity for the nematic states, contrary to what was concluded in [22]. We also show that the three A 1g jumps always follow a consistency relation, similar to what was concluded for non-superconducting order parameters in URu 2 Si 2 [17], which is also a tetragonal material with point group D 4h .…”

“…This endows ultrasound with a sensitivity to some of the symmetry properties of an order parameter, such as whether it has one or two components. Here we provide a brief overview of the connection between crystal symmetry, order parameter symmetry, and ultrasound: the detailed derivations can be found in the SI, as well as in a number of theoretical papers [16,[20][21][22]. As illustrated in Figure 1, there are five unique strains ( Γ ) in Sr 2 RuO 4 (five irreducible representations of strain in D 4h ): two compressive strains transforming as the A 1g representation, and three shear strains transforming as the B 1g , B 2g and E g representations.…”

“…OPs [22], characterized by different equilibrium values of (θ, γ) = (θ 0 , γ 0 ): (π/4, ±π/2) for the chiral state, (π/4, 0) for the diagonal nematic state and (0, 0) for the horizontal nematic state. These states also have different equilibrium values of η = η 0 , which can be calculated from ∂F op /∂η| (θ,γ)→(θ 0 ,γ 0 ) = 0.…”

Possible 'symmetry-imposed' near-nodal two-dimensional p-wave pairing in Sr$_2$RuO$_4$

“…Foremost, a chiral p-wave pairing in a two-dimensional (2D) metal is generically fully gapped. The residual density of states have been attributed to anisotropic chiral p-wave pairing with deep gap minima (near-nodes) [23][24][25][26][27][28][29][30][31], or fundamentally different pairing symmetries, such as d-wave and highly anisotropic s-wave pairings, or a nematic p-wave (p x ± p y ) [29,[32][33][34]. In 2D, the nematic p-wave phase is typically less favorable than the chiral phase.…”