Spin-orbit coupling and spin-triplet pairing symmetry in Sr2RuO4

Abstract

Spin-orbit coupling (SOC) plays a crucial role in determining the spin structure of an odd parity psedospin-triplet Cooper pairing state. Here, we present a thorough study of how SOC lifts the degeneracy among different p-wave pseudospin-triplet pairing states in a widely used microscopic model for Sr2RuO4, combining a Ginzburg-Landau (GL) free energy expansion, a symmetry analysis of the model, and numerical weak-coupling renormalization group (RG) and random phase approximation (RPA) calculations. These analyses are then used to critically re-examine previous numerical results on the stability of chiral p-wave pairing. The symmetry analysis can serve as a guide for future studies, especially numerical calculations, on the pairing instability in Sr2RuO4 and can be useful for studying other multi-band spin-triplet superconductors where SOC plays an important role. I. INTRODUCTIONUnderstanding an unconventional superconductor requires identifying and understanding both its superconducting order parameter symmetry and the pairing mechanism. The two are intimately connected. In Sr 2 RuO 4 , both of these are still not well understood. Early experiments, including muon spin relaxation 1 , NMR 2 , Polar Kerr effect 3 measurements, point toward a spin-triplet chiral p-wave pairing 4,5 , which is a two-dimensional (2D) analogue the A-phase of Helium 3 6 and is potentially useful for topological quantum computing 7,8 .

Superconducting subphase and substantial Knight shift in Sr2RuO4

“…Some of these TRSB states are the superposition of the four states in Eq. 17 and in this context it is interesting to note that inclusion of SOC results in accidental or near degeneracy between pairs of the helical states above 45,59 . Such superposition states are worthy of future study as, in addition to possibly capturing the superconducting subphase described in this work, they may plausibly (i) yield a non-zero Kerr effect and finite orbital magnetic moment similar to those found in the chiral state 60,61 , and (ii) may resolve the contradiction of the absence of edge super-currents, as the occurrence of such currents for TRSB helical states is unclear (in contrast to the TRSB chiral state in which they are expected).…”

Neutron scattering studies on spin fluctuations in Sr2RuO4

“…The new studies of the Knight shift in NMR [16,17] and those of the polarized neutron diffraction [18] reveal an unambiguous drop of the electronic susceptibility that is inconsistent with spin-triplet pairs parallel to Ru layers. Since then, numerous proposals for the superconducting state were made mostly invoking some d-wave state, and the discussion of the superconducting pairing has become very active [19][20][21][22][23][24][25][26]. The observations of broken time-reversal symmetry in muon spin relaxation experiments [27,28] and in measurements of the magnetooptical Kerr effect [29] may require interpretations other than the chiral p-wave scenario.…”

“…The observations of broken time-reversal symmetry in muon spin relaxation experiments [27,28] and in measurements of the magnetooptical Kerr effect [29] may require interpretations other than the chiral p-wave scenario. Many theories discuss a superconducting state with a complex combination of components [19][20][21][22][23][24][25].…”

Momentum-resolved superconducting energy gaps of Sr 2 RuO 4 from quasiparticle interference imaging

“…This phenomenology is in sharp contradistinction to the Sr2RuO4 ancien regime, under which 17 O Knight shift 14 and spin-polarized neutron scattering 15 reported no diminution in spin susceptibility below Tc, and where muon spin rotation 16 and Kerr effect 17 indicated TRS breaking. Therefore, an extensive reassessment of the theory of Sr2RuO4 superconductivity has quickly materialized 18 - 23 .…”

“…However, although critical to testing advanced theories [18][19][20][21][22][23] for superconductivity in Sr2RuO4, the k-space structure of Δ ( ); Δ ( ); Δ ( ) has never been measured directly. Basically, this is because the maximum magnitude of any of these gaps 30,31 is |Δ| ≤ 350 so that temperature ≲ 100 and energy resolution with ≲ 100 are required to spectroscopically detect strongly anisotropic k-space gap structures and/or their gap minima.…”