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

Abstract

Sr 2 RuO 4 has stood as the leading candidate for a spin-triplet superconductor for 26 years. Recent NMR experiments have cast doubt on this candidacy, however, and it is difficult to find a theory of superconductivity that is consistent with all experiments. What is needed are symmetry-based experiments that can rule out broad classes of possible superconducting order parameters. Here we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr 2 RuO 4 through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus c 66 , requiring that the superconducting order parameter is two-component. A two-component pwave order parameter, such as p x + ip y , naturally satisfies this requirement. As this order parameter appears to be precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely {d xz , d yz } or d x 2 −y 2 , g xy(x 2 −y 2 ) , are now the prime candidates for the order parameter of Sr 2 RuO 4 .

Theory of Strain-Induced Magnetic Order and Splitting of $T_c$ and $T_{\rm TRSB}$ in Sr$_2$RuO$_4$

“…This drop in the spin susceptibility below T c is in stark contrast to the Knight shift expected for the chiral p+ip-wave phase, and suggests the realization of an even-parity condensate in this material. A standard single-component even-parity nodal superconducting order parameter is inconsistent with evidence for TRSB and conclusions of recent ultrasound measurements [7,8]. Both resonant ultrasound spectroscopy [7] and ultrasound velocity measurements [8] observe a discontinuity of the elastic constant c 66 at T c , implying a two-component superconducting order parameter.…”

“…A standard single-component even-parity nodal superconducting order parameter is inconsistent with evidence for TRSB and conclusions of recent ultrasound measurements [7,8]. Both resonant ultrasound spectroscopy [7] and ultrasound velocity measurements [8] observe a discontinuity of the elastic constant c 66 at T c , implying a two-component superconducting order parameter. With the odd-parity E u p-wave solution out of the running, attention has therefore turned to the even-parity (twodimensional) E g representation or suitable combinations of one-dimensional representations [23][24][25][26][27][28][29][30][31][32].…”

“…Recently, a proposal of an accidental degeneracy between d x 2 −y 2 and g-wave was presented [25] as a possibility to reconcile a number of experimental observations in unstrained Sr 2 RuO 4 , notably the c 66 jump in the B 2g shear modulus in ultrasound [7,8] and the concurrent absence of a jump in the B 1g channel. In general, spinfluctuation pairing based on onsite interactions U and J find a g-wave solution which is strongly suppressed compared to the other even-parity solutions, see the orange curve of Fig.…”

“…Until recently, the material was a prime suspect in the search for topological chiral triplet superconductivity. However, several new experimental results have challenged this picture [6][7][8][9], and the hunt is on to find a new consistent explanation for the panoply of observations on this fascinating material. These developments have not consigned Sr 2 RuO 4 to the junk heap of "ordinary" unconventional superconductors, but led to a discussion of new ways that the system may be extraordinary, if not odd.…”

Quantum Oscillations and the Quasiparticle Properties of Thin Film Sr$_2$RuO$_4$

“…Sr 2 RuO 4 was long thought to be a p−wave superconductor, but recently revised NMR measurements find a substantial decrease in the Knight shift across the superconducting transition temperature T c [1], essentially ruling out all spin-triplet pairing states. While the details of the superconducting state are far from settled [2,3], it appears that single-crystal Sr 2 RuO 4 is not a p x + ip y superconductor. It may be possible, however, that multiple superconducting order parameters lie nearby in energy [4,5], suggesting that the application of the right tuning parameter could push Sr 2 RuO 4 into a p−wave state.…”