Microscopic Ginzburg-Landau theory and singlet ordering in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

The long-standing quest to determine the superconducting order of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ (SRO) has received renewed attention after recent nuclear magnetic resonance (NMR) Knight shift experiments have cast doubt on the possibility of spin-triplet pairing in the superconducting state. As a putative solution, encompassing a body of experiments conducted over the years, a $(d+ig)$-wave order parameter caused by an accidental near degeneracy has been suggested [S. A. Kivelson et al., npj Quantum Mater. 5, 43 (2020)]. Here we develop a general Ginzburg–Landau theory for multiband superconductors. We apply the theory to SRO and predict the relative size of the order parameter components. The heat capacity jump expected at the onset of the second-order parameter component is found to be above the current threshold deduced by the experimental absence of a second jump. Our results tightly restrict theories of $d+ig$ order, and other candidates caused by a near degeneracy, in SRO. We discuss possible solutions to the problem.

Figure 1

Symmetries of the even-parity order parameters in channel (a) $A_{1g}$, (b) $A_{2g}$, (c) $B_{1g}$, and (d) $B_{2g}$. The black lines display the Fermi surface applicable to the three-band case of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, where the three bands are denoted by $\alpha$, $\beta$, and $\gamma$. The Fermi surface is obtained using tight-binding parameters listed in Appendix pp2, as extracted from density functional theory [13, 54].

Figure 2

The pseudospin-singlet Cooper pair operator shown diagrammatically as a two-fermion composite operator.

Figure 3

(a) Second-order and (b) fourth-order diagrams contributing to the free energy of Eq. (5). The algebraic expressions corresponding to -(a) and (b) are given in Eqs. (8) and (9), respectively. The single-component, quadratic band case resulting in Eqs. (12) and (13) corresponds to fixing $a_1=a_2=a_3=a_4=A_{1g}$ here.

Figure 4

The weight $X\left(T\right)$, with an order parameter of the form $\mathrm{\Delta }\left(\mathit{p}\right)={\mathrm{\Delta }}_0\left(T\right)[{\mathrm{\Delta }}_{B_{1g}\mu }\left(\mathit{p}\right)+iX\left(T\right){\mathrm{\Delta }}_{A_{2g}\mu }\left(\mathit{p}\right)]$ for $T_{c1}=1.48$ K and $T_{c2}=1.44$ K. This result is obtained using the GL coefficients Eqs. (A16) and (A17).

Figure 5

Heat capacity anomaly for a $B_{1g}+i{A}_{2g}$ order parameter in SRO. (a) Contour lines for the ratio of the second heat capacity jump to the primary heat capacity jump, $\eta$, from Eq. (17), with $T_{\text{TRSB}}=1.43$ K and $T_c=1.48$ K using order parameters with three lattice harmonics (see Appendix pp4). The parameter space consistent with the current experimental threshold, $\eta \lesssim 0.05$, is marked by the cross-hatched region [16, 19]. For the dispersion, we use the (2D) three-band model listed in Appendix pp2. (b) Specific heat at the point marked with “$★$” in (a). (c) Specific heat for the GL solution of Sec. 3a. The normalized specific heat per temperature is compared to the data of Refs. [8, 16].

Figure 6

(a) Fermi surface sheets resulting from the model of Eq. (B1), and (b) the Fermi velocity as a function of the in-plane angle $\theta$ (${\theta }^{\prime }$) for bands $\beta$ and $\gamma$ ($\alpha$), cf. Ref. [70].

Figure 7

Comparison of the numerical result for $X\left(T\right)$ with the order parameter combination $\mathrm{\Delta }\left(\mathit{p}\right)={\mathrm{\Delta }}_0\left(T\right)[{\mathrm{\Delta }}_{B_{1g}\mu }\left(\mathit{p}\right)+iX\left(T\right){\mathrm{\Delta }}_{A_{2g}\mu }\left(\mathit{p}\right)]$ using two different Hamiltonians. The DFT Hamiltonian is described in Appendix pp2, the ARPES Hamiltonian is described in Ref. [24] (here at $k_z=0$). The results are very similar.

Figure 8

Order parameters from Ref. [24] for $J/U=0.20$ (full lines) and lattice harmonics fits (dashed lines) with three lattice harmonics for irreps. (a) $A_{1g}$, (b) $A_{2g}$, (c) $B_{1g}$, and (d) $B_{2g}$. We note that including more lattice harmonics in the fit to the $A_{1g}$ order parameter in (a) has a negligible effect on the corresponding relative weight $X\left(T\right)$ in the Ginzburg–Landau minimization of Fig. 11. The quality of the fit for the $\beta$ and $\gamma$ sheets we ascribe to the fairly coarse $\mathit{k}$ resolution used in Ref. [24].

Figure 9

The same as described in the caption of Fig. 5 but using only the leading lattice harmonics from Table 1 for (a) a $B_{1g}+i{A}_{2g}$ order parameter, and (b) a $A_{1g}+i{B}_{2g}$ order parameter.

Figure 10

The same as described in the caption of Fig. 5 but here for an order parameter of the form $A_{1g}+i{B}_{2g}$, using the advanced order parameters with the three leading lattice harmonics from Tables 4 and 5.

Figure 11

The weight $X\left(T\right)$, with an order parameter of the form $\mathrm{\Delta }\left(\mathit{p}\right)={\mathrm{\Delta }}_0\left(T\right)[{\mathrm{\Delta }}_{A_{1g}\mu }\left(\mathit{p}\right)+iX\left(T\right){\mathrm{\Delta }}_{B_{2g}\mu }\left(\mathit{p}\right)]$ for $T_{c1}=1.48$ K and $T_{c2}=1.44$ K. This result is obtained using the GL coefficients Eqs. (A16) and (A17).