Odd-Parity Triplet Superconducting Phase in Multiorbital Materials with a Strong Spin-Orbit Coupling: Application to Doped ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$

We explore possible superconducting states in $t_{2g}$ multiorbital correlated electron systems with strong spin-orbit coupling (SOC). In order to study such systems in a controlled manner, we employ large-scale dynamical mean-field theory (DMFT) simulations with the hybridization expansion continuous-time quantum Monte Carlo (CTQMC) impurity solver. To determine the pairing symmetry, we go beyond the local DMFT formalism using parquet equations to introduce the momentum dependence in the two-particle vertex and correlation functions. In the strong SOC limit, a singlet, $d$-wave pairing state in the electron-doped side of the phase diagram is observed at weak Hund’s coupling, which is triggered by antiferromagnetic fluctuations. When the Hund’s coupling is comparable to SOC, a twofold degenerate, triplet $p$-wave pairing state with relatively high transition temperature emerges in the hole-doped side of the phase diagram, which is associated with enhanced charge fluctuations. Experimental implications to doped ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ are discussed.

Figure 1

Phase diagram of the $t_{2g}$ Hubbard model in terms of $J/U$ and filling $n$, obtained from DMFT with CTQMC and parquet formulation, where $J$ and $U$ represent the Hund’s and intraorbital Hubbard interaction, respectively. The tight-binding parameters and spin-orbit coupling strength are fixed (see main text), and the lowest temperature achieved in the simulation is $0.05t$. Symbols correspond to the parameter sets where simulations are performed. FL, $\mathrm{SC}\text{−}d$, AFM-I, AFM-M, $\mathrm{SC}\text{−}p$, and FM-M stand for Fermi liquid, $d$-wave singlet pairing, antiferromagnetic insulator, antiferromagnetic metal, $p$-wave triplet pairing, and ferromagnetic metal, respectively. The shaded areas are guides to the eye, and the two crosses highlight the two selected parameter sets where the instability analyses are presented in Figs. 4 and 5.

Figure 2

The band dispersion $E_m(\mathbf{k})$ and Fermi surface (FS) at filling $n=5$. The spin-orbit coupling $\lambda$ separates the $m=1$ and the $m=2,3$ bands. At $\lambda =2t$, the FS only crosses the $m=1$ band, as shown by the red contour line and its projection to the bottom of the BZ.

Figure 3

(a) Bethe-Salpeter equation in the particle-hole density or magnetic channels. ${\chi }_{ph}^{d/m}$ and ${\mathrm{\Gamma }}_{ph}^{d/m}$ are two-particle correlation and vertex functions, and ${\chi }_0^{ph}$ is the bare two-particle correlation function. (b) Parquet equation for the particle-particle singlet vertex, ${\mathrm{\Gamma }}_{pp}^s(P,P^{\prime },Q)$. It is decomposed into fully irreducible vertex function ${\mathrm{\Lambda }}_{pp}^s$ and cross-channel contributions from particle-hole density or magnetic vertex ladders ${\mathrm{\Phi }}_{ph}^{d/m}={\mathrm{\Gamma }}_{ph}^{d/m}\star {\chi }_{ph}^{d/m}\star {\mathrm{\Gamma }}_{ph}^{d/m}$ (complete equations are given in the Supplemental Material [31]).

Figure 4

The leading eigenvalues (LEV) of ${\mathrm{\Gamma }}^{(1)}$ as a function of temperature in various instability channels, for an electron-doped case ($n=5.2$) with a small Hund’s coupling ($J=0.2t$) and $U=12t$, $U^{\prime }=11.6t$. The upper (lower) inset shows the $d$-wave symmetry of the leading eigenvector in the singlet pairing channel for ${\mathrm{\Gamma }}^{(1)}$ (${\mathrm{\Gamma }}^{(2)}$).

Figure 5

The leading eigenvalues (LEV) of ${\mathrm{\Gamma }}^{(1)}$ as a function of temperature in various instability channels, for a hole-doped case ($n=4.9$) with a large Hund’s coupling ($J=2t$) and $U=12t$, $U^{\prime }=8t$. The upper(lower) inset shows the $p_x^{\prime }=-p_x-p_y$ symmetry of the leading eigenvector in the triplet channel for ${\mathrm{\Gamma }}^{(1)}({\mathrm{\Gamma }}^{(2)})$. The other degenerate $p$-wave component, $p_y^{\prime }=-p_x+p_y$, is not shown.