Efficient Slave-Boson Approach for Multiorbital Two-Particle Response Functions and Superconductivity

We develop an efficient approach for computing two-particle response functions and interaction vertices for multiorbital strongly correlated systems based on the rotationally invariant slave-boson framework. The method is applied to the degenerate three-orbital Hubbard-Kanamori model for investigating the origin of the $s$-wave orbital antisymmetric spin-triplet superconductivity in Hund’s metal regime, previously found in the dynamical mean-field theory studies. By computing the pairing interaction considering the particle-particle and the particle-hole scattering channels, we identify the mechanism leading to the pairing instability around Hund’s metal crossover arises from the particle-particle channel, which contains the local electron pair fluctuation between different particle-number sectors of the atomic Hilbert space. On the other hand, the particle-hole spin fluctuations induce the $s$-wave pairing instability before entering Hund’s regime. Our approach paves the way for investigating the pairing mechanism in realistic correlated materials.

Popular Summary

Two-particle response functions are essential for describing how electrons pair up in unconventional superconductors. However, the methodologies for calculating these response functions for strongly correlated materials are usually computationally intensive, hindering applications to material design. Here, we develop an efficient theoretical approach for studying the two-particle response functions in strongly correlated systems, allowing for the investigation of arbitrary phase transitions induced from local electronic correlation effects.

Our method is based on the rotationally invariant slave-boson framework, which is an efficient and reliable approach to study the electronic correlation effects in materials. We extend the method to compute two-particle response functions and apply our approach to investigate the origin of an interesting type of superconducting pairing, emerging from a bad metallic state due to strong electronic interactions.

By computing the pairing interaction between electrons, including all the scattering channels, we show that the superconducting pairing is mediated by the fluctuations (creation and annihilation) of the local electron pairs, which breaks the conservation of the local particle number. We also show that the resulting superconducting phase diagram is in good qualitative agreement with more sophisticated dynamical mean-field theory.

Our approach can be combined with ab initio electronic structure methods, such as density-functional theory, paving the way for investigating novel superconductivity and phase transitions in realistic correlated materials.

Figure 1

Schematic representation of the block-diagonalized fluctuation matrix in the charge, spin, orbital, spin-orbital, and pairing sectors for the three-orbital degenerate Hubbard-Kanamori model [see Eqs. (4) and (5)].

Figure 2

(a) Diagrammatic representation of the susceptibility [Eq. (59)]. The thick solid line indicates the Nambu fermionic propagator. The gray circle corresponds to the fluctuation basis ${\overline{\mathbf{h}}}_s$, and the gray square corresponds to the quasiparticle interaction vertex ${\tilde{\mathrm{\Gamma }}}_{\alpha \beta \gamma \delta }^s$. (b) Diagrammatic representation of the quasiparticle interaction vertex ${\tilde{\mathrm{\Gamma }}}_{\alpha \beta \gamma \delta }^s$ [Eq. (64)]. The double wavy line corresponds to the dressed bosonic propagator containing the infinite summation of the particle-particle or the particle-hole fermionic bubbles. The black circle denotes the three-leg vertices ${\tilde{\mathrm{\Lambda }}}_{\alpha \beta \mu }$ (see main text for details).

Figure 3

(a) Pairing vertex from the local particle-particle fluctuation [Eq. (70)]. (b) Pairing vertex from the particle-hole fluctuations [Eq. (74)]. The bubbles are summed to the infinite order. The solid line with arrows corresponds to the normal fermionic propagator. The wavy line corresponds to the bare bosonic propagator.

Figure 4

(a) Density plot of the $s$-wave spin-triplet superconducting order parameter $⟨{\widehat{\mathcal{O}}}_P⟩$ as a function of electron filling $n$ and Coulomb interaction $U$ with $J=U/4$ at $T=0.0005t$. The cyan line is the phase boundary determined from the instability in the pairing susceptibility ${\chi }_{\mathrm{P}}$. (b) Uniform pairing susceptibility ${\chi }_{\mathrm{P}}$ for $n=2.8$, 2.4, 2.0, 1.6. (c) Spin-triplet superconducting order parameters $⟨{\widehat{\mathcal{O}}}_P⟩$ for $n=2.8$, 2.4, 2.0, 1.6. (d) Quasiparticle weight $Z$ for $n=2.8$, 2.4, 2.0, 1.6.

Figure 5

(a) Density plot of the $s$-wave spin-triplet superconducting order parameter $⟨{\widehat{\mathcal{O}}}_P⟩$ as a function of electron filling $n$ and temperature $T$ at $U=8$ and $J=U/4$. The cyan line is the phase boundary determined from the instability of the pairing susceptibility ${\chi }_{\mathrm{P}}$. (b) Uniform pairing susceptibility ${\chi }_{\mathrm{P}}$ for $n=2.8$, 2.4, 2.0. (c) Density plot of the spin-triplet superconducting order parameter $⟨{\widehat{\mathcal{O}}}_P⟩$ as a function of Coulomb interaction $U$ and temperature $T$ with $n=2.7$ and $J=U/4$. The cyan line is the phase boundary determined from the instability of the pairing susceptibility ${\chi }_{\mathrm{P}}$. (d) Uniform pairing susceptibility ${\chi }_{\mathrm{P}}$ for $U=5t,16t,12t$.

Figure 6

The Landau parameters in the (a) charge, (b) spin, (c) orbital, (d), spin-orbital, (e) orbital*, and (f) spin-orbital* channel defined in Eq. (5) as a function of coulomb interaction $U$ and $J=U/4$ for filling $n=3.0$, 2.8, 2.6, 2.4, 2.2, 2.0, 1.6 and $T=0.0005t$.

Figure 7

(a) Irreducible particle-particle $s$-wave spin-triplet pairing vertex ${\mathrm{\Gamma }}_{\mathrm{pp}}^{\mathrm{irr}}{\chi }_{O_P}^{(0)}$ as a function of Coulomb interaction $U$ and $J=U/4$ for filling $n=2.8$, 2.6, 2.4, 2.2, 2.0, 1.8, 1.6 and temperature $T=0.0005t$. (b) Irreducible particle-hole $s$-wave spin-triplet pairing vertex ${\mathrm{\Gamma }}_{\mathrm{ph}}^{\mathrm{irr}}{\chi }_{O_P}^{(0)}$ with the same parameter settings. The vertical dashed lines indicate the critical $U_c$ determined from ${\mathrm{\Gamma }}_{\mathrm{pp}}^{\mathrm{irr}}{\chi }_0^P=-1$, signalizing the divergence of the superconducting susceptibility and the scattering amplitude.

Figure 8

Diagrammatic representation of the Dyson equation in Eq. (f5). The double wavy line and the wavy line denote the dressed bosonic propagator $\mathcal{D}$ and the bare bosonic propagator ${\mathcal{D}}_0$. The solid line denotes the Nambu propagator $\mathbf{G}$. The circle denotes the three-leg vertices $\tilde{\mathrm{\Lambda }}$.

Figure 9

Comparison of the pairing susceptibilities ${\chi }^P$ computed from Eq. (42) (solid line, without “quasiparticle constraint”) and Eq. (56) (filled circles, with “quasiparticle constraint”) for (a) $T=0.0005$ and $n=2.99$, 2.8, 2.4, 2.0, 1.6 as a function of $U$ and $J=U/4$, and (b) $n=2.0$ and $U=6$, 8, 10, 12 as a function of $T$ and $J=U/4$.

Figure 10

Bare pairing interaction in the particle-hole channel ${\mathrm{\Gamma }}_{\mathrm{ph}}^{\mathrm{bare}}$ [Eq. (k1)] as a function of Coulomb interaction $U$ and $J=U/4$ for filling $n=2.8$, 2.6, 2.4, 2.2, 2.0, 1.8, 1.6 at temperature $T=0.0005t$.

Figure 11

Comparison of the uniform pairing susceptibility ${\chi }^{\mathrm{P}}(\mathbf{q}=0,\omega =0)$ evaluated from the fluctuation approach with the pairing susceptibility evaluated from the mean-field solution ${\chi }^{\mathrm{P}}=(d⟨{\mathcal{O}}_{\mathrm{P}}⟩/d\zeta )$ with small pairing field $\zeta ={10}^{-5}$ for (a) temperature $T=0.0005t$ and filling $n=2.0$ and (b) Coulomb interaction $U=8t$ and filling $n=2.0$. We fix Hund’s coupling interaction at $J=U/4$.

Figure 12

Kinetic energy $\mathrm{\Delta }{E}_{\mathrm{k}}$, potential energy $\mathrm{\Delta }{E}_{\mathrm{pot}}$, and total energy gain $\mathrm{\Delta }{E}_{\mathrm{tot}}$ for the superconducting paring state (a) as a function of electron filling $n$ at $U=8t$ and $T={10}^{-4}t$ and (b) as a function of Coulomb interaction $U$ and $J=U/4$ at $n=2.7$ and $T={10}^{-4}t$.