Ferroelectricity-induced multiorbital odd-frequency superconductivity in $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$

We demonstrate that $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ can be a platform for observing the bulk odd-frequency superconducting state owing to its multiorbital/multiband nature. We consider a three-orbital tight-binding model for $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ in the vicinity of a ferroelectric critical point. Assuming an intraorbital spin-singlet $s$-wave superconducting order parameter, it is shown that the odd-frequency pair correlations are generated due to the intrinsic $LS$ coupling which leads to local orbital mixing. Furthermore, we show the existence of additional odd-frequency pair correlations in the ferroelectric phase, which is induced by an odd-parity orbital hybridization term proportional to the ferroelectric order parameter. We also perform a group theoretical classification of the odd-frequency pair amplitudes based on the fermionic and space group symmetries of the system. The classification table enables us to predict the dominant components of the odd-frequency pair correlations based on the symmetry of the normal state Hamiltonian that we take into account. Furthermore, we show that experimental signatures of odd-parity orbital hybridization, which is an essential ingredient for ferroelectricity-induced odd-frequency pair correlations, can be observed in the spectral functions and density of states.

Figure 1

$\mathit{k}$ dependence of the even-parity odd-frequency pair amplitudes (a) $\mathrm{Im}{\mathcal{F}}_{12}^{-+--}\left(k\right)$, (b) $\mathrm{Re}{\mathcal{F}}_{12,z}^{+++-}\left(k\right)$, and (c) $\mathrm{Re}{\mathcal{F}}_{13,y}^{+++-}\left(k\right)$ at $k_z=0$ in the PE phase. The Matsubara frequency ${\omega }_m$ is set to be 1 meV, and the values of the pair amplitudes are normalized by ${\mathcal{F}}_{\max }^{\mathrm{BCS}}\left({\omega }_m\right)=0.237808{\mathrm{meV}}^{-1}$.

Figure 2

$\mathit{k}$ dependence of the odd-parity odd-frequency pair amplitudes (a) $\mathrm{Re}{\mathcal{F}}_{13}^{--+-}\left(k\right)$, (b) $\mathrm{Im}{\mathcal{F}}_{13,z}^{+---}\left(k\right)$, and (c) $\mathrm{Im}{\mathcal{F}}_{12,x}^{+---}\left(k\right)$ at $k_z=0$ in the FE phase. The Matsubara frequency ${\omega }_m$ is set to be 1 meV, and the values of the pair amplitudes are normalized by ${\mathcal{F}}_{\max }^{\mathrm{BCS}}\left({\omega }_m\right)=0.251432{\mathrm{meV}}^{-1}$.

Figure 3

$\mathit{k}$ dependence of the even-parity odd-frequency pair amplitudes (a) $\mathrm{Re}{\mathcal{F}}_{12}^{-+--}\left(k\right)$, (b) $\mathrm{Im}{\mathcal{F}}_{11,z}^{+++-}\left(k\right)$, (c) $\mathrm{Im}{\mathcal{F}}_{33,z}^{+++-}\left(k\right)$, (d) $\mathrm{Im}{\mathcal{F}}_{12,z}^{+++-}\left(k\right)$, (e) $\mathrm{Im}{\mathcal{F}}_{13,x}^{+++-}\left(k\right)$, (f) $\mathrm{Im}{\mathcal{F}}_{13,y}^{+++-}\left(k\right)$ at $k_z=0$ in the FE phase. The Matsubara frequency ${\omega }_m$ is set to be 1 meV, and the values of the pair amplitudes are normalized by ${\mathcal{F}}_{\max }^{\mathrm{BCS}}\left({\omega }_m\right)=0.251432{\mathrm{meV}}^{-1}$.

Figure 4

Orbital-resolved spectral functions (a) ${\mathcal{A}}_1(\mathit{k},\omega )+{\mathcal{A}}_2(\mathit{k},\omega )$ in the PE phase, (b) ${\mathcal{A}}_1(\mathit{k},\omega )+{\mathcal{A}}_2(\mathit{k},\omega )$ in the FE phase, (c) ${\mathcal{A}}_3(\mathit{k},\omega )$ in the PE phase, and (d) ${\mathcal{A}}_3(\mathit{k},\omega )$ in the FE phase. $\mathrm{\Gamma }, X$, and $M$ refer to $\mathit{k}=(0,0,0), \mathit{k}=(\pi ,0,0)$, and $\mathit{k}=(\pi ,\pi ,0)$, respectively.

Figure 5

The total DOS $\mathcal{N}\left(\omega \right)$ and orbital-resolved DOS ${\mathcal{N}}_l\left(\omega \right)$ in the (a) PE phase and (b) FE phase.

Figure 6

$\mathit{k}$ dependence of the superconducting gap on the Fermi surfaces. (a) $|{\mathrm{\Delta }}_{11}^{-+}\left(\mathit{k}\right)|$ [$=|{\mathrm{\Delta }}_{11}^{+-}\left(\mathit{k}\right)|$] in the PE phase and (b) $|{\mathrm{\Delta }}_{11}^{+}\left(\mathit{k}\right)|$ and $|{\mathrm{\Delta }}_{11}^{-}\left(\mathit{k}\right)|$ in the FE phase.

Figure 7

$\gamma$ dependence of the splitting width of the coherence peaks estimated as $\delta ={\max }_{{\mathit{k}}_{\mathrm{F}}}|{\mathrm{\Delta }}_{11}^{+}\left({\mathit{k}}_{\mathrm{F}}\right)|-{\max }_{{\mathit{k}}_{\mathrm{F}}}\left|{\mathrm{\Delta }}_{11}^{-}\left({\mathit{k}}_{\mathrm{F}}\right)\right|$. The carrier density and superconducting order parameters are set to be $n=5.0\times {10}^{-5}, {\mathrm{\Delta }}_{1,2}=0.00277$, and ${\mathrm{\Delta }}_3=0.00138$, respectively. In the gray area ($\gamma >0.105t_1\simeq 29.1$ meV), the superconducting gap ${\mathrm{\Delta }}_{11}^{+}\left({\mathit{k}}_{\mathrm{F}}\right)$ is not defined because of the Lifshitz transition, which is associated with the disappearance of the inner Fermi surface.