Pairing enhanced by local orbital fluctuations in a model for monolayer FeSe

The pairing mechanism in different classes of correlated materials, including iron-based superconductors, is still under debate. For FeSe monolayers, uniform nematic fluctuations have been shown in a lattice Monte Carlo study to play a potentially important role. Here, using dynamical mean-field theory calculations for the same model system, we obtain a similar phase diagram and provide an alternative interpretation of the superconductivity in terms of local orbital fluctuations and phase rigidity. Our study clarifies the relation between the superconducting order parameter, superfluid stiffness, and orbital fluctuations, and provides a link between the spin/orbital freezing theory of unconventional superconductivity and theoretical works considering the role of nematic fluctuations.

Figure 1

(a) The two-band tight-binding model (adapted from Ref. [16]) on the square lattice, with orbitals $d_{xz}$ and $d_{yz}$ schematically shown in red and blue. The hoppings are $t_1=-1.0t, t_2=1.5t, t_3=-1.2t, t_4=-0.95t$. (b) The band structure along the indicated path in the Brillouin zone. (c) The density of states (DOS). The black dashed lines in (b) and (c) mark the chemical potentials corresponding to $n=1, n=1.56, n=2$ (half-filling), and $n=3$.

Figure 2

DMFT phase diagram of model (1) in the space of interaction $\left|U\right| (=g)$ and total filling $n$, at temperature $T=1/8$. The white region indicates the normal metallic phase and the red region the superconducting (SC) phase with order parameter $\mathrm{\Delta }\ge 0.01$. The orbital and/or sublattice symmetry breaking phases AFO, CDW, and FO are sketched by the blue, green, and yellow hashed regions, respectively. The thin dashed lines mark the same fillings as in Fig. 1.

Figure 3

Filling $n$ and interaction $\left|U\right|$ dependence of (a) the superfluid stiffness $D_S$, (b) the SC order parameter $\mathrm{\Delta }$, (c) the local orbital fluctuations $\mathrm{\Delta }{\chi }_{\mathrm{loc}}^{\mathrm{orb}}$, and (d) the effective interaction $\mathrm{Re}{\mathrm{\Sigma }}^{\mathrm{ano}}\left(i{0}^{+}\right)/\mathrm{\Delta }$ (white-red color map) for orbital symmetric phases at $T=1/8$. The thick black bars in all panels indicate the paired Mott insulating phase. The black squares in (a), circles in (b), crosses in (c), and triangles in (d) mark the corresponding peak positions at a fixed interaction $\left|U\right|$ along the axis of filling $n$. For the sake of easier comparison, we also indicate the peak positions from panel (d) in panel (c), and similarly those from panel (c) in panel (b). The blue dashed lines link the peak positions in $\mathrm{\Delta }$ and $D_S$. Due to a large error bar in determining the SC phase near $n=2$, we truncate the data in (d) with a cutoff $\mathrm{\Delta }>0.03$. The stiffness is $D_S=D_S^{xx}$ with units $e^2/{\hslash }^2$.

Figure 4

Effective interaction $|U_{\mathrm{eff}}^{\text{DMFT}}|=|\mathrm{Re}{\mathrm{\Sigma }}^{\mathrm{ano}}\left(i{0}^{+}\right)/\mathrm{\Delta }|$ as a function of bare interaction $\left|U\right|$ (points) and parabolic fits $a\left|U\right|+{b\left|U\right|}^2$ to the small-$\left|U\right|$ data (dashed lines) for indicated fillings. The black dot-dashed line plots $\left|U\right|$.

Figure 5

Filling ($n$) dependence of (a) the superfluid stiffness $D_S$, (b) the SC order parameter $\mathrm{\Delta }$, (c) the local orbital fluctuations $\mathrm{\Delta }{\chi }_{\mathrm{loc}}^{\mathrm{orb}}$, and (d) the effective interaction $\mathrm{Re}{\mathrm{\Sigma }}^{\mathrm{ano}}\left(i{0}^{+}\right)/\mathrm{\Delta }$ at a series of $\left|U\right|$ values as indicated in panel (b). The black crosses in (c) and triangles in (d) mark the corresponding peak positions. For a better comparison between the peak positions, we also indicate the peak positions from panel (d) in panel (c) by the gray triangles. Similarly, we reproduce the maxima of (c) by the gray crosses in (b) on the hole-doped side. Due to large error bars in determining the SC phase near $n=2$, we truncate the data in panel (d) with a cutoff $\mathrm{\Delta }>0.03$. The curves in panel (a) [(b,c,d)] are shifted by multiples of 0.2 [(0.02,0.125,2)] along the vertical axis for a better presentation. The thin dashed lines in panels (a,b) mark the filling $n=1.56$.

Figure 6

(a) Orbital correlation function in the normal (green) and superconducting (red) state, plotted on the imaginary-time axis. (b) The spectra of the local orbital correlation function (red, green) and the anomalous Green's function (black). The red (green) curves are for the SC (normal) phase. The filling is $\approx 2.33$ for all subpanels and the interaction strengths are indicated in panel (b).

Figure 7

Energy of the peak in $A^{\mathrm{ano}}\left(\omega \right)$ (empty triangles) and of the peak in $\mathrm{Im}{\chi }_{\mathrm{orb}}^{\mathrm{loc}}\left(\omega \right)$ for the SC phase (red solid squares) and the normal metal (NM) phase (orange empty squares) at (a) $\left|U\right|=2.5$, (b) $\left|U\right|=3.0$, (c) $\left|U\right|=3.75$, and (d) $\left|U\right|=5.5$.