Domes of $T_c$ in single-band and multiband superconductors with finite-range attractive interactions

The rise and fall of the superconducting transition temperature $T_c$ upon tuning carrier density or external parameters, such as pressure or magnetic field, is ubiquitously observed in a wide range of quantum materials. In order to investigate such domes of $T_c$, we go beyond the prototypical attractive Hubbard model and consider a lattice model of electrons coupled via instantaneous, spatially extended, attractive interactions. By numerically solving the mean-field equations, as well as going beyond mean field theory using a functional renormalization group approach, we find that for a characteristic interaction range $\ell$, there exists a dome in $T_c$ around $k_F\ell \sim \mathcal{O}\left(1\right)$. For multiband systems, our mean field theory shows the presence of additional domes in the vicinity of Lifshitz transitions. Our results hold in both two and three dimensions and can be intuitively understood from the geometric relation between the Fermi surface and the interaction range. Our model may be relevant for domes of $T_c$ in dilute weakly coupled superconductors or in engineered cold atom systems.

Figure 1

(a) Schematic picture showing a reference momentum point on different FSs of increasing sizes (blue, orange, green). Thick lines indicate the geometrically accessible parts of the FS within the scattering circle of radius $1/\ell$. (b) Same as previous panel, but for larger Fermi surfaces (blue) where Umklapp scattering becomes relevant near the BZ boundary. (c) Arc length $\mathcal{R}\left(\overline{n}\right)$ of the accessible part of the FS as a function of electron density $\overline{n}$, plotted for $\ell =1$. Inset: $\mathcal{R}$ at low density showing a scaling $\sim \sqrt{\overline{n}}$. (d) $T_c$ as a function of $\overline{n}$ for a 2D square lattice for a fixed interaction range $\ell =5$, showing peaks at the densities $\overline{n}\approx 0.04,1,1.96$ (see text for details).

Figure 2

Value of $k_F\ell$ at the geometric peak of $T_c$ as a function of coupling $g_0$ in (a) 2D and (b) and 3D. Extrapolation (dashed lines) to the weak-coupling limit $g_0\to 0$ suggests that $k_F^{★}\ell \to 0.20$ in 2D and $k_F^{★}\ell \to 0.86$ in 3D. In 3D, $k_F^{★}\ell$ is larger than in 2D, indicating that the dome of $T_c$ is shifted towards lower densities, whereas the peak value of $T_c$ remains comparable in both dimensions. This is in part due to the difference in density of states at low filling.

Figure 3

Zero temperature solution $\left|\mathrm{\Delta }\right(\mathbf{k}\left)\right|$ (in units of $t_1=1$) to the nonlinear gap equation at different densities. (a) In the electron dilute limit ($\overline{n}=0.04$), the gap is peaked at the $\mathrm{\Gamma }$ point while (b) at intermediate densities ($\overline{n}=0.6$) it peaks around the FS. (c) At half-filling, the gap reaches its maximum at the van Hove points ($\overline{n}=1$) and (d) in the dilute hole regime ($\overline{n}=1.96$) the maximum is at the M points.

Figure 4

Superconducting transition temperature $T_c$ for a two-orbital model showing a double geometric peak in the low-density regime. Colored stars correspond to the FSs in the insets. Vertical dashed line marks the Lifshitz transition when the second band appears at the Fermi level. (a) Two hybridized elliptical FSs (b) two circular FSs with a finite potential difference (black curve). For comparison, we also show the result from a single orbital model (red curve) which reproduces the first peak in $T_c$.

Figure 5

Effective interaction in the patching approximation. Normalized color code shows the value of the flowing interaction vertex $u_T(n_1,n_2,1)$, where $(n_1,n_2)$ enumerate momentum patches around the FS. (a) Density $\overline{n}=0.17$ at $T=T_{\max }$, (b) $\overline{n}=0.17$ at $T=T_{\min }$, (c) $\overline{n}=0.94$ at $T=T_{\min }$.

Figure 6

Characteristic temperature $T_{\min }$ for the Gaussian interaction potential with $\ell =1$, as determined from FRG calculations which include only particle-particle scattering (blue curve) or all interaction channels (orange curve). Curves plotted in opaque colors are computed using $N_p=96$ momentum patches, curves in lighter colors are for $N_p=72$ and $N_p=48$. In regimes I and II, the effective vertex at $T_{\min }$ captures superconducting pairing, while in regime III the interaction becomes increasingly localized in momentum space leading to a breakdown of SC. Inset shows the geometric dome of $T_c$ at low density.

Figure 7

Eigenvectors of $\mathcal{M}$ with their corresponding eigenvalues at $T=T_c\approx 0.164$ for $(t_1,t_2,g_0,\ell ,\overline{n})=(1,1,1,5,0.36)$. The Fermi surface is shown in black, while the color map goes from blue (negative) to red (positive), with white being zero intensity. The largest eigenvalue corresponds to lowest energy state.

Figure 8

Critical temperature $T_c$ as a function of electron density $\overline{n}$ for (a): $\ell =3$ (blue), $\ell =5$ (orange), and $\ell =10$ (green). The dashed vertical line at $\overline{n}=1$ shows, in contrast to the geometric peak, that the location of the van Hove peak doesn't depend on $\ell$. In (b), we observe the two peaks merge as the FS is made increasingly anisotropic, going from $t_2=t_1$ (blue) to $t_2=0.5t_1$ (orange) to $t_2=0.1t_1$ (green).

Figure 9

Partitioning of the Brillouin zone into a set of patches ${\mathcal{P}}_1,\cdots ,{\mathcal{P}}_N$, illustrated for $N=24$. All momentum points within a single patch ${\mathcal{P}}_m$ are projected onto the representative momentum ${\mathbf{p}}_m$ which lies on the Fermi surface.

Figure 10

Diagrammatic representation of contributions to the flow of the effective interaction. Singling out particle-particle scattering (first term) is equivalent to a Bethe-Salpeter like resummation and generates results on the level of self-consistent mean-field theory. The inclusion of additional direct particle-hole (terms two to four) and crossed particle-particle (last term) scattering allows to model the interplay of competing interaction channels.

Figure 11

Benchmark of the characteristic temperature $T_{\min }$, as determined in FRG calculations (including only particle-particle scattering channel), with the mean-field critical temperature of the Hubbard model. The Hubbard limit $\ell =0$ shows excellent agreement for fillings $\overline{n}\gtrsim {10}^{-3}$. The generalized Gaussian potential ($\ell =1$) approaches the Hubbard limit in the very dilute regime.