Unconventional superconductivity in the extended Hubbard model: Weak-coupling renormalization group

We employ the weak-coupling renormalization group approach to study unconventional superconducting phases emerging in the extended, repulsive Hubbard model on paradigmatic two-dimensional lattices. Repulsive interactions usually lead to higher-angular momentum Cooper pairing. By considering not only longer-ranged hoppings, but also nonlocal electron-electron interactions, we are able to find superconducting solutions for all irreducible representations on the square and hexagonal lattices, including extended regions of chiral topological superconductivity. For the square, triangular and honeycomb lattices, we provide detailed superconducting phase diagrams as well as the coupling strengths which quantify the corresponding critical temperatures depending on the band-structure parameters, band filling, and interaction parameters. We discuss the sensitivity of the method with respect to the numerical resolution of the integration grid and the patching scheme. Eventually, we show how to efficiently reach a high numerical accuracy.

Figure 1

Relevant Feynman diagrams for on-site interaction in the Hubbard Hamiltonian (1), for the cases considered in this paper, when treated in the weak-coupling limit. Diagrams 1, 2a, and 2b refer to the singlet channel and diagram 2c to the triplet channel. Solid lines represent electron propagators and dashed lines interactions. For clarity, the spins are explicitly given as up and down arrows.

Figure 2

General Feynman diagram that produces an additional logarithmic divergence, independent of the behavior of ${\mathrm{\Gamma }}^{\left(n\right)}$ and ${\mathrm{\Gamma }}^{\left(m\right)}$.

Figure 3

All irreps of the group $D_4$ with their corresponding basis functions. Columns from left to right: label of the irrep, basis function, contour plot of the basis function, plot of the basis function (orange, dashed) and example form factors (blue, solid) along the Fermi surface, and plot of the same form factor on the Fermi surface. Note that the eigenvectors of $g$ may include linear combinations with higher harmonics. $x$ refers to $sin\left(x\right)$ and $x^2$ to $cos\left(x\right)$. The basis vectors that span the 2D space of $E$, i.e., $x$ and $y$, are rotated by $\pi /4$, such that the nodes and maxima of the form factors are arranged on the Fermi surface to maximize the condensation energy.

Figure 4

(a) Phase diagram $t_2/t_1$ vs band filling $n$ of the square lattice. The van Hove filling $n_{\text{vH}}$ is shown as a red dotted line. Note that there is also perfect nesting for $t_2/t_1=0$ at half filling. Due to the inherent particle-hole symmetry, it is sufficient to consider only negative values for $t_2/t_1$. (b) Effective interaction $V_{\text{eff}}$ as a function of the band filling $n$ and the second-neighbor hopping $t_2/t_1$. (c) Difference ${\delta }_{\lambda }$ of the two lowest eigenvalues corresponding to the superconducting ground state and the solution which is closest to it. Large differences indicate stable phases; the smaller the difference, the more fragile the superconducting ground state.

Figure 5

(a) Effective interaction, $V_{\text{eff}}$, of the most negative eigenvalue of the irrep $B_1$ as a function of the filling $n$ for different nearest-neighbor interaction strengths $\alpha$. The filling $n\approx 0.853$, at which the other figures are shown, is indicated as a red, dashed line. (b)–(d) Plot of the form factor corresponding to the eigenvalues plotted in (a). (e)–(g) Form factor of the pure $d$ and $i$ waves for (e) and (f), respectively, and of the linear combination of the two in (g). (h)–(j) Plot of the form factors (e)–(g) now shown on the Fermi surface.

Figure 6

Phase diagram showing the irrep of the leading superconducting instability in the square lattice for $0.05\le \alpha \le 0.15, 0.4\le n\le 1.45$ and $-0.7\le t_2/t_1\le 0$.

Figure 7

All irreps of the group $D_6$ with their corresponding basis functions with the lowest appearing angular momentum. Columns from left to right: label of the irrep, basis function, contour plot of the basis function, plot of the basis function (orange, dashed) and example form factors (blue, solid) along the Fermi surface, and plot of the form factor on the Fermi surface. The representative examples shown are taken from the triangular and honeycomb lattices. Here, $x$ and $x^2$ are again symbolic short notations for the respective $sin$ and $cos$ functions that appear in the periodic lattice system. The basis vectors that span the 2D spaces of $E_1$ and $E_2$ are rotated by $\pi /4$, such that the nodes and maxima of the form factors are arranged on the Fermi surface to maximize the condensation energy.

Figure 8

(a) Phase diagram $t_2/t_1$ vs band filling $n$ for the triangular lattice. The van Hove filling $n_{vH}$ is drawn as a red dotted line. For fillings $n<0.5$, the effective interaction $V_{\text{eff}}$ becomes very small, such that the difference between different symmetries reaches the level of numerical noise. (b) Effective interaction $V_{\text{eff}}$ as a function of the band filling $n$ and the second-neighbor hopping $t_2/t_1$. Regions between calculated points have been interpolated. (c) Difference ${\delta }_{\lambda }$ of the two lowest eigenvalues corresponding to the superconducting ground state and the solution that is closest to it.

Figure 9

Form factor of the superconducting instability on the Fermi surface in the extended Brillouin zone of the triangular lattice. The lattice parameters are $t_2/t_1=0.1, n=1.83$, and $\alpha =0$. The Brillouin zone is drawn in black and the green dashed lines are the nodal line of the $f$-wave form factor of irrep $B_1$. Blue and orange colors denote different signs of the form factor.

Figure 10

Phase diagram showing the irrep of the leading superconducting instability in the triangular lattice for $0.05\le \alpha \le 0.2, 0.7\le n\le 1.8$ and $-0.3\le t_2/t_1\le 0.3$.

Figure 11

(a) Phase diagram $t_2/t_1$ vs band filling $n$ for the honeycomb lattice. The van Hove fillings $n_{\text{vH}}$ are drawn as a red dotted line, the filling $n_D$ at which the Dirac points are located is drawn as a black dotted line. (b) Effective interaction $V_{\text{eff}}$ as a function of the band filling $n$ and the second-neighbor hopping $t_2/t_1$. Regions between calculated points have been interpolated. (c) Difference ${\delta }_{\lambda }$ of the two lowest eigenvalues corresponding to the superconducting ground state and the solution which is closest to it.

Figure 12

Phase diagram showing the irrep of the leading superconducting instability in the honeycomb lattice for $0.1\le \alpha \le 0.4, 0.5\le n\le 1.7$ and $0\le t_2/t_1\le 0.4$.

Figure 13

Effects of low accuracy of the integration grid $N_{\text{int}}$ and of the number of patching points $N_p$ on the effective interaction $V_{\text{eff}}$. The irrep with the largest $V_{\text{eff}}$ is the leading superconducting instability. The figures shown are for the square lattice with $t_2=\alpha =0$ and $n\approx 0.54$. (a) Effective interaction $V_{\text{eff}}$ of the $B_2$ and $E$ irreps as a function of $N_{\text{int}}$. (b) $V_{\text{eff}}$ as a function of $N_p$.

Figure 14

Progression of the dynamic grid after 1, 3, and 6 iterations for a starting grid size $N_0=10$. First row: square lattice with large $|\vec{k}|, t_2=0$ and $n\approx 0.54$. Second row: square lattice with the same parameters and $|\vec{k}|$ close to zero. Third row: honeycomb lattice with large $|\vec{k}|, t_2=0$, and $n\approx 1.4$. Fourth row: honeycomb lattice with the same system parameters and $|\vec{k}|$ close to zero.

Figure 15

Relative runtime, $T_n/T_0$, as a function of the number of iterations, $n$, while leaving the effective grid-size constant at $N_{\text{eff}}=1280$. $T_n$ denotes the runtime when using $n$ iterations to calculate $g$ from Eq. (41) for a fixed set of parameters in each curve.