Superconductivity and pair density waves from nearest-neighbor interactions in frustrated lattice geometries

TL;DR

The paper addresses how nearest-neighbor pairing can drive superconductivity and PDWs in frustrated lattices, highlighting that both DOS and the quantum geometry of Bloch states crucially shape pairing. It introduces a general pairing susceptibility χ that couples DOS and Bloch-state form factors, and computes the superfluid weight D_s to determine the Berezinskii-Kosterlitz-Thouless temperature, ensuring phase stiffness is accounted for beyond mean-field pairing. Applying the framework to Lieb (flat bipartite) and kagome (flat and van Hove) lattices, it finds that NN pairing is suppressed on flat bands in bipartite geometries but can yield PDWs in Lieb, while kagome features NN pairing on the flat band and PDWs at vHs, though the latter can have vanishing stiffness due to Fermi-surface geometry; at kagome vHs, χ may predict PDW but D_s → 0, underscoring the need to assess phase stiffness. Overall, the results demonstrate that high DOS alone does not guarantee robust superconductivity; the orbital structure (quantum geometry) and multiband contributions to the superfluid weight decisively determine PDW stability and observable T_BKT.

Abstract

We consider superconductivity and pair density waves (PDWs) arising from off-site pairing in frustrated lattice geometries. We express the pair susceptibility in a generic form that highlights the importance of both the density of states, and the quantum geometry of the eigenstates and calculate the superfluid weight (stiffness) as well as the Berezinskii-Kosterlitz-Thouless (BKT) temperature. Paradigmatic bipartite (Lieb) and non-bipartite (kagome) lattices are studied as examples. For bipartite lattices, nearest-neighbor pairing vanishes in a flat band. In the Lieb lattice flat band, we find a PDW at a finite interaction and show that its pair wave vector is determined by the quantum geometry of the band. In the kagome flat band, nearest-neighbor pairing is possible for infinitesimal interactions. At the kagome van Hove singularity, the pair susceptibility predicts a PDW due to sublattice interference, however, we find that its stiffness is zero due to the shape of the Fermi surface. Our results indicate that nearest-neighbor pairing at flat band and van Hove singularities is strongly influenced by the geometric properties of the eigenfunctions, and it is crucial to determine the superfluid weight of the superconducting and PDW orders as it may contradict the predictions by pairing susceptibility.

Figures (11)

• Figure 1: The Lieb lattice structure (a) and dispersion (b) showing a flat band and one of the two dispersive bands. The color indicates the orbital composition of each state. The orange and red circles highlight regions of large and small gradients in orbital composition near the flat band center and corners, respectively. Only the dispersive band with negative energies (bandwidth $2\sqrt{2}$) is shown; there is another, mirror image of it with positive energies (App. \ref{['sec: supp_lieb_numerical_results']} Fig. \ref{['fig:suppl_lieb_results']}(e)). (c) The critical interaction strength $J_c$ as a function of the wave vector $2\bm{q}$ with $T = 10^{-6}$. The minimum is found at the $\bm{M}$-point (with all four $\bm{M}$-points being equivalent), implying the formation of a PDW state. (d) The spatial structure of the PDW order parameter. A PDW with $2\bm{q} = \bm{M}$ preserves the $C_{4v}$ symmetry of the lattice, and this order parameter structure belongs to the irreducible representation (irrep) $B_1$, when the origin is placed on one of the central $A$ orbitals.
• Figure 2: The structure (a) and dispersion (b) of the kagome lattice. The color indicates the orbital composition of each state. (c) A phase diagram for the flat band with two uniform BCS phases ($\bm{q} = 0$, one with $s^*$-wave and another with $d$-wave pairing symmetry) and two different PDW phases ($2 \bm{q} = \bm{M}, \bm{K})$. The black dashed line shows the BKT temperature.
• Figure 3: (a) The Fermi surface of the kagome lattice at $\mu = 0$ is a hexagon, with the vHs's at the corner points $\pm \bm{M}_\alpha$. The different $\bm{M}$-points are completely localized to orbital $\alpha$, shown in color. For a particle at $-\bm{M}_B$, typical $\bm{q} = 0$ Cooper pairing with $\bm{M}_B$ is impossible with nearest-neighbor interactions due to localization to the same orbital. When $\bm{q} = 2 \bm{M}_C$, pairs can instead be formed between $-\bm{M}_B$ and $\bm{M}_A$. (b) The critical interaction strength $J_c$ as a function of the PDW wave vector $2\bm{q}$ with $T = 10^{-6}$. The minimum is found at the $\bm{M}$-point due to sublattice interference. Only one-sixth of the Brillouin zone is shown; the other regions are equivalent due to symmetry. (c) A phase diagram for the vHs. Here we see mainly a single PDW state with $2\bm{q} = \bm{M}$, with an eventual transition to $2\bm{q} = \bm{K}$ for very strong interactions. At $T=0$, the $2\bm{q} = \bm{M}$ state exists for any positive $J$, but its critical temperature is very small for $J \lesssim 1.5$. The BKT temperature is zero, or at least vanishingly small throughout the $2\bm{q} = \bm{M}$ phase. The $2\bm{q} = \bm{M}$ phase breaks the $C_{6v}$ symmetry of the lattice to $C_{2v}$, and the solution belongs to irrep $A_1$. For the $2\bm{q} = \bm{K}$ phase, the symmetry is broken to $C_{3v}$, and the solution belongs to the two-dimensional irrep $E$. (d) The real-space structure of the $2\bm{q} = \bm{M}_C$ PDW state, in which case pairing occurs between orbitals $A$ and $B$. The other permutations are equivalent.
• Figure 4: BKT temperatures for the Lieb and kagome lattices with both on-site (for which $J$ should be understood as the Hubbard $U$) and nearest-neighbor (NN) interactions, and with the chemical potential set to the flat band (FB) and the vHs. The inset shows the flatness of the grand potential $\Omega$ as a function of $2\bm{q}$ near its minimum at $\bm{M}$ for the kagome vHs with NN interactions for $J = 2.5$ and $T = 10^{-6}$, indicating a vanishing superfluid weight and thus $T_{\text{BKT}}$ (the dashed orange line).
• Figure 5: Phase diagrams for the Lieb lattice flat band (a) and vHs (b). The black dashed line indicates the BKT temperature. For the Lieb lattice, the $2\bm{q} = \bm{M}$ and $\bm{q} = 0$ are the only superconducting phases we find for the FB and vHs, respectively. As discussed in the main text, the $2\bm{q} = \bm{M}$ phase preserves the $C_{4v}$ symmetry of the lattice, and the solution belongs to irrep $B_1$. On the other hand, the $\bm{q} = 0$ phase at the vHs has $s^*$-wave symmetry, i.e. it is fully homogeneous in real space and belongs to irrep $A_1$. In (a), the finite critical interaction strength of $J_c \approx 2.9$ for the flat band is also visible. (c) and (d) show how the values of the grand potential $\Omega$ vary as a function of the PDW wave vector $2\bm{q}$. (e) The full dispersion relation of the Lieb lattice. The colors indicate orbital composition of the Bloch states (see Fig. \ref{['fig:lieb_results']} in the main text for the naming convention).