Obstructed Cooper Pairs in Line-Graph Lattices

TL;DR

This work addresses the conventional view that strong coupling suppresses Cooper-pair mobility by showing a disorder-free mechanism for interaction-driven localization on line-graph lattices. By starting from an attractive Hubbard model on the checkerboard line-graph and deriving a strong-coupling bosonic theory, the authors identify obstructed Cooper pairs whose motion is frustrated by destructive interference, causing the leading-order superfluid stiffness to vanish and yielding a flat, localized pair band. They uncover two complementary routes to localization: at low density, compact localized states generate extensive ground-state degeneracy and zero stiffness; at quarter filling, the system maps exactly to a quantum dimer model at the Rokhsar–Kivelson point, realizing a d-wave RVB spin liquid with topological degeneracy and deconfined holons upon doping. The results reveal a novel, disorder-free mechanism for localization driven by strong pairing and lattice geometry, with potential experimental signatures in edge-weight imaging and two-particle spectroscopy, and broader implications for unconventional pairing in flat-band systems.

Abstract

Doping a Mott insulator is widely expected to produce mobile Cooper pairs whose kinetic energy sets the superfluid stiffness. Here we demonstrate a striking counterexample. We show that when doped charges propagate on the line-graph of a lattice with strong pairing interaction, they bind into obstructed Cooper pairs whose motion is frustrated by destructive interference. As a result, the leading-order pair kinetic energy vanishes identically in the strong-coupling expansion, producing a flat bosonic band of compact localised pair states and a vanishing superfluid stiffness at leading order. The low-energy Hilbert space is dominated by these localised pairs, yielding an extensively degenerate ground-state manifold and a superfluid stiffness that vanishes anomalously as the third inverse power of the pairing strength. At quarter filling, the frustrated pair dynamics maps onto a quantum dimer model at the Rokhsar-Kivelson point, realizing a d-wave resonating-valence-bond spin liquid with topological ground-state degeneracy and deconfined holon excitations. Our results establish a disorder-free mechanism for interaction-driven localization, in which strong pairing collapses the kinetic energy of Cooper pairs.

Figures (10)

• Figure 1: Sketch of superconducting $T_c$ and superfluid stiffness in attractive Hubbard models on line-graphs: When pairing interactions are much stronger than the narrow bandwidth of a flat band, $T_c$ no longer follows the mean-field estimate of the pair-binding energy $T_c\sim \exp(-1/N_0 U)$ where $N_0$ is the large density of states at the Fermi level. Instead, $T_c$ follows the superfluid stiffness which is linear in the Hubbard attraction when its strength is intermediate between the narrow bandwidth and the full bandwidth. When the strength of attraction significantly exceeds the full bandwidth, conventional wisdom predicts a vanishing superfluid stiffness of $\mathcal{O}(U^{-1})$ coming from a second-order pair-hopping process. For some unconventional pairs on line-graphs, dubbed obstructed pairs, this leading order contribution is zero and the asymptotic strong-pairing stiffness of a uniform condensate instead scales as $\mathcal{O}(U^{-3})$. This implies that such unconventional superconductivity on a line-graph would have a stiffness and critical temperature orders of magnitude lower than other pairs whose hopping is not obstructed.
• Figure 2: Three distinct regimes of a model of hard-core bosons where vanishing superfluid stiffness is established by different techniques. At low-density, compact localised bosonic states lead to a ground state degeneracy. The Heisenberg limit of equal hopping and repulsion emerges from the strong-coupling expansion of a pairing Hamiltonian which has no leading-order stiffness. Finally, at quarter-filling in the Ising limit, the model maps to a quantum dimer model at the exactly solvable Rokhsar-Kivelson point whose ground state is a d-wave resonating valence bond spin liquid.
• Figure 3: Anomalous strong-pairing asymptote of mean-field superfluid stiffness: Stiffness of a uniform ($q=0$) condensate for sublattice-symmetric (s-wave) and antisymmetric (d-wave) form factors on the checkerboard lattice for fixed density $n=1.95$ for (a) $\Delta<2$ (b) Strong pairing asymptote ($\Delta=2-10000$) of the superfluid stiffness showing the conventional $1/\Delta$ scaling for s-wave pairing, and a faster $1/\Delta^{3}$ scaling for obstructed (d-wave) pairs. The absence of stiffness at the leading order numerically supports the central result of this work. ($\beta=10000,t=1$ on a 40x40 lattice)
• Figure 4: Pair-hopping integrals (a), band structure (b) of the pair-Hamiltonian $H_{\rm hop}$ in \ref{['eq:boson']} with $W_b=0.05t_{b}$. Also shown in (a) is a compacted localised state (CLS) on the lower left plaquette that is prevented from hopping to the neighbouring plaquette by destructive interference in the limit $W_b\to 0$. The color in (b) indicates the orbital content of the band eigenfunctions, which winds twice around the band-touching point at $M$.
• Figure 5: Many-body ground state degeneracies in strong-coupling effective Hamiltonian: Eigenvalues of $m$-boson eigenstates from exact diagonalization of \ref{['eq:boson']} for $t_b=1,\mu=0$ on a $5\times 5$ supercell with periodic boundary conditions. The dashed vertical lines indicate the expected fraction of degenerate eigenstates $(N+1)/(2N)$ for $m=1$ and $(N-1)(N-6)/(2N(2N-1))$ for $m=2$. Note that the eigenvalues are shifted by a constant value $3(N-m)t_b/2$ and then normalized by $m$ to highlight the degeneracy from CLS with energy $-2t_b$ each.