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arXiv:1608.00125 · Physical review. B./Physical review. B

Superconductivity in repulsively interacting fermions on a diamond chain: Flat-band-induced pairing

Keita Kobayashi, Masahiko Okumura, Susumu Yamada, Masahiko Machida, Hideo Aoki

Published Dec 1, 2016• 77 citations
arXivHTMLPDF

Abstract

To explore whether a flat-band system can accommodate superconductivity, we consider repulsively interacting fermions on the diamond chain, a simplest possible quasi-one-dimensional system that contains a flat band. Exact diagonalization and the density-matrix renormalization group are used to show that we have a significant binding energy of a Cooper pair with a long-tailed pair-pair correlation in real space when the total band filling is slightly below 1/3, where a filled dispersive band interacts with the flat band that is empty but close to ${E}_{F}$. Pairs selectively formed across the outer sites of the diamond chain are responsible for the pairing correlation. At exactly 1/3-filling an insulating phase emerges, where the entanglement spectrum indicates the particles on the outer sites are highly entangled and topological. These come from a peculiarity of the flat band in which ``Wannier orbits'' are not orthogonalizable.

DOI
10.1103/physrevb.94.214501

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To explore whether a flat-band system can accommodate superconductivity, we consider repulsively interacting fermions on the diamond chain, a simplest possible quasi-one-dimensional system that contains a flat band. Exact diagonalization and the density-matrix renormalization group are used to show that we have a significant binding energy of a Cooper pair with a long-tailed pair-pair correlation in real space when the total band filling is slightly below 1/3, where a filled dispersive band interacts with the flat band that is empty but close to EF{E}_{F}EF​. Pairs selectively formed across the outer sites of the diamond chain are responsible for the pairing correlation. At exactly 1/3-filling an insulating phase emerges, where the entanglement spectrum indicates the particles on the outer sites are highly entangled and topological. These come from a peculiarity of the flat band in which ``Wannier orbits'' are not orthogonalizable.
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In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.
Understanding superconductivity emerging from repulsive fermions remains a major challenge in condensed matter physics. In this paper, we investigate the pairing tendencies in a one-dimensional, three component repulsive Hubbard model, using the density matrix renormalization group method. At half-filling, the system exhibits density wave ground state due to strong Hubbard repulsions. Upon doping, we find that Cooper pairs can emerge, whose fluctuations predominate the long-range physics in the system across a wide parameter range. The effective attractions between Cooper pairs are mediated by the particle-hole fluctuations in the third non-pairing component, resembling an excitonic mechanism of superconductivity. The coexistence of multiple density waves and superconductivity at different fermion fillings is explored. We also present an analytical study of the pairing mechanism in both weak and strong coupling limits. Our results provide a new perspective for understanding and exploring unconventional superconductivities in strongly correlated fermionic systems.
The physics of strongly correlated quantum particles within a flat band was originally explored as a route to itinerant ferromagnetism and, indeed, a celebrated theorem by Lieb rigorously establishes that the ground state of the repulsive Hubbard model on a bipartite lattice with an unequal number of sites in each sublattice must have nonzero spin SSS at half filling. Recently, there has been interest in Lieb geometries due to the possibility of topological insulator, nematic, and Bose-Einstein condensed (BEC) phases. In this paper, we extend the understanding of the attractive Hubbard model on the Lieb lattice by using determinant quantum Monte Carlo to study real space charge and pair correlation functions not addressed by the Lieb theorems. Specifically, our results show unusual charge and charge transfer signatures within the flat band, and a reduction in pairing order at $\ensuremath{\rho}=2/3$ and $\ensuremath{\rho}=4/3$, the points at which the flat band is first occupied and then completely filled. We compare our results to the case of flat bands in the Kagome lattice and demonstrate that the behavior observed in the two cases is rather different.
Motivated by recent experiments realizing correlated phenomena and superconductivity in two-dimensional (2D) van der Waals devices, we consider the general problem of whether correlation effects may be enhanced by modifying band structure while keeping a fixed weak interaction strength. Using determinantal quantum Monte Carlo, we study the 2D Hubbard model for two different band structures: a regular nearest-neighbor tight-binding model and a partially flat-band structure containing a nondispersing region, with identical total noninteracting bandwidth Wtot{W}_{\mathrm{tot}}Wtot​. For both repulsive and attractive weak interactions ($|U|\ensuremath{\ll}{W}_{\mathrm{tot}}$), correlated phenomena are significantly stronger in the partially flat model. In the repulsive case, even with UUU being an order of magnitude smaller than Wtot{W}_{\mathrm{tot}}Wtot​, we find the presence of a Mott insulating state near half filling of the flat region in momentum space. In the attractive case, where generically the ground state is superconducting, the partially flat model exhibits significantly enhanced superconducting transition temperatures. These results suggest the possibility of engineering correlation effects in materials by tuning the noninteracting electronic dispersion.
Superconductivity in a partially filled flat band presents a vexing conceptual hurdle because the absence of a Fermi surface precludes a weak-coupling regime where one can extend insights from the Bardeen-Cooper-Schrieffer picture of a Fermi surface instability. We approach the strongly correlated problem of flat band superconductivity from the strong coupling limit of local attractive interactions on line-graph lattices, whose non-interacting bandstructures host exactly flat bands due to frustrated hopping. In this limit, the pair kinetic energy which sets the superfluid stiffness is expected to scale inversely with the pair binding interaction. Here we demonstrate a striking counterexample. We show that when doped charges propagate on the line-graph of a lattice with strong pairing interaction and broken time-reversal symmetry, 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, zero superfluid stiffness at leading order, and an extensively degenerate many-body ground state manifold. At quarter filling, the frustrated pair dynamics maps onto a quantum dimer model which has a ddd-wave resonating-valence-bond ground state when time-reversal is broken. The pairing Hamiltonian in this limit thus has a topologically ordered spin liquid ground state which becomes exact at the analytically solvable Rokhsar-Kivelson point with long-range entanglement and deconfined holon excitations. Interestingly, we find exact compact localised eigenstates and extensive degeneracies in the many-body eigenstates of this emergent dimer model. Our results establish a disorder-free mechanism for interaction-driven localisation, in which strong pairing collapses the kinetic energy of Cooper pairs.