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 $d$-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.
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 d-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.
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.
In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Nevertheless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher-lying bands via the perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian, if a simple uniform pairing condition on the single-particle states is satisfied, and that the compressibility is diverging as a consequence of an emergent SU(2) symmetry. This symmetry is broken by second-order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensional ladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality $D\ensuremath{\ge}{\mathcal{W}}^{2}$ between the Drude weight D and the winding number W, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.
Cooper pairing instability in a Fermi liquid is well understood by the BCS theory, but pairing mechanism for doped Mott insulators still remains elusive. Previously it has been shown by density matrix renormalization group (DMRG) method that a single doped hole is always self-localized due to the quantum destructive interference of the phase string signs hidden in the t-J ladders. Here we report a DMRG investigation of hole binding in the same model, where a novel pairing-glue scheme beyond the BCS realm is discovered. Specifically, we show that, in addition to spin pairing due to superexchange interaction, the strong frustration of the phase string signs on the kinetic energy gets effectively removed by pairing the charges, which results in strong binding of two holes. By contrast, if the phase string signs are "switched off" artificially, the pairing strength diminishes significantly even if the superexchange coupling remains the same. In the latter, unpaired holes behave like coherent quasiparticles with pairing drastically weakened, whose sole origin may be attributed to the resonating-valence-bond (RVB) pairing of spins. Such non-BCS pairing mechanism is therefore beyond the RVB picture and may shed important light on the high-T(c) cuprate superconductors.
Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.
Motivated by recent experiments on correlated van der Waals materials, including twisted and rhombohedral graphene and twisted WSe 2 , we perform an analytical and numerical study of the effects of strong on-site and short-range interactions in fractionally filled ideal Chern bands. We uncover an extensive nontrivial ground state manifold within the band filling range 0 < ν < 1 and introduce a general principle, the “three-rule,” for combining flat band wave functions, which governs their zero-energy property on the torus geometry. Based on the structure of these wave functions, we develop a variational approach that reveals distinct phases under different perturbations: metallic behavior emerges from a finite dispersion, and superconductivity is induced by attractive Cooper channel interactions. Our approach, not reliant on the commonly applied mean-field approximations, provides an analytical expression for the macroscopic wave function of the off-diagonal long-range order correlator, attributing pairing susceptibility to the set of nontrivial zero-energy ground state wave functions. Extending to finite screening lengths and beyond the ideal limit using exact diagonalization simulations, we demonstrate the peculiar structure in the many-body wave function's coefficients to be imprinted in the low-energy spectrum of the topologically ordered Halperin spin-singlet state. Our findings also make connections to frustration-free models of noncommuting projector Hamiltonians, potentially aiding the future construction of exact ground states for various fractional fillings. Published by the American Physical Society 2024
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? In an ordinary dispersive band, charge stiffness is closely related to kinetic energy. In a perfectly flat band, ordinary band velocity vanishes, so superconducting or superfluid stiffness can instead depend on quantum geometry and interactions. In the quantum-geometric-nesting (QGN) class of frustration-free flat-band models, previous work established a rigorous bound relating the many-body superfluid stiffness to the inverse mass of a Cooper pair and reported numerical saturation at approximately the 0.1% level. This raises a natural question: Is perfect quantum geometric nesting by itself sufficient to make the pair-mass bound an exact equality? This work does not assume that the answer is yes and does not attempt to prove a universal saturation theorem. Instead, it calculates the finite-size static charge-twist response exactly and identifies the structural obstruction that decides whether saturation occurs. MAIN RESULT: TWO EXACT RESPONSE CHANNELS For the isolated QGN condensates studied here, the finite-size second derivative of the ground-state energy under a charge twist has the exact structure E_0^(N)''(0) = alpha C_3(N) + gamma C_2(N), where C_3(N) and C_2(N) are fixed hypergeometric filling factors. The amplitudes alpha and gamma are independent of pair number N and can be determined entirely within the one-pair sector. The second amplitude satisfies gamma >= 0, so its contribution has a fixed sign. This converts the question of saturation from a numerical observation into a structural condition: The familiar one-channel pair-mass form is saturated exactly when the second channel vanishes. The paper then determines how this second channel can arise. The spin trace first decides whether a generator contributes at all. Traceless spin structures make the relevant contribution vanish identically. For the contributing charge-type sector of the on-site construction, the remaining finite-size term is an aliasing contribution associated with nonzero lattice Fourier or Poisson modes rather than the continuum mode. EXPLICIT FINITE-SIZE COUNTEREXAMPLE The paper constructs an explicit admissible QGN model with a numerically certified unique gapped ground state for which gamma / alpha = 0.846 at N_k = 4. Relative to the one-channel saturating form, the finite-size departure is approximately 42% at half filling and rises to approximately 169% at N = 3, where the reference one-channel contribution is itself small. These percentages are finite-size departures from the one-channel form. They are not claimed to be thermodynamic violations of the rigorous pair-mass bound. THERMODYNAMIC RECOVERY The same analysis also explains why near-perfect saturation may appear in larger regular systems. Under the stated joint analyticity and regularity assumptions, uniform normalization assumptions, and a uniform charge-gap condition, the finite-size aliasing contribution decays and the conjectured one-channel relation is recovered in the thermodynamic limit. The work therefore separates two logically different statements: 1. perfect QGN alone does not force exact finite-size saturation; 2. with additional regularity and gap assumptions, the finite-size obstruction disappears in the thermodynamic limit. This distinction is important because an exact identity at finite system size and a thermodynamic definition of stiffness are not interchangeable statements. ADDITIONAL STRUCTURAL RESULTS The paper derives further constraints on the pairing form factor. When the generators that actually enter the Hamiltonian act irreducibly on each flat-band fibre and the momentum graph generated by them is connected, the pairing form factor has a degenerate, momentum-independent singular-value spectrum and the condensate is unique up to an overall scale. Counterexamples to converses are given, so these properties are not claimed to follow from QGN alone. The admissible band-space bilinears are also organized by a Z2-graded real structure. In an appropriate real basis, the even sector is real symmetric and the odd sector is i times a real antisymmetric matrix, with dimensions m(m+1)/2 and m(m-1)/2, respectively. This grading clarifies which algebraic sectors can carry the finite-size obstruction. The work additionally checks that, in the translation-invariant setting considered, optimization over the relevant all-spin matrix-valued antisymmetrized-geminal-power variational family reproduces the exact finite-size curvature to second order. The additional response channel is therefore not merely an artifact of evaluating a single fixed trial vector. WHY THIS MATTERS The scientific point is not simply that one numerical conjecture has a finite-size exception. The stronger result is that the apparent equality between a collective many-body charge response and a two-particle pair-mass quantity has an identifiable structural condition. The exact two-channel decomposition explains both: - why many QGN examples may appear almost perfectly saturated, and - how a valid QGN system can depart strongly from the one-channel form at finite size. For a broader reader, the work illustrates an important problem in many-body physics: When can information obtained from a low-particle sector determine a collective many-body response? In this QGN class, the one-pair sector carries unexpectedly strong information: it determines both response amplitudes. However, an additional algebraic and finite-size aliasing condition decides whether the response collapses to the commonly assumed pair-mass form. The lesson is therefore not that few-particle information is generally insufficient, but that its sufficiency must be established structurally rather than inferred from numerical agreement. SCOPE AND NONCLAIMS The observable treated here is the static charge-twist curvature in the conventions stated in the manuscript. The work does not claim that the 42% or 169% finite-size departures survive as thermodynamic violations. The thermodynamic recovery theorem requires the regularity, normalization, and gap hypotheses stated in the paper. Finite-range or infinite-volume situations outside those hypotheses are separate questions. The surjectivity theorem in release v1.1 explicitly assumes a momentum-independent, invertible singlet nesting matrix. That hypothesis is essential and should not be omitted when quoting the theorem. The relation between the graded algebraic sectors and the complete QGN interaction class also requires the stated orbital-coverage conditions; the manuscript gives cases in which those conditions fail. VERSION CLARIFICATION Release v1.1 tightens the formal scope of the v1.0 statements. Most importantly, it explicitly adds the momentum-independence hypothesis required by the surjectivity argument and separates analytically proved statements from numerically certified properties of the explicit counterexample. The finite-size two-channel response result and the thermodynamic scope should be quoted in the v1.1 form given above. REPRODUCIBILITY The deposit includes the manuscript and a fail-closed reproducibility package containing exact or symbolic checks, numerical certificates, figure generators, deterministic reference outputs, and a frozen manifest. The explicit finite-size counterexample, response formulas, convention factors, gap diagnostics, and auxiliary algebraic claims are mapped to executable verification routines. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; quantum geometric nesting; QGN; superfluid stiffness; static charge curvature; Kohn twist; Cooper-pair mass; pair-mass bound; quantum metric; quantum geometry; exact pairing; antisymmetrized geminal power; AGP; finite-size aliasing; Poisson summation; thermodynamic limit; frustration-free Hamiltonian; Z2 grading.