Preprint and reproducibility archive; not peer reviewed.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.
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.
We introduce the concept of “quantum geometric nesting” (QGN) to characterize the idealized ordering tendencies of certain flat-band systems implicit in the geometric structure of the flat-band subspace. Perfect QGN implies the existence of an infinite class of local interactions that can be explicitly constructed and give rise to solvable ground states with various forms of possible fermion bilinear order, including flavor ferromagnetism, density waves, and superconductivity. For the ideal Hamiltonians constructed in this way, we show that certain aspects of the low-energy spectrum can also be exactly computed including, in the superconducting case, the phase stiffness. Examples of perfect QGN include flat bands with certain symmetries (e.g., chiral or time reversal) and non-symmetry-related cases exemplified with an engineered model for pair-density wave. Extending this approach, we obtain exact superconducting ground states with nontrivial pairing symmetry. Published by the American Physical Society 2024
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.
Beyond the frozen-pairing contribution, the superfluid weight of a multiband superconductor contains a correction from the self-consistent relaxation of the pairing amplitudes under a phase twist. We show that this correction has a quantum-geometric origin, not in the space of Bloch states but in the space of collective pairing fields. For an attractive multiband Hubbard model at the mean-field level, the thermodynamic Hessian governing the pairing response coincides with the static Gaussian pair-fluctuation kernel at zero momentum at every temperature below Tc. Since gauge invariance relates a phase twist to a finite center-of-mass momentum of the pair field, the superfluid weight is set by the small-momentum curvature of the eigenvalue branch that evolves from the Goldstone mode, and the relaxation correction is a stiffness-weighted quantum metric of the corresponding soft eigenvector on the manifold of pairing configurations. The same construction yields the Ginzburg-Landau gradient coefficient with the identical geometric correction, hence the coherence-length tensor, and, with the low-frequency dynamics of the kernel, the pair-mass tensor. Lieb-lattice calculations verify the equivalence of the thermodynamic, fluctuation-kernel, and soft-mode formulations and the independence of the superfluid weight from the orbital embedding.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? Quantum phases are often characterized through properties of their many-body states: order parameters, correlations, reduced density matrices, entanglement spectra, and local-unitary equivalence classes. Transport, however, is a response to a specified physical probe. This paper asks a sharp version of that distinction: Can two exact condensates be indistinguishable by every spatial reduced-density spectrum, yet exhibit different charge stiffness when the same physical gauge field is applied to both? The answer is yes. The paper constructs an exactly solvable finite-range flat-band pairing model in which two number-projected condensates are related by onsite product unitaries and therefore have identical spatial reduced-density spectra for every bipartition, while their response to the same fixed physical U(1) Peierls twist is qualitatively different. MAIN EXACT RESULT The model contains an exactly degenerate manifold of number-projected condensates. Two branches in this manifold are related by onsite product unitaries. For every spatial region R, their reduced density matrices are therefore unitarily conjugate and have exactly identical spectra. Consequently, all spatial entanglement spectra and all Rényi entropies coincide between the two branches. Nevertheless, under the same fixed physical U(1) twist, one branch is exactly soft while the other has positive static charge curvature. The difference is exact rather than a small perturbative correction. The soft tower has zero curvature, while the rigid branch satisfies an exact all-filling response law K_N^r(epsilon) = [N(M-N)/(M-1)] K_1(epsilon), with M = 2L, and retains a nonzero response density in the fixed-filling density limit evaluated in the paper. An arbitrarily weak symmetry-allowed selector epsilon > 0 uniquely chooses the rigid texture while the odd-charge gap remains open. The odd-charge gap is obtained exactly: Delta_odd(epsilon) = 3/4 + epsilon/2. Thus the switch between zero and finite static curvature is not produced by closing the protected odd-charge gap. WHAT INFORMATION IS MISSING? The two condensates are equivalent as static states under onsite local unitary transformations. However, the external gauge coupling is held physically fixed rather than being transformed together with the state. The paper formulates this distinction as an equivariance problem for the twist-defect map. Static onsite-local-unitary orbit invariants determine the charge curvature only if the physical twist-defect map transforms equivariantly along the orbit. In the constructed model, it does not. The conceptual conclusion is therefore more precise than the familiar observation that condensate fraction and superfluid density are different quantities. Here the two states agree on an exceptionally strong collection of static state information: every spatial reduced-density spectrum. They nevertheless respond differently to the same physical probe. The missing datum is the relative orientation between the state orbit and the physical gauge embedding. In other words: state equivalence does not automatically imply response equivalence when the probe is specified independently of the equivalence transformation. The result does not deny that a full microscopic wavefunction written in a fixed physical basis, together with the complete physical coupling to the external field, determines the response. Rather, it identifies exactly which information is lost when one reduces the system to probe-independent state-orbit data. TWIST-INDUCED UNSELECTION AND NONCOMMUTING LIMITS The work also studies competition between the infinitesimal selector and a finite gauge twist. In the joint scaling limit epsilon -> 0+ with A_x^2/epsilon fixed, the leading texture energies cross at lim_(epsilon -> 0+) A_(x,c)^2(epsilon) / epsilon = 24/7. The twist can therefore drive the selected rigid texture back toward the soft texture while the states remain within the same static onsite-local-unitary orbit. This provides an explicit example of noncommuting limits: selecting the state with epsilon > 0 and then taking epsilon -> 0+ does not reproduce every feature of setting epsilon = 0 from the outset. WHY THIS MATTERS For readers outside flat-band superconductivity, the broader issue is: What information about a quantum state is sufficient to determine its response to an external probe? Entanglement spectra, reduced-density spectra, and local-unitary orbit data are powerful state descriptors. This construction shows that even the complete collection of spatial reduced-density spectra does not, by itself, determine static charge response when the physical gauge coupling supplies additional orientation information. The distinction is conceptually relevant to: - quantum many-body response; - quantum geometry; - transport inferred from wavefunction information; - exact paired states; - entanglement-based classification; - local-unitary invariants; - gauge coupling and probe dependence. The value of the construction is that this separation is realized exactly, without disorder, uncontrolled effective theory, or approximate numerical diagonalization. SCOPE AND NONCLAIMS Throughout this work, “stiffness” means finite-volume static charge curvature K_ij^(N) = [d^2 E_(0,N)(A) / dA_i dA_j]_(A=0) and the explicitly evaluated selected density limit. The result is not identified with an unspecified dynamical order of limits, finite-frequency optical conductivity, or a universal dynamical definition of superfluid weight. The thermodynamic sequence used in the manuscript is an L x 1 torus, periodic in both directions. It is not claimed to be a fully two-dimensional thermodynamic family in which both linear dimensions diverge. Neutral collective excitations can soften with increasing size, so finite-twist uniformity and dynamical response limits remain separate problems. VERSION CLARIFICATION The current release should be quoted using the asymptotic scaling statement A_(x,c)^2(epsilon) / epsilon -> 24/7 as epsilon -> 0+. Earlier wording could be read as claiming the stronger finite-epsilon identity A_x^2 = (24/7) epsilon. That stronger interpretation is not supported. The current manuscript explicitly shows finite-epsilon crossing values displaced from 24/7 and states the scaling limit correctly. This clarification does not alter the paper’s main exact result: identical spatial reduced-density spectra can coexist with zero versus finite static charge curvature under the same physical twist. It also does not alter the exact odd-charge-gap formula. REPRODUCIBILITY The deposit contains the Letter, Supplemental Material, exact and numerical certificates, figure generators, manuscript-build scripts, SHA-256 manifests, and a claim ledger. The archive retains refuted or quarantined claims instead of silently removing them and separates exact identities from finite-precision numerical stress tests. The assumptions behind the selected-state, curvature, gap, and scaling-limit statements are explicitly recorded. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; static charge stiffness; Kohn curvature; Peierls twist; exact condensates; eta pairing; quantum geometric nesting; reduced density matrix; entanglement spectrum; Rényi entropy; local-unitary orbit; onsite product unitary; response from wavefunction data; gauge embedding; physical probe orientation; twist-defect map; transport switching; noncommuting limits; quantum many-body response.
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.
We study the size of two-body bound states and Cooper pairs within a multiband Hubbard model that features time-reversal symmetry and uniform pairing on a generic lattice. Our analysis involves (i) an exact calculation of the localization tensor to determine the size of lowest-lying two-body bound state in vacuum, and (ii) an evaluation of the analogous tensor to estimate the average size of Cooper pairs within the mean-field Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover theory at zero temperature. Beyond the conventional intraband contribution that depends on Bloch bands, we show that pair size also has a geometric contribution governed by the quantum-metric tensor of the Bloch states and their band-resolved quantum-metric tensors. As a concrete example, we investigate the pyrochlore-Hubbard model numerically and demonstrate that, while the pair size diverges in the weakly interacting BCS regime of dispersive bands, it remains finite and relatively small in the flat-band regime, even for infinitesimal interaction, perfectly matching the exact two-body result in the dilute limit.