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
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
Preprint and reproducibility archive; not peer reviewed.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.
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.
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
In this thesis, superconductivity of fermions in two-dimensional Hubbard lattice models is theoretically studied. Each of the studied lattice structures features intriguing properties that provide valuable insight to understand superconductivity in general. While real condensed matter systems are too complicated to tackle with theoretical tools, the Hubbard model is one of the simplest quantum many-body models to possess a wide spectrum of physically relevant phenomena, from superconductivity to magnetism and topological order. Hubbard models are also interesting as they can be experimentally realized and accurately controlled by utilizing ultracold gases.
The first part of the thesis explains the required theoretical background of superconductivity in the level of mean-field and linear response theories. Specifically, we primarily focus on the superfluid weight which implies the existence of the Meissner effect and dissipationless current, defining properties of superconductors. We further discuss how the Berezinskii-Kosterlitz-Thouless (BKT) transition, related to the loss of superconductivity due to the thermal fluctuations of the superconducting order parameter in high enough temperatures, can be accessed via the superfluid weight.
In the second part, the main results are discussed. We show how the superconductivity of lattice models featuring dispersionless Bloch bands, so-called flat bands, can be large and originating from the quantum geometric properties of the Bloch states of the flat bands. This is in contrast to conventional dispersive bands where superconductivity is determined by the effective mass of the particles, i.e. by the dispersion relation of the Bloch bands alone and not by the Bloch states. This is well demonstrated in the Lieb lattice geometry, studied in Publication I, which features both a strictly flat and two dispersive Bloch bands, allowing direct comparison of superconductivity in dispersive and flat bands. We show how the superconductivity of the flat band is more robust than that of the dispersive bands and how the superfluid weight behaves non-monotonically as a function of the interaction strength. This non-monotonic behavior should be experimentally accessible. Flat band superconductivity is also the topic of Publication III, where the experimentally realized twisted bilayer graphene (TBG) is studied with local and non-local attractive interactions. We predict qualitative differences between local and non-local interaction schemes which could be distinguished experimentally and reveal the quantum geometric origin of superconductivity in TBG systems in case of both the assumed interaction schemes.
Exotic Fulde-Ferrell (FF) superfluid phases, characterized by the non-zero momentum of Cooper pairs, were explored in Publication II. We show how the FF phases can be stabilized against thermal fluctuations in the presence of spin-orbit coupling (SOC) and that topologically non-trivial FF states, created with SOC and Zeeman fields, can be found at finite temperatures.; Tassa…
We construct quasi-one-dimensional topological and nontopological three-band lattices with a tunable band gap and winding number of the flat band. Using full multiband mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We obtain excellent agreement between MF and DMRG calculations. The SC weight Ds on the gapped topological, $W\ensuremath{\ne}0$, flat band increases linearly with interaction strength U for low values and with a slope that depends on the details of the compact localized state at U=0. As $U\ensuremath{\rightarrow}0$ for the gapped nontopological flat band (W=0), Ds decays with a power law faster than quadratically but slower than exponentially. In the gapless case (flat band touching the band above it), we find at low U (for both W=0 and $W\ensuremath{\ne}0$) that ${D}_{s}\ensuremath{\propto}{U}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}<1$. In other words, Ds increases faster than linearly for low U, thus favoring SC at weak interaction more than the gapped case. Our results reestablish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.
Quantum geometry, a nascent field, seeks to formalize the geometric properties of quantum systems through the lens of topology. This paper explores the development of a novel mathematical framework for defining and analyzing these geometric properties, explicitly incorporating topological concepts to enhance understanding and provide a more robust and insightful approach compared to traditional geometric methods. We propose a new methodology predicated on the observation that certain geometric features exhibit topological behavior, offering a fundamentally different perspective on the structure of quantum systems. The paper outlines the core mechanism, detailing how topological aspects will be integrated into the definition of geometric properties and how this will lead to a deeper comprehension of quantum phenomena. The investigation will focus on establishing a rigorous mathematical foundation for this approach, demonstrating its potential for improving the analysis of complex quantum systems.