To explore the influence of quantum-geometric effects on the Ginzburg-Landau coherence length in a dilute flat-band superconductor, we adopt a BCS-BEC crossover approach to the multiband pyrochlore-Hubbard model near the critical temperature for superconductivity. Our self-consistent formulation for this three-dimensional lattice benchmarks very well against the so-called zero-temperature coherence length, demonstrating the monotonic decay of the coherence length to zero as the interaction strength increases. Additionally, we show that the effective mass of the many-body bound states (i.e., Cooper pairs) is nearly identical to that of the lowest-lying two-body bound states in the dilute flat-band limit.
To explore the influence of quantum-geometric effects on the Ginzburg-Landau coherence length in a dilute flat-band superconductor, we adopt a BCS-BEC crossover approach to the multiband pyrochlore-Hubbard model near the critical temperature for superconductivity. Our self-consistent formulation for this three-dimensional lattice benchmarks very well against the so-called zero-temperature coherence length, demonstrating the monotonic decay of the coherence length to zero as the interaction strength increases. Additionally, we show that the effective mass of the many-body bound states (i.e., Cooper pairs) is nearly identical to that of the lowest-lying two-body bound states in the dilute flat-band limit.
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.
Superconductivity in flat-band systems, governed by quantum metric of Bloch states rather than the BCS framework, exhibits unique phenomena due to the vanishing electron group velocity. Here, we propose the vortex states and vortex size as direct probes to explore the quantum geometry effects in flat-band superconductors. We show that flat-band vortex bound states are sharply localized near the vortex core, and the energy gap between the lowest two bound states is on the order of the bulk superconducting gap. Both the spatial spread and energy scales of bound states are controlled by the flat-band's quantum metric length. Moreover, the vortex size at zero temperature, set by the quantum metric length, is atomic in scale and independent of interaction strength. Near Tc, the vortex size corresponds to the Ginzburg-Landau coherence length, diverges as ξ∼Tc/(Tc−T)ξ0, where ξ0 depends linearly on the quantum metric length. Thus, the quantum metric serves as the lower bound for vortex state spread and vortex size. We also introduce perturbations to make the flat band dispersive, and distinguish flat-band vortices from BCS-like vortices. Our results establish vortices as universal probes of quantum geometry in flat-band superconductors.
We consider multiband BCS superconductors that exhibit time-reversal symmetry and uniform pairing, and analyze their dynamic density and spin structure factors using linear-response theory within the mean-field BCS-BEC crossover framework at zero temperature. Our results for the multi-orbital Hubbard model satisfy the associated f-sum rules in several limits. In particular, in the strong-coupling limit, they coincide with those of a weakly-interacting Bose gas of Cooper pairs, where the low-energy collective Goldstone modes serve as Bogoliubov phonons. We further reveal that the quantum-geometric origin of the low-energy structure factors, along with related observables such as the superfluid-weight tensor and the effective-mass tensor of Cooper pairs, can be traced all the way back to the effective-mass theorem for Bloch bands in this limit. As an illustration, we investigate the pyrochlore-Hubbard model numerically and demonstrate that the Goldstone modes are the only relevant collective degrees of freedom in the flat-band regime.
We investigate the interplay between pairing correlations and quantum geometry in the spin-orbit-coupled BCS-BEC crossover. Working in the helicity basis, we derive the exact pair-size tensor for the two-body bound states and show that it separates into intraband and quantum-geometric interband contributions, with the latter determined by the quantum metric. We extend this decomposition to two distinct many-body length scales at zero temperature: the Cooper-pair size, obtained from mean-field BCS theory, and the coherence length, obtained from the Gaussian fluctuation theory. Although these quantities almost coincide in the BCS regime, they describe distinct physical properties away from it, with the pair size characterizing the internal extent of a pair and the coherence length characterizing the long-wavelength response of the order parameter. We evaluate both quantities for three-dimensional Rashba, three-dimensional Weyl, and two-dimensional Rashba spin-orbit coupling models. We find that the pair size decreases monotonically with increasing spin-orbit coupling as the pair localizes, whereas the coherence length develops a pronounced minimum in the crossover regime before growing again in the BEC regime as the composite bosons become weakly interacting. This minimum is tied to the collapse of the noninteracting Fermi surface onto the ring or sphere of helicity-band minima. The quantum-geometric contributions reach up to nearly one third of the pair size and about 40\% of the coherence length, tracking these same changes in the underlying helicity Fermi-surface topology.
The possibility of an unconventional form of high temperature superconductivity in flat band (FB) material does not cease to challenge our understanding of the physics in correlated systems. Here, we calculate the normal and anomalous one-particle correlation functions in various one and two dimensional FB systems and systematically extract the characteristic lengthscales. When the Fermi energy is located in the FB, it is found that the coherence length ( \xi ξ ) is of the order of the lattice spacing and weakly sensitive to the strength of the electron-electron interaction. Recently, it has been argued that in FB compounds \xi ξ could be decomposed into a conventional part of BCS type ( \xi_{BCS} ξ B C S ) and a geometric contribution which characterises the FB eigenstates, the quantum metric ( \langle g \rangle ⟨ g ⟩ ). However, by calculating the coherence length in two possible ways, our calculations show that \xi ≠ \sqrt{\langle g \rangle.} ξ ≠ ⟨ g ⟩ . This may suggest that the link between QM and coherence length is more complex, and leaves us with the open question: what is the appropriate definition of the coherence length in flat-band systems?
Current theories of high-temperature superconductivity in flat-band systems predict a linear dependence of the transition temperature on the attractive interaction, Tc(U)=c∣U∣. However, the value of c and the full nonlinear Tc(U) curve---with a maximum at large ∣U∣---are known beyond mean-field and quantum geometry estimates only for systems with isolated flat bands. Using a controlled diagrammatic Monte Carlo technique, we trace the onset of superfluid response in the Lieb lattice with attractive Hubbard interaction. Focusing on the half-filled flat-band case, where the ordering mechanism differs fundamentally from both BCS and BEC (preformed Cooper pair) scenarios, we find that the pairing response diverges linearly with decreasing temperature over a broad range of U, leading to a sharp crossover to long-range correlations at a characteristic temperature T∗, which provides a controlled upper bound on Tc. The highest T∗ occurs when all three bands touch at a single momentum point, potentially corresponding to high Tc values.