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?
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?
In flat-band systems with quenched kinetic energy, most of the conventional length scales related to the band dispersion become ineffectual. Although a few geometric length scales, such as the quantum metric length, can still be defined, because of their embedding dependence, i.e., the dependence on the choice of orbital positions used to construct the tight-binding model, they cannot serve as a universal length scale of the flat-band systems. Here, we introduce an embedding independent length scale ξ_{flat} of a flat band that is defined as the localization length of an in-gap state proximate to the flat band. Because ξ_{flat} is derived from the intrinsic localization of compact localized states, it is solely determined by the Hamiltonian and provides a robust foundation for embedding-independent observables. We show analytically that the superconducting coherence length in a flat-band superconductor is given by ξ_{flat} in the weak-coupling limit, thereby identifying ξ_{flat} as the relevant length scale for many-body phenomena. Numerical simulations on various lattice models confirm all theoretical predictions, including the correspondence between ξ_{flat} and the superconducting coherence length. Our results highlight ξ_{flat} as an intrinsic, embedding-independent length scale governing exponential localization associated with flat bands.
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.
Over the past years, one witnesses a growing interest in flat band (FB) physics which has become a playground for exotic phenomena. In this study, we address the FB superconductivity in one-dimensional stub chain. In contrast to the sawtooth chain or the Creutz ladder, for a given strength of the attractive electron-electron interaction, the stub chain allows the tuning of the real space spreading of the FB eigenstates (quantum metric or QM). We study in detail the interplay between the interaction strength and the mean value of the QM $\ensuremath{\langle}g\ensuremath{\rangle}$ on the pairings and on the superfluid weight Ds. Our calculations reveal several interesting and intriguing features. For instance, in the weak coupling regime, Ds with respect to $\ensuremath{\langle}g\ensuremath{\rangle},$ exhibits two different types of behavior. Despite the fact that the pairings differs drastically, Ds scales linearly with the QM only when its $\ensuremath{\langle}g\ensuremath{\rangle}$ is large enough (small gap limit). On the other hand, when the QM is of small amplitude an unusual power law is found, more precisely ${D}_{s}\ensuremath{\propto}{\ensuremath{\langle}g\ensuremath{\rangle}}^{\ensuremath{\nu}}$, where $\ensuremath{\nu}\ensuremath{\rightarrow}2$ in the limit of large single particle gap. In addition to the numerical calculations, we have provided several analytical results which shed light on the physics in both the weak and strong coupling regime. Finally, we have addressed the impact of the thermal fluctuations on the superfluid weight.
In flat-band systems with quenched kinetic energy, most of the conventional length scales related to the band dispersion become ineffectual. Although a few geometric length scales, such as the quantum metric length, can still be defined, because of their embedding dependence, i.e., the dependence on the choice of orbital positions used to construct the tight-binding model, they cannot serve as a universal length scale of the flat-band systems. Here, we introduce an embedding independent length scale ξflat of a flat band that is defined as the localization length of an in-gap state proximate to the flat band. Because ξflat is derived from the intrinsic localization of compact localized states, it is solely determined by the Hamiltonian and provides a robust foundation for embedding independent observables. We show analytically that the superconducting coherence length in a flat-band superconductor is given by ξflat in the weak-coupling limit, thereby identifying ξflat as the relevant length scale for many-body phenomena. Numerical simulations on various lattice models confirm all theoretical predictions, including the correspondence between ξflat and the superconducting coherence length. Our results highlight ξflat as a universal length scale for flat bands and open a pathway to embedding independent characterization of interacting flat-band materials.
Recently, the possibility of high-temperature superconductivity (SC) in flat-band (FB) systems has been the focus of a great deal of activity. This study reveals that unlike conventional intra-band SC for which disorder has a dramatic impact, that associated with FBs is surprisingly robust to disorder-induced fluctuations and quasi-particle localization. In particular, for weak off-diagonal disorder, the critical temperature decreases linearly with disorder amplitude for conventional SC, whereas it is only quadratic in the case of SC in FBs. Our findings could have a major impact on the research and development of new compounds whose high purity will no longer be a critical barrier to their synthesis.
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.