Flat conduction bands quench the group velocity and thus challenge conventional, dispersion-driven pictures of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, where localized moments are coupled via an effective exchange mediated by conduction electrons. Here we show that RKKY interactions in the flat-band limit are not extinguished by the vanishing group velocity but are instead mediated by the quantum geometry of Bloch states. Starting from a microscopic RKKY derivation, we demonstrate that the Brillouin-zone-averaged quantum metric controls the long-wavelength structure of the static susceptibility, thereby determining the magnetic correlation length and the spin stiffness. As a result, the finite spatial spread of Wannier functions provides an effective long-range coupling channel even when single-particle dispersion is absent. Furthermore, we establish the general principle that the ordering temperature is governed by the quantum metric in finite and low-dimensional samples, effectively circumventing the thermodynamic-limit constraint of the Mermin-Wagner theorem. Specifically, our theoretical investigation reveals that increasing the quantum metric enhances magnetic rigidity and leads to a corresponding rise in the critical temperature within finite-sized systems.
Flat conduction bands quench the group velocity and thus challenge conventional, dispersion-driven pictures of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, where localized moments are coupled via an effective exchange mediated by conduction electrons. Here we show that RKKY interactions in the flat-band limit are not extinguished by the vanishing group velocity but are instead mediated by the quantum geometry of Bloch states. Starting from a microscopic RKKY derivation, we demonstrate that the Brillouin-zone-averaged quantum metric controls the long-wavelength structure of the static susceptibility, thereby determining the magnetic correlation length and the spin stiffness. As a result, the finite spatial spread of Wannier functions provides an effective long-range coupling channel even when single-particle dispersion is absent. Furthermore, we establish the general principle that the ordering temperature is governed by the quantum metric in finite and low-dimensional samples, effectively circumventing the thermodynamic-limit constraint of the Mermin-Wagner theorem. Specifically, our theoretical investigation reveals that increasing the quantum metric enhances magnetic rigidity and leads to a corresponding rise in the critical temperature within finite-sized systems.
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.
The quantum geometry of Bloch states fundamentally affects a wide range of physical phenomena. The quantum Hall effect, for example, is governed by the Chern number, and flat-band superconductivity by the distance between the Bloch states: the quantum metric. While understanding quantum geometry phenomena in the context of fermions is well established, less is known about the role of quantum geometry in bosonic systems where particles can undergo Bose-Einstein condensation (BEC). In conventional single-band or continuum systems, excitations of a weakly interacting BEC are determined by the condensate density and the interparticle interaction energy. In contrast to this, we discover here fundamental connections between the properties of a weakly interacting BEC and the underlying quantum geometry of a multiband lattice system. We show that, in the flat-band limit, the defining physical quantities of BEC, namely, the speed of sound and the quantum depletion, are dictated solely by the quantum geometry. We find that the speed of sound becomes proportional to the quantum metric of the condensed state. Furthermore, the quantum distance between the Bloch functions forces the quantum depletion and the quantum fluctuations of the density-density correlation to obtain finite values for infinitesimally small interactions. This is in striking contrast to dispersive bands where these quantities vanish with the interaction strength. Additionally, we show how in the flat-band limit the supercurrent is carried by the quantum fluctuations and is determined by the Berry connections of the Bloch states. Our results reveal how nontrivial quantum geometry allows reaching strong quantum correlation regime of condensed bosons even with weak interactions. This is highly relevant, for example, for polariton and photon BECs where interparticle interactions are inherently small. Our predictions can be experimentally tested with flat-band lattices already implemented in ultracold gases and various photonic platforms.
A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size ξK∼ℏvF/(kBTK). This kinematic picture collapses in flat bands, where vF=0 and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor v(k), while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound ξK2≥∑kρ(k)Trg(k), where ρ(k)=∣v(k)∣2/∑k∣v(k)∣2. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.
Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.
On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.
In a flat Bloch band, the kinetic energy is quenched and single particles cannot propagate since they are localized due to destructive interference. Whether this remains true in the presence of interactions is a challenging question because a flat dispersion usually leads to highly correlated ground states. Here we compute numerically the ground-state energy of lattice models with completely flat band structure in a ring geometry in the presence of an attractive Hubbard interaction. We find that the energy as a function of the magnetic flux threading the ring has a half-flux quantum ${\mathrm{\ensuremath{\Phi}}}_{0}/2=hc/(2e)$ period, indicating that only bound pairs of particles with charge 2e are propagating, while single quasiparticles with charge e remain localized. For some one-dimensional lattice models, we show analytically that in fact the whole many-body spectrum has the same periodicity. Our analytical arguments are valid for both bosons and fermions, for generic interactions respecting some symmetries of the lattice and at arbitrary temperatures. Moreover, for the same one-dimensional lattice models, we construct an extensive number of exact conserved quantities. These conserved quantities are associated to the occupation of localized single quasiparticle states and force the single-particle propagator to vanish beyond a finite range. Our results suggest that in lattice models with flat bands preformed pairs dominate transport even above the critical temperature of the transition to a superfluid state.