Recent studies have revealed that the quantum geometry of electronic bands determines the electromagnetic properties of noninteracting insulators and semimetals. However, the role of quantum geometry in the optical responses of interacting electron systems remains largely unexplored. Here, we examine the interplay between Coulomb interactions and Bloch-band quantum geometry in clean metals. We demonstrate that the low-frequency optical conductivity of a correlated metal encodes the structure of Bloch wave functions at the Fermi surface. This response originates from integrating out highly off-resonant interband scattering processes enabled by Coulomb interactions. The resulting quantum-geometric contribution appears generically in multiband systems but becomes the dominant effect in the optical conductivity for a parabolic band. We consider a dilute correlated metal near a topological band inversion and show that the doping dependence of optical absorption can measure how the orbital character of Bloch wave functions changes at the Fermi surface. Our results illustrate how the confluence of quantum geometry and Coulomb interactions can enable optical processes and enrich the physics of Fermi liquids.
Recent studies have revealed that the quantum geometry of electronic bands determines the electromagnetic properties of noninteracting insulators and semimetals. However, the role of quantum geometry in the optical responses of interacting electron systems remains largely unexplored. Here, we examine the interplay between Coulomb interactions and Bloch-band quantum geometry in clean metals. We demonstrate that the low-frequency optical conductivity of a correlated metal encodes the structure of Bloch wave functions at the Fermi surface. This response originates from integrating out highly off-resonant interband scattering processes enabled by Coulomb interactions. The resulting quantum-geometric contribution appears generically in multiband systems but becomes the dominant effect in the optical conductivity for a parabolic band. We consider a dilute correlated metal near a topological band inversion and show that the doping dependence of optical absorption can measure how the orbital character of Bloch wave functions changes at the Fermi surface. Our results illustrate how the confluence of quantum geometry and Coulomb interactions can enable optical processes and enrich the physics of Fermi liquids.
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.
Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, [Formula: see text]. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) [Formula: see text] scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron-electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, [Formula: see text], and show that a factor of [Formula: see text] is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that [Formula: see text] must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, [Formula: see text], differs significantly from b = 4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b = 1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T = 0: [Formula: see text] at low frequencies and [Formula: see text] at higher frequencies. We also discuss [Formula: see text] scaling of the conductivity and show that [Formula: see text] in the same model scales in a non-Fermi-liquid way, as [Formula: see text].
We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a T=0 Ising-nematic quantum critical point (QCP) and extrapolate the results to a QCP. We employ direct perturbation theory up to the two-loop order to elucidate how the Fermi surface's geometry (convex vs concave) and fermionic dispersion (parabolic vs nonparabolic) affect the scaling of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ with frequency $\ensuremath{\omega}$ and correlation length $\ensuremath{\xi}$. We find that for a convex Fermi surface the leading terms in the optical conductivity cancel out, leaving a subleading contribution $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}{\ensuremath{\xi}}^{4}\mathcal{L}$, where L=const for a parabolic dispersion and $\mathcal{L}\ensuremath{\propto}ln\ensuremath{\omega}{\ensuremath{\xi}}^{3}$ in a generic case. For a concave Fermi surface, the leading terms do not cancel, and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\xi}}^{2}$. We extrapolate these results to a QCP and obtain $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$ for a convex Fermi surface and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}1/{\ensuremath{\omega}}^{2/3}$ for a concave Fermi surface.
The geometry of Bloch bands affects many physical properties of crystalline solids and other spatially periodic systems. Direct experimental determination of such geometry is an active area of research. In this work, we focus on the fundamental connection between optical excitations and the relative geometry of pairs of Bloch bands, as characterized by the interband Berry connection. We simulate the response of electrons in solids to optical excitation by the response of ultracold fermionic atoms in optical lattices to periodic modulation of the lattice position. The strength of resonant excitation between bands, measured at each quasimomentum and for various lattice-shaking polarizations, directly maps out the interband Berry connection. We apply this method to the optical honeycomb lattice, driving excitations between the ground n=1 band and the excited n′={2,3,4} bands. We observe transparency lines of quasimomenta at which the response to excitation of specific polarization is zero. Further, the interband Berry connection between bands 1 and 3 shows irreducible Dirac strings connecting the K and K′ points in the Brillouin zone, lines along which the interband Berry connection abruptly changes orientation. Our work establishes optical response as a powerful tool for characterizing geometrical and topological properties of band structure.
We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
Quantum geometry has emerged as a unifying concept across condensed matter physics, underlying phenomena from nonlinear topological response to flat-band superconductivity. While usually formulated within band theory, quantum geometry remains meaningful in disordered interacting systems~\cite{resta1999electron}. Here we show that the first negative moment of the optical conductivity -- proportional to the zero temperature quantum Fisher information as a bound on the multipartite entanglement -- provides an experimental probe of quantum geometry across the three-dimensional metal-insulator quantum phase transition in phosphorus-doped silicon. We extract a quantum geometric length ℓ that characterizes the local wavefunctions. Far from the transition, this length is almost coincident with the Bohr radius of the hydrogenic phosphorus donors, reflecting their atomic-scale quantum geometry. Approaching the transition, ℓ is enhanced, but does not diverge continuously like a correlation length; it jumps discontinuously to infinity at the critical point. This reflects the UV domination of the sum rule in three dimensions that renders it insensitive to the critical fluctuations driving the diverging dielectric constant and correlation length. Its enhancement demonstrates a ``puffing" of the donor polarizability volume of quantum geometric origin, which yields a quantum geometric corrected Clausius-Mossotti description in closer agreement with the diverging dielectric response and provides a quantum mechanical foundation for the century-old Herzfeld metallization criterion.