Probing the Quantum Geometry of Correlated Metals using Optical Conductivity

Abstract

Recent studies have revealed that the quantum geometry of electronic bands determines the electromagnetic properties of non-interacting 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 wavefunctions 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 wavefunctions 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.

Figures (5)

• Figure 1: (a) Schematic of the optical conductivity of a correlated metal. At low frequencies the conductivity is dominated by a Drude peak, while in the hydrodynamic regime the conductivity scales quadratically (with a possible logarithmic correction) for an isotropic 2D metal. (b) Momentum scales that govern dilute metals near a band inversion: Fermi wavevector $k_F$, band inversion scale $\lambda$ where the orbital character changes, and wavevector $k_v$ for nonparabolic corrections.
• Figure 2: (a) Schematic of an optical absorption process which results in a quantum geometric contribution to $\mathop{\mathrm{\textrm{Re}}}\nolimits \sigma(\omega)$. (b) Diagram for an intraband excitation from a velocity-dependent intraband current vertex $\mathbf{j}^{0,0}_\mathbf{k} \sim \mathbf{v}_\mathbf{k}$. (c) Quantum-geometric diagram with an interband-excited intermediate state, corresponding to the processes depicted in (a). The interband current $\mathbf{j}^{0,n}_k$ is proportional to the excitation gap $\Delta^{0,n}_\mathbf{k}$, and the interband Berry connection $\mathbf{A}^{0,n}_\mathbf{k}$.
• Figure 3: Optical conductivity of dilute correlated metals near a band inversion with angular momentum $m$. Dots are scaling coefficients extracted from the numerically-computed optical conductivity. Lines depict analytical quantum-geometric Fermi surface contributions, where $\alpha_0 = \frac{1}{(2 \pi)^6} \hbar e^2 U^2 k_F^4/\epsilon_F^4$ is an overall scale. The insets are schematics of the relevant scattering channels. (a) Coefficient of $\omega^2$ scaling for spinful fermions where both the exchange and pairing channels contribute. (b) Coefficient of $\omega^2 \log(\hbar\omega/\epsilon_F)$ scaling for spinful fermions, arising from opposite spin scattering between $\mathbf{k}$ and $-\mathbf{k}$. Insets: dependence of scaling coefficients a $m=1$ band inversion on the Thomas-Fermi screening wavevector $\kappa$.
• Figure 4: Scaling contributions for the optical conductivity from non-parabolicity ($k_v$) and quantum geometry ($\lambda$) for spinful fermions, for an isolated band [$m=0$; (a), (d)] and for Fermi surfaces near an angular-momentum $m=1,2$ band inversion [(b),(c),(e),(f)]. (a-c) and (d-f) parameterize $\omega^2$ and $\omega^2 \log(\omega)$ scaling, respectively. In materials, the ratio $k_v/\lambda$ is fixed; however, optical absorption exhibits a maximum as a function of doping [(g-i)], a signature of quantum-geometric correlated metals.
• Figure 5: Diagrams for the real part of the optical conductivity computed via Fermi's golden rule, with $(\mathbf{k},n,\sigma)$ denoting momentum, band index, and spin. The virtual intermediate state includes intraband ($n=0$) and interband ($n \neq 0$) excitations.