Quantum Geometry and the Hidden Scales in Materials

Abstract

Electronic properties of quantum materials solids are often well understood via the low energy dispersion of Bloch bands, motivating single band approximations in many metals and semiconductors. However, a closer look reveals length and time scales introduced by quantum dipole fluctuations due to interband mixing, which are reflected in the momentum space textures of the electronic wavefunctions. This structure is usually referred to as quantum geometry. These new scales not only qualitatively modify the linear and nonlinear responses of a material but can also have a vital role in determining the many-body ground state at low temperatures. In this Perspective, we explore how quantum geometry impacts properties of materials and outline recent experimental advances that have begun to explore quantum geometric effects in various condensed matter platforms. We discuss the separation of scales that can allow us to estimate the significance of quantum geometry in various response functions.

Figures (3)

• Figure 1: Dipole fluctuations lead to uncertainty in electron position. The corresponding spatial spread is characterized by the length scale $\ell_g$ determined by the quantum geometry of the ground state. a. The hydrogen atom, where the ground state wavefunction is an $s$ orbital with spread $\ell_a^2= 3 a_B^2/4$ where $a_B$ is the Bohr radius. b. Monolayer ${\rm MoTe}_2$, a semiconductor with $\approx 1\rm eV$ band gap and a topological obtruction. The localized electrons are shared between neighboring sites with a large spread spanning multiple unit cells due to nontrivial orbital interference. The geometric scale $\ell_g$ is comparable to the lattice constant.
• Figure 2: Choice of projected manifold in defining quantum metric. a. The Kohn metric is defined using the ground-state projector $\hat{P}^0$ and captures dipole transitions between occupied and unoccupied states. b. The quantum metric for a single isolated band $m$ is defined using the band-resolved projector $\hat{P}^m$, incorporating dipole transitions between band $m$ and all other bands $n \neq m$. The infinitesimal overlap between Bloch states is given by $|\langle u_{m, \boldsymbol{k}} | u_{m, \boldsymbol{k} + d\boldsymbol{k}} \rangle|^2 = 1 - g^m_{\mu\nu} \, dk_\mu \, dk_\nu$, where $g^m_{\mu\nu} = \sum_{n \neq m} \langle u_{m, \boldsymbol{k}} | \hat{r}_\mu | u_{n, \boldsymbol{k}} \rangle \langle u_{n, \boldsymbol{k}} | \hat{r}_\nu | u_{m, \boldsymbol{k}} \rangle$. The Kohn metric appears in optical sum rules, while the single-band metric governs the superfluid stiffness in flat-band systems. The two are formally distinct, in that they differ in their choice of the projected manifold, and are thus not interchangeable.
• Figure 3: Separation of scales and the emergence of a geometric length scale from lattice interference. a. Band structure for a free particle in a periodic honeycomb lattice potential. The tight-binding approximation is valid when the potential $V$ is deep enough to spectrally isolate a few orbitals, here the two lower connected bands. The arrows indicate optical transitions within a single band (orange), within the low-energy band manifold (blue), and between the low- and high-energy bands (black). b. Optical conductivity for the same system, where low-frequency spectral weight features a Drude weight due to the Fermi surface (orange) in addition to a geometric spectral weight (blue), due to optical transitions between the same orbital across distinct lattice sites. The higher-frequency features reflect dipole transitions that correspond to fluctuations at the scale of an atomic orbital, truncated away in the tight-binding approximation. Looking at the spectral weight at low energies (blue) we can define an effective geometric length scale, $\ell_g$, that governs dipole fluctuations due to lattice interference, as well as a resonant energy $\mathcal{E}$ associated with its local dynamics. The resonant energy $\mathcal{E}$ is defined by a weighted average of band energy differences $\varepsilon_{n{\bm{k}}}-\varepsilon_{m{\bm{k}}}$ by geometric factors and can often be approximated by the first dominant peak in optical conductivity, comparable to the hopping energy. The boxes indicate the characteristic length scales associated with the dipole fluctuations contributing to the long-wavelength (orange), intermediate (blue), and atomic-scale (black) spectral weight.