Light-matter coupling and quantum geometry in moiré materials

Quantum geometry has been identified as an important ingredient for the physics of quantum materials and especially of flat-band systems, such as moiré materials. On the other hand, the coupling between light and matter is of key importance across disciplines and especially for Floquet and cavity engineering of solids. Here we present fundamental relations between light-matter coupling and quantum geometry of Bloch wave functions, with a particular focus on flat-band and moiré materials, in which the quenching of the electronic kinetic energy could allow one to reach the limit of strong light-matter coupling more easily than in highly dispersive systems. We show that, despite the fact that flat bands have vanishing band velocities and curvatures, light couples to them via geometric contributions. Specifically, the intraband quantum metric allows diamagnetic coupling inside a flat band; the interband Berry connection governs dipole matrix elements between flat and dispersive bands. We illustrate these effects in two representative model systems: (i) a sawtooth quantum chain with a single flat band and (ii) a tight-binding model for twisted bilayer graphene. For (i) we highlight the importance of quantum geometry by demonstrating a nonvanishing diamagnetic light-matter coupling inside the flat band. For (ii) we explore the twist-angle dependence of various light-matter coupling matrix elements. Furthermore, at the magic angle corresponding to almost flat bands, we show a Floquet-topological gap opening under irradiation with circularly polarized light despite the nearly vanishing Fermi velocity. We discuss how these findings provide fundamental design principles and tools for light-matter-coupling-based control of emergent electronic properties in flat-band and moiré materials.

Figure 1

Illustration of the main concept of this paper. Moiré engineering has been demonstrated as a flexible route towards two-dimensional flat-band engineering (top row). From the band structure one might naïvely expect that the coupling to light in flat bands vanishes. However, a geometric contribution (present, e.g., in TBG), can contribute to light-matter coupling and dominates in flat band systems (bottom row).

Figure 2

The sawtooth chain exhibiting a flat and a dispersive band, and its light-matter couplings. (a) Sketch of the model. We illustrate the destructive interference that causes the flat band by plotting its Bloch wave function on three adjacent sites. The amplitude is denoted by the radius of the colored circle (the amplitude on the full black dot is $\sqrt{2}$ times that of the amplitude on the empty circle). The sign is determined by the color: red for positive and blue for negative. (b) The bare band structure exhibiting a completely flat and a dispersive band separated by a gap. (c) Linear intraband coupling. This quantity vanishes identically in the completely flat band. (d) Linear interband coupling. Note that this is a gauge dependent quantity. We have chosen the same gauge for the plot as in the text, Eq. (30). (e) Quadratic intraband coupling. Importantly, this is nonzero even in the completely flat band. (f) Quadratic interband coupling. We have again chosen the same gauge as in the text Eq. (36).

Figure 3

Low-energy electronic band structure of the magic-angle twisted bilayer graphene (MATBG) model (${\mathrm{\Theta }}_{\text{M}}=1.{05}^{\circ }$) along the two Dirac points of the mini Brillouin zone. Black dashed lines show the monolayer Graphene (MG) Dirac bands. The red and turquoise dotted lines indicate the two different momenta considered in Sec. 4b.

Figure 4

LMCs of the Peierls-substituted MATBG Hamiltonian. (a)–(c) show the LMCs at $\widetilde{{\mathrm{K}}_1}$ in the linear regime of the first Dirac cone around ${\mathrm{K}}_1$. (d)–(f) show the LMCs at the $\mathrm{\Gamma }$ point. Coloured dots indicate the magic-angle values that are presented on a finer grid in the insets. Colored arrows indicate the monolayer graphene reference values. (a),(d) provides a measure of the linear-field intraband coupling (band velocity) summed over the four flat bands. (b),(e) show the linear-field interband couplings. The red and turquoise lines indicate the coupling between the four flat bands. The orange and black line provide a measure of the coupling of the four flat bands to the two higher and two lower lying dispersive bands. (c),(f) measure of the quadaratic-field intraband coupling.

Figure 5

Floquet band effects as function of the driving amplitude $A_0$ at a driving frequency of $\mathrm{\Omega }=3$ eV. The colored curves illustrate the amplitude dependency of the dominant contribution to the gap renormalization obtained from the effective HFA Floquet Hamiltonian, $H_{\text{eff}}$. The dashed black lines indicate the values obtained from the band-truncated Floquet matrix ${\mathcal{H}}^{\alpha \beta }$ up to quadratic field order. Blue crosses indicate reference values obtained from the full Floquet matrix without band truncation with full exponential LMC. (a) shows a quadratic gap opening at the Dirac point $K_1$ as function of the driving amplitude. The dominant contribution clearly stems from the linear-field interband matrix elements between the flat bands of the effective Floquet Hamiltonian [lin-inter, see Eq. (46) with $m\ne n\in \{-2,1,0,1\}$]. The black solid line indicates the gap value obtained from a naïve two-band approximation with the Moire renormalized Fermi velocity of $v_F=0.23\mathrm{eV}/\AA$. We find excellent agreement between the HFA and the full Floquet calculations. (b) shows the bandwidth renormalization of the flat band manifold at the $\mathrm{\Gamma }$ point. For small driving amplitudes $A_0\le 0.004{\AA }^{-1}$ the bandwidth is dominated by the quadratic intraband coupling [quad-intra, see Eq. (48) with $m=n\in \{-2,1,0,1\}$], which leads to a reduction of the bandwidth. For higher driving amplitudes the bandwidth is dominated by the linear-field interband coupling between the flat bands [lin-inter, see Eq. (46) with $m\ne n\in \{-2,1,0,1\}$], which induces a trend towards an increasing bandwidth. The yellow curve shows the bandwidth contribution from the unperturbed Hamiltonian [zero, see Eq. (44)]. As reference, we show the sum of all three dominant contributions from $H_{\text{eff}}$ (purple curve). Despite a small overestimation of the renormalized bandwidth, the HFA (the purple curve) qualitatively reproduces the trajectories of full Floquet calculations (blue crosses).

Figure 6

Convergence of Floquet effects as function of considered photon sectors for a driving frequency $\mathrm{\Omega }=3$ eV and a driving amplitude $A_0=0.01{\text{Å}}^{-1}$. Black solid lines indicate the values computed with the truncated Floquet Hamiltonian ${\mathcal{H}}^{\alpha \beta }$. For both quantities, the Floquet gap at $K_1$ (a) and the bandwidth renormalization at $\mathrm{\Gamma }$ (b), we find good convergence at linear photon order $\left|\text{max}\right(\alpha \left)\right|=1$, which we use throughout this work.

Figure 7

Convergence of Floquet effects as function of band cutoff $N_c$ for a driving frequency $\mathrm{\Omega }=3$ eV and a driving amplitude $A_0=0.01{\text{Å}}^{-1}$. Black solid lines indicate the values computed with the truncated Floquet Hamiltonian ${\mathcal{H}}^{\alpha \beta }$. The blue dashed line indicates the reference values extracted from the full Floquet Hamiltonian without band truncation and with full exponential LMC. Red dashed lines indicate the values computed via the effective Floquet Hamiltonian $H_{\text{eff}}$ in the HFA. For both quantities, the Floquet gap at $K_1$ (a) and the bandwidth renormalization at $\mathrm{\Gamma }$ (b), we find good convergence for a band cutoff $N_c=512$. We use this cutoff for all numerical Floquet calculations presented in Sec. 4c. The truncated Floquet Hamiltonian (black) and the full Floquet Hamiltonian (blue) show consistent results at $N_c$, rendering field effects beyond second order insignificant in the chosen driving regime. While the high frequency approximation works well at the K point, it overestimates the bandwidth renormalization at the $\mathrm{\Gamma }$ point. However, it should be noted that the HFA is predominantly introduced for means of a profound understanding of the underlying physical mechanism, not to provide quantitatively exact results.