Measuring non-Abelian quantum geometry and topology in a multi-gap photonic lattice

TL;DR

This work directly probes non-Abelian quantum geometry and Euler topology in a six-band photonic lattice by fully reconstructing the Bloch Hamiltonian via orbital-resolved polarimetry. The authors measure the non-Abelian QGT, Euler curvature, and quaternion charges, revealing Dirac strings and patch-dependent Euler invariants that encode braiding of band nodes. They show two-band phase windings consistent with the measured charges and develop both experimental and numerical pipelines for computing reduced two-band Hamiltonians. The results establish a versatile platform for exploring multi-gap topology and portend extensions to driven-dissipative, non-Hermitian, and Moiré-based multi-band systems.

Abstract

Recent discoveries in semi-metallic multi-gap systems featuring band singularities have galvanized enormous interest in particular due to the emergence of non-Abelian braiding properties of band nodes. This previously uncharted set of topological phases necessitates novel approaches to probe them in laboratories, a pursuit that intricately relates to evaluating non-Abelian generalizations of the Abelian quantum geometric tensor (QGT) that characterizes geometric responses. Here, we pioneer the direct measurement of the non-Abelian QGT. We achieve this by implementing a novel orbital-resolved polarimetry technique to probe the full Bloch Hamiltonian of a six-band two-dimensional (2D) synthetic lattice, which grants direct experimental access to non-Abelian quaternion charges, the Euler curvature, and the non-Abelian quantum metric associated with all bands. Quantum geometry has been highlighted to play a key role on macroscopic phenomena ranging from superconductivity in flat-bands, to optical responses, transport, metrology, and quantum Hall physics. Therefore, our work unlocks the experimental probing of a wide phenomenology of multi-gap systems, at the confluence of topology, geometry and non-Abelian physics.

Figures (12)

• Figure 1: Illustration of momentum space braiding in a multi-gap system (two gaps in between three bands). Band nodes carry non-Abelian charges labeled as $\pm i,\pm k$. In the left panel, nodes come in pairs of opposite charges (empty/filled circles) which can annihilate each other and open a gap. This is captured by vanishing Euler class within the colored momentum patches ($\mathcal{P}_{1,2}$). Moving band nodes in adjacent gaps around each other (red and blue solid lines) produces a modified band structure with similarly charged nodes in each gap (right panel). Such braiding mechanism is illustrated in the momentum space projections in the bottom panels (dashed arrow). After braiding, the nodes cannot annihilate within $\mathcal{P}_{1,2}$, which is captured by a nonzero Euler class $\chi(\mathcal{P}_{1,2})$.
• Figure 1: a. Signed modulus square of the modes in a circular pillar in the $\{ \ket{s}, \ket{p_x}, \ket{p_y} \}$ basis (top panels), and in the $\{ \ket{sp^2_{\sigma}} \}$ basis (bottom panels). These modes are computed solving the 2D Schrödinger equation. b. Representation of the unit cell sectors chosen to select the main lobe of the $\ket{sp^2_\sigma}$ orbitals using the SLM. Each sector is offset from the pillar center by a vector $d^\sigma$ of length $|d^\sigma| = ({2-\sqrt{3}})a$ aligned along the segment connecting the pillar to one of its neighbors.
• Figure 2: a. Scanning electron microscope image of the honeycomb lattice of coupled micropillars probed in this work. We schematically represent a unit cell (black diamond shape) containing $A$ and $B$ sites. The vectors $\bm{\delta}_i$ connect an $A$ site to its three $B$ neighboring sites. b. Representation of the lattice unit cell showing the $\ket{sp^2}$ modes with positive (negative) lobes in blue (red) color. c. Measured band dispersion, obtained by applying our eigenstate reconstruction method to determine the band energies with sub-linewidth precision. Arrows point toward the different nodes in the $p$-bands. We also indicate their measured quaternion charges. d. Reconstruction of the $\bm{k}$-dependent off-diagonal matrix elements of the Hamiltonian in the $\left\{ \ket{sp^2_{\sigma}} \right\}$ basis. Each plot shows the real (upper triangle) and imaginary (lower triangle) part of one of the Hamiltonian components, as a function of $k_x$ and $k_y$. In each panel, dashed lines show iso-energy contours of the difference between bands 5 and 4.
• Figure 2: a-d: Exemplary SLM input masks used to encode four different configurations of the ${\bm m}$ vector: a. $\bm{m} = [1,0,0,0,1,0]$, b. $\bm{m} = [1,0,0,0,i,0]/\sqrt{2}$, c. $\bm{m} = [1,0,0,0,0,0]$ and d. $\bm{m} = [0,0,0,0,1,0]$. (e-h): Fourier space emission measured along $k_y = 0$ when applying the four masks shown in a-d. We clearly observe that the choice of $\bm{m}$ alters the Fourier space intensity distribution, due to the modification of the interference conditions between modes.
• Figure 3: Experimentally reconstructed Euler curvature ${\rm Eu}_{n,n+1} ({\bm k})$ for the pair of bands $p$-bands a. (3,4), b. (4,5), and c. (5,6). In each panel, nodes in adjacent gaps are represented with gray full circles. The Euler class calculated over different patches marked by dashed lines are indicated. Non-zero values of $\chi(\mathcal{P})$ in (a,c) indicates non-Abelian charges with the same sign within these gaps, $-i$ and $+k$ respectively. According to the measured Euler class we represent positive (negative) quaternion charges in each gap by full (empty) circles. Insets show the tight-binding simulations of the Euler curvature using the following parameters given in units of $t_p$: $\epsilon_s = 0,\: \epsilon_{p} = 5,\: \epsilon_{\rm el} = 0.07,\: t_s = 0.2,\: t_p = 1,\: t_{sp} = 0.2,\: \beta = 1.06$.