From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory

TL;DR

The paper surveys how quantum geometry of Bloch bands, encoded by the quantum geometric tensor $Q_{ab}^n$, links Berry curvature and quantum metric to linear and nonlinear optical responses. It explains delicate and multigap topology, exemplified by the Hopf insulator and non-Abelian Dirac-point braiding, and shows how these unstable invariants influence non-linear phenomena such as a quantized shift current and boundary anomalies. It introduces bundle gerbes and the Kalb-Ramond 3-form curvature $\mathcal{H}_{xyz}^{nm}$, whose integral yields the Dixmier-Douady invariant $\mathcal{DD}^{nm}$ and underpins a quantized contribution to the integrated circular shift photoconductivity. The Perspective outlines experimental pathways (e.g., momentum-resolved optical probes, ARPES) and computational strategies to identify material candidates, and highlights open questions on symmetry extensions, disorder, and many-body generalizations.

Abstract

Topological principles constitute at present an integral component of condensed matter physics, permeating the modern characterization of electronic states while also guiding materials design. In this brief Perspective, I highlight three research threads in single-particle topological band theory that have recently gained momentum: (i) the rise of the quantum geometric tensor, whose components can at present be directly accessed with optical probes; (ii) the notions of delicate and multigap topology, which fall outside the scope of tenfold way and symmetry-based indicators yet leave robust physical fingerprints; and (iii) the consideration of bundle gerbes, which capture formerly overlooked higher-form topological aspects of energy bands. These distinct directions have been elegantly woven together: delicate and multigap topological insulators have peculiar features in quantum geometry that can be conveniently captured by bundle gerbes. This viewpoint exposes the recently identified quantization of a non-linear optical response and provides outlooks for its realization in crystalline solids.

Figures (6)

• Figure 1: Elements of the quantum geometric tensor. (a) Given an evolution of a Bloch state $\left|{u_n}\right>$ (state vector, represented by green arrows) on a closed loop (circular gray arrow) inside the momentum space (plane of $k_x$ and $k_y$), Berry curvature ($F_{xy}$) quantifies the complex phase (blue-to-red arrows within the attached disks) accumulated by the state. (b) Quantum metric (here only components $g_{xx}$ and $g_{yy}$ explicitly considered) quantifies the rate of change (blue and red line segments) of the Bloch state upon variation of the momentum coordinates.
• Figure 2: Accessing the quantum geometric tensor. (a,b) Theoretical value of the band Drude weight and quantum metric in kagome material $\mathrm{CoSn}$, and (c,d) their experimental estimates obtained through optical probes. (Image source: Ref. Kang:2025).
• Figure 3: Multigap and delicate topological insulators. (a) Conventionally, topological invariants are based on partitioning of the energy bands with a single energy gap determined by the Fermi level. (b,c) Recently, partitioning of energy bands by multiple gaps led to valuable insights into topological invariants and non-linear optical responses. Alternatively, if a topological invariant persists only for a small total number of energy bands, the topology is described as delicate. (d) Chern insulator in two dimensions exhibits energy bands characterized by opposite Chern numbers $\pm C$ which reside in the same physical space but are separated by an energy gap ($\Delta E$). (e) In constrast, Hopf insulator in a three-dimensional slab exhibits surface features with opposite Chern numbers $\pm C$ that are physically separated by the system size ($\Delta z$).
• Figure 4: Photovoltaic shift current. (a) Elementary schematic of the shift current, where a dc-current flows in a non-centrosymmetric material under illumination (without external bias). (Image source: Ref. Xin:2024.) (b) Various microscopic processes contribute to the shift current. The discussed quantized results relate to the 'excitation' component of the shift current. (Image adapted from Ref. Zhu:2024.)
• Figure 5: Comparing Berry quantities against the higher-form topological structures. (a) Flowchart for computing the Chern number vs. the Dixmier-Douady invariant. (b) Weyl points (enclosed dots) in 3D crystals act as sources and drains of a quantum of Berry curvature (red arrows). (c) Opening of an energy gap in Dirac points in 2D crystals generates a large concentration of Berry curvature. (d) What are the analogous three-band objects that acts as monopoles or concentrations of the Kalb-Ramond curvature?