Quantum geometry in quantum materials

npj | quantum materials Review Published in partnership with Nanjing University https://doi.org/10.1038/s41535-025-00801-3 Quantum geometry in quantum materials Check for updates 1234567890():,; 1234567890():,; Jiabin Yu1,2 , B. Andrei Bernevig2,3,4, Raquel Queiroz5, Enrico Rossi6, Päivi Törmä7 & Bohm-Jung Yang8,9,10 Quantum geometry, characterized by the quantum geometric tensor, plays a central role in diverse physical phenomena in quantum materials. This pedagogical review introduces the concept and highlights its implications across multiple domains, including optical responses, Landau levels, fractional Chern insulators, superfluid weight, spin stiffness, exciton condensates, and electronphonon coupling. By integrating these topics, we emphasize the broad significance of quantum geometry in understanding emergent behaviors in quantum systems and conclude with an outlook on open questions and future directions. Quantum materials can be loosely defined as materials for which quantum mechanical effects manifest on a macroscopic scale. Two classes of quantum materials are paradigmatic: superconductors, and quantum Hall systems. For superconductors, electron-electron interaction is the key ingredient that leads to a macroscopic manifestation of quantum mechanics: such interaction causes the electrons to form phase-coherent Cooper pairs and this results in the Meissner effect and the dissipationless transport of charge current. For a two-dimensional (2D) electron gas in the integer quantum Hall regime, the perfect quantization of the Hall conductivity can be understood without explicitly taking any effects of electron-electron interactions into account. The integer quantum Hall effect (QHE) can be attributed to the unique topology of the free-electrons’ ground state1. Such topology is encoded by the Chern number, C, given by the integral over the Brillouin zone of the Berry curvature that measures the change of the eigenstate’s phase as the momentum k is varied. The Berry curvature is part of the quantum geometry of a material. The QHE is the archetypical demonstration that quantum geometry is one of the key quantities that make a material a quantum material. As we will discuss in the remainder of the review, the Berry curvature turns out to be the antisymmetric part of a tensor Q, the quantum geometric tensor (QGT)2. In recent years it has become apparent that the symmetric part of this tensor, the quantum metric, g, also plays a key role in making a material, quantum. In a loose sense, the quantum metric appears to be the key quantity to understand the properties of materials in which both interactions and quantum geometry lead to macroscopic manifestations of quantum mechanics. Quantum geometry is the geometric structure that naturally arises in the space of quantum states when such states depend on continuous parameters. One classic example of quantum geometry is the geometric phase of a quantum state under adiabatic evolution, in which case the continuous parameter is time. Within condensed matter physics, the continuous parameters are the components of the crystal momentum k, and quantum geometry refers to the geometric properties
of the Bloch states, more precisely, the periodic part of the Bloch states ∣uk , which is the focus of this review. In this context, quantum geometry is also called band geometry, which includes long-known concepts such as the spread of the possible Wannier basis and the parallel transport of the electronic states. The QGT (also called the Fubini-Study metric3,4) has components: Qij ðkÞ ¼ h∂ki uk ∣½1  ∣uk ihuk ∣∣∂kj uk i; ð1Þ where ki is the ith component of the Bloch momentum k. For simplicity in writing Eq. (1) we have considered the case of a well-isolated band. The antisymmetric part of Qij(k) is iBij(k) = [Qij(k) − Qji(k)]/2 is related to the well known Berry curvature5–7 Fij(k) as Bij(k) = −Fij(k)/2, and the symmetric part gij(k) = [Qij(k) + Qji(k)]/2 is the quantum metric g given that corresponds to the metric for infinitesimal distances of the Hilbert-Schmidt quantum qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distance d HS ðk; k0 Þ  1  ∣huk juk0 i∣2 : ds2 = ∑i,jgij(k)dkidkj. In two dimensions (2D) the integral over the Brillouin Zone (BZ) of Bxy(k)/π for the states of an occupied band is quantized and equal to the Chern number C. Conversely, the integral of gij(k) over the BZ is in general not quantized. However, in 2D, the positive semidefinite nature of Q (combined with inequality between trace and determinant) implies the following inequalities8 TrgðkÞ ≥ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det gðkÞ ≥ 2jBxy ðkÞj ð2Þ We can introduce the tensor M ≡ (1/π)∫BZddkQ(k). Because M is a sum of positive semidefinite tensors, it is itself positive semidefinite, and so det M ≥ 0. In 2D this leads to the inequality detðReðMÞÞ ≥ detðImðMÞÞ, that can be seen as the integral equivalent of Eq. (2), and can be written as det  Z   Z  1 1 d 2 k gðkÞ ≥ det d 2 k BðkÞ ¼ C 2 : π π ð3Þ 1 Department of Physics, University of Florida, Gainesville, FL, USA. 2Department of Physics, Princeton University, Princeton, NJ, USA. 3Donostia International Physics Center, Sebastian, Spain. 4IKERBASQUE, Basque Foundation for Science, Bilbao, Spain. 5Department of Physics, Columbia University, New York, NY, USA. 6Department of Physics, William & Mary, Williamsburg, VA, USA. 7Department of Applied Physics, Aalto University School of Science, Aalto, Finland. 8 Department of Physics and Astronomy, Seoul National University, Seoul, South Korea. 9Center for Theoretical Physics (CTP), Seoul National University, e-mail: yujiabin@ufl.edu Seoul, South Korea. 10Institute of Applied Physics, Seoul National University, Seoul, South Korea. npj Quantum Materials | (2025)10:101 1 Review https://doi.org/10.1038/s41535-025-00801-3 Eq. (3) is a classic example of topology bounding quantum geometry from below9. The generalization of Eq. (3) leads to the lower bound of quantum geometry due to the Euler number10–13 (the generalization is most natural in the Chern gauge for the Euler bands), and the lower bound has also been derived for obstructed atomic limits14 and chiral winding number15. Recently, the lower bound of quantum geometry has also been derived16 for the time-reversal protected Z2 topology17–20. These topological bounds allow us to put a lower bound to the geometric contribution to quantities such as the superfluid weight, as discussed in section “Superconductivity and superfluidity”. The quantization in 2D of the integral of Bij(k)/(2π) over the BZ, and its direct relation to the off-diagonal conductivity σxy1 made the study of the physical consequences of the anti-symmetric part of Q(k) one of the most active areas of research in condensed matter physics for the past 20 years. It has led to several discoveries, such as topological insulators (TIs) and superconductors21–23, Weyl and Dirac semimetals (SMs)24–26, and, more recently, higher order topological materials27–32. Conver (...truncated)