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Quantum geometric tensors from sub-bundle geometry

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Researchers have generalized the quantum geometric tensor using differential geometry, unifying Berry curvature and quantum metric in a vector bundle framework. This mathematical advancement extends quantum state analysis to parameter-dependent systems. The team introduces a sub-bundle projector and connection, revealing quantum state transport parallels to classical submanifold geometry. Generalized Gauss-Codazzi-Mainardi equations describe this relationship, offering new geometric insights. A novel quantum geometric tensor definition emerges, incorporating additional curvature contributions from state transport laws. This refinement enhances accuracy in describing complex quantum systems. The work applies these concepts to Dirac fields in curved spacetime, demonstrating how spatial curvature alters quantum geometry. Hyperbolic plane examples show measurable effects on confined fermions. This framework bridges high-energy physics and condensed matter, enabling exploration of quantum systems in curved geometries. Potential applications span from relativistic dynamics to topological materials.

Quantum geometric tensors from sub-bundle geometry

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AbstractThe geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.Popular summaryIn this paper, we provide a natural description of the differential geometry of quantum states. The central object is the quantum geometric tensor, which unifies two familiar concepts: Berry curvature, underpinning geometric phases and Hall-type effects, and the quantum metric, which gives a notion of distance between nearby quantum states that depend on external parameters. Our starting point is to treat parameter-dependent quantum states geometrically as living in a vector bundle over the parameter space, equipped with a chosen rule for transporting states (a connection) and a projector selecting physically relevant subspaces (for instance, a family of energy bands). In this language, quantum geometry becomes closely analogous to the classical geometry of surfaces embedded in a higher-dimensional space. This analogy also extends in a mathematical sense, with the geometry described by generalized Gauss-Codazzi-Mainardi relations and a shape operator that captures how the subspace bends. This perspective yields a refined definition of the quantum geometric tensor that accounts for curvature arising from the chosen transport law for the quantum states. We show that this additional contribution is relevant in general relativistic settings, such as the semiclassical dynamics of electrons described by the Dirac equation in curved spacetime, as well as in the context of condensed matter systems, where the quantum geometry of electrons confined to a hyperbolic plane is directly influenced by spatial curvature.► BibTeX data@article{Oancea2026quantumgeometric, doi = {10.22331/q-2026-01-14-1965}, url = {https://doi.org/10.22331/q-2026-01-14-1965}, title = {Quantum geometric tensors from sub-bundle geometry}, author = {Oancea, Marius A. and Mieling, Thomas B. and Palumbo, Giandomenico}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1965}, month = jan, year = {2026} }► References [1] M. V. Berry ``Quantal Phase Factors Accompanying Adiabatic Changes'' Proceedings of the Royal Society of London Series A 392 (1984). https://doi.org/10.1098/rspa.1984.0023 [2] Dariusz Chruścińskiand Andrzej Jamiołkowski ``Geometric phases in Classical and Quantum Mechanics'' Springer Science+Business Media (2004). https://doi.org/10.1007/978-0-8176-8176-0 [3] Barry Simon ``Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase'' Physical Review Letters 51 (1983). https://doi.org/10.1103/PhysRevLett.51.2167 [4] R. Resta ``The insulating state of matter: a geometrical theory'' European Physical Journal B 79 (2011). https://doi.org/10.1140/epjb/e2010-10874-4 arXiv:1012.5776 [5] Päivi Törmä ``Essay: Where Can Quantum Geometry Lead Us?'' Physical Review Letters 131 (2023). https://doi.org/10.1103/PhysRevLett.131.240001 arXiv:2312.11516 [6] Tianyu Liu, Xiao-Bin Qiang, Hai-Zhou Lu, and X C Xie, ``Quantum geometry in condensed matter'' National Science Review 12 (2024). https://doi.org/10.1093/nsr/nwae334 arXiv:2409.13408 [7] Jiabin Yu, B. Andrei Bernevig, Raquel Queiroz, Enrico Rossi, Päivi Törmä, and Bohm-Jung Yang, ``Quantum geometry in quantum materials'' npj Quantum Materials 10 (2025). https://doi.org/10.1038/s41535-025-00801-3 [8] Yiyang Jiang, Tobias Holder, and Binghai Yan, ``Revealing quantum geometry in nonlinear quantum materials'' Reports on Progress in Physics 88 (2025). https://doi.org/10.1088/1361-6633/ade454 [9] Min Yu, Pengcheng Yang, Musang Gong, Qingyun Cao, Qiuyu Lu, Haibin Liu, Shaoliang Zhang, Martin B Plenio, Fedor Jelezko, Tomoki Ozawa, Nathan Goldman, and Jianming Cai, ``Experimental measurement of the quantum geometric tensor using coupled qubits in diamond'' National Science Review 7 (2019). https://doi.org/10.1093/nsr/nwz193 [10] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, ``Measurement of the quantum geometric tensor and of the anomalous Hall drift'' Nature 578 (2020). https://doi.org/10.1038/s41586-020-1989-2 [11] Mo Chen, Changhao Li, Giandomenico Palumbo, Yan-Qing Zhu, Nathan Goldman, and Paola Cappellaro, ``A synthetic monopole source of Kalb-Ramond field in diamond'' Science 375 (2022). https://doi.org/10.1126/science.abe6437 arXiv:2008.00596 [12] Mingu Kang, Sunje Kim, Yuting Qian, Paul M. Neves, Linda Ye, Junseo Jung, Denny Puntel, Federico Mazzola, Shiang Fang, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Jun Fuji, Ivana Vobornik, Jae-Hoon Park, Joseph G. Checkelsky, Bohm-Jung Yang, and Riccardo Comin, ``Measurements of the quantum geometric tensor in solids'' Nature Physics 21 (2025). https://doi.org/10.1038/s41567-024-02678-8 arXiv:2412.17809 [13] J. P. Provostand G. Vallee ``Riemannian structure on manifolds of quantum states'' Communications in Mathematical Physics 76 (1980). https://doi.org/10.1007/BF02193559 [14] Paolo Zanardi, Paolo Giorda, and Marco Cozzini, ``Information-Theoretic Differential Geometry of Quantum Phase Transitions'' Physical Review Letters 99 (2007). https://doi.org/10.1103/PhysRevLett.99.100603 [15] Michael Kolodrubetz, Vladimir Gritsev, and Anatoli Polkovnikov, ``Classifying and measuring geometry of a quantum ground state manifold'' Physical Review B 88 (2013). https://doi.org/10.1103/PhysRevB.88.064304 arXiv:1305.0568 [16] Balázs Hetényiand Péter Lévay ``Fluctuations, uncertainty relations, and the geometry of quantum state manifolds'' Physical Review A 108 (2023). https://doi.org/10.1103/PhysRevA.108.032218 arXiv:2309.03621 [17] Javier Alvarez-Jimenez, Diego Gonzalez, Daniel Gutiérrez-Ruiz, and Jose David Vergara, ``Geometry of the Parameter Space of a Quantum System: Classical Point of View'' Annalen der Physik 532 (2020). https://doi.org/10.1002/andp.201900215 arXiv:1909.05925 [18] O. Bleu, G. Malpuech, Y. Gao, and D. D. Solnyshkov, ``Effective Theory of Nonadiabatic Quantum Evolution Based on the Quantum Geometric Tensor'' Physical Review Letters 121 (2018). https://doi.org/10.1103/PhysRevLett.121.020401 [19] Shunji Matsuuraand Shinsei Ryu ``Momentum space metric, nonlocal operator, and topological insulators'' Physical Review B 82 (2010). https://doi.org/10.1103/PhysRevB.82.245113 [20] Giandomenico Palumbo ``Momentum-space cigar geometry in topological phases'' European Physical Journal Plus 133 (2018). https://doi.org/10.1140/epjp/i2018-11856-8 arXiv:1707.00559 [21] Junyeong Ahn, Guang-Yu Guo, and Naoto Nagaosa, ``Low-Frequency Divergence and Quantum Geometry of the Bulk Photovoltaic Effect in Topological Semimetals'' Physical Review X 10 (2020). https://doi.org/10.1103/PhysRevX.10.041041 arXiv:2006.06709 [22] Christian Northe, Giandomenico Palumbo, Jonathan Sturm, Christian Tutschku, and Ewelina. M. Hankiewicz, ``Interplay of band geometry and topology in ideal Chern insulators in the presence of external electromagnetic fields'' Physical Review B 105 (2022). https://doi.org/10.1103/PhysRevB.105.155410 arXiv:2112.15559 [23] Ansgar Grafand Frédéric Piéchon ``Berry curvature and quantum metric in $N$-band systems: An eigenprojector approach'' Physical Review B 104 (2021). https://doi.org/10.1103/PhysRevB.104.085114 arXiv:2102.09899 [24] Adrien Bouhon, Abigail Timmel, and Robert-Jan Slager, ``Quantum geometry beyond projective single bands'' (2023). https://doi.org/10.48550/arXiv.2303.02180 arXiv:2303.02180 [25] Daniel Kaplan, Tobias Holder, and Binghai Yan, ``Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry'' Physical Review Letters 132 (2024). https://doi.org/10.1103/PhysRevLett.132.026301 arXiv:2211.17213 [26] Antimo Marrazzoand Raffaele Resta ``Local Theory of the Insulating State'' Physical Review Letters 122 (2019). https://doi.org/10.1103/PhysRevLett.122.166602 arXiv:1807.01063 [27] Matheus S. M. de Sousa, Antonio L. Cruz, and Wei Chen, ``Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker'' Physical Review B 107 (2023). https://doi.org/10.1103/PhysRevB.107.205133 arXiv:2301.06493 [28] Yu-Quan Ma, Shu Chen, Heng Fan, and Wu-Ming Liu, ``Abelian and non-Abelian quantum geometric tensor'' Physical Review B 81 (2010). https://doi.org/10.1103/PhysRevB.81.245129 arXiv:1003.4040 [29] Titus Neupert, Claudio Chamon, and Christopher Mudry, ``Measuring the quantum geometry of Bloch bands with current noise'' Physical Review B 87 (2013). https://doi.org/10.1103/PhysRevB.87.245103 arXiv:1303.4643 [30] Bruno Meraand Johannes Mitscherling ``Nontrivial quantum geometry of degenerate flat bands'' Physical Review B 106 (2022). https://doi.org/10.1103/PhysRevB.106.165133 arXiv:2205.07900 [31] Hai-Tao Ding, Yan-Qing Zhu, Peng He, Yu-Guo Liu, Jian-Te Wang, Dan-Wei Zhang, and Shi-Liang Zhu, ``Extracting non-Abelian quantum metric tensor and its related Chern numbers'' Physical Review A 105 (2022). https://doi.org/10.1103/PhysRevA.105.012210 arXiv:2201.01086 [32] Alexander Avdoshkinand Fedor K. Popov ``Extrinsic geometry of quantum states'' Physical Review B 107 (2023). https://doi.org/10.1103/PhysRevB.107.245136 arXiv:2205.15353 [33] Alexander Avdoshkin ``Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians'' (2024). https://doi.org/10.48550/arXiv.2404.03234 arXiv:2404.03234 [34] Kai Chen, Bishnu Karki, and Pavan Hosur, ``The Effect of the Non-Abelian Quantum Metric on Superfluidity'' (2025). https://doi.org/10.48550/arXiv.2501.16965 arXiv:2501.16965 [35] Wojciech J. Jankowski, Robert-Jan Slager, and Michele Pizzochero, ``Enhancing the Hyperpolarizability of Crystals with Quantum Geometry'' Physical Review Letters 135 (2025). https://doi.org/10.1103/z7lp-pqp6 [36] Johannes Mitscherling, Alexander Avdoshkin, and Joel E. Moore, ``Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals'' Physical Review B 112 (2025). https://doi.org/10.1103/qscv-qxqt [37] Sebastiano Peottaand Päivi Törmä ``Superfluidity in topologically nontrivial flat bands'' Nature Communications 6 (2015). https://doi.org/10.1038/ncomms9944 arXiv:1506.02815 [38] Long Liang, Tuomas I. Vanhala, Sebastiano Peotta, Topi Siro, Ari Harju, and Päivi Törmä, ``Band geometry, Berry curvature, and superfluid weight'' Physical Review B 95 (2017). https://doi.org/10.1103/PhysRevB.95.024515 arXiv:1610.01803 [39] Jonah Herzog-Arbeitman, Valerio Peri, Frank Schindler, Sebastian D. Huber, and B. Andrei Bernevig, ``Superfluid Weight Bounds from Symmetry and Quantum Geometry in Flat Bands'' Physical Review Letters 128 (2022). https://doi.org/10.1103/PhysRevLett.128.087002 arXiv:2110.14663 [40] Kristian Hauser A.

Villegasand Bo Yang ``Anomalous Higgs oscillations mediated by Berry curvature and quantum metric'' Physical Review B 104 (2021). https://doi.org/10.1103/PhysRevB.104.L180502 arXiv:2010.07751 [41] Giandomenico Palumboand Nathan Goldman ``Revealing Tensor Monopoles through Quantum-Metric Measurements'' Physical Review Letters 121 (2018). https://doi.org/10.1103/PhysRevLett.121.170401 arXiv:1805.01247 [42] Hai-Tao Ding, Yan-Qing Zhu, Zhi Li, and Lubing Shao, ``Tensor monopoles and the negative magnetoresistance effect in optical lattices'' Physical Review A 102 (2020). https://doi.org/10.1103/PhysRevA.102.053325 arXiv:2007.02093 [43] Xinsheng Tan, Dan-Wei Zhang, Wen Zheng, Xiaopei Yang, Shuqing Song, Zhikun Han, Yuqian Dong, Zhimin Wang, Dong Lan, Hui Yan, Shi-Liang Zhu, and Yang Yu, ``Experimental Observation of Tensor Monopoles with a Superconducting Qudit'' Physical Review Letters 126 (2021). https://doi.org/10.1103/PhysRevLett.126.017702 arXiv:2006.11770 [44] Wojciech J. Jankowski, Arthur S. Morris, Adrien Bouhon, F. Nur Ünal, and Robert-Jan Slager, ``Optical manifestations and bounds of topological Euler class'' Physical Review B 111 (2025). https://doi.org/10.1103/PhysRevB.111.L081103 arXiv:2311.07545 [45] Wojciech J. Jankowski, Arthur S. Morris, Zory Davoyan, Adrien Bouhon, F. Nur Ünal, and Robert-Jan Slager, ``Non-Abelian Hopf-Euler insulators'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.075135 arXiv:2405.17305 [46] Soonhyun Kwonand Bohm-Jung Yang ``Quantum geometric bound and ideal condition for Euler band topology'' Physical Review B 109 (2024). https://doi.org/10.1103/PhysRevB.109.L161111 arXiv:2311.11577 [47] Da-Jian Zhang, Qing-hai Wang, and Jiangbin Gong, ``Quantum geometric tensor in $\mathscr{PT}$-symmetric quantum mechanics'' Physical Review A 99 (2019). https://doi.org/10.1103/PhysRevA.99.042104 arXiv:1811.04638 [48] Yan-Qing Zhu, Wen Zheng, Shi-Liang Zhu, and Giandomenico Palumbo, ``Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric'' Physical Review B 104 (2021). https://doi.org/10.1103/PhysRevB.104.205103 arXiv:2106.09648 [49] Bar Alon, Roni Ilan, and Moshe Goldstein, ``Quantum metric dependent anomalous velocity in systems subject to complex electric fields'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.245103 arXiv:2402.01312 [50] C. Martínez-Strasser, M. A. J. Herrera, A. García-Etxarri, G. Palumbo, F. K. Kunst, and D. Bercioux, ``Topological Properties of a Non-Hermitian Quasi-1D Chain with a Flat Band'' Advanced Quantum Technologies 7 (2024). https://doi.org/10.1002/qute.202300225 arXiv:2307.08754 [51] Chao Chen Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. Morais Smith, ``Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems'' Physical Review Research 6 (2024). https://doi.org/10.1103/PhysRevResearch.6.023202 arXiv:2305.17675 [52] Grazia Salerno, Nathan Goldman, and Giandomenico Palumbo, ``Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric'' Physical Review Research 2 (2020). https://doi.org/10.1103/PhysRevResearch.2.013224 [53] Davide Rattacaso, Patrizia Vitale, and Alioscia Hamma, ``Quantum geometric tensor away from equilibrium'' Journal of Physics Communications 4 (2020). https://doi.org/10.1088/2399-6528/ab9505 arXiv:1912.02677 [54] Longwen Zhou ``Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.054310 arXiv:2408.05525 [55] Mohit Pandey, Pieter W. Claeys, David K. Campbell, Anatoli Polkovnikov, and Dries Sels, ``Adiabatic Eigenstate Deformations as a Sensitive Probe for Quantum Chaos'' Physical Review X 10 (2020). https://doi.org/10.1103/PhysRevX.10.041017 arXiv:2004.05043 [56] Rahul Roy ``Band geometry of fractional topological insulators'' Physical Review B 90 (2014). https://doi.org/10.1103/PhysRevB.90.165139 [57] Martin Claassen, Ching Hua Lee, Ronny Thomale, Xiao-Liang Qi, and Thomas P. Devereaux, ``Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators'' Physical Review Letters 114 (2015). https://doi.org/10.1103/PhysRevLett.114.236802 arXiv:1502.06998 [58] Bruno Meraand Tomoki Ozawa ``Kähler geometry and Chern insulators: Relations between topology and the quantum metric'' Physical Review B 104 (2021). https://doi.org/10.1103/PhysRevB.104.045104 arXiv:2103.11583 [59] Jie Wang, Jennifer Cano, Andrew J. Millis, Zhao Liu, and Bo Yang, ``Exact Landau Level Description of Geometry and Interaction in a Flatband'' Physical Review Letters 127 (2021). https://doi.org/10.1103/PhysRevLett.127.246403 arXiv:2105.07491 [60] Zhao Liu, Bruno Mera, Manato Fujimoto, Tomoki Ozawa, and Jie Wang, ``Theory of Generalized Landau Levels and Its Implications for Non-Abelian States'' Physical Review X 15 (2025). https://doi.org/10.1103/1zg9-qbd6 [61] G. Salerno, T. Ozawa, and P. Törmä, ``Drude weight and the many-body quantum metric in one-dimensional Bose systems'' Physical Review B 108 (2023). https://doi.org/10.1103/PhysRevB.108.L140503 [62] Wei Chenand Gero von Gersdorff ``Measurement of interaction-dressed Berry curvature and quantum metric in solids by optical absorption'' SciPost Physics Core 5 (2022). https://doi.org/10.21468/SciPostPhysCore.5.3.040 arXiv:2202.03494 [63] Pavlo Sukhachov, Niels Henrik Aase, Kristian Mæland, and Asle Sudbø, ``Effect of the Hubbard interaction on the quantum metric'' Physical Review B 111 (2025). https://doi.org/10.1103/PhysRevB.111.085143 arXiv:2412.02753 [64] Ahmed Abouelkomsan, Kang Yang, and Emil J. Bergholtz, ``Quantum metric induced phases in Moiré materials'' Physical Review Research 5 (2023). https://doi.org/10.1103/PhysRevResearch.5.L012015 arXiv:2202.10467 [65] Ohad Antebi, Johannes Mitscherling, and Tobias Holder, ``Drude weight of an interacting flat-band metal'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.L241111 arXiv:2407.09599 [66] Patrick J. Ledwith, Ashvin Vishwanath, and Daniel E. Parker, ``Vortexability: A unifying criterion for ideal fractional Chern insulators'' Physical Review B 108 (2023). https://doi.org/10.1103/PhysRevB.108.205144 arXiv:2209.15023 [67] Manato Fujimoto, Daniel E. Parker, Junkai Dong, Eslam Khalaf, Ashvin Vishwanath, and Patrick Ledwith, ``Higher Vortexability: Zero-Field Realization of Higher Landau Levels'' Physical Review Letters 134 (2025). https://doi.org/10.1103/PhysRevLett.134.106502 [68] Isaac Tesfayeand André Eckardt ``Quantum geometry of bosonic Bogoliubov quasiparticles'' Physical Review Research 7 (2025). https://doi.org/10.1103/1sz1-nwzf [69] Wojciech J. Jankowski, Robert-Jan Slager, and Gunnar F. Lange, ``Quantum geometric bounds in spinful systems with trivial band topology'' Physical Review Research 7 (2025). https://doi.org/10.1103/zlxq-fxgc [70] Yu-Ping Linand Giandomenico Palumbo ``Geometric semimetals and their simulation in synthetic matter'' Physical Review B 109 (2024). https://doi.org/10.1103/PhysRevB.109.L201107 arXiv:2309.06442 [71] Hector Silva, Bruno Mera, and Nikola Paunković, ``Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions'' Physical Review B 103 (2021). https://doi.org/10.1103/PhysRevB.103.085127 arXiv:2010.06629 [72] Zheng Zhou, Xu-Yang Hou, Xin Wang, Jia-Chen Tang, Hao Guo, and Chih-Chun Chien, ``Sjöqvist quantum geometric tensor of finite-temperature mixed states'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.035404 arXiv:2403.06944 [73] Shoshichi Kobayashi ``Differential geometry of complex vector bundles'' Princeton University Press (1987). https://doi.org/10.1515/9781400858682 [74] Jean-Pierre Demailly ``Complex Analytic and Differential Geometry'' (2012). https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf [75] Michael Spivak ``A Comprehensive Introduction to Differential Geometry'' Publish or Perish (1999). [76] Michael Spivak ``A Comprehensive Introduction to Differential Geometry'' Publish or Perish (1999). [77] Marcos Dajczerand Ruy Tojeiro ``Submanifold Theory'' Springer Science+Business Media (2019). https://doi.org/10.1007/978-1-4939-9644-5 [78] John M. Lee ``Introduction to Riemannian Manifolds'' Springer (2018). https://doi.org/10.1007/978-3-319-91755-9 [79] Michael V Berry ``The quantum phase, five years after'' World Scientific (1989). https://doi.org/10.1142/9789812798381_0001 [80] Di Xiao, Junren Shi, and Qian Niu, ``Berry Phase Correction to Electron Density of States in Solids'' Physical Review Letters 95 (2005). https://doi.org/10.1103/PhysRevLett.95.137204 [81] Di Xiao, Ming-Che Chang, and Qian Niu, ``Berry phase effects on electronic properties'' Reviews of Modern Physics 82 (2010). https://doi.org/10.1103/RevModPhys.82.1959 arXiv:0907.2021 [82] Lars Anderssonand Marius A. Oancea ``Spin Hall effects in the sky'' Classical and Quantum Gravity 40 (2023). https://doi.org/10.1088/1361-6382/ace021 arXiv:2302.13634 [83] Pierre Gosselin, Alain Bérard, and Hervé Mohrbach, ``Spin Hall effect of photons in a static gravitational field'' Physical Review D 75 (2007). https://doi.org/10.1103/PhysRevD.75.084035 [84] Marius A. Oancea, Jérémie Joudioux, I. Y. Dodin, D. E. Ruiz, Claudio F. Paganini, and Lars Andersson, ``Gravitational spin Hall effect of light'' Physical Review D 102 (2020). https://doi.org/10.1103/PhysRevD.102.024075 arXiv:2003.04553 [85] Abraham I. Harteand Marius A. Oancea ``Spin Hall effects and the localization of massless spinning particles'' Physical Review D 105 (2022). https://doi.org/10.1103/PhysRevD.105.104061 arXiv:2203.01753 [86] Marius A.

Oanceaand Tiberiu Harko ``Weyl geometric effects on the propagation of light in gravitational fields'' Physical Review D 109 (2024). https://doi.org/10.1103/PhysRevD.109.064020 arXiv:2305.01313 [87] Naoki Yamamoto ``Spin Hall effect of gravitational waves'' Physical Review D 98 (2018). https://doi.org/10.1103/PhysRevD.98.061701 arXiv:1708.03113 [88] Lars Andersson, Jérémie Joudioux, Marius A. Oancea, and Ayush Raj, ``Propagation of polarized gravitational waves'' Physical Review D 103 (2021). https://doi.org/10.1103/PhysRevD.103.044053 arXiv:2012.08363 [89] Marius A. Oancea, Richard Stiskalek, and Miguel Zumalacárregui, ``Frequency- and polarization-dependent lensing of gravitational waves in strong gravitational fields'' Physical Review D 109 (2024). https://doi.org/10.1103/PhysRevD.109.124045 arXiv:2209.06459 [90] Marius A. Oancea, Richard Stiskalek, and Miguel Zumalacárregui, ``Probing general relativistic spin-orbit coupling with gravitational waves from hierarchical triple systems'' Monthly Notices of the Royal Astronomical Society: Letters 535 (2024). https://doi.org/10.1093/mnrasl/slae084 arXiv:2307.01903 [91] Pierre Gosselin, Alain Bérard, and Hervé Mohrbach, ``Semiclassical dynamics of Dirac particles interacting with a static gravitational field'' Physics Letters A 368 (2007). https://doi.org/10.1016/j.physleta.2007.04.022 [92] Marius A.

Oanceaand Achal Kumar ``Semiclassical analysis of Dirac fields on curved spacetime'' Physical Review D 107 (2023). https://doi.org/10.1103/PhysRevD.107.044029 arXiv:2212.04414 [93] Tomoya Hayataand Yoshimasa Hidaka ``Kinetic derivation of generalized phase space Chern-Simons theory'' Physical Review B 95 (2017). https://doi.org/10.1103/PhysRevB.95.125137 arXiv:1610.01879 [94] Nishchhal Vermaand Raquel Queiroz ``Instantaneous response and quantum geometry of insulators'' Proceedings of the National Academy of Sciences 122 (2025). https://doi.org/10.1073/pnas.2405837122 [95] Bogar Díaz, Diego Gonzalez, Marcos J Hernández, and J David Vergara, ``Time-dependent quantum geometric tensor and some applications'' Physica Scripta 100 (2025). https://doi.org/10.1088/1402-4896/ade8af [96] Alicia J. Kollár, Mattias Fitzpatrick, and Andrew A. Houck, ``Hyperbolic lattices in circuit quantum electrodynamics'' Nature 571 (2019). https://doi.org/10.1038/s41586-019-1348-3 arXiv:1802.09549 [97] Sunkyu Yu, Xianji Piao, and Namkyoo Park, ``Topological Hyperbolic Lattices'' Physical Review Letters 125 (2020). https://doi.org/10.1103/PhysRevLett.125.053901 arXiv:2003.07002 [98] Weixuan Zhang, Hao Yuan, Na Sun, Houjun Sun, and Xiangdong Zhang, ``Observation of novel topological states in hyperbolic lattices'' Nature Communications 13 (2022). https://doi.org/10.1038/s41467-022-30631-x arXiv:2203.03214 [99] David M. Urwyler, Patrick M. Lenggenhager, Igor Boettcher, Ronny Thomale, Titus Neupert, and Tomáš Bzdušek, ``Hyperbolic Topological Band Insulators'' Physical Review Letters 129 (2022). https://doi.org/10.1103/PhysRevLett.129.246402 arXiv:2203.07292 [100] Tomáš Bzdušekand Joseph Maciejko ``Flat bands and band-touching from real-space topology in hyperbolic lattices'' Physical Review B 106 (2022). https://doi.org/10.1103/PhysRevB.106.155146 arXiv:2205.11571 [101] Anffany Chen, Yifei Guan, Patrick M. Lenggenhager, Joseph Maciejko, Igor Boettcher, and Tomáš Bzdušek, ``Symmetry and topology of hyperbolic Haldane models'' Physical Review B 108 (2023). https://doi.org/10.1103/PhysRevB.108.085114 arXiv:2304.03273 [102] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, ``Dirac equation for cold atoms in artificial curved spacetimes'' New Journal of Physics 13 (2011). https://doi.org/10.1088/1367-2630/13/3/035002 arXiv:1010.1716 [103] Alex Westströmand Teemu Ojanen ``Designer Curved-Space Geometry for Relativistic Fermions in Weyl Metamaterials'' Physical Review X 7 (2017). https://doi.org/10.1103/PhysRevX.7.041026 arXiv:1703.10408 [104] Giandomenico Palumbo, Roberto Catenacci, and Annalisa Marzuoli, ``Topological Effective Field Theories for Dirac Fermions from Index Theorem'' International Journal of Modern Physics B 28 (2014). https://doi.org/10.1142/S0217979213501932 arXiv:1303.6468 [105] Alfredo Iorioand Gaetano Lambiase ``Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that'' Physical Review D 90 (2014). https://doi.org/10.1103/PhysRevD.90.025006 arXiv:1308.0265 [106] Eran Lustig, Moshe-Ishay Cohen, Rivka Bekenstein, Gal Harari, Miguel A. Bandres, and Mordechai Segev, ``Curved-space topological phases in photonic lattices'' Physical Review A 96 (2017). https://doi.org/10.1103/PhysRevA.96.041804 [107] Ygor Pará, Giandomenico Palumbo, and Tommaso Macrì, ``Probing non-Hermitian phase transitions in curved space via quench dynamics'' Physical Review B 103 (2021). https://doi.org/10.1103/PhysRevB.103.155417 arXiv:2012.07909 [108] Giandomenico Palumbo ``Geometric model of topological insulators from the Maxwell algebra'' Annals of Physics 386 (2017). https://doi.org/10.1016/j.aop.2017.08.018 arXiv:1610.04734 [109] Matthew D. Horner, Andrew Hallam, and Jiannis K. Pachos, ``Chiral Spin-Chain Interfaces Exhibiting Event-Horizon Physics'' Physical Review Letters 130 (2023). https://doi.org/10.1103/PhysRevLett.130.016701 arXiv:2207.08840 [110] Andreas Haller, Suraj Hegde, Chen Xu, Christophe De Beule, Thomas L. Schmidt, and Tobias Meng, ``Black hole mirages: Electron lensing and Berry curvature effects in inhomogeneously tilted Weyl semimetals'' SciPost Physics 14 (2023). https://doi.org/10.21468/SciPostPhys.14.5.119 arXiv:2210.16254 [111] Glenn Wagner, Fernando de Juan, and Dung Nguyen, ``Landau levels in curved space realized in strained graphene'' SciPost Physics Core 5 (2022). https://doi.org/10.21468/SciPostPhysCore.5.2.029 arXiv:1911.02028 [112] Sara Laurilaand Jaakko Nissinen ``Torsional Landau levels and geometric anomalies in condensed matter Weyl systems'' Physical Review B 102 (2020). https://doi.org/10.1103/PhysRevB.102.235163 [113] Giandomenico Palumboand Jiannis K. Pachos ``Holographic correspondence in topological superconductors'' Annals of Physics 372 (2016). https://doi.org/10.1016/j.aop.2016.05.005 arXiv:1507.03236 [114] Maximilian Fürst, Denis Kochan, Ioachim-Gheorghe Dusa, Cosimo Gorini, and Klaus Richter, ``Dirac Landau levels for surfaces with constant negative curvature'' Physical Review B 109 (2024). https://doi.org/10.1103/PhysRevB.109.195433 arXiv:2307.09221 [115] N.W. Ashcroftand N.D. Mermin ``Solid State Physics'' Holt, Rinehart Winston (1976). [116] Robert G. Littlejohnand William G. Flynn ``Geometric phases in the asymptotic theory of coupled wave equations'' Physical Review A 44 (1991). https://doi.org/10.1103/PhysRevA.44.5239 [117] C. Emmrichand A. Weinstein ``Geometry of the transport equation in multicomponent WKB approximations'' Communications in Mathematical Physics 176 (1996). https://doi.org/10.1007/BF02099256 [118] Ali Mostafazadeh ``Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian'' Journal of Mathematical Physics 43 (2002). https://doi.org/10.1063/1.1418246 [119] Ali Mostafazadeh ``Pseudo-Hermitian Representation of Quantum Mechanics'' International Journal of Geometric Methods in Modern Physics 07 (2010). https://doi.org/10.1142/S0219887810004816 arXiv:0810.5643 [120] Jorge Romero, Carlos A. Velasquez, and J. David Vergara, ``N-bein formalism for the parameter space of quantum geometry'' Journal of Physics A: Mathematical and Theoretical 57 (2024). https://doi.org/10.1088/1751-8121/ad6f7f arXiv:2406.19468 [121] E. Dobardžić, M. Dimitrijević, and M. V. Milovanović, ``Effective description of Chern insulators'' Physical Review B 89 (2014). https://doi.org/10.1103/PhysRevB.89.235424 arXiv:1403.4000 [122] Ryan Requist ``Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders'' Journal of Physics A Mathematical General 56 (2023). https://doi.org/10.1088/1751-8121/ad0349 arXiv:2206.01716 [123] A. Comtetand P. J. Houston ``Effective action on the hyperbolic plane in a constant external field'' Journal of Mathematical Physics 26 (1985). https://doi.org/10.1063/1.526781 [124] E. V. Gorbarand V. P. Gusynin ``Gap generation for Dirac fermions on Lobachevsky plane in a magnetic field'' Annals of Physics 323 (2008). https://doi.org/10.1016/j.aop.2007.11.005 arXiv:0710.2292 [125] Matthias Ludewigand Guo Chuan Thiang ``Gaplessness of Landau Hamiltonians on Hyperbolic Half-planes via Coarse Geometry'' Communications in Mathematical Physics 386 (2021). https://doi.org/10.1007/s00220-021-04068-0 arXiv:2009.07688 [126] Tarun Tummuru, Anffany Chen, Patrick M. Lenggenhager, Titus Neupert, Joseph Maciejko, and Tomáš Bzdušek, ``Hyperbolic Non-Abelian Semimetal'' Physical Review Letters 132 (2024). https://doi.org/10.1103/PhysRevLett.132.206601 arXiv:2307.09876 [127] Kazuki Ikeda, Shoto Aoki, and Yoshiyuki Matsuki, ``Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface'' Journal of Physics Condensed Matter 33 (2021). https://doi.org/10.1088/1361-648X/ac24c4 arXiv:2104.13314 [128] Bitan Roy ``Magnetic catalysis in weakly interacting hyperbolic Dirac materials'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.245117 arXiv:2305.11174 [129] Noble Gluscevich, Abhisek Samanta, Sourav Manna, and Bitan Roy, ``Dynamic mass generation on two-dimensional electronic hyperbolic lattices'' Physical Review B 111 (2025). https://doi.org/10.1103/PhysRevB.111.L121108 [130] Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, and Margaret Dillard-Bleick, ``Analysis, Manifolds and Physics'' North-Holland (1982). [131] Vladimir Igorevich Arnold ``Mathematical Methods of Classical Mechanics'' Springer Science+Business Media (1989). https://doi.org/10.1007/978-1-4757-2063-1 [132] Michael Stone, Vatsal Dwivedi, and Tianci Zhou, ``Berry phase, Lorentz covariance, and anomalous velocity for Dirac and Weyl particles'' Physical Review D 91 (2015). https://doi.org/10.1103/PhysRevD.91.025004 arXiv:1406.0354 [133] Anffany Chen, Hauke Brand, Tobias Helbig, Tobias Hofmann, Stefan Imhof, Alexander Fritzsche, Tobias Kießling, Alexander Stegmaier, Lavi K. Upreti, Titus Neupert, Tomáš Bzdušek, Martin Greiter, Ronny Thomale, and Igor Boettcher, ``Hyperbolic matter in electrical circuits with tunable complex phases'' Nature Communications 14 (2023). https://doi.org/10.1038/s41467-023-36359-6 arXiv:2205.05106 [134] Lei Huang, Lu He, Weixuan Zhang, Huizhen Zhang, Dongning Liu, Xue Feng, Fang Liu, Kaiyu Cui, Yidong Huang, Wei Zhang, and Xiangdong Zhang, ``Hyperbolic photonic topological insulators'' Nature Communications 15 (2024). https://doi.org/10.1038/s41467-024-46035-y arXiv:2401.16006 [135] Canon Sun, Anffany Chen, Tomas Bzdusek, and Joseph Maciejko, ``Topological linear response of hyperbolic Chern insulators'' SciPost Physics 17 (2024). https://doi.org/10.21468/SciPostPhys.17.5.124 arXiv:2406.08388 [136] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, ``Two-dimensional gas of massless Dirac fermions in graphene'' Nature 438 (2005). https://doi.org/10.1038/nature04233 [137] Jiannis K. Pachos, Michael Stone, and Kristan Temme, ``Graphene with Geometrically Induced Vorticity'' Physical Review Letters 100 (2008). https://doi.org/10.1103/PhysRevLett.100.156806 arXiv:0710.0858 [138] A. Marzuoliand G. Palumbo ``BF-theory in graphene: A route toward topological quantum computing?'' Europhysics Letters 99 (2012). https://doi.org/10.1209/0295-5075/99/10002 arXiv:1111.1593 [139] Giandomenico Palumboand Jiannis K. Pachos ``Abelian Chern-Simons-Maxwell Theory from a Tight-Binding Model of Spinless Fermions'' Physical Review Letters 110 (2013). https://doi.org/10.1103/PhysRevLett.110.211603 arXiv:1301.2625 [140] Ai-Lei He, Lu Qi, Yongjun Liu, and Yi-Fei Wang, ``Hyperbolic fractional Chern insulators'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.195113 arXiv:2407.05706 [141] Benoit Estienne, Nicolas Regnault, and Valentin Crépel, ``Ideal Chern bands as Landau levels in curved space'' Physical Review Research 5 (2023). https://doi.org/10.1103/PhysRevResearch.5.L032048 arXiv:2304.01251 [142] T. Can, M. Laskin, and P. Wiegmann, ``Fractional Quantum Hall Effect in a Curved Space: Gravitational Anomaly and Electromagnetic Response'' Physical Review Letters 113 (2014). https://doi.org/10.1103/PhysRevLett.113.046803 arXiv:1402.1531 [143] Giandomenico Palumbo ``Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields'' Journal of Physics A Mathematical General 56 (2023). https://doi.org/10.1088/1751-8121/ad018b arXiv:2304.04002 [144] Santanu Dey, Anffany Chen, Pablo Basteiro, Alexander Fritzsche, Martin Greiter, Matthias Kaminski, Patrick M. Lenggenhager, René Meyer, Riccardo Sorbello, Alexander Stegmaier, Ronny Thomale, Johanna Erdmenger, and Igor Boettcher, ``Simulating Holographic Conformal Field Theories on Hyperbolic Lattices'' Physical Review Letters 133 (2024). https://doi.org/10.1103/PhysRevLett.133.061603 arXiv:2404.03062 [145] Patrick M. Lenggenhager, Santanu Dey, Tomá šBzdu šek, and Joseph Maciejko, ``Hyperbolic Spin Liquids'' Physical Review Letters 135 (2025). https://doi.org/10.1103/s25y-s4fj [146] Sangyun Lee, Junhee Shin, Hogyun Jeong, and Hosub Jin, ``Quantum geometric tensor from real-space twists of chiral chains'' Physical Review B 110 (2024). https://doi.org/10.1103/PhysRevB.110.195139 [147] Junyeong Ahn, Guang-Yu Guo, Naoto Nagaosa, and Ashvin Vishwanath, ``Riemannian geometry of resonant optical responses'' Nature Physics 18 (2022). https://doi.org/10.1038/s41567-021-01465-z arXiv:2103.01241 [148] Wojciech J. Jankowskiand Robert-Jan Slager ``Quantized Integrated Shift Effect in Multigap Topological Phases'' Physical Review Letters 133 (2024). https://doi.org/10.1103/PhysRevLett.133.186601 [149] Ze-Min Huang, Longyue Li, Jianhui Zhou, and Hong-Hao Zhang, ``Torsional response and Liouville anomaly in Weyl semimetals with dislocations'' Physical Review B 99 (2019). https://doi.org/10.1103/PhysRevB.99.155152 arXiv:1807.09183 [150] Giandomenico Palumbo ``Weyl Geometry in Weyl Semimetals'' (2024). https://doi.org/10.48550/arXiv.2412.04743 arXiv:2412.04743Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-14 16:30:16: Could not fetch cited-by data for 10.22331/q-2026-01-14-1965 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-14 16:30:17: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.Popular summaryIn this paper, we provide a natural description of the differential geometry of quantum states. The central object is the quantum geometric tensor, which unifies two familiar concepts: Berry curvature, underpinning geometric phases and Hall-type effects, and the quantum metric, which gives a notion of distance between nearby quantum states that depend on external parameters. Our starting point is to treat parameter-dependent quantum states geometrically as living in a vector bundle over the parameter space, equipped with a chosen rule for transporting states (a connection) and a projector selecting physically relevant subspaces (for instance, a family of energy bands). In this language, quantum geometry becomes closely analogous to the classical geometry of surfaces embedded in a higher-dimensional space. This analogy also extends in a mathematical sense, with the geometry described by generalized Gauss-Codazzi-Mainardi relations and a shape operator that captures how the subspace bends. This perspective yields a refined definition of the quantum geometric tensor that accounts for curvature arising from the chosen transport law for the quantum states. We show that this additional contribution is relevant in general relativistic settings, such as the semiclassical dynamics of electrons described by the Dirac equation in curved spacetime, as well as in the context of condensed matter systems, where the quantum geometry of electrons confined to a hyperbolic plane is directly influenced by spatial curvature.► BibTeX data@article{Oancea2026quantumgeometric, doi = {10.22331/q-2026-01-14-1965}, url = {https://doi.org/10.22331/q-2026-01-14-1965}, title = {Quantum geometric tensors from sub-bundle geometry}, author = {Oancea, Marius A. and Mieling, Thomas B. and Palumbo, Giandomenico}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1965}, month = jan, year = {2026} }► References [1] M. V. Berry ``Quantal Phase Factors Accompanying Adiabatic Changes'' Proceedings of the Royal Society of London Series A 392 (1984). https://doi.org/10.1098/rspa.1984.0023 [2] Dariusz Chruścińskiand Andrzej Jamiołkowski ``Geometric phases in Classical and Quantum Mechanics'' Springer Science+Business Media (2004). https://doi.org/10.1007/978-0-8176-8176-0 [3] Barry Simon ``Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase'' Physical Review Letters 51 (1983). https://doi.org/10.1103/PhysRevLett.51.2167 [4] R. Resta ``The insulating state of matter: a geometrical theory'' European Physical Journal B 79 (2011). https://doi.org/10.1140/epjb/e2010-10874-4 arXiv:1012.5776 [5] Päivi Törmä ``Essay: Where Can Quantum Geometry Lead Us?'' Physical Review Letters 131 (2023). https://doi.org/10.1103/PhysRevLett.131.240001 arXiv:2312.11516 [6] Tianyu Liu, Xiao-Bin Qiang, Hai-Zhou Lu, and X C Xie, ``Quantum geometry in condensed matter'' National Science Review 12 (2024). https://doi.org/10.1093/nsr/nwae334 arXiv:2409.13408 [7] Jiabin Yu, B. Andrei Bernevig, Raquel Queiroz, Enrico Rossi, Päivi Törmä, and Bohm-Jung Yang, ``Quantum geometry in quantum materials'' npj Quantum Materials 10 (2025). https://doi.org/10.1038/s41535-025-00801-3 [8] Yiyang Jiang, Tobias Holder, and Binghai Yan, ``Revealing quantum geometry in nonlinear quantum materials'' Reports on Progress in Physics 88 (2025). https://doi.org/10.1088/1361-6633/ade454 [9] Min Yu, Pengcheng Yang, Musang Gong, Qingyun Cao, Qiuyu Lu, Haibin Liu, Shaoliang Zhang, Martin B Plenio, Fedor Jelezko, Tomoki Ozawa, Nathan Goldman, and Jianming Cai, ``Experimental measurement of the quantum geometric tensor using coupled qubits in diamond'' National Science Review 7 (2019). https://doi.org/10.1093/nsr/nwz193 [10] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, ``Measurement of the quantum geometric tensor and of the anomalous Hall drift'' Nature 578 (2020). https://doi.org/10.1038/s41586-020-1989-2 [11] Mo Chen, Changhao Li, Giandomenico Palumbo, Yan-Qing Zhu, Nathan Goldman, and Paola Cappellaro, ``A synthetic monopole source of Kalb-Ramond field in diamond'' Science 375 (2022). https://doi.org/10.1126/science.abe6437 arXiv:2008.00596 [12] Mingu Kang, Sunje Kim, Yuting Qian, Paul M. Neves, Linda Ye, Junseo Jung, Denny Puntel, Federico Mazzola, Shiang Fang, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Jun Fuji, Ivana Vobornik, Jae-Hoon Park, Joseph G. Checkelsky, Bohm-Jung Yang, and Riccardo Comin, ``Measurements of the quantum geometric tensor in solids'' Nature Physics 21 (2025). https://doi.org/10.1038/s41567-024-02678-8 arXiv:2412.17809 [13] J. P. Provostand G. Vallee ``Riemannian structure on manifolds of quantum states'' Communications in Mathematical Physics 76 (1980). https://doi.org/10.1007/BF02193559 [14] Paolo Zanardi, Paolo Giorda, and Marco Cozzini, ``Information-Theoretic Differential Geometry of Quantum Phase Transitions'' Physical Review Letters 99 (2007). https://doi.org/10.1103/PhysRevLett.99.100603 [15] Michael Kolodrubetz, Vladimir Gritsev, and Anatoli Polkovnikov, ``Classifying and measuring geometry of a quantum ground state manifold'' Physical Review B 88 (2013). https://doi.org/10.1103/PhysRevB.88.064304 arXiv:1305.0568 [16] Balázs Hetényiand Péter Lévay ``Fluctuations, uncertainty relations, and the geometry of quantum state manifolds'' Physical Review A 108 (2023). https://doi.org/10.1103/PhysRevA.108.032218 arXiv:2309.03621 [17] Javier Alvarez-Jimenez, Diego Gonzalez, Daniel Gutiérrez-Ruiz, and Jose David Vergara, ``Geometry of the Parameter Space of a Quantum System: Classical Point of View'' Annalen der Physik 532 (2020). https://doi.org/10.1002/andp.201900215 arXiv:1909.05925 [18] O. Bleu, G. Malpuech, Y. Gao, and D. D. Solnyshkov, ``Effective Theory of Nonadiabatic Quantum Evolution Based on the Quantum Geometric Tensor'' Physical Review Letters 121 (2018). https://doi.org/10.1103/PhysRevLett.121.020401 [19] Shunji Matsuuraand Shinsei Ryu ``Momentum space metric, nonlocal operator, and topological insulators'' Physical Review B 82 (2010). https://doi.org/10.1103/PhysRevB.82.245113 [20] Giandomenico Palumbo ``Momentum-space cigar geometry in topological phases'' European Physical Journal Plus 133 (2018). https://doi.org/10.1140/epjp/i2018-11856-8 arXiv:1707.00559 [21] Junyeong Ahn, Guang-Yu Guo, and Naoto Nagaosa, ``Low-Frequency Divergence and Quantum Geometry of the Bulk Photovoltaic Effect in Topological Semimetals'' Physical Review X 10 (2020). https://doi.org/10.1103/PhysRevX.10.041041 arXiv:2006.06709 [22] Christian Northe, Giandomenico Palumbo, Jonathan Sturm, Christian Tutschku, and Ewelina. M. Hankiewicz, ``Interplay of band geometry and topology in ideal Chern insulators in the presence of external electromagnetic fields'' Physical Review B 105 (2022). https://doi.org/10.1103/PhysRevB.105.155410 arXiv:2112.15559 [23] Ansgar Grafand Frédéric Piéchon ``Berry curvature and quantum metric in $N$-band systems: An eigenprojector approach'' Physical Review B 104 (2021). https://doi.org/10.1103/PhysRevB.104.085114 arXiv:2102.09899 [24] Adrien Bouhon, Abigail Timmel, and Robert-Jan Slager, ``Quantum geometry beyond projective single bands'' (2023). https://doi.org/10.48550/arXiv.2303.02180 arXiv:2303.02180 [25] Daniel Kaplan, Tobias Holder, and Binghai Yan, ``Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry'' Physical Review Letters 132 (2024). https://doi.org/10.1103/PhysRevLett.132.026301 arXiv:2211.17213 [26] Antimo Marrazzoand Raffaele Resta ``Local Theory of the Insulating State'' Physical Review Letters 122 (2019). https://doi.org/10.1103/PhysRevLett.122.166602 arXiv:1807.01063 [27] Matheus S. M. de Sousa, Antonio L. Cruz, and Wei Chen, ``Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker'' Physical Review B 107 (2023). https://doi.org/10.1103/PhysRevB.107.205133 arXiv:2301.06493 [28] Yu-Quan Ma, Shu Chen, Heng Fan, and Wu-Ming Liu, ``Abelian and non-Abelian quantum geometric tensor'' Physical Review B 81 (2010). https://doi.org/10.1103/PhysRevB.81.245129 arXiv:1003.4040 [29] Titus Neupert, Claudio Chamon, and Christopher Mudry, ``Measuring the quantum geometry of Bloch bands with current noise'' Physical Review B 87 (2013). https://doi.org/10.1103/PhysRevB.87.245103 arXiv:1303.4643 [30] Bruno Meraand Johannes Mitscherling ``Nontrivial quantum geometry of degenerate flat bands'' Physical Review B 106 (2022). https://doi.org/10.1103/PhysRevB.106.165133 arXiv:2205.07900 [31] Hai-Tao Ding, Yan-Qing Zhu, Peng He, Yu-Guo Liu, Jian-Te Wang, Dan-Wei Zhang, and Shi-Liang Zhu, ``Extracting non-Abelian quantum metric tensor and its related Chern numbers'' Physical Review A 105 (2022). https://doi.org/10.1103/PhysRevA.105.012210 arXiv:2201.01086 [32] Alexander Avdoshkinand Fedor K. Popov ``Extrinsic geometry of quantum states'' Physical Review B 107 (2023). https://doi.org/10.1103/PhysRevB.107.245136 arXiv:2205.15353 [33] Alexander Avdoshkin ``Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians'' (2024). https://doi.org/10.48550/arXiv.2404.03234 arXiv:2404.03234 [34] Kai Chen, Bishnu Karki, and Pavan Hosur, ``The Effect of the Non-Abelian Quantum Metric on Superfluidity'' (2025). https://doi.org/10.48550/arXiv.2501.16965 arXiv:2501.16965 [35] Wojciech J. Jankowski, Robert-Jan Slager, and Michele Pizzochero, ``Enhancing the Hyperpolarizability of Crystals with Quantum Geometry'' Physical Review Letters 135 (2025). https://doi.org/10.1103/z7lp-pqp6 [36] Johannes Mitscherling, Alexander Avdoshkin, and Joel E. Moore, ``Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals'' Physical Review B 112 (2025). https://doi.org/10.1103/qscv-qxqt [37] Sebastiano Peottaand Päivi Törmä ``Superfluidity in topologically nontrivial flat bands'' Nature Communications 6 (2015). https://doi.org/10.1038/ncomms9944 arXiv:1506.02815 [38] Long Liang, Tuomas I. Vanhala, Sebastiano Peotta, Topi Siro, Ari Harju, and Päivi Törmä, ``Band geometry, Berry curvature, and superfluid weight'' Physical Review B 95 (2017). https://doi.org/10.1103/PhysRevB.95.024515 arXiv:1610.01803 [39] Jonah Herzog-Arbeitman, Valerio Peri, Frank Schindler, Sebastian D. Huber, and B. Andrei Bernevig, ``Superfluid Weight Bounds from Symmetry and Quantum Geometry in Flat Bands'' Physical Review Letters 128 (2022). https://doi.org/10.1103/PhysRevLett.128.087002 arXiv:2110.14663 [40] Kristian Hauser A.

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