• Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article 
    CAS 

    Google Scholar
     

  • Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article 
    CAS 

    Google Scholar
     

  • Nielsen, H. B. & Ninomiya, M. A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105, 219–223 (1981).

    Article 

    Google Scholar
     

  • Tian, W., Yu, W., Shi, J. & Wang, Y. The property, preparation and application of topological insulators: a review. Materials 10, 814 (2017).

    Article 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Breunig, O. & Ando, Y. Opportunities in topological insulator devices. Nat. Rev. Phys. 4, 184–193 (2021).

    Article 

    Google Scholar
     

  • Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).

    Article 
    PubMed 

    Google Scholar
     

  • He, M., Sun, H. & He, Q. L. Topological insulator: spintronics and quantum computations. Front. Phys. 14, 43401 (2019).

    Article 

    Google Scholar
     

  • Huáng, N. J. et al. Quantum anomalous Hall effect for metrology. Appl. Phys. Lett. 126, 040501 (2025).

    Article 

    Google Scholar
     

  • Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater. 11, 409–416 (2012).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Xu, N., Xu, Y. & Zhu, J. Topological insulators for thermoelectrics. npj Quantum Mater. 2, 51 (2017).

    Article 

    Google Scholar
     

  • Bardarson, J. H. & Moore, J. E. Quantum interference and Aharonov–Bohm oscillations in topological insulators. Rep. Prog. Phys. 76, 056501 (2013).

    Article 
    PubMed 

    Google Scholar
     

  • Gao, A., Nagaosa, N., Ni, N. & Xu, S.-Y. Quantum geometry phenomena in condensed matter systems. Preprint at https://arxiv.org/abs/2508.00469 (2025).

  • Verma, N., Moll, P. J. W., Holder, T. & Queiroz, R. Quantum geometry: revisiting electronic scales in quantum matter. Preprint at https://arxiv.org/abs/2504.07173 (2025).

  • Ahn, J., Guo, G.-Y. & Nagaosa, N. Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals. Phys. Rev. X 10, 041041 (2020).

    CAS 

    Google Scholar
     

  • Ahn, J., Guo, G.-Y., Nagaosa, N. & Vishwanath, A. Riemannian geometry of resonant optical responses. Nat. Phys. 18, 290–295 (2022).

    Article 
    CAS 

    Google Scholar
     

  • Jiang, Y., Holder, T. & Yan, B. Revealing quantum geometry in nonlinear quantum materials. Rep. Prog. Phys. 88, 076502 (2025).

    Article 

    Google Scholar
     

  • Yu, J. et al. Quantum geometry in quantum materials. npj Quantum Mater. 10, 101 (2025).

    Article 

    Google Scholar
     

  • Liu, T., Qiang, X.-B., Lu, H.-Z. & Xie, X. C. Quantum geometry in condensed matter. Natl Sci. Rev. 12, nwae334 (2025).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Gao, A. et al. Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure. Science 381, 181–186 (2023).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Wang, N. et al. Quantum-metric-induced nonlinear transport in a topological antiferromagnet. Nature 621, 487–492 (2023).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Kim, S. et al. Direct measurement of the quantum metric tensor in solids. Science 388, 1050–1054 (2025).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Kang, M. et al. Measurements of the quantum geometric tensor in solids. Nat. Phys. 21, 110–117 (2025).

    Article 
    CAS 

    Google Scholar
     

  • Sala, G. et al. The quantum metric of electrons with spin-momentum locking. Science 389, 822–825 (2025).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Zhao, T.-Y. et al. Magnetic field induced quantum metric dipole in Dirac semimetal Cd3As2. Phys. Rev. Lett. 135, 026601 (2025).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Chen, W. Quantum geometrical properties of topological materials. J. Phys. Condens. Matter 37, 025605 (2025).

    Article 
    CAS 

    Google Scholar
     

  • Mera, B., Zhang, A. & Goldman, N. Relating the topology of Dirac Hamiltonians to quantum geometry: when the quantum metric dictates Chern numbers and winding numbers. SciPost Phys. 12, 018 (2022).

    Article 

    Google Scholar
     

  • Mercaldo, M. T., Cuoco, M. & Ortix, C. Nonlinear planar magnetotransport as a probe of the topology of surface states. Phys. Rev. B 111, 155442 (2025).

    Article 
    CAS 

    Google Scholar
     

  • Liu, C.-X. et al. Model Hamiltonian for topological insulators. Phys. Rev. B 82, 045122 (2010).

    Article 

    Google Scholar
     

  • Fu, L. Hexagonal warping effects in the surface states of the topological insulator Bi2Te3. Phys. Rev. Lett. 103, 266801 (2009).

    Article 
    PubMed 

    Google Scholar
     

  • Kaplan, D., Holder, T. & Yan, B. Unification of nonlinear anomalous Hall effect and nonreciprocal magnetoresistance in metals by the quantum geometry. Phys. Rev. Lett. 132, 026301 (2024).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Das, K., Lahiri, S., Atencia, R. B., Culcer, D. & Agarwal, A. Intrinsic nonlinear conductivities induced by the quantum metric. Phys. Rev. B 108, L201405 (2023).

    Article 
    CAS 

    Google Scholar
     

  • Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).

    Article 
    CAS 

    Google Scholar
     

  • Rimoldi, M. et al. Epitaxial and large area Sb2Te3 thin films on silicon by MOCVD. RSC Adv. 10, 19936–19942 (2020).

    Article 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Shafiei, A. et al. Wafer-scale synthesis of topological insulator Sb2Te3 thin films. Adv. Mater. Interfaces 12, 2400961 (2025).

    Article 
    CAS 

    Google Scholar
     

  • Anderson, T. L. & Krause, H. B. Refinement of the Sb2Te3 and Sb2Te2Se structures and their relationship to nonstoichiometric Sb2Te3−ySey compounds. Acta Crystallogr. B 30, 1307–1310 (1974).

    Article 
    CAS 

    Google Scholar
     

  • Locatelli, L. et al. Magnetotransport and ARPES studies of the topological insulators Sb2Te3 and Bi2Te3 grown by MOCVD on large-area Si substrates. Sci. Rep. 12, 3891 (2022).

    Article 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Longo, E. et al. Spin–charge conversion in Fe/Au/Sb2Te3 heterostructures as probed by spin pumping ferromagnetic resonance. Adv. Mater. Interfaces 8, 2101244 (2021).

    Article 
    CAS 

    Google Scholar
     

  • Longo, E. et al. Large spin-to-charge conversion at room temperature in extended epitaxial Sb2Te3 topological insulator chemically grown on silicon. Adv. Funct. Mater. 32, 2109361 (2022).

    Article 
    CAS 

    Google Scholar
     

  • Hikami, S., Larkin, A. I. & Nagaoka, Y. Spin–orbit interaction and magnetoresistance in the two dimensional random system. Prog. Theor. Phys. 63, 707–710 (1980).

    Article 

    Google Scholar
     

  • Steinberg, H., Laloë, J.-B., Fatemi, V., Moodera, J. S. & Jarillo-Herrero, P. Electrically tunable surface-to-bulk coherent coupling in topological insulator thin films. Phys. Rev. B 84, 233101 (2011).

    Article 

    Google Scholar
     

  • He, P. et al. Bilinear magnetoelectric resistance as a probe of three-dimensional spin texture in topological surface states. Nat. Phys. 14, 495–499 (2018).

    Article 
    CAS 

    Google Scholar
     

  • He, P. et al. Nonlinear planar Hall effect. Phys. Rev. Lett. 123, 016801 (2019).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Fu, Y. et al. Bilinear magnetoresistance in HgTe topological insulator: opposite signs at opposite surfaces demonstrated by gate control. Nano Lett. 22, 7867–7873 (2022).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Tang, J. et al. Electrical detection of spin-polarized surface states conduction in (Bi0.53Sb0.47)2Te3 topological insulator. Nano Lett. 14, 5423–5429 (2014).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Che, X. et al. Strongly surface state carrier-dependent spin–orbit torque in magnetic topological insulators. Adv. Mater. 32, 1907661 (2020).

    Article 
    CAS 

    Google Scholar
     

  • Binda, F., Fedel, S., Alvarado, S. F., Noël, P. & Gambardella, P. Spin–orbit torques and spin Hall magnetoresistance generated by twin-free and amorphous Bi0.9Sb0.1 topological insulator films. Adv. Mater. 35, 2304905 (2023).

    Article 
    CAS 

    Google Scholar
     

  • Korzhovska, I. et al. Spin memory of the topological material under strong disorder. npj Quantum Mater. 5, 39 (2020).

    Article 
    CAS 

    Google Scholar
     

  • Pauly, C. et al. Probing two topological surface bands of Sb2Te3 by spin-polarized photoemission spectroscopy. Phys. Rev. B 86, 235106 (2012).

    Article 

    Google Scholar
     

  • Hong, S. S., Cha, J. J., Kong, D. & Cui, Y. Ultra-low carrier concentration and surface-dominant transport in antimony-doped Bi2Se3 topological insulator nanoribbons. Nat. Commun. 3, 757 (2012).

    Article 
    PubMed 

    Google Scholar
     

  • Steinberg, H., Gardner, D. R., Lee, Y. S. & Jarillo-Herrero, P. Surface state transport and ambipolar electric field effect in Bi2Se3 nanodevices. Nano Lett. 10, 5032–5036 (2010).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Yoshimi, R. et al. Quantum Hall effect on top and bottom surface states of topological insulator (Bi1−xSbx)2Te3 films. Nat. Commun. 6, 6627 (2015).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Chen, J. et al. Gate-voltage control of chemical potential and weak antilocalization in Bi2Se3. Phys. Rev. Lett. 105, 176602 (2010).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Feng, X. et al. Quantum metric nonlinear spin–orbit torque enhanced by topological bands. Mater. Today Quantum 6, 100040 (2025).

    Article 

    Google Scholar
     

  • Wang, H. et al. Intrinsic nonlinear spin Hall effect and manipulation of perpendicular magnetization. Phys. Rev. Lett. 134, 056301 (2025).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Ma, Q., Krishna Kumar, R., Xu, S.-Y., Koppens, F. H. L. & Song, J. C. W. Photocurrent as a multiphysics diagnostic of quantum materials. Nat. Rev. Phys. 5, 170–184 (2023).

    Article 

    Google Scholar
     

  • Bhalla, P., Das, K., Culcer, D. & Agarwal, A. Resonant second-harmonic generation as a probe of quantum geometry. Phys. Rev. Lett. 129, 227401 (2022).

    Article 
    CAS 
    PubMed 

    Google Scholar
     

  • Rimoldi, M. et al. Effect of substrates and thermal treatments on metalorganic chemical vapor deposition-grown Sb2Te3 thin films. Cryst. Growth Des. 21, 5135–5144 (2021).

    Article 
    CAS 

    Google Scholar
     

  • Longo, E. et al. Exploiting the close-to-Dirac point shift of the Fermi level in the Sb2Te3/Bi2Te3 topological insulator heterostructure for spin–charge conversion. ACS Appl. Mater. Interfaces 15, 50237–50245 (2023).

    Article 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Ren, Z., Taskin, A. A., Sasaki, S., Segawa, K. & Ando, Y. Fermi level tuning and a large activation gap achieved in the topological insulator Bi2Te2Se by Sn doping. Phys. Rev. B 85, 155301 (2012).

    Article 

    Google Scholar
     

  • Kölling, S., Westerhof, F. R. & Brinkman, A. Gate-electrode-induced nonreciprocal resistance in topological insulators. Phys. Rev. Appl. 24, 014024 (2025).

    Article 

    Google Scholar
     

  • Lin, J. J. & Bird, J. P. Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures. J. Phys. Condens. Matter 14, R501 (2002).

    Article 
    CAS 

    Google Scholar
     

  • Xu, G. et al. Weak antilocalization effect and noncentrosymmetric superconductivity in a topologically nontrivial semimetal LuPdBi. Sci. Rep. 4, 5709 (2014).

    Article 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Shrestha, K. et al. Extremely large nonsaturating magnetoresistance and ultrahigh mobility due to topological surface states in the metallic Bi2Te3 topological insulator. Phys. Rev. B 95, 195113 (2017).

    Article 

    Google Scholar
     

  • Sala, G. Probing the quantum metric of 3D topological insulators. Zenodo https://doi.org/10.5281/zenodo.16910820 (2026).