Narang, P., Garcia, C. A. C. & Felser, C. The topology of electronic band structures. Nat. Mater. 20, 293–300 (2021).
Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 1, 126–143 (2019).
Kiczynski, M. et al. Engineering topological states in atom-based semiconductor quantum dots. Nature 606, 694–699 (2022).
de Léséleuc, S. et al. Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms. Science 365, 775–780 (2019).
Sompet, P. et al. Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders. Nature 606, 484–488 (2022).
Drost, R., Ojanen, T., Harju, A. & Liljeroth, P. Topological states in engineered atomic lattices. Nat. Phys. 13, 668–671 (2017).
Mishra, S. et al. Observation of fractional edge excitations in nanographene spin chains. Nature 598, 287–292 (2021).
Kempkes, S. N. et al. Robust zero-energy modes in an electronic higher-order topological insulator. Nat. Mater. 18, 1292–1297 (2019).
Meier, E. J., An, F. A. & Gadway, B. Observation of the topological soliton state in the Su–Schrieffer–Heeger model. Nat. Commun. 7, 13986 (2016).
St-Jean, P. et al. Lasing in topological edge states of a one-dimensional lattice. Nat. Photon. 11, 651–656 (2017).
Rizzo, D. J. et al. Topological band engineering of graphene nanoribbons. Nature 560, 204–208 (2018).
Gröning, O. et al. Engineering of robust topological quantum phases in graphene nanoribbons. Nature 560, 209–213 (2018).
Schneider, L. et al. Topological Shiba bands in artificial spin chains on superconductors. Nat. Phys. 17, 943–948 (2021).
Khajetoorians, A. A., Wegner, D., Otte, A. F. & Swart, I. Creating designer quantum states of matter atom-by-atom. Nat. Rev. Phys. 1, 703–715 (2019).
Choi, D.-J. et al. Colloquium: atomic spin chains on surfaces. Rev. Mod. Phys. 91, 041001 (2019).
Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).
Liebhaber, E. et al. Quantum spins and hybridization in artificially-constructed chains of magnetic adatoms on a superconductor. Nat. Commun. 13, 2160 (2022).
Soldini, M. O. et al. Two-dimensional Shiba lattices as a possible platform for crystalline topological superconductivity. Nat. Phys. 19, 1848–1854 (2023).
Yang, K. et al. Probing resonating valence bond states in artificial quantum magnets. Nat. Commun. 12, 993 (2021).
Zhao, Y. et al. Quantum nanomagnets in on-surface metal-free porphyrin chains. Nat. Chem. 15, 53–60 (2023).
Zhao, C. et al. Tunable topological phases in nanographene-based spin-1/2 alternating-exchange Heisenberg chains. Preprint at (2024).
Yang, K. et al. Engineering the eigenstates of coupled spin-1/2 atoms on a surface. Phys. Rev. Lett. 119, 227206 (2017).
Bae, Y. et al. Enhanced quantum coherence in exchange coupled spins via singlet–triplet transitions. Sci. Adv. 4, eaau4159 (2018).
Hida, K. Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain. Phys. Rev. B 45, 2207–2212 (1992).
Yang, K. et al. Tuning the exchange bias on a single atom from 1 mT to 10 T. Phys. Rev. Lett. 122, 227203 (2019).
Haghshenas, R., Langari, A. & Rezakhani, A. T. Symmetry fractionalization: symmetry-protected topological phases of the bond-alternating spin-1/2 Heisenberg chain. J. Phys. Condens. Matter 26, 456001 (2014).
Luo, Q., Zhao, J., Wang, X. & Kee, H.-Y. Unveiling the phase diagram of a bond-alternating spin-1/2 K–Γ chain. Phys. Rev. B 103, 144423 (2021).
Lado, J. L., Ortiz, R. & Fernández-Rossier, J. in Graphene Nanoribbons (eds. Brey, L., Seneor, P. & Tejeda, A.) Ch. 4 (IOP Publishing, 2019).
Willke, P. et al. Magnetic resonance imaging of single atoms on a surface. Nat. Phys. 15, 1005–1010 (2019).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).
Willke, P. et al. Probing quantum coherence in single-atom electron spin resonance. Sci. Adv. 4, eaaq1543 (2018).
Hatsugai, Y. Quantized Berry phases for a local characterization of spin liquids in frustrated spin systems. J. Phys. Condens. Matter 19, 145209 (2007).
Pollmann, F., Turner, A. M., Berg, E. & Oshikawa, M. Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439 (2010).
Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).
Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018).
Yang, K. et al. Coherent spin manipulation of individual atoms on a surface. Science 366, 509–512 (2019).
Wang, Y. et al. An atomic-scale multi-qubit platform. Science 382, 87–92 (2023).
Broholm, C. et al. Quantum spin liquids. Science 367, eaay0668 (2020).
Baumann, S. et al. Electron paramagnetic resonance of individual atoms on a surface. Science 350, 417–420 (2015).
