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

Article 
ADS 

Google Scholar
 

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

Article 
ADS 

Google Scholar
 

Haldane, F. D. M. Nonlinear field theory of large-spin heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis néel state. Phys. Rev. Lett. 50, 1153 (1983).

Article 
ADS 
MathSciNet 

Google Scholar
 

Chen, X., Gu, Z.-C. & Wen, X.-G. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B 83, 035107 (2011).

Article 
ADS 

Google Scholar
 

Schuch, N., Pérez-García, D. & Cirac, I. Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B 84, 165139 (2011).

Article 
ADS 

Google Scholar
 

Pollmann, F., Turner, A. M., Berg, E. & Oshikawa, M. Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439 (2010).

Article 
ADS 

Google Scholar
 

Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 1604 (2012).

Article 
ADS 
MathSciNet 

Google Scholar
 

Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry protected topological orders and the group cohomology of their symmetry group. Phys. Rev. B 87, 155114 (2013).

Article 
ADS 

Google Scholar
 

Senthil, T. Symmetry-protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299 (2015).

Article 
ADS 

Google Scholar
 

Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

Article 

Google Scholar
 

Ringel, Z., Kraus, Y. E. & Stern, A. Strong side of weak topological insulators. Phys. Rev. B 86, 045102 (2012).

Article 
ADS 

Google Scholar
 

Mong, R. S. K., Bardarson, J. H. & Moore, J. E. Quantum transport and two-parameter scaling at the surface of a weak topological insulator. Phys. Rev. Lett. 108, 076804 (2012).

Article 
ADS 

Google Scholar
 

Fu, L. & Kane, C. L. Topology, delocalization via average symmetry and the symplectic anderson transition. Phys. Rev. Lett. 109, 246605 (2012).

Article 
ADS 

Google Scholar
 

Fulga, I., Van Heck, B., Edge, J. & Akhmerov, A. Statistical topological insulators. Phys. Rev. B 89, 155424 (2014).

Article 
ADS 

Google Scholar
 

Wang, J.-H., Yang, Y.-B., Dai, N. & Xu, Y. Structural-disorder-induced second-order topological insulators in three dimensions. Phys. Rev. Lett. 126, 206404 (2021).

Article 
ADS 

Google Scholar
 

Ma, R. & Wang, C. Average symmetry-protected topological phases. Phys. Rev. X 13, 031016 (2023).


Google Scholar
 

Lee, J. Y., You, Y.-Z. & Xu, C. Symmetry protected topological phases under decoherence. Quantum 9, 1607 (2025).

Article 

Google Scholar
 

Zhang, J.-H., Qi, Y. & Bi, Z. Fidelity strange correlators for average symmetry-protected topological phases. Sci. Bull. 71, 688–691 (2026).

Article 

Google Scholar
 

Ma, R., Zhang, J.-H., Bi, Z., Cheng, M. & Wang, C. Topological phases with average symmetries: the decohered, the disordered, and the intrinsic. Phys. Rev. X 15, 021062 (2025).


Google Scholar
 

Chirame, S., Burnell, F. J., Gopalakrishnan, S. & Prem, A. Stable symmetry-protected topological phases in systems with heralded noise. Phys. Rev. Lett. 134, 010403 (2025).

Article 
ADS 
MathSciNet 

Google Scholar
 

Ma, R. & Turzillo, A. Symmetry-protected topological phases of mixed states in the doubled space. PRX Quantum 6, 010348 (2025).

Article 
ADS 

Google Scholar
 

Guo, Y. & Yang, S. Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order. Phys. Rev. B 111, L201108 (2025).

Article 
ADS 

Google Scholar
 

Guo, Y., Zhang, J.-H., Zhang, H.-R., Yang, S. & Bi, Z. Locally purified density operators for symmetry-protected topological phases in mixed states. Phys. Rev. X 15, 021060 (2025).


Google Scholar
 

Sun, S., Zhang, J.-H., Bi, Z. & You, Y. Holographic view of mixed-state symmetry-protected topological phases in open quantum systems. PRX Quantum 6, 020333 (2025).

Article 
ADS 

Google Scholar
 

Konig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766 (2007).

Article 
ADS 

Google Scholar
 

Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970 (2008).

Article 
ADS 

Google Scholar
 

Hsieh, D. et al. Observation of unconventional quantum spin textures in topological insulators. Science 323, 919 (2009).

Article 
ADS 

Google Scholar
 

Tanaka, Y. et al. Experimental realization of a topological crystalline insulator in SnTe. Nat. Phys. 8, 800 (2012).

Article 

Google Scholar
 

Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167 (2013).

Article 
ADS 

Google Scholar
 

Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795 (2013).

Article 

Google Scholar
 

Jotzu, G. et al. Experimental realization of the topological haldane model with ultracold fermions. Nature 515, 237 (2014).

Article 
ADS 

Google Scholar
 

De Léséleuc, S. et al. Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms. Science 365, 775 (2019).

Article 
ADS 
MathSciNet 

Google Scholar
 

Sompet, P. et al. Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders. Nature 606, 484 (2022).

Article 
ADS 

Google Scholar
 

Mishra, S. et al. Observation of fractional edge excitations in nanographene spin chains. Nature 598, 287 (2021).

Article 
ADS 

Google Scholar
 

Zhao, C. et al. Tunable topological phases in nanographene-based spin-1/2 alternating-exchange Heisenberg chains. Nat. Nanotechnol. 19, 1789 (2024).

Article 
ADS 

Google Scholar
 

Kiczynski, M. et al. Engineering topological states in atom-based semiconductor quantum dots. Nature 606, 694 (2022).

Article 
ADS 

Google Scholar
 

Wang, H. et al. Construction of topological quantum magnets from atomic spins on surfaces. Nat. Nanotechnol. 19, 1782 (2024).

Article 
ADS 

Google Scholar
 

Li, J., Chu, R.-L., Jain, J. K. & Shen, S.-Q. Topological anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).

Article 
ADS 

Google Scholar
 

Li, K., Wang, J.-H., Yang, Y.-B. & Xu, Y. Symmetry-protected topological phases in a Rydberg glass. Phys. Rev. Lett. 127, 263004 (2021).

Article 
ADS 

Google Scholar
 

Stützer, S. et al. Photonic topological anderson insulators. Nature 560, 461 (2018).

Article 
ADS 

Google Scholar
 

Dai, T. et al. A programmable topological photonic chip. Nat. Mater. 23, 928 (2024).

Article 
ADS 

Google Scholar
 

Chen, X.-D. et al. Realization of time-reversal invariant photonic topological Anderson insulators. Phys. Rev. Lett. 133, 133802 (2024).

Article 
ADS 

Google Scholar
 

Liu, G.-G. et al. Topological anderson insulator in disordered photonic crystals. Phys. Rev. Lett. 125, 133603 (2020).

Article 
ADS 

Google Scholar
 

Ren, M. et al. Realization of gapped and ungapped photonic topological anderson insulators. Phys. Rev. Lett. 132, 066602 (2024).

Article 
ADS 

Google Scholar
 

Zangeneh-Nejad, F. & Fleury, R. Disorder-induced signal filtering with topological metamaterials. Adv. Mater. 32, 2001034 (2020).

Article 

Google Scholar
 

Liu, H. et al. Acoustic spin-chern topological Anderson insulators. Phys. Rev. B 108, L161410 (2023).

Article 
ADS 

Google Scholar
 

Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929 (2018).

Article 
ADS 

Google Scholar
 

Li, X. et al. Mapping the topology-localization phase diagram with quasiperiodic disorder using a programmable superconducting simulator. Phys. Rev. Res. 6, L042038 (2024).

Article 

Google Scholar
 

Barredo, D., de Léséleuc, S., Lienhard, V., Lahaye, T. & Browaeys, A. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science 354, 1021 (2016).

Article 
ADS 

Google Scholar
 

Endres, M. et al. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays. Science 354, 1024 (2016).

Article 
ADS 

Google Scholar
 

Kim, H. et al. In situ single-atom array synthesis using dynamic holographic optical tweezers. Nat. Commun. 7, 13317 (2016).

Article 
ADS 

Google Scholar
 

Chen, C. et al. Continuous symmetry breaking in a two-dimensional Rydberg array. Nature 616, 691 (2023).

Article 
ADS 

Google Scholar
 

Browaeys, A., Barredo, D. & Lahaye, T. Experimental investigations of dipole–dipole interactions between a few Rydberg atoms. J. Phys. B 49, 152001 (2016).

Article 
ADS 

Google Scholar
 

Weber, S. et al. Calculation of Rydberg interaction potentials. J. Phys. B 50, 133001 (2017).

Article 
ADS 

Google Scholar
 

Urban, E. et al. Observation of Rydberg blockade between two atoms. Nat. Phys. 5, 110 (2009).

Article 

Google Scholar
 

Semeghini, G. et al. Probing topological spin liquids on a programmable quantum simulator. Science 374, 1242 (2021).

Article 
ADS 

Google Scholar
 

Ebadi, S. et al. Quantum optimization of maximum independent set using Rydberg atom arrays. Science 376, 1209 (2022).

Article 
ADS 

Google Scholar
 

Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Modern Phys. 88, 035005 (2016).

Article 
ADS 

Google Scholar
 

Resta, R. Quantum-mechanical position operator in extended systems. Phys. Rev. Lett. 80, 1800 (1998).

Article 
ADS 

Google Scholar
 

Manousakis, E. The spin-\(\frac{1}{2}\) Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Modern Phys. 63, 1 (1991).

Article 
ADS 

Google Scholar
 

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 (1992).

Article 
ADS 

Google Scholar
 

Nakamura, M. & Todo, S. Order parameter to characterize valence-bond-solid states in quantum spin chains. Phys. Rev. Lett. 89, 077204 (2002).

Article 
ADS 

Google Scholar
 

Tasaki, H. Topological phase transition and \({{\mathbb{z}}}_{2}\) index for s = 1 quantum spin chains. Phys. Rev. Lett. 121, 140604 (2018).

Article 
ADS 

Google Scholar
 

Liang, X. et al. Observation of anomalous information scrambling in a Rydberg atom array. Phys. Rev. Lett. 135, 050201 (2025).

Article 
ADS 

Google Scholar
 

Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence at the edge of hot matter. Nat. Commun. 6, 7341 (2015).

Article 

Google Scholar
 

Zhang, X. et al. Digital quantum simulation of Floquet symmetry-protected topological phases. Nature 607, 468 (2022).

Article 
ADS 

Google Scholar
 

Mi, X. et al. Noise-resilient edge modes on a chain of superconducting qubits. Science 378, 785 (2022).

Article 
ADS 

Google Scholar
 

Agarwala, A. & Shenoy, V. B. Topological insulators in amorphous systems. Phys. Rev. Lett. 118, 236402 (2017).

Article 
ADS 

Google Scholar
 

Mitchell, N. P., Nash, L. M., Hexner, D., Turner, A. M. & Irvine, W. T. Amorphous topological insulators constructed from random point sets. Nat. Phys. 14, 380 (2018).

Article 

Google Scholar
 

Agarwala, A., Juričić, V. & Roy, B. Higher-order topological insulators in amorphous solids. Phys. Rev. Res. 2, 012067 (2020).

Article 

Google Scholar
 

Grushin, A. G. & Repellin, C. Amorphous and polycrystalline routes toward a chiral spin liquid. Phys. Rev. Lett. 130, 186702 (2023).

Article 
ADS 

Google Scholar
 

Cassella, G., d’Ornellas, P., Hodson, T., Natori, W. M. & Knolle, J. An exact chiral amorphous spin liquid. Nat. Commun. 14, 6663 (2023).

Article 
ADS 

Google Scholar
 

He, P., Liu, J.-X., Wu, H. & Wang, Z. D. Floquet amorphous topological orders in a Rydberg glass. Commun. Phys. 8, 237 (2025).

Article 

Google Scholar
 

Aramthottil, A. S., Sierant, P., Lewenstein, M. & Zakrzewski, J. Phenomenology of many-body localization in bond-disordered spin chains. Phys. Rev. Lett. 133, 196302 (2024).

Article 
ADS 
MathSciNet 

Google Scholar
 

Su, L. et al. Topological phase transitions and mixed state order in a Hubbard quantum simulator. Preprint at https://doi.org/10.48550/arXiv.2505.17009 (2025).

Hauschild, J. & Pollmann, F. Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes https://doi.org/10.21468/SciPostPhysLectNotes.5 (2018).

Yue, Z. et al. Data of observation of a disorder-induced many-body interacting average symmetry-protected topological phase. Figshare https://doi.org/10.6084/m9.figshare.30752111 (2026).