Callan, C. G. Jr. Disappearing dyons. Phys. Rev. D 25, 2141–2146 (1982).
Rubakov, V. A. Adler-Bell-Jackiw anomaly and fermion number breaking in the presence of a magnetic monopole. Nucl. Phys. B 203, 311–348 (1982).
van Beest, M., Smith, P., Delmastro, D., Komargodski, Z. & Tong, D. Monopoles, scattering, and generalized symmetries. J. High Energy Phys. 03, 014 (2025).
Loladze, V., Okui, T. & Tong, D. Dynamics of the fermion-rotor system. J. High Energy Phys. 2026, 52 (2026).
Loladze, V. & Okui, T. Monopole-fermion scattering and the solution to the semiton–unitarity puzzle. Phys. Rev. Lett. 134, 051602 (2025).
van Beest, M., Boyle Smith, P., Delmastro, D., Mouland, R. & Tong, D. Fermion-monopole scattering in the Standard Model. J. High Energy Phys. 2024, 4 (2024).
Maldacena, J. M. & Ludwig, A. W. W. Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species, and solution of the ‘unitarity puzzle’. Nucl. Phys. B 506, 565–588 (1997).
Affleck, I. & Sagi, J. Monopole catalyzed baryon decay: a boundary conformal field theory approach. Nucl. Phys. B 417, 374–402 (1994).
Polchinski, J. Monopole catalysis: the fermion rotor system. Nucl. Phys. B 242, 345–363 (1984).
Bultinck, N. et al. Anyons and matrix product operator algebras. Ann. Phys. 378, 183–233 (2017).
Lootens, L., Fuchs, J., Haegeman, J., Schweigert, C. & Verstraete, F. Matrix product operator symmetries and intertwiners in string-nets with domain walls. SciPost Phys. 10, 053 (2021).
Lootens, L., Delcamp, C., Ortiz, G. & Verstraete, F. Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners. PRX Quantum 4, 020357 (2023).
Lootens, L., Delcamp, C. & Verstraete, F. Dualities in one-dimensional quantum lattice models: topological sectors. PRX Quantum 5, 010338 (2024).
Aasen, D., Mong, R. S. K. & Fendley, P. Topological defects on the lattice I: the Ising model. J. Phys. A 49, 354001 (2016).
Aasen, D., Fendley, P. & Mong, R. S. K. Topological defects on the lattice: dualities and degeneracies. Preprint at https://arxiv.org/abs/2008.08598 (2020).
Lootens, L., Delcamp, C., Williamson, D. & Verstraete, F. Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models. Phys. Rev. Lett. 134, 130403 (2025).
Vanhove, R., Ravindran, V., Stephen, D. T., Wen, X.-G. & Chen, X. Duality via sequential quantum circuit in the topological holography formalism. Phys. Rev. B 112, 035173 (2025).
Fuchs, J. & Schweigert, C. Symmetry breaking boundaries I. General theory. Nucl. Phys. B 558, 419–483 (1999).
Petkova, V. & Zuber, J.-B. Generalised twisted partition functions. Phys. Lett. B 504, 157–164 (2001).
Fuchs, J., Runkel, I. & Schweigert, C. TFT construction of RCFT correlators I: partition functions. Nucl. Phys. B 646, 353–497 (2002).
Fröhlich, J., Fuchs, J., Runkel, I. & Schweigert, C. Kramers-Wannier duality from conformal defects. Phys. Rev. Lett. 93, 070601 (2004).
Fröhlich, J., Fuchs, J., Runkel, I. & Schweigert, C. Duality and defects in rational conformal field theory. Nucl. Phys. B 763, 354–430 (2007).
Bachas, C. & Brunner, I. Fusion of conformal interfaces. J. High Energy Phys. 02, 085 (2008).
Quella, T., Runkel, I. & Watts, G. M. Reflection and transmission for conformal defects. J. High Energy Phys. 2007, 095 (2007).
Hauru, M., Evenbly, G., Ho, W. W., Gaiotto, D. & Vidal, G. Topological conformal defects with tensor networks. Phys. Rev. B 94, 115125 (2016).
Sinha, M., Yan, F., Grans-Samuelsson, L., Roy, A. & Saleur, H. Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model. J. High Energy Phys. 2024, 225 (2024).
Nussinov, Z. & Ortiz, G. A symmetry principle for topological quantum order. Ann. Phys. 324, 977–1057 (2009).
Gaiotto, D., Kapustin, A., Seiberg, N. & Willett, B. Generalized global symmetries. J. High Energy Phys. 2015, 172 (2015).
McGreevy, J. Generalized symmetries in condensed matter. Annu. Rev. Condens. Matter Phys. 14, 57–82 (2023).
Shao, S.-H. What’s done cannot be undone: TASI lectures on non-invertible symmetries. Preprint at https://arxiv.org/abs/2308.00747 (2023).
Cobanera, E., Ortiz, G. & Nussinov, Z. The bond-algebraic approach to dualities. Adv. Phys. 60, 679–798 (2011).
Vanhove, R. et al. Mapping topological to conformal field theories through strange correlators. Phys. Rev. Lett. 121, 177203 (2018).
Vanhove, R., Lootens, L., Tu, H.-H. & Verstraete, F. Topological aspects of the critical three-state Potts model. J. Phys. A: Math. Theor. 55, 235002 (2022).
Haegeman, J. et al. Variational matrix product ansatz for dispersion relations. Phys. Rev. B 85, 100408 (2012).
Vanderstraeten, L., Haegeman, J. & Verstraete, F. Tangent-space methods for uniform matrix product states. SciPost Phys. Lect. Notes 7, 1 (2019).
Van Damme, M., Vanderstraeten, L., De Nardis, J., Haegeman, J. & Verstraete, F. Real-time scattering of interacting quasiparticles in quantum spin chains. Phys. Rev. Res. 3, 013078 (2021).
Milsted, A., Liu, J., Preskill, J. & Vidal, G. Collisions of false-vacuum bubble walls in a quantum spin chain. PRX Quantum 3, 020316 (2022).
Papaefstathiou, I., Knolle, J. & Bañuls, M. C. Real-time scattering in the lattice Schwinger model. Phys. Rev. D 111, 014504 (2025).
Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. & Verstraete, F. Unifying time evolution and optimization with matrix product states. Phys. Rev. B 94, 165116 (2016).
Kramers, H. A. & Wannier, G. H. Statistics of the two-dimensional ferromagnet. part I. Phys. Rev. 60, 252–262 (1941).
Fradkin, E. H. & Susskind, L. Order and disorder in gauge systems and magnets. Phys. Rev. D 17, 2637–2658 (1978).
Schütz, G. Duality twisted boundary conditions in n-state Potts models. J. Phys. A: Math. Gen. 26, 4555–4563 (1993).
Oshikawa, M. & Affleck, I. Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line. Nucl. Phys. B 495, 533–582 (1997).
Greenberger, D. M., Horne, M. A. & Zeilinger, A. in Bell’s Theorem, Quantum Theory and Conceptions of the Universe (ed. Kafatos, M.) 69–72 (Springer, 1989).
Haldane, F. D. M. Continuum dynamics of the 1-D Heisenberg antiferromagnetic identification with the O(3) nonlinear sigma model. Phys. Lett. A 93, 464–468 (1983).
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, 1153–1156 (1983).
Gu, Z.-C. & Wen, X.-G. Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B 80, 155131 (2009).
Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 1604–1606 (2012).
Senthil, T. Symmetry protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Kennedy, T. & Tasaki, H. Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains. Commun. Math. Phys. 147, 431–484 (1992).
Kennedy, T. & Tasaki, H. Hidden \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry breaking in Haldane-gap antiferromagnets. Phys. Rev. B 45, 304–307 (1992).
Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987).
Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 85, 075125 (2012).
den Nijs, M. & Rommelse, K. Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. Rev. B 40, 4709–4734 (1989).
Brehm, E. M. & Brunner, I. Entanglement entropy through conformal interfaces in the 2D Ising model. J. High Energy Phys. 2015, 80 (2015).
Eck, L. & Fendley, P. From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects. SciPost Phys. 16, 127 (2024).
Bhardwaj, L., Bottini, L. E., Schäfer-Nameki, S. & Tiwari, A. Illustrating the categorical Landau paradigm in lattice models. Phys. Rev. B 111, 054432 (2025).
Lootens, L., Delcamp, C. & Verstraete, F. Entanglement and the density matrix renormalization group in the generalized Landau paradigm. Nat. Phys. 21, 1657–1663 (2025).
Haegeman, J., Lootens, L., Mortier, Q., Stottmeister, A., Ueda, A. & Verstraete, F. Interacting chiral fermions on the lattice with matrix product operator norms. Preprint at https://arxiv.org/abs/2405.10285 (2024).
Zakharov, V. A. et al. Luttinger liquid tensor network: sine versus tangent dispersion of massless Dirac fermions. Phys. Rev. Res. 6, 043059 (2024).
Zakharov, V. A., Tworzydło, J., Beenakker, C. W. J. & Pacholski, M. J. Helical Luttinger liquid on a space-time lattice. Phys. Rev. Lett. 133, 116501 (2024).
Haegeman, J., Van Acoleyen, K., Schuch, N., Cirac, J. I. & Verstraete, F. Gauging quantum states: from global to local symmetries in many-body systems. Phys. Rev. X 5, 011024 (2015).
Fechisin, C., Tantivasadakarn, N. & Albert, V. V. Noninvertible symmetry-protected topological order in a group-based cluster state. Phys. Rev. X 15, 011058 (2025).