Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group G G in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the \mathrm{Rep}(G) R e p ( G ) symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.
A bstract Quantum systems of physical interest are often local, but there are at least three competing perspectives on how “locality” should be formalized: an algebraic framework, a path-integral framework, and a lattice framework. One puzzle in this competition is that systems with higher-form symmetries, which are perfectly local from the path-integral and lattice perspectives, can violate the algebraic principle of “additivity”. In this paper, we propose a resolution to this puzzle by introducing a weaker locality principle, disjoint additivity, which together with Haag duality should always be satisfied in local quantum systems. As evidence, we give examples in which disjoint additivity is preserved when ordinary additivity is violated; we show that Haag duality and disjoint additivity are satisfied in rather general lattice systems with local symmetry constraints; we give examples of nonlocal theories in which either disjoint additivity or Haag duality is violated; and finally we give examples of systems with nonlocal symmetry constraints in which disjoint additivity is violated, but can be restored by passing to a local “SymTFT” system in one higher dimension.
We study algebraic locality principles on a 2+1D closed lattice in the presence of a Gauss law for a non-invertible symmetry. Prior work in arXiv:2509.03589 showed that when enforcing the Gauss law of an invertible symmetry, the principle of "Haag duality" is preserved exactly, and "disjoint additivity" is preserved after appropriate treatment of discreteness artifacts. Here we show that for a large class of non-invertible on-site symmetries, Haag duality is preserved exactly only for sufficiently nice, "cuspless" regions. For cusped regions, we instead have a weak form of Haag duality that requires adding a collar. Our results apply to double models with a purely magnetic constraint, and to the more general framework of constraints induced by the on-site action of a Hopf algebra. In particular, we treat a class of extended string-net models explicitly. We also demonstrate disjoint additivity for double models based on a group, and a weakened form of disjoint additivity in the setting of a general Hopf algebra.
A bstract The Kramers-Wannier self-duality of critical quantum chains is examined from the perspective of model wave functions. We demonstrate, using the transverse-field Ising chain and the 3-state Potts chain as examples, that the symmetry operator for the Kramers-Wannier self-duality follows in a simple and direct way from a ‘generalised’ translation symmetry of the model wave function in the anyonic fusion basis. This translation operation, in turn, comprises a sequence of F -moves in the underlying fusion category. The symmetry operator thus obtained naturally admits the form of a matrix product operator and obeys non-invertible fusion rules. The findings reveal an intriguing connection between the (non-invertible) translation symmetry on the lattice and topological aspects of the conformal field theory describing the scaling limit.
The Kramers-Wannier transformation of the 1+1d transverse-field Ising model exchanges the paramagnetic and ferromagnetic phases and, at criticality, manifests as a non-invertible symmetry. Extending such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces has, however, remained challenging. In this work, we construct a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra
H that enjoys an anomaly-free non-invertible symmetry
Rep(H). We provide an intuitive diagrammatic formulation of both the Hamiltonian and the symmetry operators using a non-(co)commutative generalization of ZX-calculus built from Hopf-algebraic data. When
H is self-dual, we further construct a generalized Kramers-Wannier duality operator that exchanges the paramagnetic and ferromagnetic phases and becomes a non-invertible symmetry at the self-dual point. This enlarged symmetry mixes with lattice translation and, in the infrared, flows to a weakly integral fusion category given by a
Z2 extension of
Rep(H). Specializing to the Kac-Paljutkin algebra
H8, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six
Rep(H8)-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six
Rep(H8)-symmetric gapped phases on the lattice via the
H-comodule algebra formalism, in agreement with the module-category classification of
Rep(H8). Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.
We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L} ( − 1 ) F L emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.