In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) and p p -ality (order p p ) mapping on the \mathbb{Z}_p× \mathbb{Z}_p ℤ p × ℤ p symmetric Hamiltonians, where p p is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.
Realizing triality and p-ality by lattice twisted gauging in (1+1)d quantum spin systems
In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) and p p -ality (order p p ) mapping on the \mathbb{Z}_p× \mathbb{Z}_p ℤ p × ℤ p symmetric Hamiltonians, where p p is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.
Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in {1+1} 1 + 1 dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
Given a two-dimensional quantum lattice model with an Abelian gauge theory interpretation, we investigate a duality operation that amounts to gauging its invertible 1-form symmetry, followed by gauging the resulting 0-form symmetry in a twisted way via a choice of discrete torsion. Using tensor networks, we introduce explicit lattice realisations of the so-called condensation defects, which are obtained by gauging the 1-form symmetry along submanifolds of spacetime, and employ the same calculus to realise the duality operators. By leveraging these tensor network operators, we compute the non-trivial interplay between symmetry-twisted boundary conditions and charge sectors under the duality operation, enabling us to construct isometries relating the dual Hamiltonians. Whenever a lattice gauge theory is left invariant under the duality operation, we explore the possibility of promoting the self-duality to an internal symmetry. We argue that this results in a symmetry structure that encodes the 2-representations of a 2-group.
Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 × ℤ 2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 ⊂ ℤ 2 × ℤ 2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZ X Y Z chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n} ℤ n clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n} ℤ n × ℤ n global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n} ℤ n charge and dipole symmetries.
We extend non-invertible duality concepts from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. For the Z2×Z2 subsystem symmetry on the square lattice, we build two complementary dualities: a map that sends spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (the analogue of the Kramers-Wannier (KW) duality in 1+1D), and a generalized subsystem Kennedy-Tasaki (KT) transformation that maps SSSB phases to subsystem symmetry-protected topological (SSPT) phases while leaving the trivial phase invariant. These dualities are boundary-sensitive. On open lattices, both subsystem KW and KT transformations act as unitary, invertible operators. In particular, the KT map not only matches the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase onto the protected boundary degeneracy of the SSPT phase. On closed manifolds, however, both maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We demonstrate this non-invertibility via ground-state degeneracy matching (in two copies of the Xu-Moore/Ising-plaquette model), analysis of symmetry-twist sectors mapping, and the fusion algebra of the duality operator. Enlarging the Hilbert space to include twisted sectors allows the subsystem KW map to be formulated as a projective unitary preserving quantum transition probabilities, consistent with generalized Wigner-theorem-based constructions. We also show that the KT map faithfully transmits the algebraic content of bulk and edge invariants diagnosing strong SSPT order: although strictly local SSPT repair operators map to highly nonlocal objects in the dual SSSB phase, the essential commutation algebra and the bulk-edge correspondence remain intact.
We provide a general prescription for gauging finite non-invertible symmetries in 1+1d lattice Hamiltonian systems. Our primary example is the Rep(D _8 8 ) fusion category generated by the Kennedy-Tasaki transformation, which is the simplest anomaly-free non-invertible symmetry on a spin chain of qubits. We explicitly compute its lattice F-symbols and illustrate our prescription for a particular (non-maximal) gauging of this symmetry. In our gauging procedure, we introduce two qubits around each link, playing the role of “gauge fields” for the non-invertible symmetry, and impose novel Gauss’s laws. Similar to the Kramers-Wannier transformation for gauging an ordinary \mathbb{Z}_2 ℤ 2 , our gauging can be summarized by a gauging map, which is part of a larger, continuous non-invertible cosine symmetry.
Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized 1+1 1 + 1 D quantum XY model based on group-valued qudits. This model is specified by a finite group G G and enjoys a projective \mathsf{Rep}(G)× Z(G) 𝖱 𝖾 𝗉 ( G ) × Z ( G ) and translation symmetry, where symmetry operators obey a projective algebra in the presence of symmetry defects. For invertible symmetries, such projective algebras imply Lieb-Schultz-Mattis (LSM) anomalies. However, this is not generally true for non-invertible symmetries, and we derive a condition on G G for the existence of an LSM anomaly. When this condition is not met, we prove an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak symmetry protected topological (SPT) state with non-trivial entanglement, for which we construct an example fixed-point Hamiltonian. The projectivity also affects the dual symmetries after gauging \mathsf{Rep}(G)× Z(G) 𝖱 𝖾 𝗉 ( G ) × Z ( G ) sub-symmetries, giving rise to non-Abelian and non-invertible dipole symmetries, as well as non-invertible translations. We complement our analysis with the SymTFT, where the projectivity causes it to be a topological order non-trivially enriched by translations. Throughout the paper, we develop techniques for gauging \mathsf{Rep}(G) 𝖱 𝖾 𝗉 ( G ) symmetry and inserting its symmetry defects on the lattice, which are applicable to other non-invertible symmetries.