We construct a family of lattice models which possess subsystem non-invertible symmetry-protected topological (SPT) order and analyze their interface modes protected by the symmetry, whose codimension turns out to be more than one. We also propose 2+1d lattice models which belong to two different weak SPT phases distinguished by a combination of translational symmetry and non-invertible symmetry. We show that the interface between them exhibits an exotic Lieb-Schultz-Mattis (LSM) anomaly associated with a modulated symmetry, which cannot be factorized into a direct product of internal and translational symmetries.
We construct a family of lattice models which possess subsystem non-invertible symmetry-protected topological (SPT) order and analyze their interface modes protected by the symmetry, whose codimension turns out to be more than one. We also propose 2+1d lattice models which belong to two different weak SPT phases distinguished by a combination of translational symmetry and non-invertible symmetry. We show that the interface between them exhibits an exotic Lieb-Schultz-Mattis (LSM) anomaly associated with a modulated symmetry, which cannot be factorized into a direct product of internal and translational symmetries.
We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.
Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.
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.
We construct 2+1-dimensional lattice models of symmetry-protected topological (SPT) phases with non-invertible symmetries and investigate their properties using tensor networks. These models, which we refer to as generalized cluster models, are constructed by gauging a subgroup symmetry H⊂G in models with a finite group 0-form symmetry G. By construction, these models have a non-invertible symmetry described by the group-theoretical fusion 2-category C(G;H). After identifying the tensor network representations of the symmetry operators, we study the symmetry acting on the interface between two generalized cluster states. In particular, we will see that the symmetry at the interface is described by a multifusion category known as the strip 2-algebra. By studying possible interface modes allowed by this symmetry, we show that the interface between generalized cluster states in different SPT phases must be degenerate. This result generalizes the ordinary bulk-boundary correspondence. Furthermore, we construct parameterized families of generalized cluster states and study the topological charge pumping phenomena, known as the generalized Thouless pump. We exemplify our construction with several concrete cases, and compare them with known phases, such as SPT phases with 2Rep((Z2[1]×Z2[1])⋊Z2[0]) symmetry.
Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in 2+1 dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a dth-order SPT phase in d+1 dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in 3+1 dimensions exhibiting hinge modes protected by a noninvertible symmetry.
The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that they can both be described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it were an internal symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly that is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gaplessness local in the parameter space.