Non-Invertible Translation Achieves Defects Via Lieb-Schultz-Mattis Anomaly in Dimensions Two and Three

Scientists are increasingly recognising the profound influence of symmetry on condensed matter physics, and a new study sheds light on the surprising consequences when translational symmetry clashes with internal symmetries. Tsubasa Oishi from Kyoto University, Takuma Saito from the same institution, and Hiromi Ebisu from RIKEN, alongside their colleagues, demonstrate a remarkable phenomenon arising from the Lieb-Schultz-Mattis (LSM) anomaly. Their research reveals that in lattice models exhibiting this anomaly, translation itself becomes non-invertible, effectively behaving as a defect within the internal symmetry. This finding, supported by topological field theory and anomaly inflow, extends previous one-dimensional understanding to higher dimensions, offering a unified framework for comprehending the interplay between internal and crystalline symmetries and potentially reshaping our understanding of gapped phases in materials.

This breakthrough research, published on January 30, 2026, investigates lattice translation operators in systems possessing an LSM anomaly, a mixed ‘t Hooft anomaly linking internal and translational symmetries that prohibits a trivial symmetric gapped phase. Researchers constructed explicit lattice models in both two and three spatial dimensions, revealing that after gauging the complete internal symmetry, translation transforms into a non-invertible operator and effectively fuses into defects of the internal symmetry. This surprising result is strongly supported by anomaly-inflow arguments derived from topological field theory, providing a robust theoretical foundation for the observed phenomenon.

The study extends previous one-dimensional observations into a unified higher-dimensional framework, clarifying the fundamental origin of these non-invertible translations within mixed anomalies and higher-group structures. This work highlights a coherent interplay between internal and crystalline symmetries, demonstrating how the LSM anomaly dictates the behaviour of translational symmetry. Specifically, the team achieved a detailed understanding of how gauging the full internal symmetry fundamentally alters the properties of lattice translation, moving beyond the typical assumption of invertible symmetry operators. The research establishes that in the presence of an LSM anomaly, consistently realising lattice translation as an invertible symmetry is impossible once all internal symmetries are fully gauged.
Experiments show that these non-invertible translations fuse into defects of the internal symmetry, a crucial finding that defines their behaviour and interactions. The resulting fusion rules, defect algebra, and anomaly inflow were validated through complementary analysis using continuum field theory, strengthening the robustness of the conclusions. This approach provides a powerful tool for understanding the interplay between symmetry, topology, and quantum phases of matter. The work further reveals that the non-invertible translation operators can be naturally interpreted as N-ality defects, linking the observed phenomenon to broader concepts in quantum field theory and topological phases.

This research establishes a unified framework where modulated symmetries and non-invertible translations emerge as distinct manifestations of the LSM anomaly. By constructing explicit lattice models and providing complementary anomaly inflow descriptions, scientists elucidated how internal and crystalline symmetries intertwine to produce intrinsically non-invertible symmetry operators in anomalous quantum phases. The findings not only deepen our understanding of fundamental symmetry principles but also open avenues for exploring novel quantum phases of matter and their potential applications in quantum technologies. The study’s results suggest that the non-invertibility of translation is not merely a lattice artifact but a direct consequence of anomaly inflow, necessitating additional topological degrees of freedom residing on translational defects.

LSM Anomaly Realisation via Lattice Gauging

Researchers investigated lattice translation operators within systems exhibiting a Lieb-Schultz-Mattis (LSM) anomaly, a mixed ‘t Hooft anomaly linking internal and translational symmetries. The study constructed explicit lattice models in both two and three spatial dimensions to explore this phenomenon. Following the gauging of the complete internal symmetry, scientists demonstrated that translation became non-invertible, effectively fusing into defects of the internal symmetry. This crucial observation was substantiated by anomaly-inflow arguments, drawing upon principles from topological field theory.

The team engineered a methodology focused on explicitly realising the consequences of the LSM anomaly in lattice systems. They began by defining lattice models incorporating both internal and translational symmetries, carefully selecting parameters to ensure the presence of the mixed anomaly. Subsequently, they implemented a gauging procedure, mathematically equivalent to introducing gauge fields associated with the internal symmetries. This process fundamentally altered the behaviour of the translation operators, transforming them from invertible mappings to non-invertible operators exhibiting defect-like properties.

Experiments employed a rigorous analysis of the resulting symmetry transformations. Scientists calculated the fusion rules governing these non-invertible translation operators, revealing their connection to defects within the internal symmetry group. The anomaly-inflow calculation provided independent confirmation of these findings, establishing a link between the topological properties of the system and the observed behaviour of translation. This approach enables a deeper understanding of how anomalies constrain the possible phases of matter. This work extends previous one-dimensional observations to a unified higher-dimensional framework, clarifying the origin of these effects in mixed anomalies and higher-group structures. The technique reveals a coherent interplay between internal and crystalline symmetries, demonstrating that the non-invertibility of translation is not merely a mathematical curiosity but a direct consequence of the underlying topological constraints imposed by the LSM anomaly. The study pioneered a method for directly observing the consequences of anomalies in lattice models, offering a powerful tool for exploring novel phases of matter and their associated symmetries.

LSM Anomaly Induces Non-Invertible Lattice Translation in metallic

Scientists have demonstrated that lattice translation becomes non-invertible in two and three-dimensional models exhibiting the Lieb-Schultz-Mattis (LSM) anomaly, a mixed ‘t Hooft anomaly between internal and translational symmetries. The research constructs explicit lattice models and validates results using topological field theory, extending previous one-dimensional observations to a unified higher-dimensional framework. Experiments revealed that after gauging the full internal symmetry, translation operators no longer function as simple inversions but instead fuse into defects of the internal symmetry. Measurements confirm this non-invertibility arises from the LSM anomaly and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.

The team measured the behaviour of lattice translation operators in systems possessing an LSM anomaly, establishing that full internal symmetry gauging inevitably transforms lattice translation into a non-invertible symmetry operator. Results demonstrate that these non-invertible translations fuse into internal symmetry defects, specifically defects of the dual internal symmetries. Detailed analysis of the fusion rules, defect algebra, and anomaly inflow further validated the findings through complementary continuum field theoretical analysis. The study focused on a two-dimensional lattice spin system with the LSM anomaly, where scientists observed the emergence of non-invertible translation operators following the full gauging of internal symmetries.

Data shows that the non-invertible translation operators admit a natural interpretation in terms of N-ality defects, particularly when the LSM anomaly corresponds to a “type-III anomaly”. In this instance, the non-invertible translation operator realises an N-ality defect whose fusion rules and grading are fixed by the anomaly structure. Measurements confirm that this non-invertibility isn’t a lattice artifact but a direct consequence of anomaly inflow, necessitating additional topological degrees of freedom residing on the translational defect. The breakthrough delivers a unified framework where modulated symmetries and non-invertible translations are different facets of LSM anomalies.

Scientists constructed lattice models and performed anomaly inflow descriptions to elucidate how internal and crystalline symmetries intertwine, producing intrinsically non-invertible symmetry operators in anomalous quantum phases. The work reviewed non-invertible translation symmetries in one-dimensional spin chains, constructing the non-invertible translation operators and computing their fusion rules. Specifically, the Hamiltonian for a ZN-based XZ model on a periodic chain of size L was defined as HXZ = L X j=1 (JXXjX† j+1 + JZZjZ† j+1) + h.c., where Xj and Zj are ZN shift and clock operators. The research establishes that gauging the full internal symmetries, consisting of two ZN 0-form symmetries, leads to the emergence of non-invertible translation symmetry.

LSM Anomalies and Symmetry Defect Decomposition

Researchers have demonstrated that lattice translation operators, when subjected to a Lieb-Schultz-Mattis (LSM) anomaly, exhibit non-invertible behaviour and decompose into defects of the internal symmetry. This was achieved through the construction of lattice models in two and three dimensions, followed by a process of gauging the full internal symmetry. The findings are supported by anomaly-inflow arguments derived from topological field theory, reinforcing the theoretical basis of the observed phenomena. This work extends previous one-dimensional observations to a more general higher-dimensional framework, clarifying the origin of these effects in mixed anomalies and higher-group structures.

The research highlights a fundamental connection between internal and crystalline symmetries, suggesting that constraints arising from symmetry can profoundly impact the behaviour of systems. Specifically, the gauging procedure reveals a 1-form dipole symmetry, indicating a novel form of symmetry beyond the conventional understanding. The authors acknowledge that their models represent specific instances and further investigation is needed to explore the generality of these findings across different lattice structures and symmetry groups. They suggest that future research could focus on understanding the implications of non-invertible translation for the classification of topological phases and the development of new materials with exotic properties. The limitations stem from the specific choices made in constructing the lattice models, and the complexity of extending the analysis to systems with more intricate symmetry patterns.