Hidden Symmetries Emerge from Quantum Systems, Reshaping Our Understanding of Matter

Tsubasa Oishi and Hiromi Ebisu of the Kyoto University present a lattice model displaying ‘t Hooft anomalies, revealing the emergence of complex higher symmetry structures including 2-group and non-invertible symmetries. The model clarifies the connection between these anomalies and systems with Lieb-Schultz-Mattis anomalies, showing how modulated symmetries appear when internal symmetries become translational. The study uncovers that these symmetries can depend on the presence of defects, offering a key understanding of how they constrain the behaviour of quantum phases of matter

Simulating quantum anomalies using a simplified lattice structure

Lattice gauging served as a simplified, grid-like model to simulate complex quantum systems. Much like building with LEGO bricks, a manageable system was constructed to explore intricate quantum behaviours. Directly observing anomalies and emergent symmetries in real materials is exceptionally difficult, demanding a controlled environment to isolate and study their effects. The inherent complexity of many-body quantum systems necessitates the use of computational techniques to bridge the gap between theoretical predictions and experimental observations. By implementing the anomaly within this lattice structure, scientists systematically ‘gauged’ subgroups of the symmetry, effectively focusing on specific quantum interactions and observing the resulting higher symmetries. This process involves introducing local gauge degrees of freedom, effectively promoting global symmetries to local ones, and then analysing the resulting constraints on the system’s behaviour.

A lattice model featuring four global symmetries and a mixed anomaly was employed to investigate emergent symmetry structures. This methodical approach allowed for the identification of 2-group symmetries, where the order of applying symmetry operations is significant, and non-invertible symmetries, a more exotic form of symmetry previously predicted theoretically. These non-invertible symmetries represent a departure from conventional symmetry concepts, where every symmetry transformation has a corresponding inverse transformation. Gauging different combinations of these initial symmetries revealed 2-group, non-invertible, and higher fusion categorical symmetries. This technique provides a thorough method for understanding how anomalies constrain quantum phases of matter. The use of a lattice allows for precise control over the interactions and symmetries within the system, enabling a detailed analysis of the emergent behaviour. The specific anomaly investigated is characterised by a term proportional to $a_1\wedge a_2\wedge a_3\wedge a_$4, where the $a_i$ denote background gauge fields for the global symmetries, representing a particular type of incompatibility between the symmetries.

Realisation of a type IV anomaly and emergence of higher symmetries

Researchers at Kyoto University have, for the first time, demonstrated a lattice model realising a type IV anomaly, represented by SIV = Z M4A ∧B ∧C ∧D. This surpasses previous limitations restricted to type III anomalies and opens avenues for exploring more complex quantum behaviours. ‘t Hooft anomalies are topological obstructions to consistently defining a quantum field theory with certain symmetries. Type IV anomalies are particularly challenging to realise, requiring a specific combination of symmetries and interactions. The realization allows for the systematic gauging of subgroups, revealing emergent symmetry structures previously inaccessible. This systematic approach is crucial for understanding the interplay between anomalies, symmetries, and the resulting quantum phases of matter. Through this simplified modelling technique, a concrete method for understanding how anomalies constrain quantum phases of matter was established, moving beyond theoretical predictions. The ability to systematically gauge subgroups of the symmetry allows researchers to explore the full range of possible emergent symmetries and their associated constraints.

Applying this framework to systems exhibiting Lieb-Schultz-Mattis anomalies, arising when internal symmetries are linked to translational symmetries, revealed that modulated, or spatially varying, symmetries directly correspond to those found in purely internal type-IV anomalies. The Lieb-Schultz-Mattis theorem states that certain systems with internal symmetries cannot have a gapped, trivial ground state. The emergent symmetry structure was found to depend on the presence of symmetry defects within the material, a feature not observed in previously studied modulated symmetries. These defects, such as domain walls or vortices, can break the symmetry locally and influence the overall behaviour of the system. This finding suggests a deeper connection between spatial variations and internal symmetries within quantum materials, potentially leading to new insights into the nature of quantum phases of matter. The dependence on defects highlights the importance of considering disorder and imperfections when studying quantum materials.

Defect-dependent higher symmetries explain complex quantum material behaviour

Scientists are increasingly focused on understanding how fundamental constraints, known as anomalies, dictate the behaviour of quantum materials. These anomalies arise from the interplay between symmetries and quantum dynamics, and can have profound consequences for the material’s properties. This latest work establishes a concrete model for a particularly complex type of anomaly, revealing how it generates higher symmetries, symmetries where the order of operations matters, or which lack simple inverses. However, these symmetries are not necessarily fixed properties of the material itself. Understanding these ‘anomalies’, fundamental constraints on how quantum materials behave, is vital for designing novel materials with tailored properties. The ability to predict and control these anomalies could lead to the development of materials with enhanced functionality and performance.

This work provides a concrete pathway to explore how these constraints manifest as symmetries, even if those symmetries are not absolute, offering a more nuanced picture of quantum behaviour. The traditional view of symmetry assumes that it is a fixed property of a system, but this research demonstrates that symmetry can be emergent and dependent on external factors. Researchers have demonstrated that symmetries within quantum materials aren’t always inherent, instead arising from, and depending on, imperfections within the material itself. This nuanced understanding of fundamental constraints will likely shape the design of future quantum systems. The realization that defects can play a crucial role in determining the symmetry of a material opens up new avenues for materials design and manipulation.

A concrete lattice model to explore mixed anomalies and their connection to emergent symmetry has been established. By systematically gauging these anomalies, higher symmetries, including 2-group and non-invertible symmetries, which exhibit unusual properties compared to conventional symmetries, were revealed. Key to this research is the demonstration that these symmetries are not always inherent to a material, but can instead depend on the presence of defects within its structure, a novel finding with implications for understanding complex quantum phases. The identification of defect-dependent higher symmetries provides a new framework for understanding the behaviour of quantum materials and could lead to the discovery of new quantum phases of matter. This work represents a significant step forward in our understanding of the interplay between symmetry, anomalies, and quantum matter.

Researchers demonstrated that fundamental constraints, known as anomalies, can give rise to emergent symmetries in quantum materials. This means the symmetries observed aren’t necessarily fixed properties, but instead arise from and depend on imperfections within the material itself. Through a lattice model and explicit gauging, they revealed higher symmetries, including 2-group and non-invertible symmetries, that can change based on the presence of defects. The authors suggest this framework clarifies the role of anomalies in constraining quantum phases of matter and provides a pathway to explore these relationships.