New Index Links Symmetry Rules to Predictable Behaviour in Physical Systems

Researchers are increasingly focused on understanding non-invertible symmetries, a complex area of theoretical physics with implications for classifying topological phases of matter. Kansei Inamura, from both the Mathematical Institute and the Rudolf Peierls Centre for Theoretical Physics at the University of Oxford, alongside Kansei Inamura, proposes a novel index to characterise these symmetries in one plus one dimensions and explores its connection to their physical realisability on lattice systems. This work significantly extends the established Gross-Nesme-Vogts-Werner index for invertible symmetries and demonstrates that the fusion rules governing these non-invertible symmetries are constrained to align with those of weakly integral fusion categories under certain conditions. By introducing a general class of matrix product operators, termed topological injective MPOs, the authors provide a framework for constructing and analysing these symmetries, including well-known examples like Kramers-Wannier symmetries, and establish conditions for consistent fusion behaviour, potentially paving the way for a comprehensive index theory of non-invertible symmetries.

Researchers have developed a new framework for understanding symmetries in quantum systems, addressing a long-standing challenge in theoretical physics: defining and classifying non-invertible symmetries, which arise in various condensed matter and quantum field theory models. This work moves beyond traditional notions of symmetry that require operators to be invertible. The study introduces an ‘index’ to characterise these non-invertible symmetry operators in one spatial and one time dimension, building upon earlier work concerning invertible symmetries represented by quantum cellular automata. By analysing the constraints imposed by the structure of quantum many-body systems, specifically those described by tensor products of local Hilbert spaces, the researchers demonstrate a surprising connection between the realizability of these symmetries and a mathematical property known as ‘weak integrality’ of the associated fusion categories. The core achievement lies in establishing a link between the proposed index and the fusion rules governing how symmetry operators combine. Assuming a uniformity condition on the index, the research proves that the fusion rules on a tensor product Hilbert space can only correspond to weakly integral fusion categories, even when combined with quantum cellular automata. This result offers significant support for a recent conjecture regarding the conditions under which non-integral fusion categories can be realised in physical systems. To facilitate this analysis, the researchers propose a novel class of tensor network operators, termed ‘topological injective matrix product operators’ (MPOs), which encompass both invertible symmetries and more exotic non-invertible forms like Kramers-Wannier symmetries found in systems exhibiting duality. These MPOs serve as the central methodological tool, employed to describe non-invertible symmetries on a tensor product Hilbert space. These matrix product operators are not merely mathematical constructs; the work details how to construct ‘defect Hilbert spaces’ and map them onto sequential quantum circuits, providing a concrete pathway for simulating and potentially implementing these symmetries in quantum hardware. Crucially, the researchers demonstrate that the homogeneity of the index, the consistency of its value across different fusion channels, is linked to the existence of specific ‘fusion and splitting tensors’ that govern the combination and decomposition of symmetry operators. While proving the general existence of these tensors remains an open problem, the team has explicitly constructed them for all the examples considered, bolstering the validity of their approach and paving the way for further exploration of non-invertible symmetries in complex quantum systems. Defect Hilbert spaces, fundamental to describing non-invertible symmetries, are established under specific physical assumptions regarding symmetry operator products and sums. These spaces, denoted as Hl DX and Hr DX, represent local modifications to the original Hilbert space H in the presence of a defect X oriented upwards or downwards respectively. The dimensions of these defect spaces are independent of defect position due to translational invariance inherent in the system. The research demonstrates that these spaces can be generally written as tensor products, specifically Hl DX ∼= LX ⊗ H ⊗N−nl X o and Hr DX ∼= RX ⊗ H ⊗N−nr X o, where LX and RX are finite dimensional Hilbert spaces and nl X and nr X are positive integers defining the extent of the local modification. The left and right dimensions, ldim(DX) and rdim(DX), are defined as the ratios of the defect Hilbert space dimensions to the original Hilbert space dimension, providing a measure of the local impact of the symmetry operator. These dimensions are central to characterising the realizability of non-invertible symmetries on tensor product Hilbert spaces. The work establishes that a unitary fusion category symmetry can be realised without mixing with quantum cellular automata only when every object within that category possesses an integral quantum dimension. This condition, rigorously proven through a different method in a concurrent study, highlights a crucial link between symmetry realizability and the underlying mathematical structure of the fusion category. Further analysis reveals that the defect Hilbert spaces are determined by replacing a finite region around the defect with a different Hilbert space, a process quantified by the integers nl X and nr X, which represent the number of sites affected by the modification and are independent of system size N. Scientists are increasingly focused on symmetries beyond those traditionally considered in physics, specifically those that are ‘non-invertible’. For decades, the difficulty lay in extending the well-established mathematical tools for handling invertible symmetries to this new realm. This work offers a significant step forward by proposing a way to index and classify these non-invertible symmetries in simplified, one-dimensional systems. The implications extend beyond purely mathematical curiosity, as understanding non-invertible symmetries is crucial for describing novel phases of matter, particularly those exhibiting unusual topological properties and entanglement. These phases could potentially underpin more robust quantum technologies, less susceptible to the errors that plague current quantum computing efforts. The ability to reliably characterise these symmetries is therefore a prerequisite for designing and controlling such materials. However, the current framework relies on specific conditions regarding the ‘fusion’ of symmetry operators, and proving these conditions generally remains an open problem. Scaling these techniques to more complex, three-dimensional systems, where the true potential for topological materials lies, presents a formidable hurdle. Looking ahead, the convergence of tensor network techniques with this emerging index theory is particularly promising, potentially providing a computational pathway to explore non-invertible symmetries in realistic materials. Simultaneously, mathematicians are refining the underlying category theory, seeking a more complete and general framework. The ultimate goal is not merely to catalogue these symmetries, but to harness them, to engineer materials with properties previously thought impossible, and to unlock new avenues in quantum information processing.