Computing data for Levin-Wen with defects

We demonstrate how to do many computations for non-chiral topological phases with defects. These defects may be 1-dimensional domain walls or 0-dimensional point defects. Using $\operatorname{Vec}(S_3)$ as a guiding example, we demonstrate how domain wall fusion and associators can be computed using generalized tube algebra techniques. These domain walls can be both between distinct or identical phases. Additionally, we show how to compute all possible point defects, and the fusion and associator data of these. Worked examples, tabulated data and Mathematica code are provided.Comment: 17+25 pages, many tables and attached cod

Anomalies and entanglement renormalization

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.Comment: 12+18 pages, 6+5 figures, 0+2 tables, v2 published versio

Fusing Binary Interface Defects in Topological Phases: The Vec⁡(Z/pZ)\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z}) case

A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's $\mathbb{Z}/p\mathbb{Z}$ model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the $\mathbb{Z}/p\mathbb{Z}$ model. These physical interpretations provide a new graphical calculus which can be used to compute defect fusion.Comment: 27+10 pages, 2+5 tables, comments welcom

Detecting Topological Order with Ribbon Operators

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for $\mathbb{Z}_d$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting nonabelian topological order.Comment: 15 pages, 8 figures, comments welcom

Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state.Comment: 11 pages, 6 figures, 2 appendices, v2 published versio

Computing Defects Associated to Bounded Domain Wall Structures: The Vec⁡(Z/pZ)\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z}) Case

A domain wall structure consists of a planar graph with faces labeled by fusion categories/topological phases. Edges are labeled by bimodules/domain walls. When the vertices are labeled by point defects we get a compound defect. We present an algorithm, called the domain wall structure algorithm, for computing the compound defect. We apply this algorithm to show that the \emph{bimodule associator}, related to the $O_3$ obstruction of [Etingof et al., Quantum Topol. 1, 209 (2010), arXiv:0909.3140], is trivial for all domain walls of $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$. In the language of this paper, the ground states of the Levin-Wen model are compound defects. We use this to define a generalization of the Levin-Wen model with domain walls and point defects. The domain wall structure algorithm can be used to compute the ground states of these generalized Levin-Wen type models.Comment: 16+10 pages, 7 tables, comments welcom

Domain walls in topological phases and the Brauer-Picard ring for Vec⁡(Z/pZ)\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})

We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer-Picard ring for the fusion category $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$. Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer-Picard ring from a physical perspective.Comment: 16 pages, 1 figure, 3 tables, comments welcom

Gauging Defects in Quantum Spin Systems

The goal of this work is to build a dynamical theory of defects for quantum spin systems. A kinematic theory for an indefinite number of defects is first introduced exploiting distinguishable Fock space. Dynamics are then incorporated by allowing the defects to become mobile via a microscopic Hamiltonian. This construction is extended to topologically ordered systems by restricting to the ground state eigenspace of Hamiltonians generalizing the golden chain. We illustrate the construction with the example of a spin chain with $\mathbf{Vec}(\mathbb{Z}/2\mathbb{Z})$ fusion rules, employing generalized tube algebra techniques to model the defects in the chain. The resulting dynamical defect model is equivalent to the critical transverse Ising model

Medium-term mortality after hip fractures and COVID-19: A prospective multi-centre UK study

Purpose The COVID-19 pandemic has caused 1.4 million deaths globally and is associated with a 3–4 times increase in 30-day mortality after a fragility hip fracture with concurrent COVID-19 infection. Typically, death from COVID-19 infection occurs between 15 and 22 days after the onset of symptoms, but this period can extend up to 8 weeks. This study aimed to assess the impact of concurrent COVID-19 infection on 120-day mortality after a fragility hip fracture. Methods A multi-centre prospective study across 10 hospitals treating 8% of the annual burden of hip fractures in England between 1st March and 30th April, 2020 was performed. Patients whose surgical treatment was payable through the National Health Service Best Practice Tariff mechanism for “fragility hip fractures” were included in the study. Patients’ 120-day mortality was assessed relative to their peri-operative COVID-19 status. Statistical analysis was performed using SPSS version 27. Results A total of 746 patients were included in this study, of which 87 (11.7%) were COVID-19 positive. Mortality rates at 30- and 120-day were significantly higher for COVID-19 positive patients relative to COVID-19 negative patients (p < 0.001). However, mortality rates between 31 and 120-day were not significantly different (p = 0.107), 16.1% and 9.4% respectively for COVID-19 positive and negative patients, odds ratio 1.855 (95% CI 0.865–3.978). Conclusion Hip fracture patients with concurrent COVID-19 infection, provided that they are alive at day-31 after injury, have no significant difference in 120-day mortality. Despite the growing awareness and concern of “long-COVID” and its widespread prevalence, this does not appear to increase medium-term mortality rates after a hip fracture