Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess a fractional-spin conserved current. These currents have been constructed by utilising quantum-group algebras, fermionic and parafermionic operators, and ideas from ``discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. Such a current amounts to terminating a lattice topological defect, and I will touch on related work on such done with Aasen and Mong. I show how requiring a current be conserved yields simple constraints on the Boltzmann weights, and that all of the many models known to satisfy these constraints are integrable. This procedure therefore gives a linear construction for ``Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

t.b.a. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

I give an overview of work with Aasen and Mong on topological defects in two-dimensional classical lattice models, quantum spin chains and tensor networks. The partition function in the presence of a topological defect is invariant under any local deformation of the defect. By using results from fusion categories, we construct topological defects in a wide class of lattice models, and show how to use them to derive exact properties of field theories by explicit lattice calculations. In the Ising model, the fusion of duality defects allows Kramers-Wannier duality to be enacted on the torus and higher genus surfaces easily, implementing modular invariance directly on the lattice. In other models, the construction leads to generalised dualities previously unknown. A consequence is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization and for critical theories, the spin of the associated conformal field. Other universal quantities we compute exactly on the lattice are the ratios of g-factors for conformal boundary conditions

Traditionally, most studies of quantum many-body systems have been mainly concerned with properties of the states of low-lying energy. Recently, however, fascinating features of the full energy spectrum have been uncovered. Among these are eigenstate phase transitions, where sharp transitions occur not only in the ground state, but in all the states. I describe a simple example of such, a transition for a strong zero mode in the XYZ spin chain. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in parafermionic systems and strongly disordered many-body localized phases. My proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure – despite being a rather elaborate operator, it squares to the identity.

I discuss how systems with non-abelian anyons can be used to build a topological quantum computer. Operations are performed by braiding the anyons, because the outcome of braiding is a purely topological property, such quantum computers should be robust against local errors. I will give several examples of how such anyons arise in fractional quantum Hall systems and in quantum loop models. Mathematical byproducts of this work are algebraic proofs and extensions of Tutte's identities for the chromatic polynomial (the zero-temperature Potts-model partition function).