Simulation of Anyons Using Symmetric Tensor Network Algorithms

The study of anyons offers one of the most exciting challenges in contemporary physics. Anyons are exotic quasiparticles with non-trivial exchange statistics, which makes them difficult to simulate. However, they are of great interest as some species offer the prospect of a highly fault-tolerant form of universal quantum computation, and it has been suggested that the simplest such species may appear in the fractional quantum Hall state with filling fraction nu = 12/5. Despite the current strong interest in the development of practical quantum computing, our ability to study the collective behaviour of systems of anyons remains limited. Meanwhile, tensor network algorithms are a relatively recent development in the field of condensed matter physics. They consist of Ansaetze for the low-energy states of a lattice system, whose number of free parameters scale at most polynomially in the system size, and algorithms for their optimisation, manipulation, and analysis. However, many condensed matter systems possess a high degree of symmetry, which may be exploited to yield an even more efficient description of the low-energy subspace, and when I began work on my Thesis these algorithms (with the exception of DMRG) did not in general take advantage of these symmetries. In this Thesis I develop a formalism, based on the frameworks of spin networks and category theory, whereby a tensor network acting on any lattice model exhibiting a mathematical structure corresponding to a Unitary Braided Tensor Category (UBTC) may be represented in a particularly compact and efficient manner corresponding to the exploitation of this structure. This permits the exploitation of global Abelian and non-Abelian internal group symmetries, both to facilitate the study of particular symmetry sectors of the model, and for computational gain. Furthermore, the formalism also naturally admits the study of models possessing non-trivial exchange statistics (e.g. fermions, Abelian anyons) and models possessing a UBTC structure which is not associated with a group (some Abelian and non-Abelian anyons), all for a computational cost polynomial in the system size. In addition I also describe the development of a tensor network algorithm to exploit the spatial symmetry of scale invariance present in quantum critical lattice systems. The resulting Ansatz provides a remarkably efficient description of the low-energy subspace of an infinite quantum critical lattice model, naturally yielding the polynomial correlators typical of such a system, and providing easy access to the majority of the conformal data which describe its behaviour in the continuum limit. Combining this Ansatz with the UBTC formalism for tensor networks provides a demonstration of the flexibility of these techniques, computing the conformal data associated with the continuum limit of two non-Abelian anyonic quantum critical lattice models. In summary, this Thesis provides a new Ansatz for the study of quantum critical lattice models, and a formalism permitting the exploitation of Abelian and non-Abelian symmetries of lattice models, allowing the analysis of many fermionic and anyonic systems in polynomial time.