PhD Defense Boris Bolvig Kjær
Title: Mathematics of Topological Order - Superselection Sector Theory of Levin-Wen Models
Abstract:
In this thesis, we study mathematical aspects of topological order in two-dimensional quantum spin systems. This is a phenomenon in models of quantum matter at very low temperatures with extraordinary properties. Theoretically, we aim to establish topological order as a well-defined feature of phases of matter, and to classify phases according to different kinds of topological order. Central mathematical challenges towards this aim are to give a definition of phases, to define topological order as an invariant of a phase, and to compute the invariant. Recent years have seen much progress on the former two challenges. In particular, superselection sector theory adapted to spin systems in the thermodynamic limit has been shown to define an invariant of topologically ordered phases with respect to quasi-local automorphism,
corresponding to local perturbations of the dynamics. In this thesis, we compute this invariant for the class of Levin-Wen models which is widely regarded to be an exhaustive set of models of gapped phases admitting a gapped boundary. The invariant is a unitary modular tensor category, also known as a theory of anyons.
As a first case, we study the double semion state, the first example of a group-based quantum double model with a cohomological twist studied using superselection sector theory. This theory is shown to yield the representation category of the twisted Drinfeld double algebra $D^\phi(\mathbb Z_2)$. The string operators of this model can be defined by elementary constructions on the lattice and give a representation of the anyon theory on the observable algebra.
This is no longer the case, when we proceed to study the Levin-Wen model based on arbitrary unitary fusion categories $\mathcal C$, with obstruction coming from K-theory. Here, the analysis of the ground state and its excitation as well as the construction of string operators requires a detailed understanding of the local Hilbert spaces of the Levin-Wen model in terms of skein theory. This relies on the connection between the Levin-Wen model and the Turaev-Viro topological quantum field theory, and enables the construction of string operators producing anyon representations. The main result is the unitary braided monoidal equivalence, SSS$_f ≃Z(\mathcal C)$, between the full subcategory of superselection sectors of the Levin-Wen model with finite-dimensional endomorphism spaces, and the Drinfeld center of the category $\mathcal C$.
Supervisor: Professor Albert Werner, Department of Mathematical Sciences
Assessment committee:
Chair Professor Nathalie Wahl, Department of Mathematical Sciences
Associate professor Clément Delcamp, Laboratoire Alexander Grothendieck Institut des Hautes Études Scientifiques