Over the past 30 years experimental observations have demonstrated the existence of a variety of quantum phases of matter not admissible to a classification in terms of the Landau theory of symmetry breaking. Examples include, but are not limited to the fractional quantum
Hall states and frustrated quantum magnets. Theoretical evidence supports the idea that such phases can exist in a large class of zero temperature strongly correlated condensed matter systems.
In this thesis we study a particular case of such systems called topological phases of matter. Such phases are characterised by the presence of non-local correlations which are manifest in properties such as degenerate groundstates that depend on the global topology of the
system and the emergence of topological excitations. Remarkably the classification of such materials is profoundly tied to the mathematical construction of topological quantum field theories (TQFT).
In this thesis we utilise this connection to explore possible candidate Hamiltonian models for topological phases of matter. Our methodology is that of reverse engineering effective local Hamiltonians from a class of discrete TQFT’s called state sums.
In chapter 5 we develop a construction to canonically associate to any state-sum TQFT a corresponding local Hilbert space and Hamiltonian defining a candidate model for a topological phase of matter.
In chapter 9 we develop a candidate model of topological phases using ideas from higher gauge theory and higher category theory. In particular we define a Hamiltonian realisation of the Yetter Homotopy 2-type TQFT which describes a topological gauge theory, where the
gauge symmetry is given by a finite 2-group and relate a class of such models to the construction of Walker-Wang.
Building on the Hamiltonian construction for state-sum TQFT’s, in Part III of this thesis we develop an algebraic approach to understanding the topological excitations of such theories, we call tube-algebras. In chapter 10 we develop a general construction for defining tube-algebras for any unitary state-sum TQFT and describe the general features. In chapter 12 we apply this construction to the
Dijkgraaf-Witten TQFT in 1+1, 2+1 and 3+1D. In chapter 13 we
apply this construction to topological higher lattice gauge theories and compare the results the Dijkgraaf-Witten TQFT.
