  We describe how to construct generalized string-net models, a class of
exactly solvable lattice models that realize a large family of 2D topologically
ordered phases of matter. The ground states of these models can be thought of
as superpositions of different "string-net configurations", where each
string-net configuration is a trivalent graph with labeled edges, drawn in the
$xy$ plane. What makes this construction more general than the original
string-net construction is that, unlike the original construction, tetrahedral
reflection symmetry is not assumed, nor is it assumed that the ground state
wave function $\Phi$ is "isotropic": i.e. in the generalized setup, two
string-net configurations $X_1, X_2$ that can be continuously deformed into one
another can have different ground state amplitudes, $\Phi(X_1) \neq \Phi(X_2)$.
As a result, generalized string-net models can realize topological phases that
are inaccessible to the original construction. In this paper, we provide a more
detailed discussion of ground state wave functions, Hamiltonians, and minimal
self-consistency conditions for generalized string-net models than what exists
in the previous literature. We also show how to construct string operators that
create anyon excitations in these models, and we show how to compute the
braiding statistics of these excitations. Finally, we derive necessary and
sufficient conditions for generalized string-net models to have isotropic
ground state wave functions on the plane or the sphere -- a property that may
be useful in some applications.
