  In this work we present some new understanding of topological order,
including three main aspects: (1) It was believed that classifying topological
orders corresponds to classifying gapped quantum states. We show that such a
statement is not precise. We introduce the concept of \emph{gapped quantum
liquid} as a special kind of gapped quantum states that can "dissolve" any
product states on additional sites. Topologically ordered states actually
correspond to gapped quantum liquids with stable ground-state degeneracy.
Symmetry-breaking states for on-site symmetry are also gapped quantum liquids,
but with unstable ground-state degeneracy. (2) We point out that the
universality classes of generalized local unitary (gLU) transformations
(without any symmetry) contain both topologically ordered states and
symmetry-breaking states. This allows us to use a gLU invariant -- topological
entanglement entropy -- to probe the symmetry-breaking properties hidden in the
exact ground state of a finite system, which does not break any symmetry. This
method can probe symmetry- breaking orders even without knowing the symmetry
and the associated order parameters. (3) The universality classes of
topological orders and symmetry-breaking orders can be distinguished by
\emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible
transformations}): small SL transformations can convert the symmetry-breaking
classes to the trivial class of product states with finite probability of
success, while the topological-order classes are stable against any small SL
transformations, demonstrating a phenomenon of emergence of unitarity. This
allows us to give a new definition of long-range entanglement based on SL
transformations, under which only topologically ordered states are long-range
entangled.
