  Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$ is one of
the classical exactly solvable models in the KPZ universality class. We study
the "slow bond" model, where TASEP on $\mathbb{Z}$ is imputed with a slow bond
at the origin. The slow bond increases the particle density immediately to its
left and decreases the particle density immediately to its right. Whether or
not this effect is detectable in the macroscopic current started from the step
initial condition has attracted much interest over the years and this question
was settled recently (Basu, Sidoravicius, Sly (2014)) by showing that the
current is reduced even for arbitrarily small strength of the defect. Following
non-rigorous physics arguments (Janowsky, Lebowitz (1992, 1994)) and some
unpublished works by Bramson, a conjectural description of properties of
invariant measures of TASEP with a slow bond at the origin was provided in
Liggett's 1999 book. We establish Liggett's conjectures and in particular show
that TASEP with a slow bond at the origin, started from step initial condition,
converges in law to an invariant measure that is asymptotically close to
product measures with different densities far away from the origin towards left
and right. Our proof exploits the correspondence between TASEP and the last
passage percolation on $\mathbb{Z}^2$ with exponential weights and uses the
understanding of geometry of maximal paths in those models.
