  We study fluctuations of the current at the boundary for the half space
asymmetric simple exclusion process (ASEP) and the height function of the half
space six vertex model at the boundary at large times. We establish a phase
transition depending on the asymptotic density of particles at the boundary,
with GSE and GOE limits as well as the Baik--Rains crossover distribution near
the critical point. This was previously known for half space last passage
percolation, and recently established for the half space log-gamma polymer and
KPZ equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto.
  The proof uses the underlying algebraic structure of these models in a
crucial way to obtain exact formulas. In particular, we show a relationship
between the half space six vertex model and a half space Hall--Littlewood
measure with two boundary parameters, which is then matched to a free boundary
Schur process via a new identity of symmetric functions. Fredholm Pfaffian
formulas are established for the half space ASEP and six vertex model,
indicating a hidden free fermionic structure.
