  We consider the $L^2$-critical nonlinear Schr\"odinger equation (NLS) with
the delta potential
  $$i\partial_tu +\partial^2_x u + \mu \delta u +|u|^{4}u=0, \, \, t\in \R, \,
x\in \R , $$
  where $ \mu \in \R$, and $\delta$ is the Dirac delta distribution at $x=0$.
Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality
and the conservation laws of mass and energy implies that the solution with
mass less than $\|Q\|_{2}$ is global existence in $H^1(\R)$, where $Q$ is the
ground state of the $L^2$-critical NLS without the delta potential (i.e.
$\mu=0$).
  We are interested in the dynamics of the solution with threshold mass
$\|u_0\|_{2}=\|Q\|_{2}$ in $H^1(\R)$. First, for the case $\mu=0$, such blow-up
solution exists due to the pseudo-conformal symmetry of the equation, and is
unique up to the symmetries of the equation in $H^1(\R)$ from
\cite{Me93:NLS:mini sol} (see also \cite{HmKe05:NLS:mini blp}), and recently in
$L^2(\R)$ from \cite{Dod:NLS:L2thrh1}. Second, for the case $\mu<0$, simple
variational argument with the conservation laws of mass and energy implies that
radial solutions with threshold mass exist globally in $H^1(\R)$. Last, for the
case $\mu>0$, we show the existence of radial threshold solutions with blow-up
speed determined by the sign (i.e. $\mu>0$) of the delta potential perturbation
since the refined blow-up profile to the rescaled equation is stable in a
precise sense. The key ingredients here including the Energy-Morawetz argument
and compactness method as well as the modulation analysis are close to the
original one in \cite{RaS11:NLS:mini sol} (see also \cite{KrLR13:HalfW:nondis,
LeMR:CNLS:blp, Mart05:Kdv:N sol, MaP17:BO:mini sol, MeRS14:NLS:blp}).
