  We consider the existence and H\"{o}lder continuity conditions for the $k$-th
order derivatives of self-intersection local time for $d$-dimensional
fractional Brownian motion, where $k=(k_1,k_2,\cdots, k_d)$. Moreover, we show
a limit theorem for the critical case with $H=\frac{2}{3}$ and $d=1$, which was
conjectured by Jung and Markowsky (2014).
