  We systematically investigate the stability of the doubly heavy tetraquark
states $[QQ][\bar{q}\bar{q}]$ ($Q=c$ and $b$, $q=u$, $d$ and $s$) within the
framework of the color flux-tube model involving a multibody confinement
potential, $\sigma$-exchange, one-gluon-exchange and
one-Goldstone-boson-exchange interactions. Our numerical analysis indicates
that the states $[bb][\bar{u}\bar{d}]$ with $01^+$ and $[bb][\bar{u}\bar{s}]$
with $\frac{1}{2}1^+$ are the most promising stable states against strong
interactions. The states $[cc][\bar{u}\bar{d}]$ with $01^+$,
$[bc][\bar{u}\bar{d}]$ with $00^+$, $01^+$, and $12^+$, and
$[bb][\bar{u}\bar{d}]$ with $01^-$ and $12^+$ as stable states are also
predicted in the color flux-tube model. The dynamical mechanism producing those
stable doubly heavy tetraquark states are discussed in the color flux-tube
model.
