  In this paper, we construct the axialvector and tensor current operators to
investigate the ground state tetraquark states and the first radially excited
tetraquark states with the quantum numbers $J^{PC}=1^{+-}$ via the QCD sum
rules systematically, and observe that there are one axialvector tetraquark
candidate for the $Z_c(3900)$ and $Z_c(4430)$, two axialvector tetraquark
candidates for the $Z_c(4020)$, three axialvector tetraquark candidates for the
$Z_c(4600)$.
