  In this Letter we will show that, in the presence of a properly modulated
Dzyaloshinskii-Moriya (DM) interaction, a $U(1)$ vortex-antivortex lattice
appears at low temperatures for a wide range of the DM interaction. Even more,
in the region dominated by the exchange interaction, a standard BKT transition
occurs. In the opposite regime, the one dominated by the DM interaction, a kind
of inverse BKT transition (iBKT) takes place. As temperature rises, the
vortex-antivortex lattice starts melting by annihilation of pairs of
vortex-antivortex, in a sort of "inverse" BKT transition.

Strongly Correlated Electrons;Statistical Mechanics;High Energy Physics - Theory