Spontaneous parity breaking of Dirac spin liquid in a magnetic field

The Dirac spin liquid was proposed to be the ground state of the spin-1/2 Kagome antiferromagnets. In a magnetic field, we show that the state with Fermi pocket is unstable to the Landau level (LL) state and the XY ordered state. We mainly focus on the fully gapped LL state, which breaks parity symmetry and thus supports a finite temperature phase transition. We discuss experimental signatures which can be used to detect the possible Dirac spin liquid phase in Herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$.