Non-Hermitian off-diagonal magnetic response of Dirac fermions

We perform a comparative study for the magnetization dynamics within linear response theory of one and two dimensional massive Dirac electrons, after switching on either a real (Hermitian) or an imaginary (non-Hermitian) magnetic field. While Hermitian dc magnetic fields polarize the spins in the direction of the external magnetic field, non-Hermitian magnetic fields induce only off diagonal response. An imaginary dc magnetic field perpendicular to the mass term induces finite magnetization in the third direction only according to the right hand rule. This can be understood by analyzing the non-Hermitian equation of motion of the spin, which becomes analogous to a classical particle in crossed electric and magnetic fields. Therein, the spin expectation value, the mass term, and imaginary magnetic field play the role of the classical momentum, magnetic, and electric field, respectively. The latter two create a drift velocity perpendicular to them, which gives rise to the off-diagonal component of the dc spin susceptibility, similarly to how the Hall effect develops in the classical description.

Figure 1

The real part of the Hermitian susceptibility (blue solid line), ${\chi }_{xx}$ and the non-Hermitian one (red dashed line), ${\chi }_{xy}$ is visualized for the one dimensional Dirac equation.

Figure 2

The real part of the Hermitian susceptibility (blue solid line), ${\chi }_{xx}$, and the non-Hermitian one (red dashed line), ${\chi }_{xy}$, is visualized for the two dimensional Dirac equation with $W/\mathrm{\Delta }=100$.