Chiral separation and chiral magnetic effects in a slab: The role of boundaries

We study the chiral separation and chiral magnetic effects in a slab of Dirac semimetal of finite thickness, placed in a constant magnetic field perpendicular to its surfaces. We utilize the Bogolyubov boundary conditions with a large Dirac mass (band gap) outside the slab. We find that, in a finite-thickness slab, the axial current density is induced by helicity-correlated standing waves and, as a consequence, is quantized. The quantization is seen in its stepped-shape dependence on the fermion chemical potential and a sawtooth-shape dependence on the thickness of the slab. In contrast to a naive expectation, there is no chiral charge accumulation anywhere in the bulk or at the boundaries of the semimetal. In the same slab geometry, we also find that a nonzero chiral chemical potential induces no electric current, as might have been expected from the chiral magnetic effect. We argue that this outcome is natural and points to the truly nonstatic nature of the latter. By taking into account a nonzero electric field of a double layer near the boundaries of the slab, we find that the low-energy modes under consideration satisfy the continuity equation for axial current density without the anomalous term.

Figure 1

The dimensionless axial current density in an infinite space (red dashed line) and in the middle of the slab (blue solid line) plotted as a function of chemical potential for the three values of the band gap: $am/v_F=0$ (left panel), $am/v_F=2$ (middle panel), and $am/v_F=6$ (right panel). To plot the figure, we fixed $v_F/a=25\text{meV}$.

Figure 2

The dimensionless axial current density in an infinite space (red dashed line) and in the middle of the slab (blue solid line) plotted as a function of the width of the slab $a$ for the three values of the band gap: $m/\mu =0$ (left panel), $m/\mu =0.25$ (middle panel), and $m/\mu =0.75$ (right panel). To plot the figure, we fixed $v_F/\mu =12.5\text{Å}$.

Figure 3

The dimensionless axial current density plotted as a function of the chemical potential and $z/a$ for the three values of the band gap: $am/v_F=0$ (left panel), $am/v_F=2$ (middle panel), and $am/v_F=6$ (right panel). To plot the figure, we fixed $v_F/a=25\text{meV}$.