Counterpropagating Fractional Hall States in Mirror-Symmetric Dirac Semimetals

The Landau bands of mirror symmetric 2D Dirac semimetals (e.g., odd layers of $ABA$ graphene) can be identified by their parity with respect to mirror symmetry. This symmetry facilitates a new class of counterpropagating Hall states. We predict the presence of a Laughlin-like correlated liquid state, at the charge neutrality point, with opposite but equal electron and hole filling factors $|{\nu }_{\pm }|=1/m$ ($m$ odd). This state exhibits fractionally charged quasiparticle-hole pair excitations and counterpropagating edge states with opposite parity. Using a bosonized one-dimensional edge state theory, we show that the fractionally quantized two-terminal longitudinal conductance, ${\sigma }_{xx}=2e^2/(mh)$, is robust to short-ranged intermode interactions.

Figure 1

Lattice structure of $ABA$-trilayer graphene (TLG) (a) and $ABA$-pentalayer graphene (b) with mirror symmetry about the middle layer denoted by the red line (plane). Under mirror symmetry $A_1\leftrightarrow A_3$, $B_1\leftrightarrow B_3$, whereas the middle layer remains invariant for $ABA\text{−}\mathrm{TLG}$.

Figure 2

(a) Energy dispersion of $ABA\text{−}\mathrm{TLG}$ at $E_{\perp }=0$ (solid line) and $E_{\perp }=40\mathrm{meV}$ (dashed line). The semimetallic phase requires $|\mathrm{\Delta }|>(m_{+}+m_{-})$. (b) Landau level spectrum of $ABA\text{−}\mathrm{TLG}$. In both figures red represents the odd parity electrons and blue denotes the even parity electrons and the black dotted line denotes the Fermi energy at the CNP. The dotted and solid lines denote different valleys (sublattices for the zLLs). At neutral charge density ($\nu =0$) the even parity filling factor is ${\nu }_{+}=2$, while the odd parity filling factor is ${\nu }_{-}=-2$.

Figure 3

(a) and (b) Edge dispersion of the $\nu =1+(-1)$ fractional parity Hall state, the counterpropagating edge currents flows due to particle-hole fluctuations in the even (blue) and odd (red) parity zLLs. Counterpropagating edge states of the fractional parity Hall effect in the two-terminal (c) and Hall bar (d) measurement setup.