Surface and 3D Quantum Hall Effects from Engineering of Exceptional Points in Nodal-Line Semimetals

We show that, under a strong magnetic field, a 3D nodal-line semimetal is driven into a topological insulating phase in which the electronic transport takes place at the surface of the material. When the magnetic field is perpendicular to the nodal ring, the surface states of the semimetal are transmuted into Landau states which correspond to exceptional points, i.e., branch points in the spectrum of a non-Hermitian Hamiltonian which arise upon the extension to complex values of the momentum. The complex structure of the spectrum then allows us to express the number of zero-energy flat bands in terms of a new topological invariant counting the number of exceptional points. When the magnetic field is parallel to the nodal ring, we find that the bulk states are built from the pairing of surfacelike evanescent waves, giving rise to a 3D quantum Hall effect with a flat level of Landau states residing in parallel 2D slices of the 3D material. The Hall conductance is quantized in either case in units of $e^2/h$, leading in the 3D Hall effect to a number of channels growing linearly with the section of the surface and opening the possibility to observe a macroscopic chiral current at the surface of the material.

Figure 1

Schematic plot of the surface electronic currents in a slab of 3D NLSM for two orthogonal directions of the magnetic field.

Figure 2

Band structure for a slab of NLSM of width $W=40\mathrm{nm}$ (lateral dimension $\mathrm{\Delta }y=40\mathrm{nm}$), for parameters $m_0=1.2\mathrm{eV}$, $m_1=1.0\mathrm{eV}{\mathrm{nm}}^2$, $v=0.5\mathrm{eV}\mathrm{nm}$, and magnetic field (perpendicular to the nodal ring) $B=30\mathrm{T}$. The energy is measured in electron-volts and the momentum in inverse nanometers.

Figure 3

Spectrum of a NLSM (with $m_0=0.2\mathrm{eV}$, $m_1=0.8\mathrm{eV}{\mathrm{nm}}^2$, $v=1.0\mathrm{eV}\mathrm{nm}$, and magnetic field perpendicular to the nodal ring $B=30\mathrm{T}$) for complex momenta $k_z+i\alpha$ ($k_z$ being perpendicular to the nodal ring) showing two branch cuts connecting pairs of exceptional points along the $\alpha$ axis. The energy is measured in electron-volts and the momentum in inverse nanometers.

Figure 4

Band structure for a slab of NLSM of width $W=60\mathrm{nm}$ (lateral dimension $\mathrm{\Delta }y=60\mathrm{nm}$), for parameters $m_0=0.5\mathrm{eV}$, $m_1=0.5\mathrm{eV}{\mathrm{nm}}^2$, $v=0.5\mathrm{eV}\mathrm{nm}$, and magnetic field (parallel to the nodal ring) $B=30\mathrm{T}$. The units are the same as in Fig. 2.

Figure 5

Plot of the wave function along the direction perpendicular to the nodal ring (zoom view) showing the imaginary part of the first component [as $\mathrm{Re}({\mathrm{\Psi }}_1)=0$], and the real part of the second component [as $\mathrm{Im}({\mathrm{\Psi }}_2)=0$] of a state with $k_x=k_y=0$ in the zero Landau level of a NLSM with $m_0=2.0\mathrm{eV}$, $m_1=1.5\mathrm{eV}{\mathrm{nm}}^2$, $v=0.5\mathrm{eV}\mathrm{nm}$, and magnetic field (parallel to the nodal ring) $B=30\mathrm{T}$. The figure also shows the pseudospin $\sigma =⟨{\sigma }_y⟩$ of the two evanescent waves making up the state.