Topological protection from exceptional points in Weyl and nodal-line semimetals

We investigate the topological protection of surface states in Weyl and nodal-line semimetals by characterizing them as evanescent states when the band structure is extended to complex momenta. We find in this way a sequence of exceptional points—that is, branch points with zero energy in the complex spectrum—allowing us to identify the set of surface states with complex momentum signaling the decay into the 3D semimetal. From this point of view, Weyl and nodal-line semimetals can be classified in two types depending on the way surface states decay. Type A semimetals have surface states with smaller penetration length and oscillating decay while type B semimetals have longer simple exponential decays. The difference between both types reflects in the way the branch cuts in the spectrum accommodate in the complex plane. The stability of the surface states stems in this approach from the complex structure that develops around the exceptional points, with a topological protection which is based on the fact that the branch cuts cannot be closed by small perturbations. We check this property when nodal-line semimetals are placed under circularly polarized light, where we observe that the exceptional points survive the effect of such a perturbation, though appropriate boundary conditions for zero-energy surface states cannot be satisfied in general due to the breakdown of time-reversal invariance by the radiation field.

Figure 1

[(a) and (b)] Representation of the absolute value of the imaginary part and the real part of the lowest eigenvalue of $H_{\mathrm{w}}$ for states with complex momentum $k_z+i\alpha$ along the Fermi arc with $0\le k_x\le \sqrt{m_0/m_1}$, for $m_0=1.0\mathrm{eV}, m_1/a^2=1.0\mathrm{eV}$, and $v/a=0.8\mathrm{eV}$ ($a$ being a microscopic length scale in the model). The energy $\varepsilon$ is measured in units of eV. [(c) and (d)] Similar representation as in (a) and (b) but for $m_0=1.0\mathrm{eV}, m_1/a^2=1.0\mathrm{eV}$, and $v/a=1.98\mathrm{eV}$. The plots map the distinctive zero-energy modes with constant $\alpha$ (corresponding to states with oscillatory decay) and the bifurcation into two zero-energy branches with variable $\alpha$ (representing the states with pure exponential decay).

Figure 2

Probability density in logarithmic scale across the $z$ direction of surface states in the Fermi arcs with $k_x=0$. We compare two different cases, one with some oscillatory dependence in the $z$ direction corresponding to a type A Weyl semimetal (red line, $m_0=0.35\mathrm{eV}, m_1=1.0 \mathrm{eV}{\mathrm{nm}}^2$, and $v=1.0\mathrm{eV}$ nm) and another with pure exponential decay corresponding to a type B Weyl semimetal (black line, $m_0=0.35\mathrm{eV}, m_1=1.0 \mathrm{eV}{\mathrm{nm}}^2$, and $v=4.0\mathrm{eV}$ nm). In the inset, the same figure is shown in normal scale.

Figure 3

Energy in log scale of surface states at the midpoint of the Fermi arc ($k_x=0$ in our model) as a function of the width $W$ in a slab geometry, comparing the behavior for a type A semimetal (red line, $m_0=0.35\mathrm{eV}, m_1=1.0 \mathrm{eV}{\mathrm{nm}}^2$, and $v=1.0\mathrm{eV}$ nm) and for a type B semimetal (black line, $m_0=0.05\mathrm{eV}, m_1=1.0 \mathrm{eV}{\mathrm{nm}}^2$, and $v=1.0\mathrm{eV}$ nm).

Figure 4

Contour plots of the absolute value of the imaginary part (a) and the real part (b) of the eigenvalue in (18) for $k_x=0$ and $m_0=1.0\mathrm{eV}, m_1/a^2=1.0\mathrm{eV}$, and $v/a=0.2\mathrm{eV}, a$ being a microscopic length scale in the model. $k_z$ and $\alpha$ are measured in units of $a^{-1}$ and the energy is in units of eV.

Figure 5

Contour plots of the absolute value of the imaginary part (a) and the real part (b) of the eigenvalue in (18) for $k_x=0$ and $m_0=0.5\mathrm{eV}, m_1/a^2=1.9\mathrm{eV}$, and $v/a=2.4\mathrm{eV}, a$ being a microscopic length scale in the model. $k_z$ and $\alpha$ are measured in units of $a^{-1}$ and the energy is in units of eV.

Figure 6

(a) 3D plot of the imaginary part of the eigenvalue in (18) for $k_x=0$ and the same parameters as in Fig. 5 ($m_0=0.5\mathrm{eV}, m_1/a^2=1.9\mathrm{eV}$, and $v/a=2.4\mathrm{eV}$). $k_z$ and $\alpha$ are measured in units of $a^{-1}$ and the energy is in units of eV. (b) Contour plot of the absolute value of the imaginary part of the eigenvalue in (18) for the same parameters as in (a) and with the same units as in Fig. 5. The red circles stand for the projection of two noncontractible loops around the branch points in the spectrum.

Figure 7

Contour plots of the absolute value of the imaginary part (a) and the real part (b) of the lowest eigenvalue in the spectrum of $H_{\mathrm{NL}}$ for evanescent states with angular momentum $m=0$ in a semi-infinite cylindrical geometry with radius $R=100a$ ($a$ being the microscopic length scale in the model), for $m_0=0.1\mathrm{eV}, m_1/a^2=4.0\mathrm{eV}$, and $v/a=0.1\mathrm{eV}$. The units are the same as in Fig. 4.

Figure 8

Plot of the absolute value of the real part (blue lines) and the imaginary part (red lines) of the lowest eigenvalue in the spectrum of $H_{\mathrm{NL}}$ for evanescent states with angular momentum $m=0$ in a semi-infinite cylindrical geometry with radius $R=100a$ ($a$ being the microscopic length scale in the model), for $m_0=0.1\mathrm{eV}, m_1/a^2=2.0\mathrm{eV}$, and $v/a=1.0\mathrm{eV}$. The units are the same as in Fig. 4. The dashed lines are used to separate the dispersion of different subbands, which correspond to the evolution of different eigenvalues in the diagonalization of $H_{\mathrm{NL}}$.

Figure 9

Contour plots of the absolute value of the imaginary part (a) and the real part (b) of the lowest eigenvalue in the spectrum of $\tilde{H}$ for evanescent states in a semi-infinite cylindrical geometry with radius $R=100a$ ($a$ being the microscopic length scale in the model) for $m_0=0.01\mathrm{eV}, m_1/a^2=0.5\mathrm{eV}, v/a=0.01\mathrm{eV}, Aa=0.05$, and $\mathrm{\Omega }=1.0\mathrm{eV}$. The units are the same as in Fig. 4.

Figure 10

Plot of the absolute value of the imaginary part (a) and the real part (b) of the lowest eigenvalue of $\tilde{H}$ for evanescent states with variable parameters $\alpha$ and $\delta$, for $k_{z}a=0.2$ and $m_0=1.0\mathrm{eV}, m_1/a^2=1.0\mathrm{eV}, v/a=0.2\mathrm{eV}, Aa=0.1$, and $\mathrm{\Omega }=0.5\mathrm{eV}$. The units are the same as in Fig. 4.

Figure 11

Plot of the absolute value of the imaginary part (a) and the real part (b) of the lowest eigenvalue of $\tilde{H}$ for evanescent states with complex momentum $k_z+i\alpha$, for $m_0=1.0\mathrm{eV}, m_1/a^2=1.0\mathrm{eV}, v/a=0.2\mathrm{eV}, Aa=0.1$, and $\mathrm{\Omega }=0.5\mathrm{eV}$. The units are the same as in Fig. 4.