Effect of an incommensurate potential on nodal-link semimetals

We explore the effect of an incommensurate potential on nodal-link semimetals. For the system with surface states, we show the nodal-link semimetal undergoes a series of phase transitions: first into a metallic phase, then to a loop semimetal phase, and back to a metallic phase with increasing the strength of incommensurate potential. The phase transitions are unveiled by analyzing the properties of energy spectra and wave functions. We also study the system without surface states and find that the Fermi surface evolves from lines to loops before the system enters into a metallic phase when increasing the incommensurate potential strength.

Figure 1

Nodal rings of our system [Eq. (3)] with $V=0$, (a) $m_0=2.5$ and (b) $m_0=3.2$. (c) The energy spectra as a function of $k_x$ under OBC along $z$ direction for fixed $k_y=0$ in the absence of the incommensurate potential and (d) the corresponding cross section of energy dispersion $E(k_x,k_y)$ at $E=0\pm 0.03$ (with considering the finite size effect). Here $L=300$.

Figure 2

DOS $\rho \left(0\right)$ versus the strength of incommensurate potential $V$ for the system with $L=300$ and the inset is that $\rho \left(0\right)$ as a function of $V$ for $L=300$ and $L=600$.

Figure 3

The energy spectra as a function of $k_x$ under OBC along $z$ direction for fixed $k_y=0$ with the incommensurate potential strength (a) $V=1$, (b) $V=1.5$, (c) $V=4$, and (d) $V=6$. The corresponding cross section of energy dispersion $E(k_x,k_y)$ at $E=0\pm 0.03$ (with considering the finite size effect) for this system with (e) $V=1$, (f) $V=1.5$, (g) $V=4$, and (h) $V=6$. Here $L=300$. For (a) and (b), to make the details clearer and increase the resolution, we fix the range of $k_x$ as $(-0.2\pi ,0.2\pi )$ and $L_x=1000$.

Figure 4

DOS as a function of energy $E$ for various values of incommensurate potential strength $V$ and system size under (a) OBC and (b) PBC in the $z$ direction. (c) $\overline{\mathrm{\Delta }E}$ versus $1/L$ for $V=1$ and $V=1.5$, respectively. The inset shows the corresponding $\overline{\mathrm{\Delta }{E}_p}$ under PBC. (d) $\mathrm{\Delta }{E}_{\text{min}}$ as a function of $1/L$ for fixed $V=4$.

Figure 5

MIPR as a function of $k_x$ and $V$ with fixed (a) $k_y=0$ and (c) $k_y=\pi$. MIPR as a function of $k_y$ and $V$ with fixed (b) $k_x=0$ and (d) $k_x=\pi$. MIPR as a function of $k_x$ and $k_y$ for fixed (e) $V=1.5$ and (f) $V=6$. Here $L=300$.

Figure 6

The energy spectra as a function of $k_y$ under OBC along $x$ direction for fixed $k_z=0$ with the incommensurate potential strength (a) $V=0$, (b) $V=1$, (c) $V=2$, (d) $V=3$ and (e) $V=4$. The corresponding cross section of energy dispersion $E(k_x,k_y)$ at $E=0\pm 0.03$ for this system with (f) $V=0$, (g) $V=1$, (h) $V=2$, (i) $V=3$ and (j) $V=4$. Here $L=300$.

Figure 7

(a) $\delta E$ as a function of $k_x$ and $V$ with $k_y=0$, (b) $\delta E$ as a function of $k_y$ and $V$ with $k_x=0.4\pi$, (c) $\delta E$ versus $V$ with fixed $k_y=0$ and $k_x=0.4\pi$, (d) the distribution of the $L\text{th}$ and $(L+1)\text{th}$ eigenstates of this system with $V=3.5$ and $V=3.7$, here we fixed $k_y=0$ and $k_x=0.4\pi$. Here $L=300$.

Figure 8

(a) MIPR as a function of $k_x$ for different sizes $L$. MIPR versus $1/L$ with fixed (b) $k_x=0.1\pi$, (c) $k_x=0.3\pi$, and (d) $k_x=0.4\pi$. IPR versus eigenvalue $E$ for fixed (e) $k_x=0.1\pi$, (f) $k_x=0.3\pi$, and (g) $k_x=0.4\pi$. Other parameters are $V=1.5$ and $k_y=0$.