Engineering topological phases with a three-dimensional nodal-loop semimetal

A three-dimensional (3D) nodal-loop semimetal phase is exploited to engineer a number of intriguing phases featuring different peculiar topological surface states. In particular, by introducing various two-dimensional gap terms to a 3D tight-binding model of a nodal-loop semimetal, we obtain a rich variety of topological phases of great interest to ongoing theoretical and experimental studies, including a chiral insulator, degenerate-surface-loop insulator, and second-order topological insulator, as well as a Weyl semimetal with tunable Fermi arc profiles. The unique concept underlying our approach is to engineer topological surface states that inherit their dispersion relations from a gap term. The results provide one rather unified principle for the creation of novel topological phases and can guide the search for new topological materials. Two-terminal transport studies are also carried out to distinguish the engineered topological phases.

Figure 1

Sketches of surface Dirac cones inherited by $H=H_{\mathrm{NL}}+H_{\mathrm{gap}}$ from $H_{\mathrm{gap}}$ and the nodal loop ${\mathbf{L}}_0$ (black solid lines) of $H_{\mathrm{NL}}$. Dashed lines represent the DSL obtained from the zero solution in Eq. (3). A DSL exists only if it lies within the regime enclosed by ${\mathbf{L}}_0$. The Dirac cones in (a) and (b) are twofold degenerate, whereas this degeneracy is lifted in (c)–(f) by a nonzero $m^{\prime }$ [41].

Figure 2

A topological phase diagram with $m=2$, with the phase boundary determined from Eq. (10). Different insulating phases are characterized by two topological invariants, $\nu$ and $C_M$, defined in the text, which are ill defined in the semimetallic NPSM phase.

Figure 3

The surface zero modes with $m=2$ and their corresponding bulk spectra with $k_z=0$ are in (a2)–(d2). The zero modes are indicated by the bright markers in (a1)–(d1). The parameters are (a) $m^{\prime }=0,\mu =0$; (b) $m^{\prime }=0.8,\mu =0$; (b) $m^{\prime }=0.8,\mu =1$; and (d) $m^{\prime }=1.1,\mu =0$.

Figure 4

Spectrum versus $k_y$ with the open boundary condition in the $x$ and $z$ directions for gap terms as in Eq. (13). The parameters are $m=2$ and ${\mu }^{\prime }=0.5,1,1.5$ for (a)–(c), respectively.

Figure 5

(a) Conductance $T$ versus Fermi energy $E_F$ for four topological phases. The inset zooms in on $0<E_F<0.5$. (b) $T$ versus $\mu$ at $E_F=0$ for a disordered sample with different strengths $W$, with the gap term given by Eq. (8) and $m^{\prime }=0.8$, averaged over 100 disorder realizations. For small ${\mu }^{\prime }$, the system is an insulator with DSL, and $T$ is not robust. For larger ${\mu }^{\prime }$, the system is a Weyl semimetal, and the conductance is robust and quantized due to edge-state conductance on the Fermi arc. (c) $T$ versus ${\mu }^{\prime }$ at $E_F=0$ for a second-order topological insulator using the gap term defined in Eq. (13), with parameters $m=2,m^{\prime }=1.5$. The transitions at ${\mu }^{\prime }=1$ and ${\mu }^{\prime }=2$ agree with our theory.