In recent years, topological semimetals have become a frontier topic in condensed matter physics because of their promising applications in electronics, spintronics, and optics [1–9]. According to the dimensions of the degenerate manifolds in k-space formed by band crossings, topological semimetals are divided into nodal point semimetals, such as Weyl [9–12], Dirac [13–17], or triple-point semimetals [18, 19]; nodal line semimetals [20–23]; and nodal surface semimetals [24–27]. Due to the non-trivial topological band structure, Weyl (Dirac) semimetals can exhibit Fermi arc surface states [5, 9, 10] connecting different Weyl node (Dirac node) projections on a two-dimensional (2D) surface Brillouin zone (BZ). Nodal line semimetals can exhibit another special surface state—drumhead surface state (DSS) [21, 22, 28–31] on a 2D surface BZ. These non-trivial topological properties are not limited to being present in semimetals because they originate from the nodal band structures and exist in other systems, such as metals [32–35], optical crystals [36–38], phononic crystals [39, 40], mechanical systems [41], and circuit systems [42–44].

For topological nodal line materials, the doubly degenerate Weyl nodal line [23, 45–47] and quadruply degenerate Dirac nodal line [48–51] have been broadly studied, and the DSS has been observed in these two types of materials. However, there is little research on the triply degenerate nodal line (TDNL). The only such research we can find is [52] by Liu et al. in 2021. Liu et al. [52] proposed two TDNL models, of which one is non-topological and the other is topological, according to the existence of Fermi arc topological surface states. However, Liu et al. did not report any DSS for the TDNL models. Accordingly, in this work, we aim to construct a tight-binding (TB) model with TDNL and investigate its DSS.

However, it is almost impossible to construct a TDNL model based on real crystalline materials because real crystalline materials are constrained by the symmetries of (magnetic) space groups, and systematic studies on the possible emergent particles from band crossings have shown that no TDNL exists under various (magnetic) space groups [53–55]. Subsequently, to construct a TDNL model, one has to get rid of the constraints by (magnetic) space groups. This can be achieved in artificial systems, such as metamaterials, circuit systems, and mechanical systems, because when described by TB models, the effective hoppings in these systems can be tuned at will, for example, adjusting the connection mode among circuit components or changing the coupling strength through springs [41, 43, 56].

In this work, we first constructed a three-band TB model hosting TDNL by designing the hoppings. Subsequently, we calculated the Berry phase and Zak phase to check the topological non-triviality of the TDNL. Surface states on two different surfaces (i.e., (010) and ( 1 ̄ 10 ) ) were studied via both semi-infinite systems and slab models. The usual DSS was found on the (010) surface. However, on the ( 1 ̄ 10 ) surface, we noticed a new type of DSS, whose drumhead is not complete but rather contracted. The tuning of this DSS with a contracted drumhead was also studied by varying the hopping parameters of the model.

The model is constructed based on a simple cubic lattice whose basis vectors a ⃗ , b ⃗ , and c ⃗ are equal in magnitude and along the x, y, and z directions, respectively, as shown in Figure 1A. Only one atom with three orbitals (here called ϕ ₁, ϕ ₂, and ϕ ₃) is considered in each cell. With the hopping between orbitals ϕ i ( r ⃗ ) and ϕ j ( r ⃗ − δ ⃗ ) denoted as h i j ( δ ⃗ ) , we choose the following hoppings for the model: h 11 0 = t 0 , h 22 0 = − t 0 , h 33 0 = 2 t 0 , (1) h 11 ± α = t 1 2 , h 22 ± α = − t 1 2 , h 33 ± α = t 1 , (2) h 23 ± b ⃗ = ± t 2 2 , h 12 ± d ⃗ i = ± t 3 8 i = 1,2,3,4 , (3) where α = a ⃗ , b ⃗ , c ⃗ and d ⃗ i ( i = 1,2,3,4 ) are shown in Figure 1A. Then, the Hamiltonian of the TB model is H = t 0 + t 1 ⁡ cos k x + t 1 ⁡ cos k y + t 1 ⁡ cos k z λ 1 + t 2 ⁡ sin k y λ 2 + t 3 ⁡ sin k x ⁡ sin k y ⁡ sin k z λ 3 , (4) where λ ₁, λ ₂, and λ ₃ are the following three matrices, respectively: 1 0 0 0 − 1 0 0 0 2 , 0 0 0 0 0 i 0 − i 0 , 0 i 0 − i 0 0 0 0 0 . (5) One can easily see that when k _y = 0, Eq. 4 becomes a diagonal matrix whose diagonal elements can be null simultaneously. This implies that a TDNL can exist in the plane k _y = 0 under suitable values of t ₀ and t ₁. The key feature of the hoppings that results in this TDNL is that h _ii (0)/h _ii (±α) keeps constant for i = 1, 2, 3. This is a special request that cannot be derived from symmetries of the (magnetic) space group.

The surface density of state (SDOS) was obtained by calculating the surface Green’s function of the semi-infinite system using the WannierTools package [57]. The input data for WannierTools were prepared using the MagneticTB package [58]. To investigate the surface states, we also constructed TB slab models of 80 layers using the PythTB package [59]. To judge whether a state is a surface state, we first define the topmost five layers on each side, A or B, of the slab model as “surface layers” and then define the following quantity η to characterize the degree to which a state is a surface state: η = w A + w B − 1 8 / 7 8 , w A + w B > 1 8 , 0 , w A + w B ≤ 1 8 , (6) where w _A∕B represents the wavefunction weight within the surface layers of side A/B of the slab. A bulk state wavefunction is periodic, and its weights are equally distributed within all the 80 layers, in which case w _A = w _B = 5/80 and η = 0. For a perfect surface state, the wavefunction is totally localized within the surface layers, leading to w _A = w _B = 1/2 and η = 1. By means of η, a state can be determined as a strong (or typical) surface state if its η is greater than a critical value η _c, and in this paper, η _c = 0.5 is adopted.

If not otherwise stated, the parameters t ₀ = 2, t ₁ = −1, and t ₂ = t ₃ = 1 are used for the model, and the unit is eV for all energies. The bulk energy bands are shown in Figure 2A, in which the k-points are defined in Figure 1B. In this model, the TDNL is actually an approximately circular nodal ring in the k _y = 0 plane, as shown in Figure 2B. To check the topological properties of the TDNL, we first calculated the Berry phase defined on a closed k-point loop enclosing the TDNL, with fully gapped energies, as shown by the small orange loop in Figure 2B. The Berry phase is calculated using the Wilson loop approach [60], and the result is π, which shows the topological non-triviality of the TDNL.

Furthermore, we calculated the Zak phase [61, 62], which is the Berry phase defined in a one-dimensional BZ along a certain direction. Two Zak phases are investigated here. The first one γ ₁ (k _x ) is defined along the line from ( k x , − 1 2 , 0 ) to ( k x , 1 2 , 0 ) , with the k _x = 0 case shown by the red dashed line in Figure 2B, in which the k-point coordinates are in unit of 2π/a. The second one γ ₂ (k _z ) is defined along the line from ( 1 2 , − 1 2 , k z ) to ( − 1 2 , 1 2 , k z ) , with the k _z = 0 case shown by the green dashed line in Figure 2B. The calculated Zak phase γ ₁ (k _x ) (γ ₂ (k _z )) is shown in the top (right) panel of Figure 2C (Figure 2D), whose k _x (k _z ) axis corresponds to the red (green) thick line in the 2D projective BZ of (010) ( 1 ̄ 10) surface shown in the corresponding bottom (left) panel. We can see that, for both γ ₁ and γ ₂, the non-trivial π Zak phase emerges only when the integral path of the Zak phase traverses the nodal ring (i.e., the TDNL here). Otherwise, the Zak phase is zero. According to the bulk-edge correspondence [63], this change of topological properties from inside to outside the nodal ring implies the existence of topological surface states inside the projected nodal ring on both (010) and ( 1 ̄ 10 ) surfaces.

The semi-infinite system terminated with that surface should be constructed to explore the topological surface states of a certain surface. Two surfaces (010) and ( 1 ̄ 10 ) are studied here, where the (010) surface is parallel to the nodal ring, but the ( 1 ̄ 10) surface is not. Figure 3A shows the SDOS of the (010) surface system, whose surface states all have a constant energy (zero) and form a flat drumhead shape. This typical DSS is clearly demonstrated by the SDOS with a constant energy slice at E = 0, as shown in Figure 3C. The green ring in Figure 3C represents the front projection of the TDNL, and its interior is full of surface states. We call this type of DSS “full DSS.” From the result that both the Zak phases γ ₁ and γ ₂ equal π inside the TDNL projections, one may expect that full DSS also exists in the ( 1 ̄ 10 ) surface system. However, the SDOS of the ( 1 ̄ 10 ) surface system shown in Figures 3B, D demonstrates results different from the expectation. In particular, the leftmost panel of Figure 3D shows the “contracted-drumhead surface state (CDSS),” in which the surface states do not fill completely the interior of the TDNL oblique projection (the green ellipse). The weak surface state feature at other energies, as shown in other panels of Figure 3D, also supports this result.

In order to further explore the CDSS in the ( 1 ̄ 10 ) surface system, a slab model of 80 layers terminated with ( 1 ̄ 10 ) surface is studied. Its energy bands are shown in Figure 4B, in which the degree of surface state η defined in Eq. 6 is also shown by both the point size and color for each state. In addition, Figure 4A shows the distribution of η for all states within the energy range [−0.5, 0.5] in the whole surface BZ and the projection of η onto the surface BZ. Figure 4B corresponds to Figure 3B, but here we can access the wavefunction of any state of the slab model. The wavefunctions of the five states marked in Figure 4B have descending η from 0.73 to 0, and the distributions of their weights with respect to layer number are given in Figure 4C. We can see that state 1 with η = 0.73 is a strong surface state with most wavefunctions localized within the surface layers. At the other extreme, state 5 with η = 0 distributes periodically; hence, it is a bulk state. As for states 2–4, they have non-zero but small η. Although they contain surface state components or may be called weak surface states, they are more like bulk states. Figures 4A, B show that strong surface states exist only near zero energy. Consequently, even if the projection of η in Figure 4A selects the largest η for each k-point within the energy range [−0.5, 0.5], it is not much different from the case considering only zero energy, and it exhibits a similar shape to the E = 0 panel in Figure 3D.

Because the weak surface states are much like bulk states, they are not efficient in most applications, which require large SDOS. Thus, only strong surface states need to be considered, and the shape of the CDSS can be revealed by the distribution of the k-points at which strong surface states exist. Figure 5 shows the distribution of the k-points of strong surface states (i.e., the shape of CDSS) under different model parameters. We can see that the shape of CDSS for the ( 1 ̄ 10 ) surface can be tuned by the hoppings t ₂ and t ₃ efficiently. Namely, increasing t ₃ makes the surface states change from a full DSS to a CDSS with smaller areas (Figure 5A), and inversely, increasing t ₂ will increase the area of CDSS from zero (Figure 5B). This provides an effective route to tune the SDOS and the shape of the surface state in topological nodal systems.

We have proposed a simple TB model hosting TDNL and studied its topological properties. Both the Berry phase and Zak phase demonstrate that the TDNL is topological non-trivial. This TDNL model not only has a full DSS as usual topological nodal line systems, but also has a CDSS, which we first noticed. The CDSS exists on the ( 1 ̄ 10 ) surface, and its area can be tuned efficiently by both model parameters t ₂ and t ₃. Our model demonstrates an effective way to tune the amount and density of the surface states, which will expand the potential applications of topological nodal line systems.