Nodal-line semimetal superlattices

Spatial modulations, such as multilayers, to realize topological materials have recently been studied in theoretical and experimental works. In this paper, we investigate properties of the multilayers of the nodal-line semimetal (NLS) and the normal insulator. We consider two types of multilayers, with the stacking direction being perpendicular or parallel to the plane where the nodal line lies. In particular, we show that when the stacking direction is parallel to the plane, the nodal lines remain but they change their shapes because of the folding of the Brillouin zone. We also study the multilayers with magnetization. One can expect that the quantum anomalous Hall (QAH) phase emerges in some cases, depending on the direction of the magnetization. If the magnetization is along the $C_2$-invariant axis, the multilayer becomes the Weyl semimetal phase if the $C_2$-invariant axis intersects the nodal lines, and otherwise it becomes the QAH phases.


I. INTRODUCTION
Recently, topology has been shown to play a crucial role in condensed matter physics [1][2][3] . Realizations of topological phases in condensed matter physics have started from the discovery of the quantum Hall effect 4 . After that, various topologically-insulating systems, such as the quantum anomalous Hall (QAH) systems [5][6][7][8] and the topological insulators [9][10][11] , have been theoretically proposed and experimentally observed over the past decade.
On the other hand, spatial modulations, such as multilayers, to realize topological materials have recently been studied in theoretical 7,10,33-38 and experimental 39,40 works. One of the pioneering works is on a multilayer of a topological insulator (TI) and a normal insulator (NI) with magnetization; it was theoretically proposed that the multilayer can realize a WSM phase 33 . Another example is a multilayer of a WSM and a NI, giving rise to QAH phases 36 . In these proposals, spatial modulations are useful means to manipulate bulk-band structure, and in particular to realize topological band structures by manipulating singularities in the momentum space. Meanwhile, multilayers of a NLS and a NI has not been studied thus far, to the authors' knowledge.
In this paper, we study multilayers of a NLS and a NI. Here, we consider the class of the spinless NLSs with inversion symmetry (IS) and time-reversal symmetry (TRS), and thereby the nodal lines are characterized by the π Berry phase. We investigate the multilayers with the stacking direction being either perpendicular or parallel to the plane where the nodal line lies, and we call them pattern A and pattern B, respectively. We find that the resulting phase diagrams include only the NLS phase and the NI phase. Furthermore, in pattern B with magnetization added, the phase diagram shows rich physics, including the QAH phases with various values of the Chern number, similarly to the WSM-NI multilayer 36 . To be more specific, when the system has C 2 symmetry and there are no intersections between the C 2 -invariant axis and the nodal lines in the NLS multilayer, the QAH phase appears when the magnetization is added along the C 2 -invariant axis.
This paper is organized as follows. In Sec. II, we investigate the properties of NLS multilayers with patterns A and B from the effective model of the NLS. In Sec. III, we also study how the QAH phases appear in the multilayers with magnetization. Furthermore, we calculate the band structure using the lattice model and compare the results of the effective model with those of the lattice model in Sec. IV. Finally, we summarize and discuss the results in Sec. V. Throughout the paper, we restrict ourselves to the cases with the IS and assume that the spin-orbit couping is negligible.

II. MULTILAYER FROM THE EFFECTIVE MODEL
A. Effective model of the NLS Here, we review an effective model of the NLS proposed in Ref. 19. It is showed that Cu 3 NZn has nodal lines around the X points [ Fig. 1 (a)]. The crystal structure of Cu 3 NZn is a cubic anti-ReO 3 structure, intercalated with Zn atoms at the body center of the cubic unit cell of Cu 3 N as shown in Fig. 1 (b). Let a be the lattice constant. Then, the Hamiltonian around the X point X z = πẑ/a can be expanded as where k ⊥ is given by k 2 ⊥ = k 2 x + k 2 y , v z , ∆ε, g ⊥ , and g z are constants, and v z , g ⊥ , g z are positive. Here, we have imposed both the TRS represented by T = K, with K being the complex conjugation operator, and the IS represented by P = τ z , together with the D 4h point group symmetries at X. The Pauli matrices τ i (i = 1, 2, 3) act on the space spanned by the A 1g and A 2u states. The energy eigenvalues are given by Henceforth, we set the Fermi energy to be E F = 0. If ∆ε < 0, the valence and conduction bands are degenerate at k 2 ⊥ = −∆ε/g ⊥ and k z = 0, and the degeneracy forms the nodal line with the radius −∆ε/g ⊥ at the Fermi energy. We note that since the nodal lines of Cu 3 NZn have a non-trivial Z 2 invariant defined from the parity eigenvalues 19 and a mirror symmetry with respect to the k z = 0 plane, the nodal lines are preserved both by topology and by the mirror symmetry. On the other hand, if ∆ε > 0, the system is in the NI phase.
In the following, we consider two patterns of multilayers shown in Fig. 2 (a) and Fig. 3 (a), separately.

B. Multilayer: Pattern A
We consider a multilayer of a NLS and a NI stacked perpendicularly to the nodal-line plane as shown in Fig. 2 (a). We call it pattern A. To realize this multilayer, we periodically modulate ∆ε in the effective Hamiltonian Eq. (1) between positive and negative values as shown in Fig. 2 (b). Thus, the Hamiltonian of the NLS multilayer can be written as where k ⊥ = k 2 x + k 2 y . Here, ∆ε (z) = −V S < 0 and ∆ε (z) = V I > 0 represent the NLS and the NI layers, respectively.
One can solve the eigenvalue problem of this model similarly to the Krönig-Penny model. After straightforward but lengthy calculation, we find that the nodal line in the multilayer appears only when V S a > V I b, and it is located at where k z is the Bloch wave number along the z direction. Thus, the multilayer is in the NLS phase only when V S a > V I b, and the nodal line disappears at V S a = V I b.
When V S a < V I b, the multilayer is in the NI phase. In Eq. (4), we consider how the nodal line changes across the phase transition between the NLS and the NI phases. When b = 0, the multilayer corresponds to the original NLS, and the nodal line with the radius V S /g ⊥ is located on the k z = 0 plane. By increasing b, the radius of the nodal line gradually decreases to , and the nodal line eventually shrinks to a point at k = 0 when b = (V S /V I ) /a. In the case b > (V S /V I ) /a, the nodal line disappears, and the multilayer becomes the NI. As a result, the phase diagram is shown in Fig. 2 (c).

C. Multilayer: Pattern B
In this section, we consider a NLS-NI multilayer with the stacking direction being parallel to the nodal-line plane as shown in Fig. 3 (a). Similarly to Sec. II B, the Hamiltonian describing the multilayer is given by where ∆ε (x) is the same as that in Sec. II B, and the schematic figures of the multilayer and spatial dependence of ∆ε (x) are shown in Fig. 3. We can calculate the eigenstates and the energies in the similar way to as in the previous subsection. Since the Hamiltonian Eq. (5) has the mirror symmetry with respect to the k z = 0 plane, we expect that the nodal lines in the multilayer lie within the plane. Therefore, we can obtain the equation for the position of the nodal lines: where k x is the Bloch wave number along the x direction, V ′ S = V S /g ⊥ , and V ′ I = V I /g ⊥ . When b is sufficiently large, the multilayer is in the NI phase, while by increasing a, the multilayer becomes the NLS phase. Here, since the original nodal lines are preserved both by the TRS and by the mirror symmetry 19 , once the nodal lines appear in the multilayer, they cannot disappear due to the π Berry phase guaranteed by symmetries. Therefore, the topologically-insulating phases cannot appear in the multilayer by a small perturbation preserving the symmetries.
We investigate how the nodal lines appear in the multilayer from Eq. (6). The nodal lines for various values of a and b are shown in Fig. 4. When a is small as shown in Fig. 4 (a), by increasing b, the nodal line shrinks. It does not cross the boundary of the Brillouin zone. On the other hand, when a is sufficiently large as shown in Figs. 4 (b) and (c), the nodal line crosses the boundary. When a = 3, the nodal lines recombine and gradually becomes straight along the k x axis. Furthermore, when a = 6, the nodal lines recombine to form three line nodes. Then, one shrinks while the others gradually become straight and parallel to the k x axis. To understand the origin of these almost straight nodal lines, we consider the b → ∞ limit which corresponds to the quantum well of the NLS. In this limit, Eq. (6) does not depend on the Bloch wave number k x as shown in Appendix A, and the gapless states exist in the quantum well when a is above some critical values. Therefore, these states corresponds to the nodal lines parallel to the k x direction.
These nodal lines shrink not only by increasing b but also by increasing the parameter V I . In the effective Hamiltonian Eq. (1), V I represents the bulk gap of the NI. When a = 6, b = 0.08a, and V I = 1, among the nodal lines, the circular one crosses the Brillouin zone boundary, and the others are almost parallel to the k x axis as shown in the red lines [ Fig. 5 (a)]. Then, by increasing V I , the circular nodal line gradually shrinks (blue lines, V I = 1.5), and eventually disappears (black line, V I = 2).
On the other hand, by increasing the parameter V S which represents the radius of the original nodal line, the almost straight nodal line disappears. This can be seen in Fig. 5 (b). When a = 6, b = 0.08a, and V S = 0.3, the nodal lines appear as shown by the red lines. By increasing V S , the nodal lines which are almost parallel to the k x axis gradually move along the k y axis, and finally disappear at the Brillouin zone boundary. On the other hand, the circular nodal line also extends, and eventually recombines to form new nodal lines shown by the black lines.

III. MULTILAYER WITH MAGNETIZATION
In this section, we investigate what happens to the NLS multilayers if the magnetization appears. It has been shown that spinless NLSs undergo a transition to spinless WSMs by breaking the TRS under some conditions 41 . Therefore, we can expect that the NLS multilayer with magnetization may behave similarly to the WSM multilayer in the previous paper 36 .  In the NLS multilayer with pattern A, we can immediately find that only the NI and the WSM phases appear when the magnetization is introduced. On the other hand, in the NLS multilayer with pattern B, we expect that the QAH phases may emerge, depending on the direction of the magnetization. We classify the pattern B into two cases; we call the case of the magnetization m parallel to the stacking direction n pattern B-1 and that of the magnetization m perpendicular to the stacking direction n pattern B-2.

A. Multilayer: Pattern B-1
We consider the NLS multilayer, with the magnetization introduced parallel to the stacking direction. For this purpose, we introduce the perturbation V T = mk y τ x to the effective Hamiltonian Eq. (1). The effective model without the perturbation V T has the TRS and D 4h symmetry, and when we introduce the perturbation V T , the IS and the C 2 symmetry around the magnetization direction remain. From this symmetry argument, this perturbation represents the magnetization along the x axis, and the parameter m represents the magnitude of the magnetization.
We can solve the energy eigenvalue problem with the perturbation, similarly to Sec. II C. By using the fact that the Weyl nodes appear along the stacking direction because of the symmetries 36 , we conclude that the band gap closes at k y = k z = 0, E = 0 and If Eq. (7) has real solutions for k x , the multilayer is in the WSM phase, and otherwise it is in a bulk-insulating phase. We note that whether the multilayer becomes the WSM phase or a bulk-insulating phase does not depend on the value of the magnetization m ( = 0) but on the parameters V S and V I . We show the phase diagram of the multilayer in Fig. 6  (a) by changing a and b. To see the physical origin for this phase diagram, we consider how the phase changes along the a = 6 line shown as the arrow in Fig. 6 (a). In Fig. 6 (b)-(d), we show the positions of the nodal lines in the original NLS multilayer without the magnetization with b = 0.5, 2, and 5 calculated from Eq. (6). In Ref. 41, it is shown that when the magnetization along the C 2 -invariant axis is added to the NLS, the system becomes a WSM with the Weyl nodes appearing at intersections between the C 2 -invariant axis and the nodal lines. It is indeed the case here. First of all, when a = 6, b = 0.5 and m = 0, the nodal lines consist of the circular one and the two almost straight ones [ Fig. 6 (b)]. Because the circular nodal line intersects with the C 2invariant axis, it is natural that the multilayer is in the WSM phase [ Fig. 6 (a)]. On the other hand, by increasing b, the circular nodal lines at m = 0 disappear and the others eventually become almost straight as shown in Figs. 6 (c) and (d). Because there is no intersection between the C 2 -invariant axis and nodal line, by adding the magnetization, no Weyl nodes appear. Then, the multilayer becomes the QAH phase [ Fig. 6 (a)], which is analogous to the QAH phase in the NI-WSM multilayer in Ref. 36. As b is increased, the pair of the Weyl nodes created at k = 0 are annihilated pairwise at the bound- ary of the Brillouin zone. Therefore, we conclude that the QAH phase has the Chern number ν = −1.
Next, we consider the phase transitions for m = 0 by increasing the values of the parameters V S or V I . For example, we fix a = 6 and b = 0.48. When we increase the value of V I , a phase transition occurs from the WSM phase [ Fig. 7 (a)] to the QAH phase with the Chern number −1 [ Fig. 7 (b)]. Meanwhile, when we increase V S , it gives rise to a transition from the WSM phase[ Fig. 7 (a)] to the QAH phase with the Chern number −2 [ Fig. 7 (c)]. Therefore, the transition from the WSM phase to the bulk-insulating phase occurs not only by increasing b but also either by increasing the gap V I of the NI layer or increasing the radius of the original nodal line V S .

B. Multilayer: Pattern B-2
In this subsection, we introduce the magnetization perpendicular to the stacking direction, i.e. the y axis, of the multilayer. It is added to the Hamiltonian Eq. (5) in the form V ′ T = mk x τ x . In this case, the WSM phase appears in the multilayer if the C 2 -invariant axis intersects the nodal lines, and otherwise the NI phase appears. In particular, when a and b are sufficiently large, the number of pairs of Weyl nodes increases since the number of the almost straight nodal  lines also increase [ Fig. 8]. This change of the number of the Weyl nodes is not seen in the model of the NI-WSM multilayer in Ref. 36. We note that topologicallyinsulating phases cannot appear since the C 2 -invariant axis always intersects the almost straight nodal lines.

IV. MULTILAYER FROM THE LATTICE MODEL
In this section, we numerically investigate the behavior of the NLS-NI multilayers with a lattice model introduced in Sec. IV A. We calculate the band structure of the multilayers with patterns A and B in Secs. IV B and Secs. IV C, respectively. Finally, we study the effect of magnetization in the NLS multilayer with pattern B in Sec. IV D.

A. NLS from the lattice model
We construct a tight-binding model describing a NLS. For this purpose, we start from the continuum Hamiltonian Eq. (1). In a simple cubic lattice consisting of an s-like orbital and a p z -like orbital per one site, we can construct the lattice Hamiltonian as where a 1 (k) = 0, , and d is the lattice constant. Here, we have ∆ε = −V S for the NLS phase, while ∆ε = V I for the NI phase.
In the following sections, we consider a lattice model of the NI-NLS multilayer by modulating ∆ε. We set N S and N I to be the numbers of the atomic layers within the NLS layer and the NI layer, respectively. Then, the thickness of the NLS layer and that of the NI layer are given by a = N S d and b = N I d, respectively.

B. Multilayer: Pattern A
In this subsection, we study the NLS multilayer with pattern A, which has the stacking direction perpendicular to the nodal-line plane. We numerically calculate the band structure with the parameters V S = −0.5 and V I = 0.1 as shown in Fig. 9. We find that the multilayer has the nodal line in the bulk when N S = 10 and N I = 5 [ Fig. 9 (a)]. By comparing this result with that of the continuum model, we find that this multilayer corresponds to the case of V S a > V I b in Fig. 2 (c). As we have seen in Sec. II B, by increasing b, the nodal line shrinks and disappears in the multilayer. It is also the case for the lattice model as shown for N S = 10 and N I = 30 in Fig. 9 (b), where the multilayer is the NI phase. There is a tiny gap throughout the whole Brillouin zone.
Here, an almost flat band around E = 0 is seen in the bulk as shown in Fig. 9 (b). We have discussed the appearance of this flat band in Ref. 36. This behavior is caused by the drumhead surface states which appear on the interfaces between the NLS layers and the NI layers. Namely, since the adjacent drumhead interface states are separated by the insulating layers, these interface states hybridize to form the states near the Fermi energy in Fig. 9 (b), with small hybridization between them.

D. Multilayer with magnetization
Similarly to Sec. III, we call the case of the magnetization m parallel to the stacking direction n pattern B-1 and that of the magnetization m perpendicular to the stacking direction n pattern B-2.

Multilayer Pattern B-2
Next, we study the multilayer with the magnetization in the k y axis, which is perpendicular to the stacking direction. To do this, we add the perturbation term V ′ T = m d sin k x d σ x to the Hamiltonian Eq. (8). In Sec. III B, we find that there exist more than one pairs of the Weyl nodes in this case. In the lattice model, we numerically confirm that it is indeed the case as shown in Fig. 12 (a). While there are more than one pairs of the Weyl nodes when the magnetization is sufficiently small, only a single pair of the Weyl nodes survives by increasing the magnetization [ Fig. 12 (b)]. Namely, the multilayer eventually behaves similarly to the WSM multilayer with pattern A in Ref. 36 by increasing the magnetization.

V. SUMMARY AND DISCUSSION
In this paper, we studied properties of the NLS multilayers using the effective model and the lattice model. First of all, we instigated the multilayers with the stacking direction being either perpendicular or parallel to the plane where the nodal line appears, and called them pattern A and pattern B, respectively. In the multilayer with pattern A, we found a phase transition from the NLS phase to the NI phase by increasing the thickness of the NI layer. On the other hand, in pattern B, we showed that single nodal line is folded in the multilayer because of the periodicity of the Brillouin zone and this nodal line change into multiple nodal lines by increasing the thickness of the NI layer. Some nodal lines run across the entire Brillouin zone and they gradually become almost straight. Nodal lines always exist since the multilayer has both the IS and the TRS.
Furthermore, we introduced magnetization into the NLS-NI multilayers, and studied their properties. We then compared the results with our previous work on the WSM-NI multilayers 36 , because the NLS phase becomes the WSM phase by adding magnetization 41 . We studied the two cases, with the magnetization m being perpendicular and with m parallel to the stacking direction n. In particular, the QAH phase can appear in the multilayer similarly to the WSM multilayer when m is parallel to n. To realize the QAH phase, we need both a gap and a band inversion. First, the magnetization can open the gap when there are no intersections between the C 2 -invariant axis and the nodal lines 41 . Second, by increasing the thickness of the NLS layer, the Brillouin zone becomes narrower, and the nodal lines are folded at the Brillouin zone boundary. When m is added, it gives rise to a movement of the Weyl nodes along the C 2invariant axis across the whole Brillouin zone, giving rise to the band inversion. This band inversion occurs multiple times by increasing the thickness of the NLS layer or the radius of the nodal line. Here, we propose that the NLS multilayer with the magnetization can realize the QAH phase with large Chern number, which has been a long-standing issue in this field 35,36,[42][43][44][45][46] .
On the other hand, when m is perpendicular to n, some pairs of the Weyl nodes appear in the NLS-NI multilayer with the magnetization. This is in contrast with the result in the WSM-NI multilayer 36 , where the WSM multilayer has only a single pair of the Weyl nodes. Namely, although these two cases can be regarded as WSM-NI multilayers, the number of pairs of Weyl nodes in the multilayer is different between them. This difference comes from the bulk-band structure of the constituent WSM. In the WSM 47 used in Ref. 36, the bulk gap is wide except for the Weyl nodes. On the other hand, in our model of the NLS with the magnetization, the gap is very narrow along the original nodal line. Therefore, multiple pairs of the Weyl nodes can appear in the multilayer along the original nodal line having the narrow gap. Thus, the distribution of the gap size in the k-space in the WSM determines how the Weyl nodes appear in the WSM-NI multilayer. expressed as (k y , k z ) perpendicular to the well direction. By continuity of the wave function, we can obtain the relation between the energy eigenvalues E and the well width a as .
(A1) To find the gapless states, we set k z = 0 since they appear on the C 2 -invariant axis, the k y axis. The energy bands are shown in Fig. 13. The gapless states are given by E = 0. We note that in the limit b → ∞, the almost straight nodal lines in Figs. 4 (b) and (c) becomes the gapless states in Fig. 13 (b) and (c), respectively .