Introduction

Distinct phases do not always differ from each other in their symmetries as expected in the Ginzburg-Landau paradigm. Two gapped phases having the same symmetry may be distinguished by a set of global quantum numbers called topological invariants1,2,3,4,5,6,7,8. These invariants are quantized numbers, whose types (integer, Boolean, and others) only depend on the symmetry group and the dimension of the system9,10. The invariants fully characterize topological properties that are unchanged under arbitrary adiabatic tuning of the Hamiltonian that preserves the relevant symmetry group. Materials having non-zero topological invariants are loosely called topological materials (whereas the technical term is symmetry protected topological states11,12,13), a new kind of quantum matter that hosts intriguing physical observables such as quantum anomaly on their boundaries14,15,16, and are considered candidate materials for new quantum devices17,18,19,20,21. Success in finding these materials largely depends on the numerical evaluation (prediction) of the topological invariants in a given candidate material. However, even for electronic materials having weak electron correlation, where the topological invariants are best understood and expressed in terms of the wave functions of the valence bands, these calculations still prove quite challenging. In fact, numerically finding a new topological material has proved so hard that a single success17,22,23,24,25 would have triggered enormous interest26,27,28,29,30,31,32.

On the other hand, mathematicians and physicists have since long developed, via the representation theory of space groups, a complete toolkit for the study of the symmetry properties of bands in solids33,34. Given any point in momentum space, each energy level corresponds to an irreducible representation (irreps) of the little group at that momentum, depending on the Blöch wave functions at the level. Modern implementations of the density functional theory output both the energy levels and their Blöch wave functions for any given crystal momenta, such that finding the irreps of all valence bands in a band structure (BS) is now considered a solved problem that can be automated.

It has been eagerly hoped that quantitative relations exist between the topological invariants and the irreps in the valence bands at high-symmetry points in the Brillouin zone, i.e., the symmetry data of valence bands. These relations, if existed, would reduce the difficult task of finding the former to a routine calculation of the latter. However, the examples are rare35,36,37,38. Fu-Kane formula35 for topological insulators protected by time-reversal symmetry (TI for short from now) is an exemplary one, mapping the four topological \({\Bbb Z}_2\)-invariants to inversion eigenvalues at eight high-symmetry points. This simple golden rule considerably expedited the search for TI in all centrosymmetric materials via first-principles numerics22. Nevertheless, for general topological states in three dimensions protected by any one of the 230 space groups with and without time-reversal, or topological crystalline insulators39 (TCIs), explicit formulae relating their topological invariants to symmetry data have so far been missing.

Recently, a solid step along this direction is made in refs. 40,41,42,43, where the authors systematically study the connectivity of bands in a general gapped BS and identify the constraints on the symmetry data in the form of linear equations called the compatibility equations. ref. 41 explicitly provides these relations for each space group and observes that if a symmetry data satisfying all compatibility relations cannot decompose into elementary band representations (sets of symmetry data of atomic insulators (AIs), given in the same paper), the material must be topologically nontrivial. ref. 40 shows that the symmetry data of any gapped BS can be compressed into a set of up to four \({\Bbb Z}_{n = 2,3,4,6,8,12}\) numbers called symmetry-based indicators (SIs) (see Methods section for a brief review of SI). The set of SI is a lossless compression of symmetry data as far as topological invariants are concerned: all topological invariants that may be extracted from symmetry data can be inferred from the corresponding SI. The theory presented in ref. 40 does not, however, relate SI to the topological invariants, the defining quantities of topological states: a BS having non-zero SI is necessarily topological, but the type of the topology in terms of invariants is unknown. The explicit expressions of the SI in terms of symmetry data are also missing in ref. 40.

This paper aims to complete the mapping between symmetry data and topological invariants in systems with time-reversal symmetry and significant spin–orbital coupling (the symplectic Wigner–Dyson class or class AII in the Altland-Zirnbauer system44). To achieve this, we first derive the explicit expression of each SI in all space groups (Supplementary Tables 1–3) and then, given any non-zero set of SI in every space group, we enumerate all possible combinations of topological invariants that are compatible with the SI (Supplementary Tables 4–8). These invariants include: three weak topological invariants δw,i=1,2,3 8, mirror Chern number Cm23,45, glide plane (hourglass) invariant δh24, rotation invariant δr21,46,47, the inversion invariant δi21,36,37, a new \({\Bbb Z}_2\) topological invariant protected by screw rotations δs, and finally a new \({\Bbb Z}_2\) topological invariant protected by S4-symmetry \(\delta _{S_4}\). The last two invariants are theoretically established in Supplementary Note 1. In the main results, the strong time-reversal invariant δt8 is assumed to vanish, so that the results are restricted to TCI only, or states that can be adiabatically brought to AIs in the absence of crystalline symmetries; the δt = 1 cases are briefly discussed in the end of the Results section. The exhaustive enumeration maps 478 sets of SI to 3133 linearly independent combinations of topological invariants, as tabulated in Supplementary Table 7. A guide for reading this table is offered in Supplementary Note 7.

Results

An example showing the usage of our results

Before entering into the derivation of the results, we use tin telluride (SnTe) crystal having space group \(Fm\bar 3m\) (#225) to illustrate how the results should be used in Fig. 1. One should first compute the symmetry data of the material, finding the numbers of appearances for each irrep in the valence bands at the high-symmetry momenta, namely, Γ, X, L, and W. This can be done in any modern implementation of first-principles numerics and here we use Vienna ab-initio simulation package48,49. From the symmetry data obtained in the top of Fig. 1, we apply the formulae given in Supplementary Tables 1 and 2 to find the SI, which in this case is a single \({\Bbb Z}_8\) number, and we find z8 = 4. After this, we can use Supplementary Table 7 and find that z8 = 4 corresponds to two and only two possible sets of topological invariants shown on the bottom of Fig. 1: it either has non-zero mirror Chern number Cm(001) = 4 (mod 8) for the kz = 0 plane (and symmetry partners) or has mirror Chern number Cm(110) = 2 (mod 8) for the kx + ky = 0 plane (and symmetry partners). It is impossible, however, to distinguish these two cases using symmetry data, but advanced tools such as Wilson loops must be invoked. Further analysis shows that the latter state appears in the real material23.

Fig. 1
figure 1

A demonstration of the diagnosis for tin telluride of space group \(Fm\bar 3m_{}^{}\) (#225) using our results. The table on the top shows the symmetry data obtained in the first-principles calculation (details given in text), where the numbers of appearance of each irrep in the valence bands are listed for each high-symmetry point in the face-centered-cubic Brillouin zone. From the data one finds the SI z8 = 4 using Supplementary Tables 1 and 2, and then by searching for this indicator in Supplementary Table 7, two possible sets of topological invariants are found, listed at the bottom left and bottom right, respectively. The yellow planes in the Brillouin zone are where the mirror Chern numbers Cm(001) and Cm(110) are defined. Indices in the parentheses in subscript represent the directions of the corresponding symmetry elements. The real material has been shown in ref. 23 to have the topological invariants listed on the bottom right

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Layer construction as an general approach

A remarkable feature of all known TCIs is that any TCI can be adiabatically (without gap closing) and symmetrically tuned into a simple product state of decoupled, identical layers in real space, each of which decorated with some two-dimensional (2D) topological state21,50,51,52,53,54. This form of fixed-point wave function for a TCI is called its layer construction (LC). An analogy to AIs can be drawn to help understand the physical nature of LC in the following aspects: although an AI is built from decoupled point-like atoms, the building blocks of an LC are decoupled layers. Each atom in an AI is decorated with electrons occupying certain atomic orbitals, whereas each layer is decorated with electrons forming a 2D topological state. The atomic orbitals of an atom in lattice correspond to the irreducible representations of the little group at that atomic position, whereas the possible topological states on a layer also depends on the little group leaving the layer invariant. In an LC, there are only two possible decorations: if the layer coincides with some mirror plane of the space group the state for decoration is a 2D mirror TCI with mirror Chern number Cm, and if it does not coincide it is a 2D TI. We define elementary LC (eLC) as an LC generated by a single layer in real space

$$(mnl;d) \equiv \left\{ {{\bf{r}}|(m{\bf{b}}_1 + n{\bf{b}}_2 + l{\bf{b}}_3) \cdot {\bf{r}} = 2\pi d{\kern 1pt} \mathrm{mod}{\kern 1pt} 2\pi } \right\}$$
(1)

Here (mnl) are the Miller indices, and bi’s the reciprocal lattice vectors; generation here means we take all elements g ∈ G to obtain the set of layers E(mnl; d) ≡ {g(mnl; d)|g ∈ G} by acting g on (mnl; d). Every LC is a superposition of a finite number of eLCs and, thanks to the additive nature of all known topological invariants, the topological invariants of any LC is the sum of the invariants of all constituent eLCs.

For any space group, we exhaustively find all eLCs using the method detailed in Supplementary Note 3. Although the calculation of topological invariants is difficult for an arbitrary BS, it is easy for an eLC, thanks to its simple structure. In fact, the invariants only depend on how many times each symmetry element is occupied (see Supplementary Note 1 for proof, wherein the occupation for glide plane or screw axis is subtle). A symmetry element is a manifold in real space, where each point is invariant under some symmetry operation. It could be a discrete point such as an inversion center or a center of S4 : (x, y, z) → (−y, x, −z), a line such as a rotation axis, or a plane such as a mirror plane. In this way, topological invariants for each eLC are calculated and tabulated in Supplementary Tables 5 and 6. On the other hand, the SI of an eLC are also easily calculated, detailed in Supplementary Note 5 again due to the decoupled nature of the layers. Matching the SI with invariants for each eLC, we hence find the full mapping between SI and topological invariants for TCI. For intuitive understanding, we also plot a set of figures (Supplementary Figs. 1–8) showing the invariants, SI, and phase transitions of eLCs.

From indicators to invariants

Here we take space group \(P\bar 1\) as an example to show the mapping between indicators and invariants and leave the general discussion in the Supplementary Notes 1 and 5. The space group \(P\bar 1\) has non-orthogonal lattice vectors ai=1,2,3 and inversion symmetry. Within a unit cell, there are eight inversion centers at (x1, x2, x3)/2 in the basis of lattice vectors (the red solid circles in Fig. 2), where xi = 0, 1. These inversion centers are denoted by \(V_{x_1x_2x_3}\) ≡ (x1, x2, x3)/2 mod 1. A generic layer (mnl; d) is given by L = {r|(mb1 + nb2 + lb3) · r = 2πd mod 2π}, where d ∈ [0, 1), and at least one of m, n, l is odd (or they would have a common factor). If d ≠ 0, \({\textstyle{1 \over 2}}\), we have d ≠ −d mod 1, then under inversion a generated plane L′ = (mnl; 1 − d) ≠ L is a different plane symmetric to L about the origin. In that case, the two planes L and L′ can adiabatically move towards each other without breaking any symmetry until they coincide, a process illustrated in Fig. 2a. The state decorated on L and L′ are 2D TIs, and due to the \({\Bbb Z}_2\)-nature, when L and L′ coincide, the resultant double layer becomes topologically trivial. The eLC generated by (mnl; d ≠ 0, \({\textstyle{1 \over 2}}\)) is hence a trivial insulator. For d = 0, \({\textstyle{1 \over 2}}\), L is invariant under inversion, and always passes four of the eight inversion centers, that is, Vi’s that satisfy the equation mx1 + nx2 + lx3 = 2d mod 2. For examples, if (mnl; d) = (010; 0), then V000,001,100,101 are on L (the left yellow plane in Fig. 2a). As each layer is decorated with 2D TI, eLC(mnl; d) (d = 0, \({\textstyle{1 \over 2}}\)) is the familiar weak TI having weak invariants

$$\delta _{{\mathrm{w}},1} = m{\kern 1pt} {\mathrm{mod}}{\kern 1pt} 2 \quad \delta _{{\mathrm{w}},2} = n{\kern 1pt} {\mathrm{mod}}{\kern 1pt} 2 \quad \delta _{{\mathrm{w}},3} = l{\kern 1pt} {\mathrm{mod}}{\kern 1pt} 2$$
(2)
Fig. 2
figure 2

Layer constructions for space group \(P\bar 1\) (#2). a The yellow planes are (010; 0) and (010; \({\textstyle{1 \over 2}}\)) respectively, and the two green planes are (010; d) and (010; 1 − d) with d ≠ 0, \({\textstyle{1 \over 2}}\). The arrows mean that the two green planes can move towards each other without breaking inversion. b After doubling the unit cell along x2-direction, the open dots are no longer inversion centers as they were, whereas the solid dots remain. Again the arrows mean that two green planes can move towards each other without breaking the inversion symmetry, after unit cell doubling

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Now we turn to the inversion invariant δi, which is a strong invariant robust against all inversion preserving perturbations. Let us consider a perturbation that doubles the periodicity in the (mnl)-direction, whereas preserving the inversion center at origin. After the doubling, four of the eight inversion centers satisfying mx1 + nx2 + lx3 = 1 mod 2 are no longer inversion centers, so that the plane (mnl, \({\textstyle{1 \over 2}}\)) after the doubling no longer passes through any inversion center, and the generated eLC by (mnl, \({\textstyle{1 \over 2}}\)) can be trivialized after the doubling by pairwise annihilating with its inversion partner. Therefore, eLC(mnl, \({\textstyle{1 \over 2}}\)) can be trivialized, while keeping the inversion symmetry about the origin, thus having δi = 0. In Fig. 2b, we take (mnl) = (010) as an example and doubled the unit cell. We see that the four inversion centers marked by empty circles are not inversion centers in the new cell and the blue plane at (010; \({\textstyle{1 \over 2}}\)) in the original cell becomes (010; \({\textstyle{1 \over 4}}\)) in the new cell. The two blue planes can move and meet each other at the yellow plane, denoted by (010;\({\textstyle{1 \over 2}}\)) in the new cell. The eLC generated by (mnl; 0)-plane, however, passes all eight inversion centers in the enlarged unit cell and cannot be trivialized without breaking inversion (yellow planes in Fig. 2b), so that the inversion invariant δi = 1.

After finding the invariants for all possible eLCs, we turn to the SI for each eLC. The SI group of \({\mathrm{P}}\overline {\mathrm{1}}\) takes the form \({\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_4\), wherein the first three are the weak TI indicators z2w,i=1,2,3 and the last one is the z4 indicator. The calculation method is briefly described in Supplementary Note 5 and here we only give the results. For eLC(mnl; 0) and eLC(mnl; \({\textstyle{1 \over 2}}\)), their values are found to be (m mod 2, n mod 2, l mod 2, 2) and (m mod 2, n mod 2, l mod 2, 0), respectively. For this space group, the mapping from SI set to topological invariants is therefore one-to-one: z2w,i = δw,i and z4 = 2δi.

Convention dependence of topological invariants

A subtle but important remark is due at this point. There are always eight inversion centers in a unit cell in the presence of inversion symmetry, and when translation symmetry is broken, only one, two, or four of them remain. In the definition of inversion invariant δi, one of the eight is chosen as the inversion center that remains upon translation breaking. In the above example, when the unit cell is doubled, the origin was chosen as the center that remains, but if we chose V010 = (\({\textstyle{1 \over 2}}\), 0, 0), which is a completely valid choice, the four open circles in Fig. 2b are still inversion centers but the solid circles are not after the doubling. In that case, we would find that eLC(mnl, \({\textstyle{1 \over 2}}\)) has δi = 1 but eLC(mnl, 0) has δi = 0. The inversion invariant δi hence depends on the convention which one of the eight inversion centers in the unit cell is chosen in the definition of δi. However, when we superimpose the two eLCs into an LC that passes all eight inversion centers in a unit cell, the value of δi is independent of the choice of the inversion center, as all eight are occupied in this LC. We emphasize that only if this is the case can we hope to observe the physical properties, such as the characteristic boundary states associated with the bulk invariant δi21, because physical observables should not depend on the conventions.

Moreover, similarly, as detailed in Supplementary Note 2, the rotation (screw) invariant δr = 1 (δs = 1) is convention-independent if and only if each rotation (screw) axis in unit cell is occupied by the LC for n/2 mod n times, where n = 2, 4, 6 is the order of the rotation (screw) axis. For the S4 invariant \(\delta _{S_4}\) = 1 or the hourglass invariant δh = 1 to be convention-independent, the LC should occupy each S4 center or glide plane for an odd number of times. Invariants that are convention-independent are marked blue in Supplementary Tables 7 and 8.

The one-to-many nature of the mapping

In the example of space group \(P\bar 1\), the mappings between indicators and topological invariants are one-to-one. However, this is in fact the only space group where mappings are bijective. By definition, different sets of indicators must correspond to different sets of invariants, but multiple sets of invariants may correspond to the same set of indicators, i.e., the mapping from indicators to invariants is one-to-many.

To understand the one-to-many nature of the mapping more concretely, we look at the specific group P2/m, containing two mirror planes, four C2-axes and eight inversion centers in each unit cell, all marked in Fig. 3a. Now we consider two different LCs illustrated in Fig. 3b, c: in Fig. 3b for LC1, two horizontal planes, each decorated with a 2D TI, occupy all four C2-axes and all eight inversion centers, and in Fig. 3c for LC2, two vertical planes, each decorated with a mirror Chern insulator with Cm = 1, occupy the two mirror planes and the eight inversion centers.

Fig. 3
figure 3

Two layer constructions for space group P2/m, sharing the same set of SI of (0002). a All symmetry elements of the space group in one unit cell, including eight inversion centers (red solid circles), four rotation axes (red solid lines), and two mirror planes (shaded planes). b, c LC1 and LC2 defined in the text, respectively. They have distinct topological invariants but identical indicators. d 3D Blöch wave functions in LC2 as superpositions of 2D Blöch wave functions with coefficients \(e^{ik_2x_2}\). Here we use red and blue loops to represent the 2D wave functions having mirror eigenvalues i and −i, respectively, wherein i wave functions have Chern number 1 and −i wave functions have Chern number −1. For A-eLC the 3D Blöch wave functions at k2 = 0 and k2 = π have the same mirror eigenvalues, leading to identical mirror Chern numbers at k2 = 0 and k2 = π. Although for B-eLC the Blöch wave functions at k2 = 0 and k2 = π have opposite mirror eigenvalues, leading to opposite mirror Chern numbers at k2 = 0 and k2 = π. e The two mirror-invariant planes (gray planes) in Brillouin zone

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As all inversion centers are occupied in LC1 and LC2, in both cases we have δi = 1. LC1 occupies all four C2-rotation axes, once each, thus having nontrivial rotation invariant δr = 1, whereas LC2 does not occupy any of the rotation axes, having δr = 0. On the other hand, LC2 occupies the two mirror planes, each with 2D TCI having Cm = 1. According to the calculation in Method section LC2 has mirror Chern numbers Cm = 2 at kz = 0 plane and Cm = 0 at kz = π plane; LC1, not occupying any mirror plane, has vanishing mirror Chern number. LC1 and LC2 are therefore topologically distinct states.

Now we turn to the SI of LC1 and LC2. For space group P2/m, the SI have the same group structure \({\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_4\) as that of its subgroup \(P\bar 1\). In this case, the value of each indicator remains the same as we break the symmetry down to \(P\bar 1\). Viewed as LC in \(P\bar 1\), both LC1 and LC2 are the superpositions of eLC(mnl, 0) and eLC(mnl, \({\textstyle{1 \over 2}}\)), thus having, by the additivity of SI, z2w,i = 0 and z4 = 2.

The failure in distinguishing LC1 and LC2 by indicators reveals a general ambiguity in the diagnosis of mirror Chern numbers. We can add additional even number (2p) of A-eLC and even number (2q) of B-eLC to LC1 such that the composite state has the same SI and δr with LC1 but non-zero mirror Chern numbers Cm,0 = 2p + 2q, Cm,π = 2p − 2q. On the other hand, we can also add these additional eLCs to LC2 to get a state having the same SI with LC2 but different mirror Chern numbers Cm,0 = 2 + 2p + 2q, Cm,π = 2p − 2q. The proof here can be generalized to any space group having mirror planes and perpendicular rotation axes, providing that the order of the rotation is even. In all these space groups, as shown in Supplementary Table 7, the TCIs having invariants δr = 1 and Cm,0 − Cm,π = 0 mod 2n and the TCIs having δr = 0 and Cm,0 − Cm,π = n mod 2n have the same SI, here n is the order of the rotation axis. The two possible sets of invariants shown in Fig. 1, wherein one is Cm(001) = 4 mod 8, δr(001) = 0 and the other is Cm(001) = 0 mod 8, δr(001) = 1, are the examples for n = 4.

Indicators for time-reversal topological insulators

In the SI group of each space group, except #174 and #187–190, there is one special indicator of \({\Bbb Z}_{2,4,8,12}\)-type, denoted by zt, marked red in Supplementary Tables 2 and 3. When this special indicator is odd, the system is the well-known three-dimensional (3D) time-reversal topological insulator40 (TI for short). The essential difference between a TI and a TCI is that the former only requires time-reversal symmetry, such that it remains nontrivial even when all crystalline symmetries are broken. TI does not have LCs, so that the method we use does not apply to SI having zt ∈ odd. To construct states having zt ∈ odd, we first notice that TI is consistent with all space groups, such that for each space group, we have at least one state that is a TI. Then we can superimpose this TI with all existing LCs obtained, and generate gapped states for all non-zero combinations of SI with zt ∈ odd (but with five exceptions discussed in Discussion section).

Regarding zt ∈ odd, we comment that the value of zt generally has a convention dependence on the overall signs in the definition of inversion and the rotation operators. For example, in space group \(P\bar 1\), the defining properties of the symmetry operators are \(\hat P^2\) = 1, \(\hat T^2\) = −1 and \([\hat P,\hat T]\) = 0. It is easy to check that the overall sign in front of \(\hat P\) can be freely chosen without violating any of the above relations. In other words, without external references, it is unknown a priori if, e.g., an s-orbital should be assigned with positive or negative parity. Upon redefining \(\hat P \to - \hat P\), a state having z4 = 1 goes to z4 = 3 and vice versa. In similar ways, it is proved in Supplementary Note 4 that states having the \({\Bbb Z}_8\)-indicator z8 = 1, 3, 5, 7 differ only by convention and so do the states have \({\Bbb Z}_{12}\)-indicator z12 = 1, 5, 7, 11. In these cases, the convention refers to the overall sign in front of inversion operator and the sign in front of the rotation operator. It is difficult to distinguish these states from each other experimentally. However, here we emphasize that the SI z4 = 1, 3 (so do the SI z8 = 1, 3, 5, 7 and z12 = 1, 5, 7, 11) have a relevant difference under a fixed convention, which can be detected by the anomalous boundary between the two phases. For example, suppose we have a spherical sample of z4 = 3 phase and fill the space outside the sphere with z4 = 1 phase, then as long as the geometry keeps inversion symmetry, the boundary state on the spherical surface should be identical with the boundary state between z4 = 2 and z4 = 0 phases, which is known as one-dimensional helical mode (see Supplementary Note 1 for details). This is because we can deduct a background of z4 = 1 phase both inside and outside the sphere without changing the boundary state.

The space groups #174, #187, #189, and #188, #190, where one cannot diagnose TI from SI, have the SI groups \({\Bbb Z}_3 \times {\Bbb Z}_3\) and \({\Bbb Z}_3\), respectively, and the corresponding SI z3m,0 and z3m,π are the mirror Chern numbers (mod 3) in the k3 = 0 and k3 = π planes. In #188 and #190 z3m,π is trivialized by nonsymmorphic symmetry and thus the corresponding SI groups reduce to \({\Bbb Z}_3\). In these space groups, the TI invariant is the parity of Cm,0 − Cm,π23 whereas SI have ambiguity for the parities of mirror Chern numbers; thus, TI can never be diagnosed from SI. For example, z3m,0 = 1, z3m,π = 0 can correspond to Cm,0 = 1, Cm,π = 0 (a TI), or Cm,0 = −2, Cm,π = 0 (not a TI).

Discussion

A byproduct of this study is a complete set of TCIs that can be layer-constructed in all 230 space groups (Supplementary Tables 5–8), even including groups not having SI. The abundance of the states thus obtained naturally suggests the question: are all TCI states exhausted in these layer constructions? We regret to answer it in the negative: LC cannot give us the weak topological insulator states in five space groups, namely #48, #86, #134, #201, and #224. In any one of the five, there is a weak indicator z2w, but all layer-constructed states have z2w = 0. A common character of these space groups is that they have three perpendicular glide planes {m001|\({\frac{1}{2}} \, {\frac{1}{2}}\)0} \(\left\{ {m_{010}|{\textstyle{1 \over 2}}0{\textstyle{1 \over 2}}} \right\}\) {m100|0\({\frac{1}{2}} \, {\frac{1}{2}}\)} such that any single layer having weak index z2w,i = 1 would be doubled along the i-th direction and so the generated eLC has vanishing weak index. Explicit (non-LC) tight-binding models for the z2w = 1 states are given in Supplementary Note 6, completing the proof that for any non-zero SI there is at least one corresponding gapped topological state. These corner cases are somewhat surprising as weak TI have so far been considered most akin to stacking of decoupled 2D TI.

Finally, we comment that all LCs can be used to build 3D symmetry protected topological states of bosons and fermions protected by space group G plus a local group GL. To do this one only needs to decorate each layer with a 2D SPT protected by GL instead of the 2D TI.

Towards the completion of the work, we have been aware of a similar study55. To our knowledge, the results, when overlapping, are consistent with each other.

Methods

A short review of SIs

For each momentum in the Brillouin zone, there is an associate subgroup, called the little group, of the space group G, under the action of which the momentum is invariant up to a reciprocal lattice vector. A point is a high-symmetry point, denoted Kj, if its little group is greater than the little group of any point in the neighborhood. A fundamental theorem is that each band at momentum Ki or multiplet of degenerate bands corresponds to an irreducible representation of the little group at Ki. The symmetry data of a BS is defined as the an integer vector n, each element of which, \(n\left( {\xi _i^{K_j}} \right)\), is the number of appearance of the i-th irreducible representation in the valence bands at the j-th high-symmetry momentum Kj, where i = 1, ..., rj labels the irreducible representations of the little group at Kj. One could further define the addition of two symmetry data as the addition of each entry, which corresponds to, physically, the superposition of two BSs.

For a gapped BS, the elements of its symmetry data cannot take arbitrary integers and there are constraints on the symmetry data known compatibility relations40,41,43. For example, gapped-ness requires that the occupation numbers at each Ki be the same, i.e., \(\mathop {\sum}\nolimits_i {\kern 1pt} n\left( {\xi _i^{K_j}} \right)\) = const. All compatibility relations are linear equations so that the symmetry data satisfying all these relations again form a smaller linear space, termed the BS space, denoted {BS}.

On the other hand, we consider the symmetry data of AIs. In AIs, the bands are generated by decoupled atomic orbitals placed at certain Wyckoff positions in the unit cell. By this definition, one finds that the symmetry data of AIs also form a linear space, denoted {AI} (also called the space of band representations41). Obviously a symmetry data n ∈ {AI} satisfies all compatibility relations, so {AI} ⊆ {BS}. One then naturally considers the quotient space XBS = {BS}/{AI}. XBS is always a finite group generated by several \({\Bbb Z}_{n = 2,3,4,6,8,12}\)40. Each generator of XBS is called an SI.

The following properties of indicators should be mentioned: any two gapped BSs having different sets of SI must be topologically distinct, and any two different symmetry data having the same set of SI only differ from each other by the symmetry data of an AI.

In ref. 40, the authors calculate the group structure of the indicators for all 230 space groups. However, it does not give explicit formulae for the generators. In order for application, we derive all these formulae in Supplementary Note 4.

Mirror Chern number of LC

Below we explicitly calculate the mirror Chern numbers of LC1 and LC2 in Fig. 2. As shown in Fig. 3e, in BZ of space group P2/m there are two mirror-invariant planes, i.e., the k2 = 0 and k2 = π planes; thus, we have two mirror Chern numbers Cm,0 and Cm,π. We assume there are only two occupied bands in the vertical 2D TCIs in LC2 and denote the corresponding Blöch wave functions as \(\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle\). Here, ±i represent the mirror eigenvalues, where i is the imaginary unit, k2D = (k1, k3) is the 2D momentum, and x2 is the position along a2 where the 2D TCIs are attached. We also assume that the wave functions with the mirror eigenvalue i (−i) give a Chern number 1 (−1) such that the 2D mirror Chern number Cm = 1. Under the mirror operation \(\hat M\) the 2D Blöch wave function \(\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle\) first get a mirror eigenvalue ±i and then move to the mirror position −x2

$$\hat M\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle = \pm i\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}}, - x_2} \right)} \right\rangle$$
(3)

To calculate the mirror Chern numbers of LC2, we divide it into two subsystems: the eLC generated from A layer and the eLC generated from B layer (Fig. 3c). As the total mirror Chern numbers are the sum of mirror Chern numbers of the two subsystems, we need only to analyse the two subsystems, respectively. The 3D Blöch wave functions of A- and B-eLCs can be constructed as

$$\left| {\psi _{ \pm i}^{\mathrm{A}}({\bf{k}})} \right\rangle = \mathop {\sum}\limits_{x_2 = 0, \pm 1 \cdots } {\kern 1pt} e^{ik_2x_2}\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle$$
(4)
$$\left| {\psi _{ \pm i}^{\mathrm{B}}({\bf{k}})} \right\rangle = \mathop {\sum}\limits_{x_2 = \pm \frac{1}{2}, \pm \frac{3}{2} \cdots } {\kern 1pt} e^{ik_2x_2}\left| {\phi _{ \pm i}\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle$$
(5)

Due to Eq. (3), it is direct to show that \(\left| {\psi _i^{\mathrm{A}}\left( {k_1,0,k_3} \right)} \right\rangle\) and \(\left| {\psi _i^{\mathrm{A}}\left( {k_1,\pi ,k_3} \right)} \right\rangle\), both of which are superpositions of \(\left| {\phi _i\left( {{\bf{k}}_{{\mathrm{2D}}},x_2} \right)} \right\rangle\) and thus have the Chern number 1, have the same mirror eigenvalue i (Fig. 3d). Thus, for A-eLC m the mirror Chern numbers at k2 = 0 and k2 = π are all 1. On the other hand, \(\left| {\psi _i^{\mathrm{B}}\left( {k_1,0,k_3} \right)} \right\rangle\) and \(\left| {\psi _i^{\mathrm{B}}\left( {k_1,\pi ,k_3} \right)} \right\rangle\), again both of which have the Chern number 1, have mirror eigenvalues i and −i, respectively (Fig. 3d). Thus, for B-eLC the mirror Chern numbers at k2 = 0 and k2 = π are 1 and −1, respectively. Therefore, the total mirror Chern numbers in momentum space are Cm,0 = 2 and Cm,π = 0 for LC2. It should be noticed that the values of Cm,0 and Cm,π do not depend on the two band assumption we take: as long as the 2D TCI has Cm = 1, the results remain the same. On the other hand, the mirror Chern numbers of LC1 should be zero for both k2 = 0 and k2 = π by the following argument. Without breaking mirror symmetry, each vertical plane can bend symmetrically towards the mirror plane until the two halves coincide on mirror-invariant planes in real space, due to the \({\Bbb Z}_2\)-nature of each half, the folded plane is topologically equivalent to a trivial insulator. As LC1 can be smoothly trivialized without breaking mirror symmetry, it must have vanishing mirror Chern numbers.