Symmetry indicators and anomalous surface states of topological crystalline insulators
The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.
Topological phases of free fermions protected by internal symmetries (such as time reversal) feature a gapped bulk and symmetry-protected gapless surface states Franz and Molenkamp (2013). Such bulk-boundary correspondence is an important attribute of such topological phases, and it plays a key role in their classification, which has been achieved in arbitrary spatial dimensions Kitaev (2009); Schnyder et al. (2009); Ryu et al. (2010). Crystals, however, frequently exhibit a far richer set of spatial symmetries, like lattice translations, rotations, and reflections, which can also protect new topological phases of matter Fu (2011); Hsieh et al. (2012); Tanaka et al. (2012); Slager et al. (2013); Benalcazar et al. (2014); Lu and Lee (2014); Varjas et al. (2015); Shiozaki et al. (2015); Dong and Liu (2016); Wang et al. (2016); Ezawa (2016); Varjas et al. (2017); Wieder et al. (2017); Benalcazar et al. (2017); Song et al. (2017a); Schindler et al. (2017); Langbehn et al. (2017); Benalcazar et al. (2017); Fang and Fu (2017).
However, several new subtleties arise when discussing topological crystalline phases. For one, spatial symmetries are often broken at surfaces, and when that happens, the surface could be fully gapped even when the bulk is topological. Consequentially, the diagnosis of bulk band topology is a more delicate task as, a priori, the existence of anomalous surface states in this setting is only a sufficient, but not necessary, property of a topological bulk. Furthermore, the presence of crystalline symmetries leads to a plethora of topological distinctions even when the band insulators admit atomic descriptions and are therefore individually trivial Po et al. (2017); Bradlyn et al. (2017). For example, consider trivial (atomic) insulators, where electrons are well-localized in real space and do not support any nontrivial surface states. Suppose two atomic insulators are built from orbitals of different chemical characters, which will typically lead to different symmetry representations in the Brillouin zone (BZ). These representations cannot be modified without a bulk gap closing, and therefore the two atomic insulators are topologically distinct even though they are both derived from an atomic limit.
Since the mathematical classification of topological band structures, using K-theory, is formulated in terms of the mutual topological distinction between insulators Kitaev (2009); Freed and Moore (2013); Kruthoff et al. (2017); Shiozaki et al. (2017), the mentioned distinction between trivial phases is automatically incorporated. In fact, they appear on the same footing as conventional topological indices like Chern numbers. While for a physical classification one would want to discern between these different flavors of topological distinctions, they are typically intricately related. On the one hand, the pattern of symmetry representations is oftentimes intertwined with other topological invariants, as epitomized by the Fu-Kane parity criterion Fu and Kane (2007), which diagnoses 2D and 3D topological insulators (TIs) through their inversion eigenvalues; on the other hand, the topological distinction between atomic insulators may reside fully in their wavefunction properties and, hence, is not reflected in their symmetry eigenvalues Michel and Zak (2001); Bradlyn et al. (2017).
In this work, we sidestep these issues by focusing on topological crystalline phases displaying anomalous gapless surface states, which, through bulk-boundary correspondence, are obviously topological. Unlike the strong TI, these phases require crystalline symmetries for their protection. Typically, their physical signatures are exposed on special surfaces where the protecting symmetries are preserved. Such is the case for the 3D phases like weak TI Franz and Molenkamp (2013), the mirror Chern insulator Hsieh et al. (2012); Tanaka et al. (2012), and the nonsymmorphic insulators featuring so-called hourglass Wang et al. (2016); Ezawa (2016) or wallpaper Wieder et al. (2017) fermions, which are respectively protected by lattice translation, reflection, and glide symmetries. While these conventional phases feature 2D gapless surface states on appropriate surfaces, we will also allow for more delicate, 1D “hinge” surface states, whose existence is globally guaranteed on suitably chosen sample geometries although any given crystal facet can be fully gapped Sitte et al. (2012); Zhang et al. (2013); Benalcazar et al. (2014, 2017); Hashimoto et al. (2017); Song et al. (2017a); Schindler et al. (2017); Benalcazar et al. (2017); Fang and Fu (2017); Langbehn et al. (2017). We will primarily focus on 3D time-reversal symmetric band structures with a bulk gap and significant spin-orbit coupling (class AII). However, our approach can be readily extended beyond this specific setting. In particular, in symmetry classes with particle-hole symmetry, the notion of hinge states can be generalized to protected gapless points on the surface Benalcazar et al. (2014).
We will base our analysis on two general techniques, which are respectively employed to systematically study gapless surface states and identify bulk band structures associated with them. First, we will describe a surface Dirac theory approach Franz and Molenkamp (2013); Schnyder et al. (2009); Ryu et al. (2010); Witten (2016); Fang and Fu (2017). Specifically, we will consider a surface theory with multiple Dirac cones, which may transform differently under the spatial symmetries Fang and Fu (2017). Such a theory can be viewed as the surface of a “stack of strong TIs,” which arise, for example, when there are multiple band inversions (assuming inversion symmetry) in the BZ. When an even number of Dirac cones are present at the same surface momentum, the Dirac cones can generically be locally gapped out and, naively, one expects that a trivial surface results. However, spatial symmetries can cast global constraints which obstruct the full gapping out of the Dirac cones everywhere on the surface Fang and Fu (2017). When that happens, the surface remains stably gapless along certain domain walls, and hence one can infer that the bulk is topological. We will provide a comprehensive analysis of such obstructions due to spatial symmetries, and catalogue all the possible patterns of such anomalous surface states in a space group.
Next, we will turn to bulk diagnostics that informs the presence of such surface states. In particular, we will focus on those which, like the Fu-Kane criterion Fu and Kane (2007), expose band topology using only symmetry eigenvalues Fu and Kane (2007); Turner et al. (2012); Hughes et al. (2011); Fang et al. (2012). Such symmetry-based indicators are of great practical values, since they can be readily evaluated without computing any wavefunction overlaps and integrals. In fact, a general theory of such indicators has been developed in Ref. Po et al., 2017, and the indicators for all 230 space groups (SGs) have been computed.
Here, we provide a precise physical interpretation for all the phases captured by these indicators in class AII. In particular, we discover that the majority of them can be understood in terms of familiar indices, like the strong and weak TI indices and mirror Chern numbers. However, there exists symmetry-diagnosed band topology which persists even when all the conventional indices have been silenced. The prototypical example discussed in Ref. Po et al., 2017 is a “doubled strong TI” in the presence of inversion symmetry, which, as pointed out later in Ref. Fang and Fu, 2017, actually features hinge-like surface states. Motivated by these developments, we relate the symmetry indicators to our surface Dirac theory and prove that, for spin-orbit coupled systems with time-reversal symmetry (TRS), any system with a nontrivial symmetry indicator is a band insulator with anomalous surface states on a suitably chosen boundary. This is achieved by establishing a bulk-boundary correspondence between the indicators and the surface Dirac theory. Parenthetically, we note that a nontrivial symmetry indicator constitutes a sufficient, but not necessary, condition for the presence of band topology, and we will discuss examples of topological phases which have gapless surface states predicted by our Dirac approach, although the symmetry indicator is trivial.
Although the techniques we adopted are tailored for weakly correlated materials admitting a band-theoretic description, many facets of our analysis have immediate bearing on the more general study of interacting topological crystalline phases Song et al. (2017b); Thorngren and Else (2016); Xiong (2017); Huang et al. (2017); Zhaoxi Xiong and Alexandradinata (2017). A common theme is to embrace a real-space perspective, where “topological crystals” are built by repeating motifs that are by themselves topological phases of some lower dimension, and are potentially protected by internal symmetries or spatial symmetries that leave certain regions invariant (say on a mirror plane) Song et al. (2017b); Huang et al. (2017). In particular, such picture allows one to readily deduce the interaction stability of the phases we describe Isobe and Fu (2015), and we will briefly discuss how our Dirac surface theory analysis can be reconciled with such general frameworks.
I Symmetry indicators and bulk topology
In this section, we provide a precise physical meaning for the phases captured by the symmetry-based indicators obtained in Ref. Po et al., 2017, assuming TRS and focusing on the physically most interesting case of strong spin-orbit coupling.
We start by considering a band insulator of spinful electrons symmetric under an SG and TRS (class AII). In this setting, we can define topological indices corresponding to weak and strong TI phases protected by TRS. For centrosymmetric SGs, the Fu-Kane formula Fu and Kane (2007) allows us to compute these indices using only the parities, i.e., inversion eigenvalues, of the bands. Namely, they can be determined just by multiplying the parities of the occupied bands without performing any sort of integrals. This is one particular instance of the relation between the symmetry representations in momentum space and band topology.
Recently, this relation was extended to all space groups (i.e., not restricted to inversion alone) and also to a wider class of band topology Po et al. (2017). By comparing the symmetry representations of the bands against that of trivial (atomic) insulators, the symmetry-based indicator was computed for all 230 SGs Po et al. (2017) (and even for all 1651 magnetic SGs Watanabe et al. (2017)). It was found to take the form
where and depend on the assumed SG. While has been exhaustively computed for all SGs in Ref. Po et al., 2017, the concrete physical interpretation for some of the nontrivial classes was left unclear. We will attack this problem in the following by first recasting these topological indices into more explicit forms, akin to those in Refs. Fu and Kane, 2007; Fang et al., 2012, and then clarifying their physical meanings.
i.1 Review of familiar indices
Here, we first review some well-known symmetry-based indicators and discuss their relations with the general results in Ref. Po et al., 2017. Let us start with the inversion symmetry. As mentioned above, for an SG containing the inversion symmetry , the inversion eigenvalues are related to the strong TI index and the three weak TI indices () Fu and Kane (2007). Each of the three weak TI indices gives rise to a factor in when no additional constraints are imposed. Curiously, however, the factor in corresponding to the strong TI index is , not Po et al. (2017). This indicator will be discussed in Sec. I.2.
Next, let us discuss the mirror Chern numbers, which can be defined for every mirror symmetric plane in the momentum space Hsieh et al. (2012). On any such plane, the single-particle Hamiltonian can be block-diagonalized into , defined respectively in the sectors corresponding to the mirror eigenvalues . Since is not necessarily time-reversal symmetric, can possess a Chern number , which satisfies () due to the TRS of the total Hamiltonian . To be more concrete, let us assume the mirror symmetry is about the plane. For primitive lattice systems, both and planes are mirror symmetric and can individually support and . For body-centered systems and face-centered systems, however, is the only mirror symmetric plane and there is only one mirror Chern number .
When the SG is further endowed with an -fold rotation symmetry () or screw symmetry ( rotation followed by a fractional translation along the rotation axis) whose axis is orthogonal to the mirror plane, one can diagnose the mirror Chern number modulo by multiplying the rotation eigenvalues Fang et al. (2012). (For simplicity, in the following we will often leave the direction of a rotation or screw symmetry implicit, and, whenever necessary, label that as the axis corresponding to the third momentum coordinate.) Naively, each mod produces a factor in . However, we sometimes find a “doubled” factor in . The mechanism behind this enhancement is clarified in Sec. I.4 and I.5.
Following the examination of these familiar indices, we find that in Eq. (1) can be always factorized into “weak factors” and a “strong factor”:
Every factor in can be completely characterized either by the weak TI indices () or the weak mirror Chern number , both of which can be understood as arising from stacking 2D TIs. On the other hand, for most SGs, the strong factor cannot be fully understood by these familiar indicators. We will thus focus on the strong factor in the reminder of this section.
i.2 index for inversion symmetry
Here we show that the strong TI index can actually be promoted to a index in the presence of inversion symmetry. The refined index can capture topological crystalline insulators with anomalous 1D edge state, as we discuss in detail in Sec. II.
Let () be the number of occupied bands with even (odd) parity at each time-reversal invariant momentum (TRIM) . Due to Kramers pairing, is even. The Fu-Kane formula Fu and Kane (2007) for the strong TI index may be expressed as
We now introduce a -valued index Lu and Lee (2014) that is simply the sum of the inversion parities of occupied bands (up to a pre-factor):
Using the total number of occupied bands , one can rewrite as . Comparing this with Eq. (3), we find
Although Eq. (5) suggests that mod 2, contains more information than as it is “stable” mod , in the sense that any trivial insulator has () and adding or subtracting trivial bands does not alter mod . Weak topological phases may realize mod 4, but the most interesting case is when mod 4 while all vanish. One way of achieving this phase is to stack two copies of a strong TI. The surface signature of this phase was discussed in Ref. Fang and Fu, 2017, and will be reintroduced in Sec. II.
The space group that contains only inversion in addition to translations is called (numbered No. 2 in the standard crystallographic references Hahn (2006)). The index can be defined for every supergroup of (SGs containing as a subgroup), i.e., for every centrosymmetric SG.
i.3 New index for four-fold rotoinversion
The SGs (No. 81) and (No. 82) possess neither inversion nor mirror symmetry. Hence, the indicator found in Ref. Po et al., 2017 cannot by accounted for by the Fu-Kane parity formula Fu and Kane (2007) or the mirror Chern number Hsieh et al. (2012). Here, we propose a new index in terms of the eigenvalues of four-fold rotoinversion (the four-fold rotation followed by inversion) and show that the nontrivial phase is actually a strong TI.
To introduce the invariant, we note that there are four momenta in the BZ invariant under , which we will denote by ; they are , , and for primitive lattice systems and , , , and for body-centered systems. For spinful electrons, the four possible values of -eigenvalues are (). Let us denote by the number of occupied bands with the eigenvalue at momentum .
The new -valued index is (up to a pre-factor) the sum of the -eigenvalues of occupied bands over the momenta in :
This quantity is always an integer in the presence of TRS, and is stable modulo against the stacking of trivial bands. As we show in Appendix A, agrees with . It is, however, still useful to distinguish from , as conveys more information than in the presence of additional symmetries, as we will discuss in the next section.
i.4 The combination of and
The SGs (No. 83) and (No. 87) contain both inversion and four-fold rotation with the inversion center on the rotation axis. The product of and is a mirror about the plane and one can define the mirror Chern number . The eigenvalues of the four-fold rotation determine mod 4 Fang et al. (2012). However, studied in Ref. Po et al., 2017 contains a factor, which cannot be explained by the (detected) mirror Chern number alone. Here, we argue that the combination of and is responsible for this enhanced factor.
Since these SGs have both inversion and rotoinversion , and can be defined separately using Eqs. (4) and (6). Furthermore, the inversion center coincides with the rotoinversion center. In this case, we find that the difference
is stable modulo , not only 4, against the stacking of trivial bands. Hence mod 8 should be understood as a new bulk invariant, which can be reconciled with the strong factor . To see this, recall first that the definition of and has no ambiguity and they are perfectly well-defined as -valued quantities. Furthermore, an explicit calculation verifies that, for any trivial insulator symmetric under either or , the two invariants and are not independent and mod 8 always holds.
i.5 The combination of and under six-fold rotation, screw, or rotoinversion
The case of SG (No. 175) is similar to . It has a six-fold rotation in addition to inversion . The product of and is the mirror protecting the mirror Chern number . Although the eigenvalues of the six-fold rotation can only detect mirror Chern numbers mod 6, contains a factor. In this case, the combination of mod 6 and mod 4 can fully characterize this factor.
When the rotation is replaced by the screw (i.e., followed by a half translation in ), the resulting SG is (No. 176). In this case, the product of and is also a mirror symmetry. The corresponding includes the same factor.
In contrast, (No. 174) generated by the six-fold rotoinversion does not have an inversion symmetry and is not defined. The mirror Chern numbers protected by can be diagnosed by thee-fold rotation modulo 3 and this fully explains .
i.6 Summary of
The eight SGs studied above (SG 2, 81, 82, 83, 87, 174, 175, and 176) play the role of “key” SGs, in the sense that they provide an anchor for understanding the symmetry-based indicator of any other SG. Suppose that we are interested in one of the 230 SGs . To understand the nature of a nontrivial , one should first identify its maximal subgroup among the eight key SGs. Let be the key SG. Then we know that (i) the topological indices characterizing are the same as those for , (ii) and are the same, (iii) the weak factor may be reduced from due to the constraints imposed by the additional symmetries in . In Table 1, we identify the key SGs for all the 230 SGs. This table reproduces for all 230 SGs presented in Ref. Po et al., 2017, but in a way which renders their physical properties more transparent.
Physically, the key space groups play the role of the minimal SGs for which a certain non-trivial phase is possible. This means that a non-trivial phase in a SG in the family of the key SG will remain nontrivial even if some symmetries of are broken, so long as those of are preserved. For instance, consider SG (No. 84) characterized by a 4-fold screw rotation and the inversion . The product of and gives rise to a mirror symmetry, whose mirror Chern number is diagnosed mod 4 by . (In this case, since the inversion center and the rotoinversion () center do not agree, does not enhance to .) If we break the mirror symmetry without breaking inversion, the mirror Chern number becomes undefined, but the phases corresponding to the nontrivial are still protected by the inversion symmetry, and are diagnosed by the inversion index .
|Key Space Group||Key Indices||Space Groups|
|,||2, 10, 47.|
|11, 12, 13, 49, 51, 65, 67, 69.|
|14, 15, 48, 50, 53, 54, 55, 57, 59, 63,|
|64, 66, 68, 71, 72, 73, 74, 84, 85, 86,|
|125, 129, 131, 132, 134, 147, 148, 162,|
|164, 166, 200, 201, 204, 206, 224.|
|0||52, 56, 58, 60, 61, 62, 70, 88, 126, 130, 133,|
|135, 136, 137, 138 141, 142, 163, 165, 167,|
|202, 203, 205, 222, 223, 227, 228, 230.|
|0||81, 111–118, 215, 218.|
|0||82, 119–122, 216, 217, 219, 220.|
|, ,||87, 139, 140, 229.|
|174, 187, 189.|
|,||0||176, 193, 194.|
Ii Surface states
We have seen in the previous section that some phases diagnosed by the symmetry indicator do not fall into the standard categories of strong TI, weak TIs, and mirror Chern insulators. This raises the question on the nature of the possible surface states possessed by these phases. With this question in mind, this section will be devoted to developing a general approach to studying anomalous surface states protected by crystalline symmetries. We will argue that our approach captures most, if not all, TCIs with anomalous surface states
We start by considering a sample with a specific geometry, and boundary conditions which are periodic or open along different directions. The surface is a compact manifold
We then proceed to define the notion of an “anomalous surface topological crystalline insulator (sTCI)” based on its surface properties: we say a band insulator is a sTCI if there exists a geometry with some boundary conditions such that the surface Hamiltonian
is not gapped everywhere;
cannot be realized in an independent lower dimensional system with the same symmetries; and
can be gapped everywhere by breaking the crystalline symmetry or going through a bulk phase transition.
This definition includes familiar phases such as weak TIs (protected by translation) and conventional TCIs, as well as the recently discovered TCI with surface hinge modes Song et al. (2017a); Schindler et al. (2017); Langbehn et al. (2017); Benalcazar et al. (2017); Fang and Fu (2017).
Given a gapped bulk, the surface region where the gap vanishes is necessarily boundaryless, since otherwise the spatial spread of the surface states would diverge as we approach the boundary of the gapless region, necessitating the existence of gapless bulk states as in Weyl semimetals Wan et al. (2011); Huang et al. (2015); Xu et al. (2015). This boundaryless region residing on the surface of the 3D bulk could be a 2D, 1D (collection of closed curves) or a 0D region (collection of points). We will show below that the latter is impossible for an insulator in class AII, but first, let us point out that designating a certain dimensionality to a gapless region generally depends not only on the symmetries at play, but also on the geometry of the sample. For example, the hinge insulators protected by rotation Song et al. (2017a); Fang and Fu (2017) possess 1D helical gapless modes when placed on a sphere, but exhibit gapless 2D Dirac cones when the surface is a plane normal to the rotation axis (with periodic boundary conditions along the two in-plane directions), as we illustrate Fig. 5c. The relation between 1D and 2D surface states will be explored in more detail in Sec. II.4.
To explore the types of possible surface states, we start by defining a (surface) high-symmetry point as a point left invariant by some crystalline symmetries. That is to say, the stabilizer group of this point contains elements apart from the identity. Let us first consider a generic point (that is not a high-symmetry point). Since the only symmetry at is TRS, the surface Hamiltonian at can only be gapless if it is locally identical to (i) the surface of a strong TI, or (ii) a (movable) domain wall between two different quantum-spin Hall phases. This can be understood in terms of the classification of stable topological defects Teo and Kane (2010), which implies that 1D and 2D defects are topologically stable in class AII (with a classification), whereas 0D defects are not. The latter typically describe vortices in superconductors and require particle-hole or chiral symmetry to be stable.
At a high-symmetry point, one extra complication arises from the fact that the crystalline symmetry acts locally as an on-site discrete unitary symmetry, which can be used to block-diagonalize the local Hamiltonian. Each block can potentially have less symmetries than the original Hamiltonian. As a result, we need to consider the additional possibility of class A (unitary-type) defects. Only 1D defects, representing domain walls between two phases with different Chern numbers, are stable in class A Teo and Kane (2010) and have a classification. They will only exist if the symmetry leaves a line or curve on the surface invariant, which only occurs for mirror symmetry in 3D. Such line can host an arbitrary number of 1D gapless modes due to the classification, which can be understood in terms of mirror Chern number Hsieh et al. (2012).
Our argument can be summarized as follows: a gapless surface Hamiltonian at a certain point will be a part of (i) a single domain wall, (ii) multiple intersecting domain walls (this could only happen at a high-symmetry point since an extra symmetry is required to stabilize the crossing), or (iii) the 2D gapless surface of a strong TI. The last case does not correspond to a sTCI since it does not require a crystalline symmetry for its stability, while the former two cases correspond to sTCIs with domain walls, which are either movable or are pinned to high-symmetry points or lines. Consequently, we arrive at a unified description for the surface states of all sTCIs in terms of “globally irremovable, locally stable topological defects.” Our approach is reminiscent to the approach of Ref. Song et al., 2017b, which argued that a symmetry protected topological (SPT) phase protected by a point group symmetry can be understood in terms an embedded SPT of a lower dimensionality stabilized by the point-group symmetry.
Having reduced our problem to the study of globally stable configurations of surface domain walls, the next simplification is achieved by observing that a surface domain wall in class AII can be generically constructed by first stacking together two strong TIs, and then adding a mass term which changes sign at the domain wall. This is also consistent with the analysis of Sec. I and Refs. Fang and Fu, 2017; Song et al., 2017a, and it means that the surface theory of any sTCI can be constructed by stacking strong TIs and studying the symmetry transformation properties of the possible mass terms. Our basic building block in the following sections will be the doubled strong TI (DSTI), which is constructed by stacking two strong TIs. In particular, we note that every even entry in the strong factor of any space group can be viewed as the stack of two strong TIs (e.g., by rewriting ). If the order of is even (say ), then the DSTIs correspond precisely to the even subgroup of (Table 1); however, if the order is odd, then this identification does not hold, and any given entry could correspond to either a strong or a doubled strong TI. We will elaborate further on such identification ambiguity in Sec. III.
ii.1 Stacked strong TIs
As explained in detail in Appendix B, the gapless Hamiltonian on the surface of a strong TI can be written as
Here, is the normal to the surface at point and is the vector of Pauli matrices representing the spin degrees of freedom. TRS is implemented as , where denotes complex conjugation, and it protects the gaplessness of .
A DSTI is constructed by stacking two strong TIs, whose surface Hamiltonian can be described by two copies of (8). The only possible -symmetric mass term which can gap-out the surface has the form
with denoting the Pauli matrices in the orbital space of the two copies.
To study the action of spatial symmetries on the DSTI model, we start by considering a generic spatial symmetry , where is a point-group operation and denotes the (possibly fractional) translation in . can be parameterized by three pieces of data: , , and , where and are respectively the rotation angle and axis, and indicates whether is a proper or improper rotation. We can construct a natural spin-1/2 representation of the point-group action of , given by .
which leaves the surface Hamiltonian (as a whole) invariant for any spatial symmetry . Clearly, the same conclusion holds when we generalize for any phase factor .
Time-reversal symmetry further restricts this phase to .
In addition, we demand to respect the (projective) group structure of the point group. Specifically, for a pair of symmetry elements , we demand , which fixes
Microscopically, different values of arise naturally since the rotational symmetry is reduced to a finite, discrete point group in a crystal, and both spin and orbital angular momenta contribute to the symmetry representation. Yet, when is traceless, say for , the sign can be removed through a basis transformation, and hence do not give rise to distinct representations. However, such a basis transformation will simultaneously modify the form of the surface Hamiltonian (8), which in turn amounts to a redefinition of the helicity of the surface Dirac cone, as we will discuss later. In the following, we will say is a “signed representation” when we want to emphasize the importance of the sign choice , regardless of whether this choice actually gives rise to distinct representations.
Spatial symmetries may force the mass term to vanish along some curve, which creates a domain wall hosting a propagating 1D helical gapless mode. To see this, consider a spatial symmetry operation which leaves the mass term (9) invariant. The action of the symmetry on the mass term can be generally written as
Here, is the signed representation of acting on the DSTI, where for the two strong TIs labeled by .
The invariance of the mass under symmetry action implies that
This means that, for any signed representation of the symmetry group on the DSTI, we can define its “signature” as a map , which assigns a sign to each symmetry operation according to whether or not the mass changes sign under the action of (in the specified signed representation):
The only condition which should be satisfied by is that it is a group homomorphism i.e. for . The signature is not to be confused with the sign . While the former describes a representation on the DSTI specifying the transformation properties of the mass term (9), the latter describes the sign choice of the representation on a single strong TI. The two are related by Eq. (14) given below.
The properties of the mass term on the surface are fixed by specifying the signatures of the symmetry operations, which are in turn completely fixed by the parameters and via the relation
In the following, we will usually use the terminology “a representation for symmetry ” for symmetry representations on the DSTI to indicate that the symmetry has a signature .
Let us point out that, although our DSTI model with two flavors of Dirac fermions is not sufficient for implementing all possible sTCIs of interest, say those with a high mirror Chern number, it is sufficient for constructing the “root states” which generate such states upon stacking. To see this, consider a system consisting of copies of the DSTIs. In this case, there are independent -preserving mass terms , . The crystalline symmetries act on the vector as orthogonal transformations leaving the length of the vector (which gives the magnitude of the gap at a given point) fixed (see appendix B). Any crystalline symmetry apart from mirror leaves at most two points on any given surface invariant. As a result, it will only protect anomalous surface states if it enforces the existence a domain wall between a point to its image under symmetry, . Such a domain wall will be irremovable if and only if there is no trajectory connecting and on the -sphere, establishing a correspondence between the fundamental group of the -sphere and the stable domain walls in a model with DSTIs. Since the fundamental group of the -sphere is trivial for , we deduce that we cannot build stable domain walls whenever . For , and only one mass term is possible. In this case, the fact that the 0-sphere (just two points) has two disconnected components implies the possibility of having domain walls, thereby establishing a classification for sTCIs not protected by mirror symmetry in class AII. Physically, the classification here simply descends from that of 2D TIs in class AII.
The only exception to the previous analysis is mirror symmetry. In this case, we need to consider the mass vector in the mirror plane. We find that it has to remain invariant under the action of mirror symmetry at any point in this plane, , which is only possible if one of the eigenvalues of is . In a representation which does not satisfy this condition, e.g , the mass vector will necessarily vanish in the mirror plane regardless of . Nevertheless, the state with DSTIs can always be built by stacking the “root” state implemented using a single mirror-symmetric DSTI.
Notice that, up to this point, the analysis is general and applies to any crystalline symmetry. For instance, weak TIs can be understood within the DSTI model by considering a surface with periodic boundary along the “weak” direction, and choosing a “” representation for translation along this direction. This will enforce the existence of a domain wall for every unit lattice translation along this direction, leading to the surface states obtained by stacking 2D quantum-spin Hall systems (Fig. 1). Alternatively, for a system without any weak index, i.e., all the lattice translations are assigned a “” signature, some other elements in the SG could also be in a “” representation and lead to other patterns of gapless modes on suitably chosen surfaces.
Having outlined our general framework, we now apply it to classify all sTCIs in class AII. We will proceed in two steps: first, we will focus on phases protected solely by point-group symmetries; second, we will discuss how to extend these results consistently to cover all the 230 space groups.
ii.2 Crystallographic point groups
This subsection is devoted to the study of crystalline point groups, whose associated sTCIs can be understood by considering a spherical geometry with open boundary conditions in all directions. We will describe a procedure to construct all possible sTCIs in a given symmetry group, and then use it to obtain the complete classification of sTCIs protected by point group symmetries.
We begin by reviewing the natural action of the individual point-group symmetries on the physical degrees of freedom, and discussing the types of gapless surface states they can protect. Afterwards, we will provide a general procedure to construct all sTCIs in a given point group, and use it to obtain an exhaustive classification of sTCIs for the 32 crystallographic point groups.
Crystallographic point group symmetries
Inversion: Inversion symmetry acts by inverting position and momentum while keeping the spin unchanged
According to (14), inversion can be represented with negative signature by taking its action on the DSTI to be (corresponding to ), or with positive signature by acting on the DSTI as (corresponding to ).
In the “” representation, the mass term changes sign between a point and its image under inversion, leading to a “hinge” phase with a helical gapless mode living on some inversion-symmetric curve on the sphere. The curve can be moved around but cannot be removed without breaking inversion. Such a phase is graphically illustrated in the entry in Fig. 3. Two copies of this state can be trivialized by adding the mass terms , where denote the Pauli matrices in this additional orbital space of copies. These additional mass terms, like (9), change sign under inversion; however, can be chosen so that they do not both vanish at a point where the original mass term, in Eq. (9), also vanishes. This leads to a completely gapped surface. As a result, gapless modes protected by inversion will have a classification, as anticipated in the general discussion of the previous subsection.
rotation: An -fold rotation acts by rotating the position and momentum as vectors and rotating the spin as a spinor. An -fold rotation about the -axis is implemented as
A positive signature representation is obtained by taking (corresponding to ), whereas the choice (corresponding to ) leads to a negative signature
We note that the “” representation is only possible for even , since it violates the condition whenever is odd. Thus, is only consistent with a “” representation, whereas can have of either signature. A “” representation in this case forces the mass to vanish at (closed) curves related by rotation and intersecting at the two points left invariant by rotation (the poles) (as shown in Fig. 3 PGs , and, ). We remark that these conclusions have already been drawn in Ref. Fang and Fu, 2017 using a slightly different language.
Similar to the case of inversion, two copies of the described DSTI can be gapped out by adding the mass term , which does not change sign under rotation and can be chosen to be positive everywhere. This means that rotations lead to a classification as well, consistent with our general discussion.
Mirror symmetry: Mirror symmetry acts by inverting the position and momentum components perpendicular to the mirror plane and the spin component parallel to it. For example, the action of mirror symmetry about the plane is implemented as
In the mirror plane, flips the sign of . Thus, in the “” representation, implemented by taking the mirror to act in the DSTI as (corresponding to ), the mass term has to vanish in this plane. The “” representation, on the other hand, is implemented in the DSTI as (corresponding to ), which does not impose a constraint on the mass term in the mirror plane and leads to a completely gapped surface.
As we noted in the previous subsection, the “” representation for the mirror implies that the mirror plane remains gapless regardless of the number of DSTIs stacked together. This can be seen from the fact that, for any number of copies, any mass term will have the form , where denotes matrices in the orbital space of copies, which will always vanish in the mirror plane. As discussed previously, this implies a classification corresponding to the mirror Chern number in real space Hsieh et al. (2012).
rotoinversion: As , its action on a DSTI can be readily understood through the corresponding discussions for and above. Note that a “two-fold rotoinversion” is simply a mirror symmetry, which leaves a plane invariant in 3D, and is differentiated from , which only leave the origin invariant.
Classification of sTCIs in the 32 crystallographic point groups
We are now in a position to perform a systematic investigation of sTCIs in the 32 crystallographic point groups. As we argued in the beginning of this section, all these states can be built from either the DSTI model considered in Sec. II.1, or copies thereof.
In Sec. II.1, we proposed that there is a one-to-one correspondence between the signatures of the different symmetries and the pattern of gapless modes on the surface. One part of this correspondence is obvious since surface states corresponding to different symmetry signatures cannot be deformed into each other without changing these signatures. We conjecture that the opposite is also true: two patterns of surface modes can always be deformed into each other if they correspond to the same representation signatures for all the possible symmetries. We have checked this explicitly for several examples, where seemingly different surface state patterns corresponding to the same symmetry signatures turned out to be deformable into each other.
For example, consider the point group , where the only symmetry is a 4-fold rotoinversion. In this case, there seems to be two distinct surface state patterns corresponding to the “” representation for , given by the “equator” state and the “hinge” state (cf. Fig. 2). These two can, nevertheless, be deformed into each. The reason is that, unlike rotation, rotoinversion does not leave the poles fixed. Therefore, there is no symmetry constraint on the mass terms at the poles, implying the intersection of gapless modes at the poles is spurious, i.e., it is unstable against symmetry-allowed perturbations. Thus, we can move the hinges symmetrically away from the poles, brining them to the equator. This can be seen more clearly by adding the two phases and noting that a mass term can be added to gap-out the modes at their intersection points (this is possible since rotoinversion does not enforce any local constraints on the mass). The resulting surface can be deformed to a trivial one as shown in Fig. 2. Alternatively, such correspondence can also be understood in a slightly more general language (i.e., beyond the Dirac theory analysis), as we elaborate in Sec. IV.
Having shown that the classification of sTCIs reduces to classifying distinct signatures of the different symmetries, we now proceed by constructing these phases explicitly. We first note that two symmetries related by conjugation, i.e. there is an such that , necessarily have the same signature. In the following, we will call two symmetries independent if they are not related by conjugation, so that their signatures can be assigned independently. For two rotations (mirrors), independence means that the rotation axes (mirror planes) are not related by any symmetry operation in the symmetry group . This implies that we can classify all possible signatures by specifying the signatures for a minimal set of independent generators of the group (i.e., every group element can be written as a product of powers of the elements in the set, and the size of the set is as small as possible). Notice that the generators in a minimal set can always be chosen to be independent
The number of independent mirror symmetries included as generators is as large as possible.
The number of independent rotations included as generators is as large as possible .
Note that these two conditions are consistent with each other since we cannot generally increase (decrease) the number of independent mirror symmetries by decreasing (increasing) the number of symmetries in a generator set
Once we have a minimal generator set satisfying these properties, we can read off the classification of sTCIs as follows:
To every mirror symmetry in the generator set, we assign a factor of indicating the number of gapless modes in this mirror plane. The phase which generates this factor is obtained by choosing the “” representation for the corresponding mirror symmetry in the DSTI model.
To every symmetry generator other than mirror and , we assign a factor of . Each of these phases is generated by choosing the “” representation for the corresponding symmetry generator.
Implementing these rules leads to the classification of sTCIs in all crystallographic point groups, which we tabulate under in Table 2. The “hinge” states generating the different , factors by choosing the “” representation in the corresponding symmetry are graphically illustrated in Fig. 3.
ii.3 Space groups
In this subsection, we extend the analysis of the previous section to space groups, which requires considering symmetries that do not fix any point in space (lattice translations and non-symmorphic symmetries). We first start by discussing the main complication arising from the inclusion of non-symmorphic symmetries, which requires a certain choice of the sample geometry and boundary conditions. We then provide a few examples of sTCIs protected by non-symmorphic symmetries. Next, we discuss how to consistently combine the preceding results with lattice translations, i.e., we systematically study the spatial-symmetry constraints on weak indices Varjas et al. (2017). Finally, we will provide a complete classification of sTCIs in the 230 SGs.
A non-symmorphic symmetry arises when a point-group symmetry is combined with an irremovable, fractional translation, which results in a symmetry that leaves no point in space invariant. The extension of the analysis of Sec. II.2 to include non-symmorphic symmetries is, in principle, straightforward. As we will elaborate on later, to analyze sTCIs protected by non-symmorphic symmetries, it suffices to first assume that all the weak indices of the system are trivial, i.e., all the lattice translations are given a “” representation. Microscopically, this is the case when we can stick with DSTI models with degrees of freedom arising form the point in the BZ, such that, in momentum space, the fractional translation associated to a non-symmorphic symmetry becomes irrelevant. This restriction is justified in Sec. III and Appendix C.
As we can focus on the point in the momentum space, the main conceptual difference in understanding sTCIs protected by point-group and non-symmorphic symmetries, therefore, lies in the real space. Recall that, to expose the anomalous surface states of an sTCI, one has to consider a geometry respecting the protecting spatial symmetries. In order to preserve a non-symmorphic symmetry on the surface, we need to consider a sample with periodic boundary conditions along the directions of the fractional translation vector of the symmetry. A non-symmorphic symmetry, which can either be a screw (-fold rotation followed by a fractional lattice translation along the rotation axis) or a glide (mirror reflection followed by a fractional lattice translation along a vector in the mirror plane), does not leave any point invariant. As a result, the surface states protected by non-symmorphic symmetries will have a classification (see the discussion at the beginning of this section).
The anomalous surface states protected by non-symmorphic symmetries can be understood by considering a cylinder geometry whose axis is parallel to the fractional translation axis, along which periodic boundary condition is taken. An -fold screw in a “” representation would give rise to a state with symmetry-related hinges related, along the sides of the cylinder. This surface state is very similar to the surface state protected by rotation, in that the hinges can be moved around freely as long as they preserve the screw or rotation as a set.
Extending a mirror symmetry to its non-symmorphic counterpart, a glide, leads to a more drastic modification of the physics. Picking a “” representation for a glide symmetry gives rise to a state with two hinge modes confined to the glide plane. Despite looking similar to surface states protected by mirror symmetry, this state differs in an essential aspect, as it becomes trivial when added to itself. The reason is that, unlike mirror symmetry, a mirror Chern number cannot be defined for glide symmetry, since it does not act as an on-site symmetry in the glide plane. One can also understand such modification from a momentum-space perspective: unlike a mirror, the band eigenvalues of a glide are not invariantly defined, as they are interchanged upon the addition of a reciprocal lattice vector to the crystal momentum. Consequentially, one cannot define a -valued Chern number using a glide symmetry.
As an example of surface states protected by a non-symmorphic symmetry, let us consider SG (No. 77), where the only symmetry is a 4-fold screw . On a cylinder geometry with a periodic boundary condition taken along the screw axis, a “” representation for the screw symmetry leads to the state with four hinges shown in Fig. 4. We note that a state with domain walls at every half lattice translation along the screw axis is also consistent with the “” representation for the screw symmetry. This state is gapless everywhere on the surface, and looks very similar to the surface of a weak TI shown in Fig. 1. The main difference in this case is that such a state is unstable against being gapped out everywhere except for the four hinges as in Fig. 4.
Another example is SG (No. 176), which is one of the key space groups considered in Table 1. This space group is generated by a 6-fold screw and a mirror about a plane normal to the screw axis. Choosing a cylinder geometry with a periodic boundary condition along the screw axis, we may consider a “” representation for either the screw or the mirror. In the former case, we get a phase with six hinges, whereas the latter is characterized by gapless modes protected by a mirror Chern number in the mirror planes. Notice that, in this case, we have several mirror planes related by screws which all have the same mirror Chern number. The signatures of the mentioned mirror and screw are independent, and consequentially the resulting classification is .
While we have argued quite generally that the sTCI classification for non-symmorphic symmetries can be understood from the point-group counterpart, our discussion so far has one caveat: Unlike 3-fold rotations, a three-fold screw with a lattice translation along the rotation axis, , does admit a “” representation. However, such a choice implies , a unit lattice translation, is also assigned a “” representation, leading to a nontrivial weak TI index, and hence the described phase falls outside of our present discussion. This brings us to the last piece of consideration required for our classification of sTCIs: the incorporation of weak phases.
As we have briefly addressed at the end of Sec. II.1, in our framework a weak TI is characterized by the set of lattice translations taking a “” signature. More specifically, consider a weak TI characterized by the vector , where denotes a reciprocal lattice vector and is the associated weak index. The signature of a lattice translation, characterized by the vector , is then given by .
Generally, the presence of additional spatial symmetries leads to a restriction on the possible weak indices. For instance, a cubic system cannot realize a weak index which favors a particular axis, say but . As discussed in Ref. Varjas et al., 2017, such restrictions are encoded in the SG constraints on the admissible values of , and originate either from a point-group or non-symmorphic symmetry. As we discuss below, the same problem can be analyzed through a complementary, though equivalent, perspective by studying the symmetry restrictions on the signature assignments to the lattice vectors .
Let be a symmetry in the SG, and a lattice translation
The presence of non-symmorphic symmetries can further restrict the possible weak indices. To see this, note that, for any non-symmorphic symmetry which is not a 3-fold screw, is a lattice translation for some even integer . Therefore, regardless of the signature chosen for , the translation is always taking a “” signature, and hence any weak index demanding a “” signature for is forbidden.
While we have discussed the constraints on weak phases utilizing the group structure of the SG, it is instructive to connect it to the more microscopic DSTI model we described. To this end, observe that, if the surface Dirac cone originates from degrees of freedom around a TRIM , the signed representation for the lattice translation is given by . According to Eq. (14), a DSTI built from strong TIs with effective degrees of freedom coming from the TRIMs and would then realize a weak index of .
Classification of sTCIs in the 230 space groups
Having separately described how to extend our point-group results to sTCIs protected by a lattice translation or non-symmorphic symmetry, which does not fix any point in space, we now tackle the problem of classifying all sTCI phases built from stacking strong TIs in all the 230 space groups. From the discussion on point groups, we see that the desired classification can be reconciled with the different ways to assign the signature to a generating set of the space group, which we denote by .
We first focus on a subgroup classifying sTCIs which are “not weak,” i.e., those for which the signature for all lattice translations .
We now argue that the sTCIs described by an element of can be readily studied via . Recall that, given a SG , a corresponding point group is defined by “modding out” the translation part of , i.e. , where is the translation subgroup of (which is always a normal subgroup).
This procedure reduces screws/glides in to rotations/mirrors in ,
The classification of sTCIs in any given SG is the same as the classification of , except for the fact that every mirror in which derive from a glide in should be assigned a rather than factor. More concretely, we note that, for any consistent signature assignment on satisfying , any two symmetries in with the same point-group action will be given the same signature. This induces a consistent signature assignment on . Conversely, for any consistent assignment on one can define an assignment on by composing with the canonical projection . This demonstrates the stated one-to-one correspondence, up to the modification required for mirror vs. glide
Next, we incorporate weak phases into our discussion. Recall that the computation of amounts to the identification of a minimal generating set of the SG, paying a special attention to the presence of mirror and three-fold rotation symmetries (conditions outlined in Sec. II.2). Consider a weak index, generated by assigning a “” signature to a lattice vector , which satisfies all the previously outlined SG constraints. This implies the signature of is not fixed by the part of indices contained in , and therefore must must be incorporated into the generating set for the SG. In addition, if the SG possesses a mirror about a plane normal to , the mentioned weak TI is “promoted” to a weak mirror Chern insulator, i.e., the classification is modified from to . When this happens, we should append the shifted mirror, , instead of the translation, , to our generator set. Following this procedure, one incorporates all the needed lattice translations or shifted mirrors into the generator set. Each of such additional generator for the SG then corresponds to either a -valued weak TI index or a -valued weak mirror-Chern index, which we append to to arrive at the full classification . Implementing this rule leads to Table 4 in Appendix C, which provides the sTCI classification for all the 230 SGs.
Finally, we comment on the meaning of adding two phases in , which is an abelian group representing the equivalence classes of distinct sTCI phases. The subtlety arises if there is no geometry for which both phases exhibit anomalous surface states. This will never be the case for SGs containing only symmorphic elements, but is possible in the presence of non-symmorphic symmetries. However, we have to remember elements of are distinct bulk phases despite being physically defined by their surface signatures. This means that the possibility of anomalous states on any given surface is completely fixed by the bulk, as will be discussed in detail in Sec. III. Moreover, it is always possible to distinguish two distinct sTCI phases by considering them on many different geometries. For instance, given two phases 1 and 2 and two geometries and , such that 1 (2) exhibits surface states on () but not on (), we can distinguish the sum of the two phases from either phase by the fact that it exhibits surfaces states on both geometries and .
ii.4 Surface dispersion at special planes
Although the 1D surface modes shown in Fig. 3 can be, in principle, detected by means of transport measurements, the most experimentally accessible tool to investigate such surface states is provided by the angle-resolved photoemission spectroscopy (ARPES), which probes the energy dispersion at a given surface. The dispersion measured by ARPES can be readily understood from the surface state pattern given in real space in Fig. 3. In brief, we consider what happens when we attach a tangent plane to the sphere at a given point, i.e., when we consider a flat, macroscopic crystal facet with the same surface normal as the considered point on the sphere.
We note that the surface modes can generally be freely moved on the surface except when they pass through a rotation-invariant point or lie in a mirror/glide plane. As a result, a tangent plane at a point that is not rotation-invariant or lies in a mirror/glide plane will generically have gapped dispersion. Therefore, the analysis of surfaces where 2D gapless modes may exist boils down to considering the cases where the normal to the surface (i) lies in a single mirror/glide plane, (ii) lies at the intersection of multiple mirror/glide planes, or (iii) is parallel to a rotation axis. These three cases are illustrated in Fig. 5, and we elaborate on them below.
Single mirror/glide plane
Here, we investigate the dispersion in a tangent plane whose normal lies in a single mirror plane. The glide case can be considered very similarly. The dispersion in the plane can be written as
with reflection acting as . Reflection symmetry implies that any mass term has the form , which necessarily vanishes at since it satisfies . It is, therefore, not possible to gap-out the surface but it is possible to move the two Dirac cones apart by adding the -symmetric term , which shifts the Dirac cones to Hsieh et al. (2012).
Multiple mirror/glide planes
If the normal to the tangent plane lies in the intersection of multiple mirror planes or a glide and a mirror, all in the “” representation, we can repeat the same argument of the previous section for each mirror/glide plane separately. The result is that the Dirac cones will have to be confined to the line invariant under mirror/glide symmetry for each mirror/glide, leading to the pair of Dirac cones being pinned at the intersection point of the mirror/glide planes Wieder et al. (2017).