Wallpaper Fermions and the Topological Dirac Insulator
Recent developments in the relationship between bulk topology and surface crystalline symmetries have led to the discovery of materials whose gapless surface states are protected by crystal symmetries, such as mirror topological crystalline insulators and nonsymmorphic hourglass fermion insulators. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 “wallpaper groups.” Here, we show that all possible crystalline insulators, symmorphic and nonsymmorphic, can be exhaustively characterized by considering these wallpaper groups. In particular, the two wallpaper groups with multiple glide lines, and , allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this “topological Dirac insulator” provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the AB family of materials in space group 127, finding that SrPb hosts this new topological surface Dirac fermion. Furthermore, -strained AuY and HgSr host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in BaInSb in space group 55. For orthorhombic space groups with two perpendicular glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and provide topological invariants to distinguish them. Finally, we show how in a particular limit of these systems, the crystalline phases reduce to copies of the Su-Schrieffer-Heeger model.
Topological phases stabilized by crystal symmetries have already proven to be both a theoretically and an experimentally rich set of systems. The first class of these proposed materials, mirror topological crystalline insulators (TCIs), host surface fermions protected by the projection of a bulk mirror symmetry to a particular surfaceTeo et al. (2008); Fu (2011); Kim et al. (2015). They have been observed in SnTe Hsieh et al. (2012); Tanaka et al. (2012) and related compounds Dziawa et al. (2012); Xu et al. (2012). Recent efforts to expand this analysis to nonsymmorphic systems with surface glide mirrors – operations composed of a mirror and a half-lattice translation – have yielded additional exotic free fermion topological phases, which can exhibit the so-called surface gapless “hourglass fermions,” and the glide spin Hall effect Liu et al. (2014); Alexandradinata et al. (2016); Wang et al. (2016); Shiozaki et al. (2016); Chang et al. (2016). The theoretical proposal of Refs. Alexandradinata et al., 2016; Wang et al., 2016 has recently also seen incipient experimental support Ma et al. (2016).
In addition, topological insulators (TIs) – crystalline or otherwise – provide exceptions to fermion doubling theorems. These theorems impose deep bounds on phenomena in condensed matter physics. For example, in 2D, a single Kramers degeneracy in momentum space must always have a partner elsewhere in the Brillouin Zone (BZ), otherwise the Berry phase of a loop around it suffers from ambiguity Haldane (2004). The discovery of the topological insulator provided the first exception to this theorem: in these systems 2D Kramers pairs are allowed to be isolated on a single 2D surface because they are connected to a 3D topological insulating bulk and have partners on the opposite surface Fu and Kane (2007); Moore and Balents (2007).
Higher-fold-degenerate bulk fermions, such as Dirac points, which are stabilized by crystal symmetry, may come with their own fermion doubling theorems Young et al. (2012); Steinberg et al. (2014); Wieder et al. (2016); Bradlyn et al. (2016). As noted in Ref. Young and Kane, 2015, a single fourfold-degenerate Dirac fermion cannot be stabilized by 2D crystal symmetries as the only nodal feature at a given energy; it must always have a partner or accompanying hourglass Weyl points. This is because a single Dirac point in 2D represents the quantum critical point (QCP) separating a trivial insulator (NI) from a topological insulator. Shown in more detail in section A2 of the appendix (SM A2), stabilizing just one of these Dirac points with crystal symmetries would therefore force the broken-symmetry NI and TI phases to be related by just a unitary transformation, violating their topological inequivalence. In this manuscript, we report a new class of symmetry-protected topological materials which, like the topological insulator before it, circumvents this restriction by placing a single, stable Dirac point on the surface of a 3D material.
To realize this, the crucial requirement is that the surface preserves multiple nonsymmorphic crystal symmetries111We define a nonsymmorphic symmetry to be a generating element of the maximal fixed-point-free subgroup(s), or Bieberbach subgroup(s) Watanabe et al. (2015); Wieder and Kane (2016), of a nonsymmorphic wallpaper or space group , modulo full lattice translations (SM A2).. Until now, most attention has been paid to crystal systems with surfaces that preserve only a single glide mirror. However, two of the 17 two-dimensional surface symmetry groups, called wallpaper groups, host two intersecting glide lines Conway et al. (2008). As we show in SM B, the algebra of the two glides requires that bands appear with fourfold degeneracy at a single time-reversal-invariant momentum (TRIM) at the edge of the BZ.
In this work, we study the non-interacting topological phases allowed in bulk crystals with surfaces invariant under the symmetries of these two wallpaper groups, and . We show that, in addition to generalizations of the hourglass fermions introduced in Ref. Wang et al., 2016, they host a novel topological phase characterized by a single, symmetry-enforced fourfold Dirac surface fermion, i.e., twice the degeneracy of a traditional topological insulator surface state. This Dirac fermion is symmetry-pinned to the QCP between a TI and an NI, allowing for controllable topological phase transitions of the 2D surface under spin-independent glide-breaking strain.
We classify the allowed topological phases for orthorhombic crystals with two perpendicular glides that are preserved on a single surface by considering the possible connectivities of the non-abelian Wilson loop eigenvalues Fu and Kane (2006); Ryu et al. (2010); Soluyanov and Vanderbilt (2011); Yu et al. (2011); Taherinejad et al. (2014); Alexandradinata et al. (2014a, 2016). We demonstrate that these systems can host three classes of topological phases: an hourglass phase with broken symmetry, a previously uncharacterized “double-glide spin Hall” phase, and the novel topological Dirac insulating phase mentioned above. We present topological invariants to distinguish these phases and use these invariants to predict material realizations. Using density functional theory (DFT) to calculate bulk Wilson loop eigenvalues and surface Green’s functions, we report the existence of the topological Dirac insulating phase in SrPb in space group (SG) 127. We also report the discovery of a related topological hourglass fermion phase in the narrow-gap material BaInSb in SG 55. In SM F2, we show that two additional members of the SG 127 AB family of materials, AuY and HgSr, can also realize this topological hourglass phase under -strain, giving unique promise for strain-engineering topological phase transitions in these materials. Finally, we show in SM G how these crystalline phases reduce in a particular limit to copies of the Su-Schrieffer-Heeger Model (SSH).
Ii Wallpaper Groups and
The surface of a crystal is itself a lower-dimensional crystal, which preserves a subset of the bulk crystal symmetries, and all 2D surfaces are geometrically constrained to exist in one of the 17 wallpaper groups. The set of wallpaper symmetry operations is restricted to those which preserve the surface normal vector: rotations about that vector and in-plane mirror or glide lines. Crystals with these surfaces may host quantum spin Hall (QSH) phases and rotational variations of the mirror TCI phases, which exhibit twofold-degenerate free fermions appearing along high-symmetry surface lines Hsieh et al. (2012); Tanaka et al. (2012); Dziawa et al. (2012); Xu et al. (2012).
For the four wallpaper groups with nonsymmorphic glide lines (, , , and ), this picture is enriched. Even in 2D, glide symmetries require that groups of four or more bands intersect, an effect which frequently manifests itself in hourglass-like band structures Young and Kane (2015); Watanabe et al. (2015); Wieder and Kane (2016). For the wallpaper groups with only a single glide line, and , surface bands can be connected in a topologically-nontrivial pattern of interlocking hourglasses, resulting in twofold-degenerate surface fermions along some of the glide lines Wang et al. (2016). We designate all topological 2D surface nodes “wallpaper fermions,” and find that for the remaining two wallpaper groups with multiple glide lines, and , higher-degenerate surface fermions are uniquely allowed.
We consider a -normal surface with glides , i.e., a mirror reflection through the -axis followed by a translation of half a lattice vector in the and directions (Fig. 1). When spin-orbit coupling is present, . At the corner point, (), , and . When combined with time-reversal, , this symmetry algebra requires that all states at are fourfold-degenerate. Furthermore, wallpaper groups with two glides are the only surface groups that admit this algebra, and therefore the only surface groups that can host protected fourfold degeneracies. Examination of symmetry-allowed terms reveals that fourfold points in these wallpaper groups will be linearly-dispersing, rendering them true surface Dirac fermions. In SM B, we provide proofs relating this algebra to Dirac degeneracy and dispersion.
For bulk insulators, the glide-preserving bulk topological phase and, consequently, -normal surface states, can be determined without imposing a surface by classifying the allowed connectivities of the -projection Wilson loop holonomy matrix Fidkowski et al. (2011); Alexandradinata et al. (2014a, 2016), a bulk quantity defined by:
where indicates that the integral is path-ordered, is the matrix-valued Berry connection, and is a gauge transformation relating the wavefunctions in adjacent BZs. The rows and columns of correspond to filled bands, where is the cell-periodic part of the Bloch wavefunction at momentum with band index . The eigenvalues of are gauge invariant and of the form . As detailed in SM C,D, the Wilson bands inherit the symmetries of the -normal wallpaper group and, therefore, must also exhibit the required degeneracy multiplets of wallpaper groups and . In particular, both surface and Wilson bands are twofold-degenerate along () by the combination of time-reversal and () and meet linearly in fourfold degeneracies at .
By generalizing the invariant defined in Ref. Shiozaki et al., 2016 for the single-glide wallpaper groups Alexandradinata (); Alexandradinata et al. (), we define topological invariants for double-glide systems using the (001)-directed Wilson loop eigenvalues. For in a surface BZ in wallpaper group , the quantized invariant is defined in Ref. Shiozaki et al., 2016 by integrating the Wilson phases, , along the path :
where is the number of occupied bands, the superscript indicates the glide sector, and the absence of a superscript indicates the line where is not a symmetry and the sum is over all bands. In the presence of an additional glide, , one can obtain by the transformation in Eq. (2). Though Eq. (2) appears complicated, can be easily evaluated by considering the bands within each glide sector which cross an arbitrary horizontal line in the Wilson spectrum (SM E1). Wallpaper group also has symmetry, which requires and implies the existence of the mirrors, and . These mirrors yield mirror Chern numbers, , respectively, which satisfy (SM E4).
To enumerate the allowed topological phases shown in Fig. 2, we consider possible restrictions on . Though can individually take on values ; only pairs that satisfy are permitted in bulk insulators; this can be understood as follows: if is odd, the 2D surface consisting of the four partial planes and possesses an overall Chern number, which implies the existence of a gapless point Fang et al. (2016); Alexandradinata and Bernevig (2016), contradicting our original assumption that the system is insulating. We present a rigorous proof in SM E2, and show that the remaining collection of eight insulating phases is indexed by the group .
For , the system is a strong topological insulator (STI). These four “double-glide spin Hall” phases possess the usual twofold-degenerate Kramers pairs at and as well as a fourfold-degenerate Dirac point at . The four STI phases are topologically distinct, but will appear indistinguishable in glide-unpolarized ARPES experiments. However, if two double-glide spin Hall systems with differing are coupled together, the resulting surface modes will distinguish between Shiozaki et al. (2016) (SM E2).
When , the system is in a topological crystalline phase. For or , which is only permitted in a -broken surface , a variant of the hourglass insulating phase Wang et al. (2016) is present on the surface. For example, when , either time-reversed partners of twofold-degenerate free fermions live along or both twofold-degenerate fermions live along and a fourfold-degenerate Dirac fermion exists at .
Finally, for , we find that the system exists in a previously uncharacterized “topological Dirac insulating” phase, capable of hosting just a single fourfold-degenerate Dirac surface fermion at .
Iii Materials Realizations
We apply DFT to predict the presence of topological phases stabilized by wallpaper groups and in known materials. The details of these calculations are provided in SM F. We find double-glide topological phases on the -surface (wallpaper group ) of three previously synthesized members of the SG 127 () AB family of materials: SrPb Merlo (1984); Bruzzone et al. (1981), AuY Chai and Corbett (2011), and HgSr Druska et al. (1996); Gumiński (2005). Shown in Fig. 3, we find that SrPb has a consistent, direct gap at the Fermi energy, in spite of the presence of electron and hole pockets. A Wilson loop calculation of the bands up to this gap (Fig. 3(d)) indicates that this material possesses the bulk topology of a topological Dirac insulator (SM F2). Calculating the surface spectrum through surface Green’s functions (Fig. 4), we find that the -surface of SrPb, while displaying an overall metallic character, develops gap of 45 meV at the Fermi energy at . Inside this gap, we find that there is a single, well-isolated, fourfold-degenerate surface Dirac fermion.
Unlike SrPb, AuY and HgSr in SG 127 are gapless, with bulk -protected Dirac nodes Wang et al. (2013) present near the Fermi energy. In SM F2 we show that under weak -strain, these Dirac nodes can be gapped to induce the topological hourglass phase in these two materials.
We additionally find that the -surface (wallpaper group ) of the narrow-gap insulator BaInSb in SG 55 () Cordier and Steher (1988) hosts a double-glide topological hourglass fermion. Shown in Fig. 5, we find that BaInSb develops an indirect band gap of 5 meV (direct band gap: 17 meV). The Wilson loop spectrum obtained from the occupied bands, shown in (Fig. 5(d)), demonstrates that this material is a double-glide topological hourglass insulator (SM F1). We find that the -surface of BaInSb has a projected insulating bulk gap which is spanned along by two bands of a topological surface hourglass fermion (Fig. 6).
We have demonstrated the existence of a topological Dirac insulator – a topological crystalline material with a single fourfold-degenerate surface Dirac point stabilized by two perpendicular glides. This is one of eight topologically distinct phases that can exist in insulating orthorhombic crystals with surfaces that preserve two perpendicular glides; we have classified all eight phases by topological indices that characterize the connectivity of the -projection Wilson loop spectrum. After an exhaustive study of the 17 wallpaper groups, these phases are revealed to be the final theoretically undiscovered 3D time-reversal-symmetric bulk topological insulating phases with linear surface fermions222In hexagonal crystals with very specific orbital character, it may be possible to realize a topological insulating phase with twofold-degenerate quadratically-dispersing surface fermions. However, the surface band connectivities in these materials would be topologically equivalent to those in previous works.. We report the discovery of the topological Dirac insulating phase in SrPb and of related double-glide topological hourglass phases in BaInSb, as well as in -strained AuY and HgSr.
We also find that there exists a simple intuition for the topological crystalline phases . In SM G1 we present an eight-band tight-binding model which, when half-filled, can be tuned to realize all double-glide insulating phases. In a particular regime of phase space, where the spin-orbit coupling terms at the and points are absent and bulk inversion symmetry is imposed, the Wilson loop eigenvalues at the edge TRIMs are pinned to () and each TRIM represents the end of a doubly-degenerate SSH model. In this limit, when the product of parity eigenvalues at satisfies , the bulk topology is fully characterized by the relative SSH polarizations, .
Finally, as the topological surface Dirac point is symmetry-pinned to the QCP between a 2D TI and an NI, we examine its potential for hosting strain-engineered topological physics. Consider the two-site surface unit cell in wallpaper group from Fig. 1. In the topological Dirac insulating phase, the surface Dirac fermion can be captured by the Hamiltonian near :
where is a sublattice degree of freedom, is a -odd orbital degree of freedom, and (). There exists a single, -even mass term, , which satisfies and is therefore guaranteed to fully gap . Therefore, surface regions with differing signs of will be in topologically distinct gapped phases and must be separated by 1D topological QSH surface domain walls, protected only by time-reversal symmetry Jackiw and Rebbi (1976). As the little group of has point group , and , can be considered an distortion Tinkham (1964), which could be achieved by strain in the direction and compression in the direction. These domain walls would appear qualitatively similar to those proposed in bilayer graphene Zhang et al. (2011); Ohta et al. (2006); Oostinga et al. (2007). However, whereas those domain walls are protected by sublattice symmetry and are therefore quite sensitive to disorder, domain walls originating from topological Dirac insulators are protected by only time-reversal symmetry, and therefore should be robust against surface disorder. Furthermore, under the right interacting conditions or chemical modifications, a topological Dirac insulator surface might reconstruct and self-induce regions of randomly distributed , separated by a network of 1D QSH domain walls.
Acknowledgements.We thank Aris Alexandradinata for a discussion about the invariant in Eq. (2), and Eugene Mele for fruitful discussions. BJW and CLK acknowledge support through a Simons Investigator grant from the Simons Foundation to Charles L. Kane and through Nordita under ERC DM 321031. ZW and BAB acknowledge support from the Department of Energy Grant No. DE-SC0016239, the National Science Foundation EAGER Grant No. NOA-AWD1004957, Simons Investigator Grants No. ONR-N00014-14-1-0330, No. ARO MURI W911NF-12-1-0461, and No. NSF-MRSEC DMR- 1420541, the Packard Foundation, the Schmidt Fund for Innovative Research, and the National Natural Science Foundation of China (No. 11504117). YK and AMR thank the National Science Foundation CEMRI Program for support under DMR-1120901 and acknowledge the HPCMO of the U.S. DOD and the NERSC of the U.S. DOE for computational support. BAB wishes to thank Ecole Normale Supérieure, UPMC Paris, and Donostia International Physics Center for their generous sabbatical hosting during some of the stages of this work.
Appendix A Fermion Doubling in 2D Crystals
In this section, we discuss how fermion doubling theorems in 2D constrain band structures and how apparent exceptions to them manifest on the surfaces of bulk 3D topological phases. To be precise, we will use the phrase “Dirac fermion” only to refer to fermions with point-like fourfold degeneracies and linear dispersion away from the degeneracy point, in either two or three dimensions. In this nomenclature, the surface states of a topological insulator are therefore not “Dirac points,” but are twofold-degenerate linearly dispersing fermions by another name (in a topological insulator they are linearly dispersing Kramers pairs). By making this choice of nomenclature, we preserve the designation of the linear fermions in graphene as Dirac points, which are fourfold-degenerate if we include spin, and consider spin-orbit coupling (SOC) to be negligible. Furthermore, as we will show in this appendix, Dirac points are crystalline-symmetry-pinned to the quantum critical point between a topological and a trivial insulator.
a.1 2D Fermion Doubling for Twofold-Degenerate Linear Fermions
In 2D, a system may host linearly-dispersing twofold-degenerate fermions. Such gapless fermions may exist as fine-tuned points or, if additional symmetries are present to protect them, may exist in pairs in a stable phase (i. e. spinless graphene). Here, we are interested in stable phases. Thus, we restrict ourselves to only discussing systems for which all possible symmetry-allowed hopping terms have been included. For example, the critical point separating two topologically distinct insulating phases in a two-band model features a single twofold-degenerate gapless fermion. However, without imposing additional symmetries, this fermion can be gapped, and, generically, the system will be insulating at half-filling.
The simplest example of a symmetry protecting a twofold linear fermion occurs in a crystal with time-reversal symmetry, , satisfying . This requires states to be twofold-degenerate at the Time-Reversal-Invariant-Momenta (TRIMs) by Kramers’ theorem. Twofold-degenerate fermions can also appear pairwise along high-symmetry lines and, if the system is a layer group, along planes, with symmetry stabilization coming from a combination of crystalline and time-reversal symmetries.
In 2D systems, these symmetry-protected gapless points come in pairs, a consequence of the parity anomaly Alvarez-Gaumé and Witten (1984); Redlich (1984); Jackiw (1984). We illustrate this result in the case of a single twofold-degenerate linear fermion at a TRIM protected by time-reversal symmetry, , described by the following 2D theory:
There is a single remaining Pauli matrix, , which anticommutes with all of the terms in Eq. (4) and opens a gap. The mass term , which could originate from an external magnetic field or mean-field magnetic order, breaks symmetry and gaps locally to a theory of a Chern insulator, with two bands of winding number and , respectively, for some . If Eq. (4) is a complete description of the low-energy physics, a contradiction arises: because , the two gapped phases that result from choosing opposite signs of are related to each other by transformation under . In particular, the band with Chern number when is related to the band with Chern number when . Since is odd under time-reversal, . This condition cannot be satisfied by .
Thus, the hypothesis that the stable gapless fermion (Eq. (4)) is a complete description of the low-energy physics cannot be true. For a system with a single twofold fermion, even at , time-reversal symmetry must be anomalously broken by terms beyond the level, and the system forms an anomalous Hall state with (here need not be integer since the system is gapless). In order for time-reversal to remain unbroken, there must be a compensating second degeneracy point somewhere else in the Brillouin zone, also with . The only details of time-reversal symmetry that entered into the preceding argument are that , and so the general result remains true for any symmetry that protects a gapless fermion in two dimensions333This is due to the fact that a Pauli-Villars regulator is a function of , and so this regularization must break the symmetry, and so we derive the same anomaly-generating functional as in the literature. (since any mass term must anticommute with the Hamiltonian in order to open a gap).
We can gain some further intuition about the parity anomaly by noting that the anomalous hall conductance is related to the Berry phase at the Fermi surface by Haldane (2004):
where is the path-ordering operator. Let us take a compact Brillouin zone with gapless fermions, and let the Fermi level be above all bands. By evaluating the Berry phase we find that . However, with time-reversal symmetry, we also have that is the total Chern number of all bands, and hence . Thus we conclude . In the context of three-dimensional topological insulators, we note that each surface taken in isolation is a 2D system with a single gapless fermion. From the above discussion we thus recover the well-known result that each surface of a topological insulator has a half-quantized anomalous Hall conductivity, and that only when both surfaces are connected by a bulk can the system be described in an anomaly-free way Fu and Kane (2007); Qi et al. (2008); Mulligan and Burnell (2013).
The same logic can be applied to a 3D material by replacing the mass term by ; in this case the gapless point would be a Weyl point of Chern number and the two gapped Hamiltonians of opposite-signed mass lie in the planes above and below it. This expresses the so-called “descent relation” between the parity anomaly in two dimensions and the chiral anomaly in three dimensions Alvarez-Gaumé’ et al. (1985). If is periodic, this would imply that, absent another Weyl point, two systems with different Chern number (above and below the Weyl point) could be adiabatically connected through the BZ boundary, which is impossible. To avoid this contradiction, the doubling theorem then requires that the low-energy physics cannot be described by only a single Weyl point: there must be another Weyl point or other band crossing at the Fermi level. In 3D, this is the celebrated Nielsen-Ninomiya theorem Nielsen and Ninomiya (1981).
a.2 2D Fermion Doubling for Dirac Fermions
We now extend these arguments to show why fourfold-degenerate 2D Dirac fermions cannot be stabilized as the only nodal features in a metallic phase at a given energy. As in the twofold-degenerate case, while many models might display Dirac fermions upon fine-tuning, here we are interested in robust Dirac fermion phases. Thus, we only consider systems that display Dirac fermions when all symmetry-allowed hopping terms are present. The crux of the arguments in this section was originally highlighted in Refs. Fu et al., 2007; Young and Kane, 2015.
When Dirac points occur off of the TRIMs, time-reversal requires that they come in pairs. For these -paired Dirac fermions the preceeding discussion can easily be generalized, and follows by relating the gapping of the 2D Dirac point to the index, in much the same way as we did above for Chern numbers and Weyl points Yang and Nagaosa (2014). Just as the Chern number jumps by one as a plane in 3D passes a Weyl point, the index flips when a plane passes a Dirac fermion at a low-symmetry -point Wang et al. (2012, 2013).
In other cases, the Dirac points which occur in 2D are filling-enforced and pinned to the TRIMs by crystalline symmetries and time-reversal. As shown in Ref. Wieder and Kane, 2016, these filling-enforced, high-symmetry Dirac points can only occur in 2D systems where either inversion anticommutes with a twofold nonsymmorphic symmetry or where two perpendicular nonsymmorphic symmetries anticommute, such as the two glides in .
We briefly note here that the expression “nonsymmorphic symmetry,” is a slight abuse of terminology, though one rampant in this field. For precision, we define a nonsymmorphic symmetry to be a generating element of the maximal fixed-point-free subgroup(s), or Bieberbach subgroup(s) Watanabe et al. (2015); Wieder and Kane (2016), of a nonsymmorphic wallpaper or space group , modulo full lattice translations . Consider, for example, wallpaper group , generated by and . There are two maximal fixed-point-free subgroups of : one generated by and and one generated by and , where is a full lattice translation. Both of these groups are isomorphic to the Bieberbach wallpaper group Wieder and Kane (2016), and when the generating elements are divided by the group of full lattice translations respectively give and . For a more complicated example, consider space group 14 , generated by , , and spatial inversion about the origin Bradley and Cracknell (1972). This group also has two maximal fixed-point-free subgroups, which when divided by full lattice translations respectively give (isomorphic to Bieberbach space group 4 modulo ) and (isomorphic to Bieberbach space group 7 modulo ).
Consider first a 2D spinful system with inversion symmetry and a screw, . At , and . In a time-reversal-invariant system, these symmetries require a four-dimensional representation: more generally, for any Hamiltonian invariant under two symmetries, and , in addition to , such that and , any eigenstate of the Hamiltonian, , which is also an eigenstate of , is part of a fourfold-degenerate quartet of orthogonal states, . In our example, there is a fourfold degeneracy at each TRIM with . In general, the combination of and a twofold nonsymmorphic symmetry will always mandate that there are no fewer than two Dirac points for a given filling: since anti-commutes with the nonsymmorphic symmetry at two of the TRIMs in 2D and , there are always at least two TRIM points with fourfold degeneracies.
In the case where there are two perpendicular nonsymmorphic symmetries, but no inversion symmetry, the obstruction to forming an isolated, stable Dirac fermion takes a slightly different form. Consider the trivial phase of wallpaper group , for example, which is characterized by and two glides, . At , and , and therefore a Dirac point exists. However, this condition is only met at this corner TRIM, and no other fourfold degeneracies are allowed elsewhere in this system. In this case, the Dirac point is obstructed from being alone by the filling-enforced hourglass structures also required to exist by the presence of singly degenerate eigenstates of . In these systems, the Dirac point occurs at the same filling as four 2D twofold-degenerate linear fermions, and is also prevented from being alone and stable at any filling.
We pose an explanation for this obstruction in similar terms to the resolution of the parity anomaly of the previous section. Suppose a combination of symmetries were to allow a single stable Dirac point. The model Hamiltonian at a corner TRIM in our previous system with and is described by a linear model:
where , , and . The four Dirac matrices present in span the space of symmetry-allowed matrices. Thus, the system supports a robust fourfold-degenerate gapless fermion stabilized by crystal-symmetries.
As before, we can examine the consequences of locally breaking one of the symmetries. To guarantee that the Hamiltonian is gapped everywhere, we seek a mass term to anticommute with all of the terms in . Generically, the Clifford algebra of Dirac matrices is spanned by four -odd matrices which couple to crystal momenta, and one -even matrix, here . For either sign of , the resulting phase is 2D, gapped, and -symmetric with ; a quantum spin Hall (QSH) index can thus be defined. Noting that at , 444A generic feature of Hamiltonians restricted to TRIM points, c. f. Ref.Bernevig and Hughes, 2013., we see by the Fu-Kane formula Fu and Kane (2007) for the QSH index that occupied bands for and have opposite parity indices.
To show the need for fermion doubling, we expand to the full BZ of a hypothetical system where this fermion is the only feature at the Fermi energy and show that there is a contradiction. We label the Bloch wavefunctions of the phase when by and those of the phase with by , where . The integral of the pfaffian of the matrix gives the QSH topological invariant Kane and Mele (2005). Consider relating this matrix for one gapped phase to the other by the operation of the broken symmetry, : because . Therefore, having the same matrix, the two phases have the same QSH invariant, contradicting the earlier Fu-Kane requirement that the two insulating phases are topologically distinct.
The resolution of this is a fermion doubling requirement for 2D fourfold-degenerate Dirac points. Specifically, a closed 2D crystal cannot host a single symmetry-stabilized Dirac point at the Fermi level. In the cases where the Fermi surface is gapped except for exactly two Dirac points, each Dirac point can have a single -symmetric mass term such that the overall QSH invariant, , satisfies . Under the action of the broken symmetry operation, the signs of both are flipped and is preserved Young and Kane (2015).
Like the topological insulator before it, the topological Dirac insulator that we present in this manuscript “cheats” this fermion doubling by placing each of its two Dirac points on opposite surfaces of a 3D bulk. While they each live alone on a surface, and therefore pin that surface to a 2D QSH transition, the Dirac points (or a Dirac point and four Weyl points) in the combined system of two opposing surfaces respect the fermion doubling requirement.
Appendix B Symmetries and Degeneracies of Wallpaper Groups and
The wallpaper, or plane, groups describe the 17 possible configurations of symmetries on the two-dimensional surface of a three-dimensional, time-reversal-symmetric crystal Conway et al. (2008). Of these groups, only two have multiple nonsymmorphic symmetries Bradley and Cracknell (1972) (as defined previously in A.2). Specifically, wallpaper groups and contain glide lines in the and (surface in-plane) directions; has an additional symmetry that is not present in . The group is generated entirely by the two glides:
while wallpaper group is generated by
The product of the two generators for yields two additional mirror symmetries, and . Though containing translations, these symmetries are not glides, as leaves the line invariant. In Fig. 1, we show the locations of the glides, mirrors, and rotation centers for both wallpaper groups realized with a two-site unit cell.
In crystal momentum space, the symmetry generators enforce required band groupings along lines and at points. In Figure 7, we show one quarter of the surface BZ and identify relevant lines and points with the letters (a), (b), and (c). Lines sharing the same letter obey the same symmetry restrictions, though for wallpaper group , which lacks a symmetry, they may individually display different features.
The lines designated (a) in Fig. 7 are and and host singly degenerate bands which are eigenstates of and , respectively. As detailed extensively in Refs. Young and Kane, 2015; Wieder and Kane, 2016; Wang et al., 2016; Alexandradinata et al., 2016, singly degenerate bands along lines invariant under a glide form hourglass or Quantum Spin Hall flows.
Bands along lines designated (b), and , are also eigenstates of and , respectively. However, unlike the bands along (a), they are doubly degenerate, because the symmetry operation () leaves the line invariant and squares to .
We now show that all potential band crossings along the line are avoided by utilizing the fact that bands along this line are also eigenstates of . If , then, using the commutation relation ,
Thus, the Kramers partners and have opposite eigenvalues. We note that eigenstates of along must behave in the same manner by exchange symmetry. Since all sets of Kramers pairs along line (b) have the same pair of eigenvalues, they belong to the same irreducible representation and so crossings between these bands will generically be avoided.
Finally, at the point , labeled (c), bands are fourfold-degenerate, a unique property of wallpaper groups and . This can be easily seen because the point is invariant under , and . Thus, if is an eigenstate of , , and form a degenerate quartet of states; clearly the Kramers pairs are orthogonal and since the first two states have the opposite eigenvalue as the last two states, they are also orthogonal.
This algebra can only be satisfied in the strong spin-orbit coupled wallpaper groups by two perpendicular glides. Furthermore, examination of two-dimensional filling constraints in Ref. Wieder and Kane, 2016 confirms that no other algebra in 2D can enforce fourfold point-like band degeneracies. Therefore a fourfold point degeneracy can only be hosted on the surfaces of three-dimensional objects which satisfy the symmetries of or .
We now determine the low-energy band dispersion near a fourfold crossing at (c). Returning to the two-site unit cell depicted in Fig. 1, we construct a theory around . Letting represent the sublattice degree of freedom and the local spin degree of freedom, we choose the representation and . Then, to linear order in , the most general allowed Hamiltonian is given by,
This is the equation of a linearly-dispersing fourfold-degenerate 2D fermion. Therefore bands at generically form in Dirac points. It is worth noting that unlike the 3D Dirac fermions characterized in Refs. Wang et al., 2012, 2013, which are optional degeneracies created by band inversion, all band multiplets at in and must be at least fourfold-degenerate, and therefore the Dirac fermions in these wallpaper groups are instead more closely related to the filling-enforced 2D Dirac points proposed in Refs. Young and Kane, 2015; Wieder and Kane, 2016; Young and Wieder, 2016. Like the 2D inversion-broken magnetic Dirac points in Ref. Young and Wieder, 2016, these surface Dirac points also have generically non-degenerate cones; the cones here are only degenerate along the glide lines and . In our model, this behavior emerges upon the introduction of the quadratic term .
Inducing a -symmetric mass term breaks each surface into a 2D trivial or topological insulator. On a single surface, as discussed in more detail in A.2, domain walls between regions with different signs of will host 1D QSH edge modes, which are topologically protected by time-reversal symmetry alone. These domain walls have been shown to host Luttinger liquid physics, and intersections between them can form effective quantum point contacts with switching behavior characterized by universal scaling functions and critical exponents Teo and Kane (2009). Additionally, these surface domain walls would provide an improvement over recent efforts to test this physics in qualitatively similar domain walls in gapped bilayer graphene Castro Neto et al. (2009); McCann and Fal’ko (2006); Zhang et al. (2011); Ohta et al. (2006); Oostinga et al. (2007), which have also been proposed as Luttinger liquids Killi et al. (2010); Wieder et al. (2015). Notably, as the domain walls in bilayer graphene are protected by valley index, they are quite sensitive to disorder, and thus far Luttinger liquid physics in bilayer graphene has not been observed Ju et al. (2015); Li et al. (2016). Domain wall modes on gapped topological Dirac insulator surfaces, conversely, should remain robust against nonmagnetic disorder.
Appendix C Tight-Binding Notation
Here we provide the tight-binding notation that will be used in subsequent sections, following Ref. Alexandradinata et al., 2016. In the unit cell labeled by the Bravais lattice vector, , the wavefunction corresponding to an orbital labeled by at position is denoted by . The Fourier transformed operators are given by,
The single-particle Hamiltonian, , defines a tight-binding Hamiltonian,
whose eigenstates are denoted .
The Fourier transform in Eq. (11) shows that the Hamiltonian is not necessarily invariant under a shift of a reciprocal lattice vector, . Instead,
where . Thus, we can choose the basis of eigenstates so that,
If the lattice is invariant under a spatial symmetry, , that acts in real space by taking , where is the matrix that enforces a point group operation and is a (perhaps fractional) lattice translation, then acts on states by:
where is a Bravais lattice vector and is a unitary matrix. It is convenient to define the Fourier transformed operator, , which can have explicit dependence when it acts on Bloch states, by:
Thus, separates into a product of a -dependent phase and a matrix, , that rotates the orbitals:
If is invariant under , then Eq. (16) shows that:
We can follow the same procedure for an antiunitary operator, , to find:
where is the complex conjugation operator. Similarly to Eq. (18),
c.2 Projector onto Occupied States
We define the projector onto the occupied states:
which satisfies, using Eq. (14),
Given a spatial symmetry, , Eq. (18) shows that has the same energy as and is a state at momentum . Hence, the projector onto the occupied states at momentum is given by:
For an antiunitary symmetry, , the analogous equations are:
Appendix D Wilson Loops
A precise way to distinguish the distinct surface connectivities is obtained through the eigenvalues of Wilson loops, which, as we will elaborate, also gives information about the edge state spectrum Fu and Kane (2006); Ryu et al. (2010); Soluyanov and Vanderbilt (2011); Yu et al. (2011); Taherinejad et al. (2014); Alexandradinata et al. (2014a, 2016). Here, we are interested in the Wilson loop matrix:
where indicates that the integral is path-ordered, and is a matrix whose rows and columns correspond to filled bands. The eigenvalues of are gauge invariant and of the form , i.e., they are independent of the ‘base point,’ .
In this appendix, we show that a space group with two glides, , as well as time-reversal symmetry, , has the following constraints on its -directed Wilson loop eigenvalues:
Along the lines and , the Wilson loop eigenvalues are doubly degenerate, due to the antiunitary symmetries and , respectively (see D.2).
At the point, the doubly degenerate bands meet at a fourfold band crossing (see D.3).
Along the and lines, bands can be labeled by the eigenvalue (see D.3).
The gauge invariance of the loops allows us to write down a topological invariant that distinguishes the states, as derived in E.1.
d.1 Discretized Wilson Loop
For completeness, following Ref. Alexandradinata et al., 2016, we derive a discretized version of the Wilson loop (Eq. (28)), which is useful for clarity when deriving symmetry constraints. Using the projector onto occupied states, , defined in Eq. (21), we derive the discretized Wilson loop matrix,
where and in the last line we have defined the ordered product of projectors,
Eq. (29) shows that the discretized Wilson loop can be written in the basis-invariant form,