Nonsymmorphic symmetry-required band crossings in topological semimetals
We show that for two-band systems nonsymmorphic symmetries may enforce the existence of band crossings in the bulk, which realize Fermi surfaces of reduced dimensionality. We find that these unavoidable crossings originate from the momentum dependence of the nonsymmorphic symmetry, which puts strong restrictions on the global structure of the band configurations. Three different types of nonsymmorphic symmetries are considered: (i) a unitary nonsymmorphic symmetry, (ii) a nonsymmorphic magnetic symmetry, and (iii) a nonsymmorphic symmetry combined with inversion. For nonsymmorphic symmetries of the latter two types, the band crossings are located at high-symmetry points of the Brillouin zone, with their exact positions being determined by the algebra of the symmetry operators. To characterize these band degeneracies we introduce a global topological charge and show that it is of type, which is in contrast to the local topological charge of Fermi points in, say, Weyl semimetals. To illustrate these concepts, we discuss the -flux state as well as the SSH model at its critical point and show that these two models fit nicely into our general framework of nonsymmorphic two-band systems.
pacs:03.65.Vf, 71.20.-b, 73.20.-r,71.90.+q
Since the experimental discovery of topological insulators Konig et al. (2007); Hsieh et al. (2008), symmetry protected topological phases have become a major research subject Hasan and Kane (2010); Qi and Zhang (2011); Hasan and Moore (2011); Schnyder and Brydon (2015); Senthil (2015); Chiu et al. (2015). Recent studies have been concerned with topological phases that are protected by spatial symmetries, such as topological crystalline insulators Ando and Fu (2014); Chiu et al. (2013); Shiozaki and Sato (2014); Morimoto and Furusaki (2013) and topological semimetals stabilized by reflection, inversion, or other crystal symmetries Chiu and Schnyder (2014); Chan et al. (2016); Zhao et al. (2016). Until recently, the study of these topological crystalline materials has focused on the role of point group symmetries. However, besides point group symmetries the space group of a crystal can also contain nonsymmorphic symmetries, which are combinations of point group operations with nonprimitive lattice translations. It has been shown that the presence of nonsymmorphic symmetries leads to new topological phases, which can be insulating Shiozaki et al. (2015); Fang and Fu (2015); Liu et al. (2014); Dong and Liu (2016); Lu et al. (2016); Shiozaki et al. (2016), or semimetallic with Dirac points protected by nonsymmorphic symmetries Young and Kane (2015); Bzdušek et al. (2016). In the latter case, the Dirac points possess local topological charges, which guarantees their local stability.
However, as we show in this paper, nonsymmorphic symmetries restrict the form of the band structure not only locally but also globally, which may lead to unavoidable band crossings in the bulk Michel and Zak (2001); König and Mermin (1997); Watanabe et al. (2016); Parameswaran (2015); Parameswaran et al. (2013). Indeed, the nonsymmorphic symmetries can put so strong constraints on the global properties of the band structure that the system is required by symmetry to be in a topological semimetal phase, with Fermi surfaces of reduced dimensionality. These symmetry-enforced semimetals possess low-energy excitations with unconventional dispersions and may exhibit novel topological response phenomena and unusual magneto-transport properties. In the following we consider three different types of nonsymmorphic symmetries: (i) Unitary nonsymmorphic symmetries, (ii) nonsymmorphic symmetries combined with inversion, and (iii) nonsymmorphic magnetic symmetries. We first rigorously prove that for any one-dimensional (1D) two-band system unitary nonsymmorphic symmetries enforce the existence of band crossings, due to global topological constraints on the band structure. In the presence of an additional inversion symmetry, the symmetry enforced band degeneracies are located either at the origin or at the boundary of the Brillouin zone (BZ), depending on the algebra of the symmetry operators. The same holds true for nonsymmorphic magnetic symmetries, which are composed of a unitary nonsymmorphic symmetry followed by an anti-unitary time-reversal symmetry. We present generalizations of these results to higher dimensions, for which nonsymmorphic symmetries may enforce the existence of zero- or higher-dimensional band crossings. In all of the above cases we find that the nonsymmorphic symmetries restrict the momentum space structure in the BZ both locally and globally. To characterize the global topological features we introduce a novel global topological charge, which as we show, is always of type. Hence, the global topological features exhibit a classification, which is in contrast to the local topological characteristics, which possess a classification. Finally, we illustrate these findings by considering two prototypical examples: (i) the -flux square lattice model and (ii) the SSH model at its critical point. Within our unified framework, we show that the former model can be viewed as the higher-dimensional generalization of the latter.
Ii Unitary nonsymmorphic symmetry
We start by considering a general 1D two-band Hamiltonian with the two-fold unitary nonsymmorphic symmetry
which acts on as
Since , the eigenvalues of are . Therefore, the nonsymmorphic symmetry can be viewed as an operation on internal degrees of freedom (e.g., pseudospin) followed by a half translation, as illustrated in Fig. 1(a). Observing that anti-commutes with , the Hamiltonian can be written as
Without loss of generality, we have dropped the term proportional to the identity, which only shifts the energy of eigenstates. Inserting Eqs. (3) and (1) into Eq. (1), we find that due to the non-symmorphic symmetry , must satisfy
We claim that any periodic function satisfying Eq. (4) has zeros, and thus any two-band model with the nonsymmorphic symmetry (1) is required to be gapless. To see this, we introduce with , from which it follows that . If or is nonzero on the unit circle , then
which however is impossible. This is because for , the two sides of Eq. (5) both define functions from to , but the left-hand side has winding number , while the winding of the right-hand side is even, since . Thus, must vanish at some momentum by contradiction. For the topological argument to work for multi-band theories, we may replace in Eq.(3) by the determinant of the off-diagonal entry, which is discussed in Sec.VII.
Iii Nonsymmorphic symmetry combined with inversion symmetry
We note that while a unitary nonsymmorphic symmetry guarantees the existence of a band crossing point, it does not fix the position of this degeneracy point in momentum space. However, in the presence of an additional inversion symmetry, the band crossings are pinned to either the origin or the boundary of the BZ. To demonstrate this, let us consider the inversion symmetry , where inverses the momentum. We find that
Since is a periodic function, we expand it as . It follows from Eq. (4) that which, as a recursion relation, allows us to express as
From Eq. (6) it follows that which implies or equivalently . Since are all purely imaginary, we find that
with being real numbers. We observe that independent of there always exists a band crossing point at . For example, by keeping only the zeroth term in Eq. (8), one finds as a simple concrete model,
We note that the nonsymmorphic symmetry relates seemingly independent terms to each other in the Hamiltonian. This is exemplified by Eq. (9), where all three terms (which are usually independent) have the same coefficients. Obviously, higher order terms in Eq. (8), which constitute symmetry-preserving perturbations, cannot split the band crossing point of at . That is, the gapless mode at described by the low-energy effective Hamiltonian is stable against symmetry-preserving perturbations.
The fact that the Hamiltonian given by Eq.(8) exhibits a band crossing at can directly be seen by computing the eigenstate of and . Because and commute [see Eq.(2)], they can be simultaneously diagonalized by the same set of eigenstates
where the eigenfunctions are given by
and the eigenvalues are
We find that the energy and nonsymmorphic symmetry eigenvalues of at are continuously connected to the corresponding eigenvalues of at , see Fig.2. That is, we have
We note that the eigenfunctions become degenerate in energy at [i.e., ], while their nonsymmorpic symmetry eigenvalue remains non-degenerate at [i.e., ]. Therefore the two bands must cross each other.
To see the topological features of the band structure, we first note that the eigenvalues of the nonsymmorphic symmetries form a manifold as a function of momentum . That is, the eigenvalues are multivalued functions of , with different branches being smoothly connected. Inversely, is a single-valued continuous function of the eigenvalues of the symmetry . For the two-fold nonsymmorphic symmetry (1), the momentum has winding number as a function of the eigenvalue , which indicates a nontrivial topology [see Fig.2(c)]. To better understand this nontrivial topology, it is instructive to draw the mutual dependence of the energy eigenvalues , the nonsymmorphic eigenvalues , and the momentum in terms of a trajectory in the three-dimensional space . For the two-band model (9) this is shown in Fig. 2(d). The projections of this trajectory onto the three orthogonal planes (, ), (, ), and (, ) are shown in Figs.2(a), 2(b) and 2(c), respectively. We can see that the two bands are connected as a circle in space, corresponding to the element in the homology group .
Instead of , another possible choice for is with the symmetry relations
With this choice, we find the following relations for and ,
Using Eq. (7), it follows that
Hence, there always exists a band crossing point at .
Let us now show that the algebra obeyed by the symmetry operators determines whether the band crossing point is at or . To that end, we recall that for the choice the operators at the inversion invariant point , , , and are mutually commuting, see Eq. (6). At the other inversion invariant point , however, and are anti-commuting, while commutes with and , i.e., and . It follows that the two degenerate eigenstates of at can be written as eigenstates of with different eigenvalues. Explicitly, we find that is an eigenstate of with eigenvalue , while is an eigenstate of with eigenvalue . Therefore, the band crossing, which is protected by , occurs at .
A similar analysis can be performed for the choice , i.e., the Hamiltonian given by Eq. (16). In that case, we find that at the operators , , and are mutually commuting, while and anti-commute at , where the band degeneracy is located. We conclude that the algebraic relations obeyed by the symmetry operators determine the location of the symmetry-enforced band crossing, see Table 1.
Iv Nonsymmorphic magnetic symmetry
From the discussion in the previous section it follows that not all the symmetry constraints are necessary to enforce the existence of the band crossing. As we shall see, a single nonsymmorphic antiunitary symmetry, namely a magnetic nonsymmorphic symmetry, is sufficient to ensure the existence of a band crossing at or . As illustrated in Fig. 1(b), a magnetic nonsymmorphic symmetry can be viewed as the combination of a nonsymmorphic symmetry with a time-reversal symmetry . We only require that the combined symmetry is satisfied. That is, both and may be broken individually, but the combination must be preserved. In what follows we assume that and consider two possible choices for , namely, (i) and (ii) , where denotes the complex conjugation operator. By use of Eq. (1), we find that in case (i) , while in case (ii) .
Let us start with the discussion of case (i), where the combined symmetry acts on as
We observe that due to the symmetry constraint Eq. (17), , i.e., there is a band crossing point at . Note that the term is not forbidden by the combined symmetry . However, Eq. (17) requires that is an odd function of . Hence must vanish at the high-symmetry points and , due to the periodicity of the BZ, i.e., . To summarize, the magnetic nonsymmorphic symmetry is sufficient to enforce the existence of a band crossing at .
Next, we discuss choice (ii) for , in which case the combined symmetry acts on as
Since in Eq. (20), there is an unavoidable band crossing at . As before, the term is symmetry allowed, with an odd function. Hence, vanishes at the high-symmetry points and , and therefore cannot gap out the band crossing point.
From the above discussion, one infers that the commutation relation between and determines the position of the band-degeneracy points. Namely, for [case (i)] and [case (ii)], we have and , respectively. Hence, we find that for case (i) at , while for case (ii) at . Since is an anti-unitary operator, leads to a band degeneracy, in analogy to Kramers theorem. Thus, for the band-crossing point is at , while for it is at , see Table 1.
We note that a number of recent works Shiozaki et al. (2015); Fang and Fu (2015); Shiozaki et al. (2016); Wang et al. (2016) have discussed edge band structures of two-dimensional (2D) nonsymmorphic insulators that are similar to the 1D bulk band structures studied here. In contrast to conventional topological insulators, the edge bands of these 2D nonsymmorphic insulators do not connect valence and conduction bands. Hence, our results for the bulk band structures of 1D systems, can be applied directly to the edge spectrum of these 2D nonsymmorphic insulators. This suggests, in particular, that the crossing of the edge bands of these 2D systems is, at least in some cases, enforced by the nonsymmorphic symmetry of the edge theory.
In closing this section, we note that the existence of a band crossing cannot be enforced by a nonsymmorphic particle-hole symmetry, which is discussed in detail in Appendix A.
V Topological classification of band-crossing points
Let us now derive the classification of the global topological properties of the considered Hamiltonians. By global topology, we mean a band structure is allowed to be deformed smoothly in the whole momentum space with the symmetries being preserved, in contrast to the ordinary local one, where deformations are restricted only in a open neighborhood around the band-crossing point. The group structure is given by the direct sum of the Hamiltonians. To that end, we study whether the band crossings of the doubled Hamiltonians and can be gapped out by symmetry-preserving terms. Here, ’s represent an additional set of Pauli matrices and is the identity matrix. The symmetry operators for the doubled Hamiltonians are , , and . We observe that can be continuously deformed to without breaking the symmetries and without opening a gap at or . In addition, we find that is a symmetry preserving mass term that gaps out the spectrum of in the entire BZ. It follows that the global topological features of possess a classification, namely even number of copies can be gapped with symmetries being preserved, while odd number cannot. It is noted that although in the above simple deformation the gap is fully closed at , a more carefully chosen deformation can be made such that at any intermediate stage band crossing happens only at finite number of momenta.
To infer the classification of the local topological properties of the band crossing points, we enclose the degeneracy point by an sphere (consisting of two points on the left and right of the degeneracy point) and consider adiabatic deformations that do not close the gap on the chosen . The only possible gap opening term is , which however vanishes at the high-symmetry points due to the nonsymmorphic symmetry. This also holds for multiple copies of . From this we conclude that the local topological features of the band crossing points exhibit a classification.
Vi Higher-dimensional generalizations
Our results for symmetry-enforced band crossings in 1D can be readily generalized to higher dimensions. We assume that the fractional translation is along the direction for -dimensional systems. The -dimensional Hamiltonian can then be decomposed into a family of 1D Hamiltonians , which are parametrized by the () momenta perpendicular to . Let us briefly discuss how the three different types of nonsymmorphic symmetries that we considered above constrain this -dimensional Hamiltonian. (i) If is invariant under a unitary nonsymmorphic symmetry , then there are in general several branches of Fermi surfaces (possibly of dimension ) that are parametrized by . (ii) If there exists in addition a reflection symmetry reversing , then the Fermi surfaces are pinned at or , depending on the algebraic relations obeyed by the symmetry operators, as specified in Table 1. (iii) If we consider symmetries that relate to , such as a nonsymmorphic magnetic symmetry (or an additional inversion), then there exist 1D inversion invariant subsystems () of , which are labeled by the perpendicular momenta that are invariant under . These subsystems have band degeneracies at or , as determined by the algebraic relations in Table 1. The other 1D subsystems , where is not invariant under , are generally gapped.
We illustrate our theoretical results by two examples: (i) The Su-Schrieffer-Heeger (SSH) model Heeger et al. (1988) and (ii) the -flux square lattice model. The Hamiltonian of the SSH model is given by
where , see Fig. 3(a).
At the critical point the system is invariant under a half translation followed by an exchange of and sublattices, which corresponds to the nonsymmorphic symmetry , Eq. (1), see Fig. 3. Since the SSH model also has the inversion symmetry , we find in accordance with Table 1 that there is a gapless point at . Observe that at the translation symmetry is promoted from (with translator ) to (with the translator ) and the SSH model becomes a 1D tight-binding model of free fermions with the dispersion [see Fig. 3(c)]. It is noted that in order to create a bandcrossing point protected by nonsymmorphic symmetry in a tight-binding model through dimerization, one must ensure that the original non-dimerized model has two chiral gapless modes. A nonvanishing , on the other hand, reduces the translation symmetry from to and breaks the nonsymmorphic symmetry , leading to a topological () or trivial () insulator, depending on the sign of .
The Hamiltonian of the -flux square lattice model reads in momentum space
where denotes the nearest neighbor hopping amplitude, see Fig. 4(a). The model is invariant under the nonsymmorphic magnetic symmetry , which corresponds to a time-reversal symmetry followed by a half translation along and an exchange of and sublattices. The high-symmetry 1D subsystems and have gapless Dirac points at , as shown in Fig. 4(b). This is in agreement with Table 1, since . Similar to the SSH model, the -flux state is driven into a topological or trivial insulating phase by a dimerization along , that breaks the nonsymmorphic symmetry . In closing, we observe that the -flux model can be viewed as a higher-dimensional generalization of the SSH model.
Vii Discussions about multi-band theories
It is noted that the topological arguments in this work are not limited to two-band models, which is exemplified by two cases in follows. First, if a multi-band theory has a chiral symmetry, then the Hamiltonian can be anti-diagonalized with the upper-right entry being a matrix , and equation (4) still holds for . The topological argument around Eq.(5) implies that has to vanish somewhere in momentum space, namely there exists band-crossing points enforced by the chiral and nonsymmorphic symmetry.
Secondly, let us extend the spinless time-reversal symmetry discussed in Sec.IV to the spinful one , which acts in spinful four-band theories. If both and are preserved ( acts on the space of ), equations (4) and (18) hold for , which implies vanishes at . But diagonal terms have to vanish at and as required by the symmetries, except the chemical potential term. Thus bands are enforced to cross at . Note that the vanishing of at needs only the combined symmetry , but both and are required for that of the diagonal terms.
Acknowledgements.The authors thank C.-K. Chiu for useful discussions.
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Appendix A Nonsymmorphic Particle-hole symmetry
In this section, we discuss whether the existence of a band crossing can also be enforced by a nonsymmorphic particle-hole symmetry. As it turns out this is not possible. To see this, we consider for instance an anti-unitary non-symmorphic particle-hole symmetry , which acts on as
Hence, the off-diagonal component of must satisfy , and its Fourier components obey . Therefore, can be written as
which vanishes at . However, there does not exist a symmetry protected band crossing at , since the gap opening term , with an even function, preserves the non-symmorphic particle-hole symmetry . To protect the band crossing point at , an additional symmetry is needed which forbids the term. For example, the chiral symmetry , with and , prevents the term.