Band Topology and Linking Structure of Nodal Line Semimetals with Monopole Charges
We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion and time reversal when spin-orbit coupling is negligible. In contrast to the well-known -symmetric nodal lines protected only by Berry phase in which a single nodal line can exist, the nodal lines with monopole charges should always exist in pairs. We show that a pair of nodal lines with monopole charges is created by a double band inversion (DBI) process, and that the resulting nodal lines are always linked by another nodal line formed between the two topmost occupied bands. It is shown that both the linking structure and the monopole charge are the manifestation of the nontrivial band topology characterized by the second Stiefel-Whitney class, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a -invariant two-dimensional (2D) insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator (3D weak SWI). Moreover, using first-principles calculations, we predict ABC-stacked graphdiyne as a nodal line semimetal (NLSM) with monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with monopole charges under a -breaking perturbation.
Introduction.— Topological semimetals DiracWeyl_review (); classification_review (); line_review (); Herring (); 3DDirac (); Dirac_charge (); Dirac_discovery_Cd3As2_1 (); Dirac_discovery_Cd3As2_2 (); Dirac_discovery_Na3Bi (); unconventional (); double_Dirac (); Young-Kane (); filling-enforced (); nonsymmorphic_Zhao-Schnyder (); off-centered (); hourglass (); spinless_hourglass (); Murakami (); Murakami_classification (); nodal (); Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer (); Weyl_discovery1 (); Weyl_discovery2 (); Ahn (); Park (); unified_PT-CP (); Z2WeylDirac (); realDirac (); Z2line (); Mikitik (); Chiu (); Kim (); PT_NL_ZrSiS_ncomms (); PT_NL_ZrSiS_prb (); Fang_inversion (); Bzdusek () are novel states of matter whose band structure features gap-closing points or lines. Such gapless nodal points or lines are protected by either crystalline symmetry Herring (); 3DDirac (); Dirac_charge (); Dirac_discovery_Cd3As2_1 (); Dirac_discovery_Cd3As2_2 (); Dirac_discovery_Na3Bi (); unconventional (); double_Dirac (); Young-Kane (); filling-enforced (); nonsymmorphic_Zhao-Schnyder (); off-centered (); hourglass (); spinless_hourglass () or topological invariants Murakami (); Murakami_classification (); nodal (); Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer (); Weyl_discovery1 (); Weyl_discovery2 (); Ahn (); Park (); unified_PT-CP (); Z2WeylDirac (); realDirac (); Z2line (); Mikitik (); Chiu (); Kim (); PT_NL_ZrSiS_ncomms (); PT_NL_ZrSiS_prb (); Fang_inversion (); Bzdusek (). The nodal point (Weyl point) in a Weyl semimetal Murakami (); nodal (); Murakami_classification (); Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer (); Weyl_discovery1 (); Weyl_discovery2 () is a representative example of the latter case. Due to the quantized monopole charge, Weyl points always exist in pairs nodal (); Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer (). Moreover, pair creation and annihilation of Weyl points can mediate topological phase transitions between a normal insulator (NI) and a topological insulator in three dimensions (3D) Murakami (); Murakami_classification (); Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer (); inversion_response (); Ramamurthy (). Since the origin of the monopole charge is the Berry curvature of complex electronic states, breaking either time reversal Weyl_pyrochlore1 (); Weyl_pyrochlore2 (); Weyl_multilayer () or inversion Murakami (); Murakami_classification (); Weyl_discovery1 (); Weyl_discovery2 () is a precondition to host a Weyl point nodal ().
However, recent theoretical studies have found that, in the presence of and symmetries, a nontrivial monopole charge can exist, carried by a nodal line (NL), when spin-orbit coupling is negligibly weak Z2WeylDirac (); unified_PT-CP (); realDirac (); Z2line (); Bzdusek (); Fang_inversion (). Here the monopole charge is a number originating from the topology of real electronic states Z2WeylDirac (); unified_PT-CP (); realDirac (); Z2line (), which is clearly distinct from the integer monopole charge of Weyl points originating from complex electronic states. In fact, recently, spinless fermions in -symmetric systems have received great attention due to the discovery of semimetals with NLs protected by Berry phase Mikitik (); Chiu (); Kim (); Z2line (), appearing in various forms including rings Ca3P2_1 (); Ca3P2_2 (); CaP3family (); NCA (); BP (); AEM1 (); AEM2 (); AEM3 (); CaTe (), crossings GN (); Cu3PdN (); REM (); crossing-line (), chains chain (); chain_experiment (); chain_field (), links chain_link_Hasan (); link (); link_SCZ (); double-helix (); Vanderbilt_link (); multi-link (); Ezawa (); 2pi (), knots Ezawa (); 2pi (); knot (), nexus Volovik (); Hyart (); triple_point (), and nets four-band (); net (); various (). However, all the NL belonging to this class do not carry a monopole charge. Because of this, such a NL can exist alone in the Brillouin zone (BZ), which can disappear after shrinking to a point Z2line (). No candidate material has been predicted to host -nontrivial NLs (NLs) yet. Although there are preceding theoretical studies on NLs Z2line (); Bzdusek (); Fang_inversion (), the generic feature of the associated band structure topology, which is useful to facilitate material discovery, has not been thoroughly studied.
In this work, we study topological characteristics unique to a nodal line semimetal (NLSM) with monopole charges and propose the first candidate material, ABC-stacked graphdiyne. In particular, we describe the mechanism for creating NLs and the linking structure between them, which originates from the underlying global topological characteristics of real electronic states represented by the second Stiefel-Whitney (SW) class. The linking structure exists between a NL near the Fermi energy and another NL below , similar to the linking structure predicted in 5D Weyl semimetal recently 5DWeyl (). This demonstrates that, in contrast to the common belief, the topological property of NLSM is determined not only by the local band structure near crossing points at but also by the global topological structure of all occupied bands below .
Band crossing in -invariant spinless fermion systems.— -trivial NLs can be described as follows Kim (); Z2line (). Since in the absence of spin-orbit coupling, operator can be represented by where denotes the complex conjugation. In this basis, the invariance of the Hamiltonian, , requires to be real. Then the effective two-band Hamiltonian near a band crossing point can be written as where are the Pauli matrices for the two crossing bands and are real functions of momentum =. Because closing the band gap requires only two conditions to be satisfied whereas there are three independent variables , the generic shape of band crossing points is a line.
On the other hand, to describe NLs, one needs to consider a four-band Hamiltonian as first proposed in Z2line (). When the reality condition is imposed, can include three anticommuting matrices, which indicates that a 3D massless Dirac fermion can exist. The Dirac point is stable against the gap opening because the mass terms, which are imaginary, are forbidden. However, there are other allowed real matrix terms that can deform the Dirac point into a NL. For instance, let us consider the following Hamiltonian introduced in Z2line (),
where and are Pauli matrices. The energy eigenvalues are where . One can see that the conduction and valence bands touch along the closed loop (a NL) satisfying and . Moreover, two occupied bands cross along another line along (NL), which is linked with the NL. Because of this linking, the NL is stable and distinct from trivial NLs. As , the linking requires that the NL shrinks to a Dirac point. As becomes finite after sign reversal, the size of the NL increases again. It can never disappear by itself. Because a single NL is stable, only a pair of NLs can be created by band inversions.
NLs in ABC-stacked graphdiyne.— Our first-principles calculations predict that ABC-stacked graphdiyne realizes NLs with the linking structure. ABC-stacked graphdiyne is an ABC stack of 2D graphdiyne layers composed of a sp-sp carbon network of benzene rings connected by ethynyl chains. [See Fig. 1(c).] Recently, Nomura et. al. graphdiyne () theoretically proposed ABC-stacked graphdiyne as a NLSM. Here we show that the NLs in this material are NLs. Consistent with graphdiyne (), we find NLs occurring off the high-symmetry point of the BZ. While the electronic band structure displays band gap along the high-symmetry lines as shown in Fig. 1(d), the valence and conduction bands cross off the high-symmetry points along a pair of closed NLs colored in red in Fig. 1(e). Additionally, we find that two topmost occupied bands form another NL [the orange line in Fig. 1(e)], which pierces the red NLs, manifesting the proposed linking structure. Interestingly, the effective four-band Hamiltonian describing ABC-stacked graphdiyne near is identical to Eq.(1) graphdiyne (), indicating the generality of our theory.
Double band inversion (DBI).—
Let us illustrate a generic mechanism for a pair creation of NLs in inversion symmetric systems, which is comprised of consecutive band inversions, dubbed a double band inversion (DBI). For concreteness, we describe a DBI by using the Hamiltonian in Eq. (1) after the replacement . The evolution of the band structure during the DBI is illustrated in Fig. 2(a) as a function of the parameter . As we increase from , the first band inversion occurs at between the top valence and bottom conduction bands, creating a trivial NL. Then, the inversion at between two occupied (unoccupied) bands generates another NL below (above) , which we call NL. The last band inversion at between two inverted bands near splits the trivial NL into two NLs linked by the NL below [Fig. 2(a)]. During the DBI, each occupied (unoccupied) band crosses both of two unoccupied (occupied) bands, which explains why the minimal number of bands required to create a NL is four. In ABC-stacked graphdiyne, both valence and conduction bands are doubly degenerate along the high-symmetry axis due to three-fold rotation symmetry, thus NL exists from the beginning. In such a system, a single band crossing immediately inverts two occupied and two unoccupied bands having opposite parities, generating a NL pair as shown in Fig. 2(b). In noncentrosymmetric systems, NL pair creation occurs in a similar manner, that is, by splitting a trivial NL into a NL pair which are linked with another NL below as described in supp ().
monopole charge, linking number, and the second Stiefel-Whitney class.— Here we give a formal proof for the equivalence between the monopole charge and the linking number, based on the correspondence between the monopole charge and the second Stiefel-Whitney (SW) class implied by K-theory Gomi ().
The invariant was originally defined in Z2line () as follows. First, we take real occupied states by imposing . Then we consider a sphere surrounding a NL, which is divided into two patches (the northern and southern hemispheres) overlapping along the equator as shown in Fig. 3(a). One can find smooth real states () on the northen (southern) hemisphere. On the overlapping circle, are connected by a smooth transition function in a way that , where denotes the number of occupied bands. Let us note that, since the real occupied states are orientable on a sphere, transition functions can be restricted to supp (). The homotopy group indicates that there is a -type obstruction for defining real smooth state on the sphere, which is nothing but the monopole charge of NLs. Because , the winding number of is an integer invariant when . In this case, the monopole charge is defined by the parity of the winding number.
Now we make a connection between the monopole charge and the second SW class . characterizes the obstruction to lifting transition functions of real occupied states to their double covering group pin_structure (); DeWitt-Morette_v2 (); Nakahara (). When (), the lifting is allowed (forbidden). For simplicity, let us first consider the case with so that the transition function , where are the Pauli matrices for two occupied bands. When the monopole charge on the sphere is 0 (1), the angle evolves from 0 to () with an interger , because is periodic along the equator and has an even (odd) winding number. Now let us ask whether it is possible to take a lift from SO to its double covering group U while the periodicity of is kept. To answer this, one defines a two-to-one mapping by using and . Let us note that when has an even (odd) winding number with (), is periodic (non-periodic), thus the lifting from to is well-defined (ill-defined). The same argument applies to the case with pin_structure (). The monopole charge is thus identified with .
To derive the equivalence between and the linking number, let us continuously deform the sphere wrapping a NL , by gluing the north and south poles at the center, into a thin torus completely enclosing . As long as the band gap remains finite during the deformation, is invariant since the gluing of the north and south poles does not creat a monopole, which is further confirmed numerically as shown in Fig. 3(c,d). We assume that the torus is thin enough so that all occupied bands on it are non-degenerate. In this limit, according to the Whitney sum formula Hatcher (); Hatcher_AT (), safisfies the following relations modulo two supp ()
where is the first SW class of the th occupied band along the toroidal/poloidal cycle on the torus wrapping . As shown in supp (), the first SW class corresponds to the Berry phase of the th band along calculated in a smooth complex gauge, and it characterizes the orientability of the occupied states. Through a direct calculation of the Berry phase in a Coulomb gauge, we find that supp ()
where is the linking number Ricca-Nipoti () between and another NL formed by the occupied band degeneracy. Let us notice that NLs formed between unoccupied bands do not contribute to the linking number because the monopole charge is defined by occupied bands. For the model in Eq. (1) with , , , , and , so as expected.
Wilson loop method for computing .— can be computed efficiently by using the Wilson loop technique Z2line (); Bzdusek (); Wilson_loop (); inversion_Wilson_loop (); group_cohomology (). The relation between the Wilson loop spectrum and the monopole charge can be proved by using the definition of Nakahara (); DeWitt-Morette_v2 () as explicitly shown in supp (). In general, on a 2D closed manifold with coordinates , the Wilson loop operator along at a fixed is defined by Wilson_loop (); inversion_Wilson_loop (); group_cohomology () where is the overlap matrix at with matrix elements , and . On the wrapping sphere covered by three patches shown in Fig. 3(b), the Wilson loop operator becomes , where , , and . Let us take a parallel-transport gauge defined by , where for , respectively, and is defined with smooth states within the patch . Then the Wilson loop operator becomes
where and are the Wilson loop operator and the transition function in the parallel-transport gauge. Let ue note that, in this gauge, is simply given by the product of transition functions along the cycle. Since due to the consistency condition at triple overlaps supp (), the image of the map for forms a closed loop. Then is given by the parity of the winding number of supp (), which can be obtained gauge-invariantly from its eigenvalue Wilson_loop (); Bzdusek (). We apply the Wilson loop technique to ABC-stacked graphdiyne, and find that the Z2NLs carry nontrivial monopole charges. Figure 3(c) shows the first-principles calculations of the Wilson loop spectrum computed on a sphere wrapping a NL. The single crossing on the line indicates the odd winding number, leading to . Fig. 3(d) shows that the Wilson loop spectrum computed on a torus is also nontrivial. These first-principles results confirm the NLSM phase that we proposed here hosted in ABC-stacked graphdiyne.
2D SW insulator (SWI).— Using computed on a 2D BZ torus, we can define a new -invariant 2D topological insulator characterized by when (i.e., ). To prove this, we consider a 2D BZ torus with coordinates [Fig. 3(e)]. Then is again given by the spectral degeneracy of the Wilson loop on the torus, as shown in supp ().
Let us first consider case. We calculate along an orientable cycle, because otherwise the Wilson loop spectrum has no stable crossing points such that it does not show the topological property. One can always choose such an orientable cycle supp (). Then, there are four homotopy classes of Wilson loop spectra shown in Fig. 3(f-i). They are classified by the parity of the number of linear crossing points on and . A spectrum corresponds to () when it has an even (odd) linear crossing points on . Fig. 3(f,g) and 3(h,i) are distinguished by the total number of linear crossing points, which is even (odd) since () supp ().
Notice that the topology of the spectrum in Fig. 3(h) and (i) differs only by an overall shift of the eigenvalues by , whereas those in Fig. 3(f,g) are invariant under the shift. This indicates that is independent (dependent) of the unit cell choice when , because the Wilson loop eigenvalues correspond to the Wannier centers for insulators Wilson_loop (). Indeed, the same unit cell dependence exists for any even whereas is independent of the unit cell choice for any odd supp (). Therefore, is a well-defined topological invariant when . We may call the insulator characterized by as a 2D SW insulator (SWI). This is a new kind of fragile topological phase fragile (); Disconnected (); BuildingBlock () since it can be trivialized when bands with are added.
Topological phase transition.— As a sphere wrapping a NL can be continuously deformed to two parallel 2D BZs, one with and the other with , a NL can be considered as a critical state separating a 2D NI and a 2D SWI. Accordingly, the pair creation and annihilation of NLs can mediate a topological phase transition between a 3D NI and a 3D weak SWI, a vertical stacking of 2D SWIs. The presence of two NLs formed between occupied bands clearly distinguishes a 3D weak SWI from a NI. Interestingly, first-principles calculations show that ABC-stacked graphdiyne turns into a 3D weak SWI after pair annihilation of NLs under about 3 of a uniaxial tensile strain applied along the direction. [See supp ().]
Discussion.— Let us discuss about measurable properties of NLSM with NLs. Unfortunately, its surface states are generally not robust due to breaking on the surface Z2line (). Nevertheless, our study suggests that observing the linking structure using angle-resolved photoemission spectroscopy bulkARPES () can provide strong evidence for NLs. Moreover, the bulk magnetoelectric response under magnetic field can provide another evidence. When and are individually symmetries of the system, the number of pairs of NLs () can be determined from the inversion eigenvalues of the occupied bands at inversion-invariant momenta (IIM). Since a DBI changes two inversion eigenvalues at an IIM, is given by the sum of the number of negative eigenvalue pairs over all IIM Fang_inversion (); supp (). Let us note that, in -invariant insulators with broken , two times magnetoelectric polarizability is determined by inversion eigenvalues in the same way as is inversion (). This implies that a NLSM with an odd number of NL pairs turns into an axion insulator, which can host chiral hinge modes along the domain wall RotationAnomaly (); Khalaf (); Khalaf-Po-Vishwanath-Watanabe (), when the band gap is opened due to a -breaking perturbation such as magnetic field supp (). We believe that the theoretical prediction given in the present work can be experimentally tested in ABC-stacked graphdiyne in near future.
Acknowledgment.— J.A. was supported by IBS-R009-D1. B.-J.Y. was supported by the Institute for Basic Science in Korea (Grant No. IBS-R009-D1) and Basic Science Research Program through the National Research Foundation of Korea (NRF) (Grant No. 0426-20170012, No.0426-20180011), the POSCO Science Fellowship of POSCO TJ Park Foundation (No.0426-20180002), and the U.S. Army Research Office under Grant Number W911NF-18-1-0137. Y.K. was supported by Institute for Basic Science (IBS-R011-D1) and NRF grant funded by the Korea government (MSIP) (No. S-2017-0661-000). D. K. was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-07 and Basic Science Research Program through NRF funded by the Ministry of Education (NRF-2018R1A6A3A11044335). The computational calculations were performed using the resource of Korea institute of Science and technology information (KISTI). We appreciate the helpful discussions with Yoonseok Hwang, Sungjoon Park, Eunwoo Lee, Ken Shiozaki, Haruki Watanabe, and Akira Furusaki.
Note Added.— Recently, fragile topology in -nontrivial NLSMs was also explored in moTe2 (); the results of that work is consistent with our conclusions.
- SI 1 Outline
- SI 2 The First Stiefel-Whitney Class
SI 3 The Second Stiefel-Whitney Class
- SI 3.1 Obstruction to Spin and Pin structures
- SI 3.2 Čech cohomology
- SI 3.3 Whitney sum formula
- SI 3.4 Linking number
- SI 3.5 Flux integral
- SI 4 Wilson Loop Method
- SI 5 Inversion Symmetry
- SI 6 Pair Creation of Z2 Monopoles
- SI 7 Material Realization
Si 1 Outline
In this Supplemental Material, we provide detailed derivations of the results shown in the main text. In Sec. SI 2 and SI 3, we connect the known definition of the 1D and 2D topological invariants (i.e., the Berry phase and the monopole charge) to the first and second Stiefel-Whitney classes. For a rigorous treatment of the second Stiefel-Whitney class, we introduce the concept of the Čech cohomology in Sec. SI 3, which can be used to check the Whitney sum formula. This formula is crucial for proving the equivalence between the linking number of nodal lines and the second Stiefel-Whitney class and for deriving the Wilson loop method in Sec. SI 4. In Sec. SI 4, we provide a simple formula for calculating the second Stiefel-Whitney class from inversion eigenvalues. The derivation of this formula is based on the flux integral formula proposed in Ref. realDirac, , which we review in Sec. SI 3.5. In Sec. SI 4, we demonstrate various types of pair creations of monopoles and the corresponding creation of the linking structure. We also explain the double band inversion process in more detail. Finally, in Sec. SI 7, we provide the detailed information of the first principles calculations of the ABC-stacked graphdiyne that was not presented in the main text. In particular, we show the Wilson loop spectrum which verifies that a tensile strain can induce the topological phase transition in ABC-stacked graphdiyne from a -nontrivial nodal line semimetal to a 3D weak Stiefel-Whitney insulator.
Si 2 The First Stiefel-Whitney Class
The topological phases of -symmetric one-dimensional (1D) systems are classified by the Berry phase: there are two distinct classes with 0 and Berry phase modulo . In 3D systems, the Berry phase serves as a topological invariant protecting nodal lines. As a nodal line generates Berry phase over any closed loop encircling a line segment, it is topologically protected when the system is -symmetric. Here, we show how the Berry phase is related to the global structure of quantum states. While quantum states can always be smoothly defined over a 1D submanifold in the Brillouin zone, they may not be smoothly defined after we require the reality condition because of the topological obstruction characterized by the topological invariant so-called the first Stiefel-Whitney class. In this section, we show that the Berry phase in (complex) smooth gauges is identical to the first Stiefel-Whitney class in real gauges. This result will be useful in the next section for deriving the linking number from the 2D topological invariant.
Si 2.1 Quantization of Berry phase due to symmetry
Let us first review the quantization of Berry phase in the presence of symmetry. The Berry connection satisfies the following symmetry constraint
which follows from .
Suppose a set of bands is isolated from the other bands by the band gap. By taking a trace over , we have an abelian Berry connection
where the trace and determinant are defined over . By integrating the abelian Berry connection, we have a quantized phase
where is the winding number of . Because of this quantization of the Berry phase, the Berry phase of a nodal line cannot be adiabatically changed to zero.
Si 2.2 Berry phase and the first Stiefel-Whitney class
Now we review the definition of the orientation of vector spaces and quantum states, and then we show that the Berry phase in a smooth complex gauge characterizes the orientability of quantum states in a real gauge. This property allows the Berry phase computed with smooth complex states to be interpreted as the first Stiefel-Whitney class of real quantum states.
The orientation of a real vector space refers to the choice of ordered basis. Any two ordered bases are related to each other by a unique nonsingular linear transformation. When the determinant of the transformation matrix is positive (negative), we say the bases have the same (different) orientation. After choosing an ordered reference basis , the orientation of another basis is specified to be positive (negative) when the basis have the same (different) orientation with respect to the reference basis.
Real quantum states in the Brillouin zone are real unit basis vectors defined at each momentum (In other words, quantum states have the structure of a real vector bundle over the Brillouin zone.). The basis can be defined smoothly in a local patch, but may not be smooth over a closed submanifold of our interest. We say quantum states are orientable over when local bases can be glued only with transition functions with positive determinant, i.e., all transition functions are orientation-preserving. When quantum states are orientable, they are classified into two classes with the positive and negative orientation in the same way as the real vector spaces are.
Because the orientability of real quantum states is determined by their the global structure, it is encoded in the form of a topological invariant. Over a closed 1D manifold, the topological invariant which measures the orientability of real quantum states is the first Stiefel-Whitney class Hatcher (). Real quantum states are orientable (non-orientable) when ().
Although the term the first Stiefel-Whitney class may be unfamiliar to readers, it is in fact equivalent to the well-known quantized Berry phase defined in a smooth complex basis. It can be observed by investigating how the Berry phase computed with smooth complex states affects the real states given by a gauge transformation. In order to make states real, the Berry phase should be eliminated by a local phase rotation of the states because the diagonal components of the Berry connection are zero when the states are real.
where and are the Berry connection given by real states and by smooth complex states , respectively, and is the gauge transformation matrix defined by . Integrating the term, we have
Thus, when the total Berry phase is nontrivial, the real states require an orientation-reversal to transit from to , because it follows from that . We conclude that the first Stiefel-Whitney class for a closed curve in the Brillouin zone is
where is the Berry connection calculated in a complex smooth gauge.
As an example, let us consider the Su-Schuriffer-Heeger (SSH) model SSH () in a real basis
This Hamiltonian is symmetric under . It is well-known that this system describes an insulator which is topologically trivial when and nontrivial when . Let us see how the topology manifests on the occupied state. The occupied state is
where is an arbitrary overall phase factor and is a positive normalization factor.
First, we impose the reality condition on the occupied state over the whole Brillouin zone, i.e., at each . As shown in Fig. S1(a), when , the real occupied state can be made smooth over , but the boundaries should be glued with an orientation-reversing transition function. The occupied state is thus non-orientable. In contrast, it is orientable when as shown in Fig. S1(b).
Next, we relax the reality condition on the occupied state. By taking , we can make the occupied state globally smooth even when [See Fig. S1(c)]. The cost of taking smoothness is to have a nontrivial Berry phase. In this gauge, we have and thus .
In fact, the first Stiefel-Whitney class determines the orientability of real states even in higher dimensions Hatcher (). From the analysis in 1D, we find
is globally smooth when the Berry phase is trivial over every closed cycle. Otherwise, becomes discontinuous at some points so that real states are non-orientable as in the 1D case. Thus, real states are orientable over an arbitrary dimensional closed manifold if and only if the total Berry phase, which is calculated in a smooth complex gauge, is trivial for any 1D closed loop in .
Si 3 The Second Stiefel-Whitney Class
-symmetric two-dimensional systems are topologically classified (according to the K-theory) by a invariant called the second Stiefel-Whitney class Gomi (). In three-dimensional systems, the second Stiefel-Whitney class defines the monopole charge of a gap-closing object when the invariant is defined over the 2D manifold enclosing the object. Historically, it was first discovered by Hoava Horava () using K-theory that gap-closing points in three-dimensional systems described by real quantum states are classified by a invariant. Subsequently, Dirac points in spinless -symmetric systems were found to be the gap-closing points carrying the nontrivial monopole charges by Morimoto and Furusaki Z2WeylDirac () (and also by Zhao et al. unified_PT-CP (); realDirac ()). More recently, however, Fang et al. Z2line () showed that the Dirac point is not stable against small perturbations: it deforms into a nodal line which still carries the monopole charge. In this section, we show that the known expression for the monopole charge corresponds to the definition of the second Stiefel-Whitney class as the obstruction to the spin structure. We also present a formal definition of the second Stiefel-Whitney class in terms of the Čech cohomology. This definition will be useful when we identify the monopole charge with the topology of the Wilson loop spectrum in the next section. It is also useful for demonstrating a nontrivial property of the second Stiefel-Whitney class, so-called the Whitney sum formula. Then, by using the Whitney sum formula, we show that the monopole charge of a nodal line is identical to its linking number modulo two. Finally, we comment on the flux integral form of the second Stiefel-Whitney class.
Si 3.1 Obstruction to Spin and Pin structures
Here, we begin by recalling how spinors and pinors are defined in arbitrary dimensions. The spin and pin structures for real quantum states are defined in an analogous way as physical spinors are defined, although they are independent of the physical spin. Then, we show that the nontrivial monopole charge of a nodal line induces an obstruction to the existence of the spin structure over a sphere enclosing the nodal line. This identifies the monopole charge with the second Stiefel-Whitney class , because is the topological invariant characterizing the obstruction to the existence of a spin structure of real orientable quantum states DeWitt-Morette_v2 ().
Si 3.1.1 Spin and Pin groups
Spinors in three spatial dimensions are the objects transforming under the double covering group of the orthogonal group . Let us recall that spinors are the objects that transforms only half under spatial rotations in the following sense. For an object with azimuthal angular momentum , the rotation around the -axis is represented by
A spinor has a half-integer , while a vector or a tensor has an integer . Accordingly, while the vectors and tensors are invariant under any rotation, spinors get a phase when the system is rotated by and come back to their original state after another rotation. Therefore, the group of rotations for spinors are twice larger than that for vectors. The former and the latter are and , and the two-to-one mapping is given by where , and . The double covering group of , which is , is called the spin group , because it is responsible for the transformation of spinors under rotations in real space.
Similary, spinors in arbitrary spatial dimensions are defined as the objects transforming under , the double covering group of . In general, the two-to-one mapping is given by
where , is the angle of rotation in the plane, and is the Gamma matrix satisfying the Clifford algebra , where the bracket is the greatest integer function.
We can also construct a spinor which transforms under the double covering group of not only that of . A spinor having this property is called a pinor, and the double covering group of is called 111 Notice that is written by dropping a from . [S]pinor transforms under the double covering group of . . Because can be obtained by adding a mirror operation, e.g., the -mirror operation to , it is enough for constructing to consider the lift of to its double covering group. There are two choices of the two-to-one mapping Witten_RMP ()
where and , respectively, for and (we use to denote the identity element in the double covering group, while we use 1 for the original group.). The double covering group of is called in the former and in the latter.
Si 3.1.2 Spin and Pin structures
Now we explain what are the spin and pin structures of real quantum states in the Brillouin zone (i.e., spin and pin structures on real vector bundles). Let us recall that the transition function is defined by
on an overlap of two open covers and . belongs to for real occupied states , where is the number of occupied bands. In analogy to defining a pinor, we may define pinor states . They are smooth over each cover of , and their transition function is given by lifting to its double covering group .
over , where for the two-to-one projection . We have two choices for over each overlap because we are free to choose the sign of it. A choice of the signs over all overlaps is called a pin structure Witten_RMP (). The pin structure determines the topology of pinor states. When the states are orientable, transition functions can be confined to over all overlaps. Then, we may define a spin structure rather than a pin structure. However, real occupied states may not admit a pin or spin structure due to an obstruction coming from the global structure. This obstruction for a pin or spin structure (a pin structure) is characterized by the topological invariant called the second Stiefel-Whitney class (the dual second Stiefel-Whitney class ). A pin or spin structure (a pin structure) exists if and only if () DeWitt-Morette_v2 (). Although there are two obstruction invariants and in principle, when we consider 2D closed submanifolds of the 3D Brillouin zone, we have a unique obstruction invariant because the second Stiefel-Whitney class is identical to its dual on those manifolds 222 The dual second Stiefel-Whitney class is defined by , where is the cup product DeWitt-Morette_v2 (). We have over orientable surfaces, because a closed orientable surface is a sphere, a torus with an arbitary genus, or a connected sum of them according to the classification theorem of surfaces Munkres (), and the cup product is trivial on those surfaces Hatcher_AT (). Moreover, every closed surface in the 3D Brillouin zone is orientable, because non-orientable closed surfaces cannot be embedded in the three-torus embedding (). It follows that in the 3D Brillouin zone. .
Si 3.1.3 monopole charge
As the first Stiefel-Whitney class is associated with the one-dimensional topological invariant of a nodal line, the second Stiefel-Whitney class is also associated with the topological invariant of a nodal line. It is the two-dimensional topological invariant called the monopole charge. We now show the monopole charge of a nodal line corresponds to the second Stiefel-Whitney class, because the nontrivial monopole charge forbids the existence of the spin structure on a sphere enclosing the nodal line.
The monopole charge of a nodal line is defined over a sphere enclosing the nodal line. It is defined by the winding number of the transition function at the overlapping region of two patches of the sphere Z2line (). Let be the transition function between two patches and defined over the overlapping region: for . We restrict the transition function to , which is possible because every loop on a sphere is contractible to a point such that the first Stiefel-Whitney class is trivial. Then we see that the winding number of along a loop in gives a number because for . This number is the monopole charge. When the number of occupied bands is two, the winding number is integer-valued because . In this case, the monopole charge is defined by the parity of the winding number.
We can show that this invariant characterizes the obstruction to constructing a spin structure over the wrapping sphere. For simplicity, we take a gauge where the transition function is an identity at some . Then, it evolves from the identity to a rotation ( rotation) for an integer along a loop containing in when the monopole charge is trivial (nontrivial), because the generator of the homotopy group is the path from the identity to a rotation pin_structure (); Prasolov (). While the rotation and the identity are identical as elements, they are not identical as elements. Therefore, the transition function is well-defined over the overlap only as a element when the monopole charge is nontrivial. On the other hand, no obstruction arises when the monopole charge is trivial because a rotation is identical to the identity element even as a element. Thus, the monopole charge is identical to the second Stiefel-Whitney class over the enclosing sphere.
Si 3.2 Čech cohomology
Here we review how to formally define the second Stiefel-Whitney class as a Čech cohomology class DeWitt-Morette_v2 (); Nakahara (). In other words, we define it as a two-dimensional integral over the geometric structure (semi-simplicial complex) constructed from the patches and their overlaps on the original manifold [See Fig. S2]. We consider a covering whose geometric structure is topologically equivalent to the original manifold. The value of the function to be integrated will be assigned according to whether the consistency conditions of transition functions are satisfied after they are lifted to the double covering group. After we define the second Stiefel-Whitney class with the lifted transition functions, we will connect it to the original transition functions in two examples: on a sphere and on a torus. This approach will be useful when we interpret the Wilson loop method in the next section.
In general, transition functions should satisfy the following consistency conditions.
for , where , , and are arbitrary patches. The transition functions defined by satisfy these consistency conditions automatically.
After well-defined transition functions are lifted at each overlap to the double covering group, the consistency conditions are not automatically satisfied in general. Let us write and to denote the and rotation in the double covering group. In general, after the lift , the lifted transition functions satisfy
for . The sign can be either or because both and are projected to 1 by the two-to-one map from to or from to . is gauge-invariant as one can see from the transformation of the lifted transition functions under , where is a lift of . Also, has a unique value on each triple overlap, because it is fully symmetric with respect to the permutation of , , and and is independent on within a triple overlap. We will thus omit the subscript and not care about the order of the superscripts , , and for from now on.
Let us now see when we can find a lift satisfying the consistency conditions. No obstruction arises for the first condition: we can always find a lift satisfying from an arbitrary lift at every overlap . We can see this as follows. A lift is related to the lift by
where is an integer defined modulo two (the subscript is omitted for because is uniform over each overlap.). Let us take a lift such that at each overlap . Then, we have . There is no obstruction here coming from topology because the constraint on and is local in the sense that the constraint is defined on a single overlap .
If we require the first consistency condition, however, we may not be able to find a lift satifying the second consistency condition due to the global constraint. transforms under the change of the lift by Eq. (S19) as
By this transformation, we can get on a triple overlap . However, is not a local condition because , , and are defined on different overlaps, which implies that there can be a topological obstructure to taking over every triple overlap. In fact, the product
over all triple overlaps in a closed manifold is invariant under the change of the lift.
We can observe this invariance from the geometric structure of patches shown in Fig. S2, where patches, overlaps, and triple overlaps are considered as vertices, edges, and faces, respectively. In this intepretation, if we let , is a two-dimensional integral defined by
where , and is the surface formed by the union of all faces. Because transforms under the change of the lift by Eq. (S19) as , where , changes by , where is the boundary of , and the line integral is defined by over each overlap . The Stokes’ theorem is valied here because the line integral is well-defined modulo two: modulo two, and this follows from the consistency condition . Because we consider closed manifolds, i.e., , we find . Thus, is invariant under any change of the lift modulo two.
One can see that transition functions cannot be lifted to their double covering group when modulo two, because then there is at least one triple overlap where . Therefore, the obstruction to the existence of a spin or pin structure is characterized by the invariant . The obstruction invariant is the second Stiefel-Whitney class for spin and pin structures, whereas it is the dual second Stiefel-Whitney class for pin structures DeWitt-Morette_v2 (). In our case, however, we can restrict our attension only to the second Stiefel-Whitney class, because for 2D closed submanifolds in the 3D Brillouin zone as we mentioned in the previous subsection. We will investigate more on Eq. (S21) using a sphere and a torus as examples below.
Si 3.2.1 Sphere
First we consider three patches , , and covering a sphere shown in Fig. S3. In the spherical coordinates , there are three overlaps , , and on , , and , respectively. We restrict all transition functions on the overlaps to , which is possible because every loop on a sphere is contractible to a point such that the first Stiefel-Whitney class is trivial. Then
where and denotes the polar angle .
The second Stiefel-Whitney class can be related to the winding number as follows. Let us first define
where we omit in the argument of transition functions because they are uniquely specified by the overlapping region. is smooth for because is smooth within an overlap. , and . modulo two when the image of the map is an arc connecting and , while when the image is a closed loop containing or . Next, we project by the two-to-one map . We have
which is smooth for , and . By this projection, an arc connecting and projects to a loop winding the non-contractible cycle an odd number of times, whereas a closed loop projects to a contractible loop or a non-contractible loop winding the non-contractible cycles an even number of times pin_structure (). As a result, the second Stiefel-Whitney class is given by the winding number of modulo two. This definition of the second Stiefel-Whitney class corresponds to the monopole charge enclosed by the sphere which is defined in Ref. Bzdusek, using the Wilson loop spectrum. We will discuss more about it in the next section.
We can also connect the formalism given here to the definition of the monopole charge in Ref. Z2line, , which was discussed in the previous subsection. Let us choose a gauge where , and take a lift such that and . We have
Let us define
where parametrizes the overlap between the two hemispheres. is smooth for , and it satisfies the boundary condition . The image of is a closed loop when and is an arc connecting