Protected boundary states in gapless topological phases

Protected boundary states in gapless topological phases

Shunji Matsuura, Po-Yao Chang, Andreas P. Schnyder, Shinsei Ryu Departments of Physics and Mathematics, McGill University, Montréal, Québec, Canada Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana, IL 61801, USA Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany a.schnyder@fkf.mpg.de chang153@illinois.edu
July 5, 2019
Abstract

We systematically study gapless topological phases of (semi-)metals and nodal superconductors described by Bloch and Bogoliubov-de Gennes Hamiltonians. Using K-theory, a classification of topologically stable Fermi surfaces in (semi-)metals and nodal lines in superconductors is derived. We discuss a generalized bulk-boundary correspondence that relates the topological features of the Fermi surfaces and superconducting nodal lines to the presence of protected zero-energy states at the boundary of the system. Depending on the case, the boundary states are either linearly dispersing (i.e., Dirac or Majorana states) or are dispersionless, forming two-dimensional surface flat bands or one-dimensional arc surface states. We study examples of gapless topological phases in symmetry class AIII and DIII, focusing in particular on nodal superconductors, such as nodal noncentrosymmetric superconductors. For some cases we explicitly compute the surface spectrum and examine the signatures of the topological boundary states in the surface density of states. We also discuss the robustness of the surface states against disorder.

pacs:
73.43.-f, 73.20.At, 74.25.Fy, 73.20.Fz, 03.65.Vf
: New J. Phys.

1 Introduction

The recent discovery of topological electronic phases in insulating materials with strong spin-orbit coupling [1, 2, 3, 4, 5, 6] has given new impetus to the investigation of topological phases of matter. Topological materials, such as the integer quantum Hall state and the spin-orbit induced topological insulators, are characterized by a nontrivial band topology, which gives rise to protected exotic edge (or surface) states. Many interesting phenomena, including magneto-electric effects [7] and the emergence of localized Majorana states [8], have been predicted to occur in these systems. These phenomena could potentially lead to a variety of new technical applications, including novel devices for spintronics and quantum computation.

Besides the topological insulators and the integer quantum Hall state, which have a full bulk gap, there are also gapless phases that belong to the broad class of topological materials, such as, e.g., (semi-)metals with topologically protected Fermi points and nodal superconductors with topologically stable nodal lines. These gapless topological phases also exhibit exotic zero-energy edge (or surface) states with many interesting properties. These boundary states may be linearly dispersing (i.e., of Dirac or Majorana type), or dispersionless, in which case they form either two-dimensional surface flat bands or one-dimensional arc surface states. Notable examples of gapless topological materials include, among others [9, 10, 11], graphene [12, 13, 14, 15, 16, 17], -wave superconductors [18, 19, 20, 21, 22, 23], the A phase of superfluid He [24, 25], and nodal noncentrosymmetric superconductors [26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

The topologically stable Fermi points and superconducting nodal structures in the aforementioned materials can be viewed, in a sense, as momentum-space defects, that is, as momentum-space analogues of real-space topological defects. In other words, the nodal points in -wave superconductors, the Fermi points in graphene, and the nodal points in He A can be interpreted as momentum-space point defects, i.e., as vortices and hedgehogs, respectively. The nodal lines in noncentrosymmetric superconductors, on the other hand, correspond to momentum-space line defects, i.e., vortex lines. Similar to real-space defects, the stability of these Fermi points, nodal points, and nodal lines is guaranteed by the conservation of some topological invariant, i.e., e.g., a Chern or winding number.

In this paper, building on previous works [24, 27, 28, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], we derive a classification of topologically stable Fermi surfaces in (semi-)metals and nodal lines in superconductors using K-theory arguments (Table 3 in Sec. 2) 111 By definition, a Fermi surface is a set of gapless points in the Brillouin zone. To simplify terminology, we will refer to Fermi points/lines in metals and nodal points/lines in superconductors, etc., simply as “Fermi surfaces”. 222 Hořava [36] pointed out an interesting connection between the classification of stable Fermi surfaces and the classification of stable D-branes. Making an analogy with string theory, we observe that the result by Hořava corresponds to D-branes in Type IIA string theory. Hence, one might wonder what are the gapless topological objects in condensed matter physics that correspond to D-branes in Type IIB string theory. Furthermore, in Type I or Type I’ string theory it is customary to consider besides D-branes also orientifold hyperplanes. (Note that every space-time point on an orientifold hyperplane is identified with its mirror image.) It is known that D-branes in Type I string theory are classified in terms of real K-theory [47]. Hence, one might again wonder what are the gapless topological objects in condensed matter physics that correspond to orientifold hyperplanes or D-branes in Type I or Type I’ string theory. In fact, for topological insulators and superconductors, it was found that there is a one-to-one correspondence between the K-theory classification of topological insulators/superconductors and the K-theory charges of D-branes in Type IIA and Type IIB string theory, or the K-theory charges of non-BPS D-branes in Type I and I’ string theory [48, 49]. . As it turns out, the presence of discrete symmetries, such as time-reversal symmetry (TRS) or particle-hole symmetry (PHS), plays a crucial role in the classification of gapless topological phases, a fact that has not been emphasized previously. The appearance of protected zero-energy states at the boundary of gapless topological phases is discussed, and it is shown that the existence of these boundary states is directly linked to the topological stability of the Fermi surfaces (superconducting nodal lines) in the bulk via a generalized bulk-boundary correspondence (Sec. 2.4). In particular, we demonstrate that gapless topological phases in symmetry class A or AIII with stable Fermi surfaces of codimension necessarily support zero-energy surface flat bands. Finally, in Sec. 3, we present a few examples of gapless topological phases and discuss their topological surface states.

2 Local stability of Fermi surfaces

The classification of topologically stable Fermi surfaces in terms of K-theory is closely related to the classification of topologically stable zero modes localized on real-space defects. In Sec. 2.1, we will therefore first review the stability of localized gapless modes on topological defects, before discussing the classification of topologically stable Fermi surfaces in Sec. 2.2. To denote the dimensionality of the Brillouin zone (BZ), the Fermi surfaces, and the real space defects we use the notation:

In other words, the dimension of a Fermi surface and a real space defect are and , respectively.

2.1 Real-space defects

In this subsection, we review the classification of localized gapless modes on topological defects from the K-theory point of view [45, 50]. To that end, let us consider the topology associated with gapped Hamiltonians , where denotes the -dimensional momentum coordinate, and the position-space parameters characterizing the defect. That is, are the coordinates parametrizing the surface that encloses the defect in question. For instance, a line defect in a three-dimensional system is described by the Hamiltonian , where , , and are five anticommuting matrices. In this case, and .

complex case ():
Symmetries
class
0 A 0 0 0 0
1 AIII 0 0 0 0
real case ():
Symmetries
class
0 AI 0 0 0 0

1
BDI 0 0 0 0


2
D 0 0 0 0



3
DIII 0 0 0 0



4
AII 0 0 0 0





5
CII 0 0 0 0
6 C 0 0 0 0
7 CI 0 0 0 0
Table 1: K-group from Ref. [45].

For condensed matter systems defined on a lattice, the BZ is a -dimensional torus, , and , where is a -dimensional sphere surrounding the defect in real space. If we are interested in “strong” but not in “weak” topological insulators and superconductors, we can take . Furthermore, it turns out it is enough to consider [45]. To determine the topology of the family of Hamiltonians , one needs to examine the adiabatic evolution of the wavefunctions of along a closed real-space path surrounding the defect 333It is assumed that the path is sufficiently far away form the singularity of the defect.. From this consideration, one can define a K-theory charge for and describe the stable equivalent classes of Hamiltonians in terms of the K-group

(1)

where represents one of the Altland-Zirnbauer symmetry classes [41, 42, 43] given in Table 1, () stands for the complex (real) Altland-Zirnbauer symmetry classes, and and represent the dimensions of and , respectively. An important relation used in the analysis of Ref. [45, 50] is that K-groups of different symmetry classes are related by

(2)

and

(3)

Relations (2) and (3) can be derived by considering smooth interpolations/deformations between two Hamiltonians belonging to different symmetry classes and different position-momentum dimensions (, ), thereby demonstrating that the two Hamiltonians are topologically equivalent [45, 50]. Combining Eqs. (2) and (3), one finds

(4)

which shows that the topological classifications only depend on the difference

(5)

From this, it was shown in Refs. [45, 50] that the classification of zero-energy modes localized on real-space topological defects is given by the K-groups (see Table 1) with

(6)

In other words, whenever the K-group is nontrivial (i.e., or ) the K-theory charge can take on nontrivial values, which in turn indicates the presence of one or several zero-energy modes localized on the topological defect. As a special case, the periodic table of topological insulators and superconductors [40, 44, 51, 6] is obtained from the K-groups by taking (cf. Table 2)

(7)

Representative Hamiltonians of the stable equivalent classes of can be constructed in terms of linear combinations of anticommuting Dirac matrices [45] (see also [6]). For instance, consider

(8)

with “position-type” Dirac matrices and “momentum-type” Dirac matrices , where , , and . If the Hamiltonian satisfies time-reversal symmetry , we require

(9)

while for particle-hole symmetry , we have

(10)

Under the antiunitary symmetries and the coefficients and transform in the same way as position and momentum, respectively, i.e., and . As shown in Ref. [45], a representative Hamiltonian of the real symmetry class can be constructed in terms of a linear combination of position-type matrices and momentum-type matrices , with .

0
0
Table 2: Assignment of dimensions (, ) for different classification schemes: (i) classification of insulators (fully gapped superconductors), (ii) classification of zero modes localized on real-space defects in insulators (fully gapped superconductors), and (iii) classification of Fermi surfaces.

2.2 Fermi surfaces (momentum-space defects)

The analysis of Refs. [45, 50], which we have reviewed above, can be extended to study the topological stability of Fermi surfaces. For a given Hamiltonian , we define the Fermi surface as the momentum-space manifold where  444Alternatively, the Fermi surface can be defined in terms of the poles of the single particle Green’s function.. The key observation is that topologically stable Fermi surfaces can be viewed as defects in the momentum-space structure of the wavefunctions of . Hence, in order to determine the topology of a -dimensional Fermi surface, we need to examine the adiabatic evolution of the wavefunctions of along a closed momentum-space path surrounding the Fermi surface. This closed path in momentum space is parametrized by variables, i.e., it defines a -dimensional hypersphere surrounding the Fermi surface. Hence, the topological stability of a -dimensional Fermi surface in a -dimensional BZ is describe by the K-group , with

(11)

i.e., by , where . That is, the classification (or “periodic table”) of topologically stable Fermi surfaces in symmetry class can be inferred from Table 1 together with Table 2. For two- and three-dimensional systems, the classification of -dimensional Fermi surfaces is explicitly given in Table 3 555Focusing on noninteracting systems, we study the topological stability of Fermi surfaces in terms of Bloch or Bogoliubov-de Gennes Hamiltonians. However, it is straightforward to extend our analysis to the Green’s function formalism describing weakly (or moderately weakly) interacting systems (see Refs. [52, 36, 45] and compare with Ref. [53])..

complex case ():
class
line point
A 0
AIII 0
complex case ():
class
surface line point
A 0
AIII 0 0
real case ():
class
line point
AI 0
BDI
D
DIII 0
AII 0
CII 0
C 0 0
CI 0 0
real case ():
class
surface line point
AI 0
BDI 0
D
DIII 0
AII 0
CII 0 0
C 0 0
CI 0 0 0
Table 3: (Symmetry of restricted to ) Classification of topologically stable Fermi surfaces in two- and three-dimensional systems ( and , respectively) as a function of Fermi-surface dimension and symmetry class of restricted to a hypersphere surrounding an individual Fermi surface. Ten symmetry classes are distinguished, depending on the presence or absence of time-reversal symmetry (), particle-hole symmetry (), and chiral (or sublattice) symmetry (). The labels , , and indicate the presence or absence of time-reversal, particle-hole, and chiral symmetries, respectively, as well as the types of these symmetries.

Let us construct a few simple examples of topologically stable (and unstable) Fermi surfaces in terms of Dirac Hamiltonians defined in the continuum. Examples of topological Fermi surfaces defined in terms of lattice Hamiltonians will be discussed in Sec. 3.

Class A.

We first consider single-particle Hamiltonians with Fermi surfaces in symmetry class A, i.e., Fermi surfaces that are not invariant under time-reversal (), particle-hole (), and chiral symmetry (). Below we list examples of Hamiltonians in spatial dimensions with Fermi surfaces in symmetry class A

(12)

Here, denote the three Pauli matrices, while and represent the four Dirac matrices (gamma matrices). For each example, the Fermi surface is given by the manifold ; with for , where is the dimension of the Fermi surface. In the above examples, Fermi surfaces with odd (i.e., odd) are perturbatively stable against any deformation of the Hamiltonian. Fermi surfaces with even, on the other hand, are topologically unstable (see Hořava [36]). Due to the absence of a spectral symmetry (i.e., no chiral symmetry) in class A, we can add a nonzero chemical potential term to the Hamiltonians in Eq. (12). Thus, for example, the -dimensional stable Fermi surface of can be turned into a stable Fermi surface of dimension upon inclusion of a finite chemical potential. Note that the third row in the above list, i.e., , corresponds to a Weyl semi-metal [54, 55, 46, 56, 57, 58].

Class AIII.

Second, we consider Hamiltonians with Fermi surfaces in symmetry class AIII. Recall that due to the presence of chiral symmetry in class AIII (i.e., , where is an arbitrary unitary matrix), the chemical potential is pinned at . Below we list a few examples of topologically stable (and unstable) Fermi surfaces in symmetry class AIII

(13)

Here, we find that Fermi surfaces with dimension even are topologically stable, wheres those with odd are topologically unstable.

In passing, we remark that the above analysis can also be applied to gapless Hamiltonians defined in an extended parameter space, i.e., Hamiltonians that are parametrized by momentum coordinates and some external control parameters, such as, e.g., mass terms . The topological arguments can then be used to predict the existence of extended regions of gapless phases in the topological phase diagram (see, e.g., Refs. [59, 60, 61]).

2.3 Comments on the stability of multiple Fermi surfaces

It should be stressed that the above topological stability criterion (i.e., Table 3) applies only to individual Fermi surfaces. That is, in the above analysis we considered the wavefunction evolution along a hypersphere that encloses a single Fermi surface. However, many lattice Hamiltonians exhibit multiple Fermi surfaces that are located in different regions in the BZ. In that situation, one can either consider the wavefunction evolution along hyperspheres that surround more than one Fermi surface, or study the topological stability of each Fermi surface separately. Depending on this choice of one generally finds different stability characteristics. In the following, we make a few remarks on the topological stability of these multiple Fermi surfaces.

Fermion doubling.

Due to the Fermion doubling theorem [62], certain topologically stable Fermi surfaces, which in the continuum limit are described in terms of Dirac Hamiltonians, (e.g., the Weyl semi-metal) cannot be realized as single Fermi surfaces in lattice systems. That is, on a lattice these Fermi surfaces always appear in pairs with opposite K-theory charges. In that case the Fermi surfaces are not protected against commensurate perturbations, such as charge-density-wave, spin-density-wave, or other nesting-type perturbations that connect Fermi surfaces with opposite K-theory charges. However, these Fermi surfaces are individually stable, i.e., stable against deformations that do not lead to nesting instabilities.

Effective symmetry classes.

In the presence of multiple Fermi surfaces, anti-unitary symmetries (i.e., TRS and PHS) can act in two different ways on the system: (i) the symmetry maps different Fermi surfaces onto each other, or (ii) each individual Fermi surface is (as a set) invariant under the symmetry transformation. In case (i) the symmetry class of the entire system is distinct from the symmetry class of each individual Fermi surface. Hence, the topological number describing the stability of an individual Fermi surface differs from the topological invariant characterizing the entire system.

complex case ():
class
line point
A 0
AIII 0
complex case ():
class
surface line point
A 0
AIII 0 0
real case ():
class
line point
AI 0 0
BDI 0 0
D 0
DIII
AII
CII 0
C 0
CI 0
real case ():
class
surface line point
AI 0 0
BDI 0 0 0
D 0 0
DIII 0
AII
CII 0
C 0
CI 0 0
Table 4: (Symmetry of total system) Alternative classification of topologically stable Fermi surfaces in two- and three-dimensional systems ( and , respectively) as a function of Fermi surface dimension and symmetry class of the total system.
Classification of gapless topological phases from higher-dimensional topological insulators and superconductors.

Here we derive an alternative classification of gapless topological phases in terms of symmetries of the entire system (as opposed to the symmetries of restricted to a hypersphere surrounding an individual Fermi surface as in Sec. 2.2). To that end, we apply a dimensional reduction procedure to obtain -dimensional gapless topological phases from the zero-energy boundary modes of -dimensional topological insulators (fully gapped superconductors). Namely, we observe that the surface states of -dimensional topological insulators can be interpreted as topologically stable Fermi surfaces in dimensions. In fact, as was shown in Ref. [63], the bulk topological invariant of a -dimensional topological insulator is directly related to the K-theory topological charge of the boundary Fermi surface. Hence, we argue that the two Fermi surfaces that appear on either side of a -dimensional topological insulator can be embedded in a -dimensional BZ. Moreover, we recall from Sec. 2.2 that the classification of stable Fermi surfaces only depends on the codimension of the Fermi surface, since a -dimensional stable Fermi surface in dimensions can always be converted into a -dimensional stable Fermi surface in dimensions by including an extra momentum-space coordinate. Based on these arguments we find that the classification of ()-dimensional Fermi surfaces in terms of symmetries of the total system is obtained from Table 1 with , see Table 4.

Note that in the above construction of gapless topological phases the stable Fermi surfaces always appear in pairs (one from each of the two surfaces of the topological insulator) due to the Fermion doubling theorem. Therefore, these gapless topological phases are unstable against commensurate nesting-type deformations that connect Fermi surfaces with opposite K-theory charges.

2.4 Bulk-boundary correspondence

Topological characteristics of stable Fermi surfaces can lead to the appearance of zero-energy surface states via a bulk-boundary correspondence. We discuss this phenomenon in terms of a few specific examples defined in the continuum (similar considerations can also be applied to lattice systems, cf. Sec. 3).

Fermi rings in three-dimensional systems.

Consider first the case of two topologically stable Fermi rings in a three-dimensional system described by the Hamiltonian (see Fig. 1). These rings of gapless points occur, for example, in nodal topological superconductors (e.g., class DIII, AIII, or CI, see Sec. 3), or in topological semi-metals with sublattice symmetry (class AIII) [64]. The topological characteristics of these Fermi rings are determined by the topology of the wavefunctions along a circle enclosing the Fermi ring (red circle in Fig. 1(c)). That is, the stability of the Fermi ring is guaranteed by the conservation of a topological charge, which is given in terms of the homotopy number of the map of onto the space of Hamiltonians.

Figure 1: Illustration of integration path deformation in the three-dimensional BZ. The gray planes represent the two-dimensional surface BZ parametrized by the two surface momenta and . The light and dark blue areas indicate regions in the surface BZ where there appear zero-energy states.

Let us now discuss the appearance of zero-energy states at a two-dimensional surface of this system. To that end, we define a two-dimensional surface BZ (gray planes in Fig. 1) parametrized by the two surface momenta and . The third momentum component, which is perpendicular to the surface BZ, is denoted by . The appearance of a zero-energy state at a given surface momentum ) can be understood by considering a continuous deformation of the closed path in Fig. 1(c) into a infinite semi-circle (Fig. 1(a)), such that the diameter of the semi-circle is parallel to and passes through ). This path deformation does not alter the value of the topological number (i.e., the K-theory topological charge), as long as no Fermi ring is crossed during the deformation process. Furthermore, one can show that in the limit of an infinitely large semi-circle, the topological charge of the Fermi ring is identical to the topological number of the one-dimensional system . Hence, there appear zero-energy surface states at those momenta , where the corresponding one-dimensional gapped Hamiltonian  has nontrivial topological characteristics, i.e., at momenta that lie inside the projections of the Fermi rings of the bulk system. For the complex symmetry classes (i.e., class AIII for the present case) it follows that the zero-energy surface states occur in two-dimensional regions in the surface BZ that are bounded by the projections of the Fermi rings (light and dark blue areas in Fig. 1) [29, 30, 31]. In other words, the zero-energy states form two-dimensional surface flat bands. For the real symmetry classes (i.e., for symmetries that relate to in momentum space), it follows that zero-energy states appear at certain symmetry-invariant surface momenta that lie inside the projections of the Fermi rings [29, 31].

dimension of BZ () dimension of FS () dim. of sphere surrounding FS () dimension of surface flat band
2 0 1 1
3 0 2 1
3 1 1 2
4 0 3 1
4 1 2 2
4 2 1 3
Table 5: Dimensionality of zero-energy flat bands appearing at the boundary of a -dimensional system with a -dimensional stable Fermi surface in symmetry class A or AIII.
Fermi points in three-dimensional systems.

As a second example, let us consider a three-dimensional system with two topologically stable Fermi points. Topologically stable Fermi points can be found, for example, in Weyl semi-metals (class A) [54, 55, 46, 56, 57, 58]. The stability of these Fermi points is ensured by the conservation of the homotopy number of the map of onto the space of Hamiltonians, where surrounds one of the two Fermi points. To derive the existence of surface states on a given line, say, , within the surface BZ, we consider a continuous deformation of the sphere into a half-sphere, such that the diameter of the half-sphere is perpendicular to the surface BZ and passes through . As before, one can show that in the limit of an infinitely large half-sphere, the topological charge of the Fermi point enclosed by the half-sphere is identical to the topological invariant of the two-dimensional system . Thus, there appears a linearly dispersing surface state within , whenever the fully gapped two-dimensional Hamiltonian has a nontrivial topological character. For symmetry class A, we can repeat this argument for arbitrary . Therefore there exists a line of zero-energy modes in the surface BZ (i.e., an arc surface state) connecting the two projected Fermi points.

It is straightforward to generalize the above considerations to Fermi surfaces with arbitrary codimensions, provided that . The result is summarized in Table. 5. We find that for a -dimensional topologically stable Fermi surface in symmetry classes A or AIII, there appears a -dimensional zero-energy flat band at the boundary of the system. For , however, which corresponds to stable Fermi surfaces of codimension 1 (i.e., e.g., a two-dimensional Fermi surface in a three-dimensional BZ), there is no topological state appearing at the boundary of the system. The reason for this is that a Fermi surface of codimension 1 cannot be surrounded by a hypersphere in momentum space, since the Fermi surface separates the BZ into two distinct regions.

3 Protected surface states in nodal topological superconductors

To demonstrate the usefulness of the classification scheme of Sec. 2, we study in this section a few examples of topologically stable Fermi surfaces. Specifically, we examine topologically stable nodal lines in three-dimensional time-reversal invariant superconductors with and without spin- conservation. Using the generalized bulk-boundary correspondence of Sec. 2.4, the appearance of different types of topological surface states is discussed. Before introducing the specific Bogoliubov-de Gennes (BdG) model Hamiltonians in Secs. 3.2 and 3.3, we present in Sec. 3.1 a general derivation of topological invariants that characterize the stability of nodal lines in these systems [6, 7, 29, 40, 61, 65, 66, 67]. The robustness of the nodal lines and the associated topological surfaces states against disorder is discussed in Sec. 3.4.

3.1 Topological invariants

We start from a general lattice Hamiltonian describing time-reversal invariant superconductors with bands and two spin degrees of freedom. The following derivation of topological invariants (Sec. 3.1.1) is applicable to any Hamiltonian with chiral symmetry, i.e., any that anticommutes with a unitary matrix . This includes, in particular, BdG Hamiltonians in symmetry class AIII, DIII, and CI, where chiral symmetry is realized as a combination of time-reversal and particle-hole symmetry. The presence of chiral symmetry implies that can be brought into block off-diagonal from

(14)

where is a unitary transformation that brings into diagonal form. In order to derive the topological invariants, it is convenient to adiabatically deform into a flat-band Hamiltonian with eigenvalues . This adiabatic transformation does not alter the topological characteristics of . The flat-band Hamiltonian can be defined in terms of the spectral projector

where are the negative-energy eigenfunctions of , which are obtained from the eigenequation

(16)

Here, denotes the combined band and spin index. In Eq. (3.1), it is implicitly assumed that for the considered values there is a spectral gap around zero energy with , for all . By multiplying Eq. (16) from the left by one can show that the eigenfunctions of can be expressed in terms of the eigenvectors and of and , respectively,

(17)

where and are taken to be normalized to one, i.e., , for all . That is, we have [29]

(18)

We observe that the eigenvectors of follow from via

(19)

with the normalization factor . Combining Eqs. (3.1), (18), and (19) yields [29]

(20)

In other words, the off diagonal-block of is obtained as

(21)

As shown below, both and topological invariants can be conveniently expressed in terms of the unitary matrix .

3.1.1 topological invariant (winding number)

Topologically stable Fermi surfaces (or nodal lines) in symmetry class AIII exist for even codimension (see Table 3). The stability of these nodal lines is guaranteed by the conservation of an integer-valued topological number, namely the winding number of

with the totally antisymmetric tensor and

(23)

Here, denotes a hypersphere in momentum space surrounding the Fermi surface (nodal line). The winding number characterizes the topology of the occupied wavefunctions of restricted to , i.e., it describes the topology of on . In other words, represents the homotopy number of the map . For (i.e., ), Eq. (3.1.1) simplifies to

(24)

which describes the topological stability of Fermi surfaces (nodal lines) of codimension . In particular, defines the topological charge of stable nodal lines in three-dimensional time-reversal invariant superconductors [27, 28, 29] (see Secs. 3.2 and 3.3).

3.1.2 topological invariant

For time-reversal invariant superconductors in class DIII we can define, besides the winding number (3.1.1), also topological numbers, provided the consider hypersphere surrounding the nodal line/point is left invariant under the transformations (see Table 3). In the following, we derive these numbers for the cases and , and assume that the centrosymmetric hyperspheres and contain two and four time-reversal invariant points , respectively. With these assumptions, the topological numbers can be defined in terms of the Paffian of the sewing matrix 666 The Pfaffian is an analogue of the determinant. It is defined for antisymmetric matrices and can be expressed in terms of a sum over all elements of the permutation group , i.e. [6, 68, 69, 70, 71, 72],

(25)

where the product is over the two (four) time-reversal invariant momenta in () and

(26)

with . Here, denotes the -th eigenvector of with eigenvalue , is the time-reversal symmetry operator, and represents the complex conjugation operator. indicates a topologically trivial (nontrivial) character of the enclosed Fermi surface/nodal line. Due to the block off-diagonal structure of the flat-band Hamiltonian (20), a set of eigenvectors of , with , can be constructed as

(27)

or, alternatively, as

(28)

where are momentum-independent orthonormal vectors. For simplicity we choose . Observe that both and , with , are well-defined globally over the entire hypersphere . In the following we work with the basis . Eq. (26) together with Eq. (27) gives

(29)

In going from the second to the third line in Eq. (29), we used the fact that due to time-reversal symmetry . Thus, the topological number for and is

(30)

with the two (four) time-reversal invariant momenta of ().

3.2 Nodal topological superconductors with spin- conservation

As a first example, we study a three-dimensional time-reversal invariant superconductor with spin- conservation described by the BdG Hamiltonian , with . Rotational symmetry about the -axis in spin space is implemented by , with . Hence, the Hamiltonian can be brought into block diagonal form, , where and . It follows that the topology of is fully determined by the topology of . For concreteness, we consider 777 This model is equivalent to the polar state of He [73]. A two-layer version of this model might be realized in the pnictide superconductor SrPtAs [74, 75, 76].

(31)

The normal part of this Hamiltonian, , describes electrons hopping between nearest-neighbor sites of a cubic lattice with hopping integral , chemical potential , and spin-orbit coupling strength . The superconducting order parameter contains both even-parity spin-singlet and odd-parity spin-triplet components, denoted by and , respectively. Due to the presence of time-reversal symmetry the gap functions are purely real, and hence anticommutes with , i.e., . Therefore, belongs to symmetry class AIII and we find that this system can exhibit stable nodal lines (see Table 3). Indeed, the energy spectrum of Eq. (31), , shows a nodal ring, which is located within the -plane and centered around the axis (Fig. 2(a)). The nodal line is described by the manifold

(32)

The topological stability of this nodal ring is characterized by the winding number , Eq. (24). Evaluating for Hamiltonian (31) gives

(33)

where represents a circle in momentum space. We find that , whenever interlinks with the nodal ring (32). As discussed in Sec. 2.4, topologically nontrivial nodal lines of codimension in symmetry class AIII lead to the appearance of zero-energy surface flat bands. This is demonstrated in Figs. 2(b) and 2(c), which show that zero-energy surface states appear in a two-dimensional region of the surface BZ that is bounded by the projection of the nodal ring.

Figure 2: (a) Nodal structure of the energy spectrum of Hamiltonian (31). Here, we used the following parameters , , , , and . (b) Winding number , Eq. (33), as a function of surface momentum for the (001) face. The color scale is such that purple corresponds to , whereas light gray corresponds to . (c) Surface band structure for the (111) face as a function of with .

3.3 Nodal noncentrosymmetric superconductors

As a second example, we consider nodal noncentrosymmetric superconductors. The absence of bulk inversion symmetry in these materials leads to a spin splitting of the electronic bands by spin-orbit coupling. This in turn allows for the existence of mixed-parity superconducting states with both spin-singlet and spin-triplet pairing components. Over the past few years a number of (nodal) noncentrosymmetric superconductors have been discovered [77, 78, 79, 80, 81, 82], most notably LiPtB [83, 84], BiPd [85, 86], and the heavy-fermion compounds CePtSi [87, 88], CeIrSi [89], and CeRhSi [90]. Recently, nontrivial topology characteristics of nodal noncentrosymmetric superconductors have been studied both theoretically and experimentally [29, 30, 31, 32, 33, 85, 86, 91, 92, 93, 94]. Specifically, it was found that noncentrosymmetric superconductors belong to symmetry class DIII, which, according to Table 3, implies that three-dimensional noncentrosymmetric superconductors can support topologically stable nodal lines. To exemplify the topological features of these nodal superconductors we study in this subsection a simple BdG model Hamiltonian describing a single-band nodal noncentrosymmetric superconductor with monoclinic crystal symmetry (relevant for BiPd). Implications of some of our findings for experiments on BiPd will be discussed at the end of this subsection.

Model definition.

We start from the BdG Hamiltonian