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Higher-order topological band structures

The interplay of topology and symmetry in a material's band structure may result in various patterns of topological states of different dimensionality on the boundary of a crystal. The protection of these ``higher-order'' boundary states comes from topology, with constraints imposed by symmetry. We review the bulk-boundary correspondence of topological crystalline band structures, which relates the topology of the bulk band structure to the pattern of the boundary states. Furthermore, recent advances in the K-theoretic classification of topological crystalline band structures are discussed.


I. INTRODUCTION
Traditionally, the understanding of a material's band structure requires knowledge of symmetry representation theory. During the last decades it became increasingly clear that not only symmetries, but also concepts borrowed from topology, and often a combination of the two, are required for a complete understanding of band structures 1-4 . In particular, gapped band structures can be classified into different topological classes (also referred to as "topological phases"), where two band structures are in the same class if they can be smoothly deformed into each other by changing system parameters, without closing the excitation gap and without reducing the symmetry of the band structure at an intermediate stage. This topological classification of band structures applies equally well to superconductors if these have a gapped excitation spectrum in their BCS mean-field description.
An important practical consequence of the existence of distinct topological classes of band structures is that a nontrivial topology of the bulk band structure may imply the existence of anomalous boundary states. A boundary state is called "anomalous" if it cannot exist without the presence of the topological bulk. Anomalous boundary states are immune to local perturbations and can be removed only by a perturbation that closes the excitation gap of the bulk band structure or reduces its symmetry. This connection between a nontrivial topology of the bulk band structure and the existence of anomalous boundary states is referred to as bulk-boundary correspondence. For topological phases that are subject to nonspatial symmetries only, such as time-reversal symmetry or the particle-hole antisymmetry of the superconducting Bogoliubov-de Gennes Hamiltonian, the bulk-boundary correspondence is complete: Each topological class of a d-dimensional bulk band structure is uniquely associated with an anomalous boundary state of dimension d − 1 and vice versa 5,6 .
The question of a bulk-boundary correspondence for topological phases that are also subject to spatial symmetries, such as inversion or mirror symmetries, is more subtle. In general, in this case the bulk-boundary correspondence is incomplete and requires a degree of compatibility of the crystal termination and the crystalline symmetries. Immediately after the discovery of topological crystalline band structures 7,8 , it was understood that a conventional bulk-boundary correspondence, in which the anomalous boundary states of a d-dimensional crystal have dimension d − 1, exists only for a boundary orientation that is invariant under the action of the crystalline symmetry group 2,7-12 . This condition, however, can be met for a small number of symmetry groups only -mirror symmetry being an example -, and even in those cases is restricted to selected surface orientations. Recently, following pioneering work by Schindler et al. 13 , it was realized that topological crystalline phases may also have anomalous boundary signatures of dimension less than d − 1 [13][14][15][16][17][18][19][20][21][22][23][24][25][26] , provided the crystal termination as a whole respects the crystalline symmetry group. Examples are anomalous states at hinges or corners of a three-dimensional crystal or anomalous corner states of a two-dimensional crystal, see Fig. 1. The condition that the crystal termination as a whole respects the crystalline symmetry group is a much weaker condition on the surface orientations than the condition that the orientation of individual crystal faces is invariant under the crystalline symmetry. Moreover, it is a condition that can be met for all crystalline symmetry groups. Topological phases with this type of boundary signature are called higher-order topological phases, where the order n indicates the codimension of the boundary states. In this terminology, topological phases that do not rely on crystalline symmetries, which have boundary states of dimension d − 1, are called "first-order". Boundary states of codimension n ≥ 2 may also occur as the anomalous boundary states of nontrivial topological phase located on the crystal boundary 15,[27][28][29] . Since such states do not have their origin in the topology of the bulk band structure, they are referred to as extrinsic 18 ; anomalous boundary states that are rooted in the topology of the bulk band structure are called intrinsic. Although they are a property of the crystal termination, extrinsic anomalous boundary states still have some degree of topological protection. Specifically, extrinsic corner states cannot be removed by perturbations that respect the crystalline symmetries and do not close the gaps along hinges or surfaces of the crystal. Similarly, extrinsic hinge states cannot be removed by symmetrypreserving perturbations that do not close surface gaps.
In this article we review the arguments that show how intrinsic higher-order boundary states arise as a consequence of a topologically nontrivial band structure. We discuss the simplified model systems that appeared in the original publications and that have become paradigmatic examples of higher-order topological band structures. We also discuss the bulk-boundary correspondence for topological crystalline phases. Such a bulk-boundary correspondence was formulated by us for the case of "ordertwo" crystalline symmetries that square to the identity, such as mirror, inversion, or twofold rotation in Ref. 21. Here, these ideas are extended to more general crystalline symmetry groups. Throughout the review we restrict ourselves to "strong" topological phases, which are robust to a breaking of the lattice translation symmetry (while preserving the crystalline symmetries, of course).
We note that the relevance of topology to condensed matter systems is not only limited to understanding band structures. Indeed, as the pioneering work of Thouless, Kohmoto, Nightingale, and den Nijs shows 30 , the topological classes for quantized Hall systems can be defined even in the absence of a discrete translation symmetry. The results for strong topological phases that we consider in this review, such as the bulk-boundary correspondence in presence of a crystalline symmetry, remain valid if the lattice translation symmetry is broken in a manner that preserves the crystalline symmetries. Some auxiliary results, such as the definition of topological invariants, rely on the existence of a band structure.
The remainder of this article is organized as follows: In Sec. II we briefly review the ground rules for defining topological equivalence. In Sec. III we discuss three paradigmatic examples of topological band structures in one, two, and three dimensions, their boundary signatures, and how and under what conditions the existence of these boundary signatures is rooted in the topology of the bulk band structure. In Sec. IV we formulate the formal bulk-boundary correspondence for topological band structures with a crystalline symmetry. In Sec. V we discuss a specific example to make the rather general considerations of Sec. IV more explicit. We conclude in Sec. VI.

II. TOPOLOGICAL EQUIVALENCE
For a precise topological classification of band structures and the associated boundary signatures, one has to define the "rules" of topological equivalence. When taken literally, the definition of topological equivalence stated in the first paragraph of the introduction implies that two band structures with different numbers of occupied bands belong to different classes. It is customary to relax this criterion and allow that the separate addition of (topologically trivial) occupied or empty bands or sets of bands does not change the topological class of the band structure. This topological equivalence of band structures modulo the addition of trivial occupied or empty bands is known as "stable equivalence". The complete bulk-boundary correspondence for topological band structures subject to non-spatial symmetries was derived using the rules of stable equivalence 5,6 .
More precisely, under the rules of stable equivalence one considers pairs (H, H') of band structures with equal number of bands -an approach known as the Grothendieck construction. Two pairs (H 1 , H 1 ) and (H 2 , H 2 ) are considered topologically equivalent if H 1 ⊕ H 2 can be smoothly deformed into H 2 ⊕H 1 , without closing the excitation gap and without violating the symmetry constraints. With this definition, topological classes acquire a group structure, the group operation being the direct sum of band structures: Using the pair (H, H ) to represent its topological class, the group operation is (H 1 , H 1 ) ⊕ (H 2 , H 2 ) = (H 1 ⊕ H 2 , H 1 ⊕ H 2 ). The inverse group operation is also defined: (H 1 , H 1 ) (H 2 , H 2 ) = (H 1 ⊕ H 2 , H 2 ⊕ H 1 ). In this manner, the topological phases are classified with an Abelian classifying group K. The corresponding classification scheme is known as the "K-theory classification".
For completeness, we mention that the topological classification based on stable equivalence is not the only type of classification used in the literature. One example of a different classification is the "non-stable" classification of gapped band structures with a fixed number of bands. A well studied classification of this kind concerns the Hopf insulator 31 . The Hopf insulator does not have a bulk-boundary correspondence of the type mentioned above, although generalized bulk-boundary correspondence was formulated recently 32 . Like the stable classification, the non-stable classification also has a group structure, although the group operation can not be defined by superimposing two band structures, since the number of bands is kept fixed. For band structures defined on a d-dimensional sphere instead of the Brillouin zone, the group structure is given by concatenation completely analogous to the group structure of homotopy groups 33 . Another alternative classification scheme is one that allows for the addition of bands, but only if these are separated by gaps from the existing bands. With such classification rules, the resulting group structure of the classes can be non-Abelian 34,35 . The "fragile" topological classification scheme allows the addition of non-occupied bands or set of bands, but not for the addition of occupied bands 36 . As a result, the fragile topological classification results in a monoid 37 , in which addition is defined as a direct sum, but subtraction is not defined. A bulkboundary correspondence has been formulated for fragile topological phases 38 , although its relevance to condensed matter systems is unclear at the moment. Finally, one can define a binary classification that only distinguishes between "topologically trivial" and "topologically nontrivial" 39 , where a band structure is called "trivial" if it is "Wannierizable", i.e., it can be obtained from localized states that respect the symmetry constraints. If only local symmetries are imposed, this definition of topological triviality [40][41][42] agrees with the triviality of the stable equivalence classification. The two definitions differ when non-local crystalline symmetries are imposed. In that case, Wannierizable band structures are topologically connected to "atomic-limit phases". The binary classification does not have a group structure, since superimposing two "nontrivial" band structures may or may not result in a "trivial" band structure.

III. BOUNDARY SIGNATURES OF TOPOLOGICAL BAND STRUCTURES
In this Section we first consider three examples of gapped band structures with crystalline symmetries in dimensions d = 1, 2, and 3. The examples are meant to illustrate the role of the crystalline symmetries as well as the non-spatial symmetries for the topological characterization of the band structure and the protection of eventual boundary states. We then show how and when the boundary signatures derive from the nontrivial topology of the bulk band structure.
A. One-dimensional model with inversion symmetry As a first example, we consider the one-dimensional model known as the Su-Schrieffer-Heeger model or the "Kitaev chain" in its non-superconducting or superconducting realizations, respectively 43,44 , with "Dirac gamma matrices" Γ 0 = σ 1 and Γ 1 = σ 2 . The model is invariant under particle-hole conjugation P, with U P = σ 3 . If P is not enforced, Eq. (1) represents a one-dimensional chain with a two-atom unit-cell and nearest-neighbor hopping amplitudes that alternate between t and t + m, see Fig. 2a. With particle-hole symmetry, Eq. (1) represents the Bogoliubov-de Gennes Hamiltonian of a one-dimensional superconductor with a one-atom unit-cell, be it with a non-standard form of the particle-hole conjugation operation. In addition, the Hamiltonian (1) is also invariant under time-reversal T , the "chiral antisymmetry" C, and inversion I, with U T = 1, U C = U P U T = σ 3 , and U I = Γ 0 = σ 1 . The inversion operation I commutes with T , but it anticommutes with P, so that this model represents an oddparity superconductor if P is present. The model (1) describes a one-parameter family of Hamiltonians H(k), labeled by the parameter m. The spectral gap of the model (1) closes at m = 0. To see, whether this gap closing point represents a topological phase transition, one looks for the existence of "mass terms", perturbations to H(k) that anticommute with the matrices Γ 0 and Γ 1 . If such mass terms do not exist, a gap closing can not be avoided when m is tuned through the gapless point at m = 0 and the gapless point represent a topological phase transition. If it exists, the phases below and above m = 0 are topologically equivalent.
By inspection, one easily verifies that the model (1) allows a single mass term, M = σ 3 . This mass term is, however, incompatible with P, C, or I, so that the gapless point at m = 0 represents a topological phase transitions if at least one of these three symmetries is present.
With P or C the model (1) is in a first-order (i.e., noncrystalline) topological phase for −2t < m < 0 with a (Majorana) zero mode at each end. Without P or C, but with I, the model no longer has protected zero-energy end states, but it has an anomalous half-integer "end charge" if −2t < m < 0. The existence of the end charge follows from a calculation of the bulk polarization 45 . It can also be inferred from the fact that the mid-gap end state in the presence of P or C symmetrically removes a half-integer charge from the valence and conduction bands. If P or C is broken, the end states can be removed, e.g., by a local potential at each end, but the half-integer end charge is immune to the addition of a local perturbation, since the inversion symmetry I prevents charge flow through the insulating bulk. If P, C, and I are broken the existence of the mass term M = σ 3 implies that the model is topologically trivial.
Having no anomalous end states, the model (1) with broken P or C symmetry is an example of a "Wannierizable" or "atomic limit" band structure, a band structure for which there exists a basis of localized "Wannier functions". That such atomic-limit band structures can nevertheless be topologically distinct can be seen by inspecting the model (1) in the limits t/m = −1 and t/m = 0 in which the system dimerizes and the eigen-b) c) a) FIG. 2. Without particle-hole symmetry, the model (1) describes a one-dimensional lattice with a two-atom unit-cell and nearest-neighbor hopping amplitudes that alternate between t and t + m (a). In the "topological phase" −2t < m < 0 Wannier functions are localized between unit-cells (b), whereas they are localized in the center of the unit-cells if m < −2t or m > 0 (c). The two patterns can not be smoothly deformed into each other in the presence of inversion symmetry.
states are trivially constructed. For t/m = 0 the Wannier states are localized in the center of the unit-cells, whereas for t/m = −1 the Wannier states exist at the boundary of the unit-cells, see Figs. 2b and c. Since it is not possible to continuously move Wannier states from the position in Fig. 2b to the position in Fig. 2c without breaking the inversion symmetry, the two cases represent different topological phases. As it is the presence of a crystalline symmetry that rules out a continuous transition between the two topological phases, such topologically different atomic-limit insulators are referred to as "symmetry-obstructed atomic insulators" 39,46 .
One-dimensional gapped Hamiltonians with inversion symmetry, but without particle-hole antisymmetry, have a Z classification 9-12,47 , the topological index N corresponding to the difference of the number of occupied oddparity bands at k = π and k = 0. The generator of the classifying group is the topological equivalence class of the model (1) with −2t < m < 0. There are no anomalous corner charges, nor any other boundary boundary signatures, if N is even.

B. BBH model
The first model featuring intrinsic anomalous corner states was considered by Benalcazar, Bernevig, and Hughes (BBH) 14,48 . It is a four-band model, where t and t are real parameters and Γ + = τ 1 σ 0 , Γ − = −τ 2 σ 2 , Γ 1 = −τ 2 σ 3 , Γ 2 = −τ 2 σ 1 are mutually anticommuting hermitian matrices that square to one. The Hamiltonian (4) satisfies two mirror symmetries M x,y , With fourfold rotation symmetry R4 the model exhibits zeroenergy corner states in the presence of P or C. These corner states are intrinsic: They cannot be removed by an R4compatible change of boundary termination (b). With R4, but without P and C, the corners may have anomalous halfinteger charges. These, too, cannot be removed by an R4compatible change of boundary termination (c). If R4 is broken, zero-energy corner states can be removed by suitably added decorations, regardless of the presence of the mirror symmetries Mx and My (d).
with U x = τ 1 σ 3 and U y = τ 1 σ 1 , as well as a fourfold rotation symmetry R 4 with (R 4 ) 4 = −1, with Like the previous example, the Hamiltonian (4) is invariant under time-reversal T , particle-hole conjugation P, and the chiral antisymmetry C, with U T = 1, U P = U C = τ 3 . In the presence of particlehole symmetry, the BBH model (4) may be understood as a superconductor with a two-atom unit-cell. In this case the order parameter has odd mirror parity, but it transforms trivially under rotations. Without P, Eq. (4) has an intuitive explanation in terms of a tight-binding model on the square lattice with a four-atom unit-cell, see Fig. 3. Equation (4) does not have the Dirac-like form of Eq. (1), which allows a straightforward analysis of its topological content by counting mass terms. However, it can be smoothly transformed to a Hamiltonian of that form. This is most conveniently performed in the vicinity of the gapless point at t = ±t by parameterizing t = −t + m/2 (9) and rewriting Eq. (4) as where Γ ± = −(Γ 0 ± Γ 4 )/ √ 2 and we rescaled the terms proportional to Γ 0,4 . In the vicinity of the gapless point m = 0, the term proportional to Γ 4 can be sent to zero without violating any of the symmetries or closing a spectral gap. The remaining three terms have the desired Dirac-like form [compare with Eq. (1)].
Whether the gapless point at m = 0 represents a topological phase transition can be easily decided by inspection of the "mass terms" of the Hamiltonian (10). There are two such mass terms: M 1 = Γ 4 and M 2 = τ 3 σ 0 . The first of these is compatible with all symmetries discussed above, except for R 4 , which implies that the model (4) is topologically trivial unless R 4 is present. The mass term M 2 is compatible with T and with twofold rotation With fourfold rotation symmetry the model (4) has a nontrivial topological phase for −4t < m < 0. In the presence of P or C this phase has anomalous zero-energy (Majorana) corner states. If P or C are broken, the corner states may be removed by a local perturbation. In this case, the model (4) is in an obstructed atomic-limit phase in which -analogously to the one-dimensional example discussed above -anomalous half-integer corner charges remain 14,48,49 . These corner states or corner charges are intrinsic: they are a consequence of the topology of the bulk band structure and they exist independently of the lattice termination, as long as the termination is compatible with the fourfold rotation symmetry R 4 50 . To see this, observe that the relevant changes of boundary termination correspond to "decorating" the boundaries with one-dimensional chains with a gapped excitation spectrum. Although such "decorations" may have zeroenergy (Majorana) end states (with P or C) or fractional end charges (without P and C), the requirement that the decoration be R 4 -compatible means that the net number of zero-energy states added to each corner is even (with P or C) or that the net charge added to each corner is an integer (without P or C) 49,51 , see Fig. 3.
The bulk band structure of Eq. (4) is always trivial if R 4 is broken. 52 Nevertheless, if R 4 is broken, zero-energy corner states may also exist if P or C is present. In this case the corner states are extrinsic: Their existence depends on the lattice termination and they may be removed by decorating the boundaries with one-dimensional gapped chains, see Fig. 3. The same applies to the existence of half-integer corner charges for the case that P and C are broken. The presence or absence of the two mirror symmetries M x,y does not affect these conclusions. Reference 53 refers to the BBH model with mirror symmetries M x,y as a "boundary-obstructed topological phase". Under stable equivalence this concept is the same as that of an "extrinsic higher-order anomalous boundary state", which is the language we use in this review.
Two-dimensional gapped Hamiltonians with intrinsic anomalous corner states realize a second-order topological phase. The example of the BBH model shows that the identification of the relevant non-spatial and crystalline symmetries is key to deciding whether or not a given band structure represents a second-order topological phase. Indeed, the very same model (4) with the same choice of parameters −4t < m < 0 may be a second-order phase, an obstructed atomic-limit phase, or a trivial phase depending on whether or not P or C and fourfold rotation symmetry R 4 are enforced.
As in the one-dimensional example discussed in the previous Subsection, the BBH model is part of a larger family of topological Hamiltonians. For example, twodimensional Hamiltonians with P and R 4 have a Z 3 classification, see Refs. 54-56 and the appendix, where one factor Z describes a first-order topological phase with chiral (Majorana) edge modes. The second factor Z describes a sequence of topological phases for which the topological class of the BBH model (4) with −4t < m < 0 is the generator. There are no anomalous corner states if the corresponding topological index is even, consistent with the Z 2 nature of the Majorana corner modes. The last factor Z describes additional atomic-limit phases without boundary signatures.

C. Three-dimensional example
As a third example we discuss a three-dimensional generalization of the BBH model originally proposed by Schindler et al. 13 , where Γ 0 = τ 3 σ 0 , Γ k = τ 1 σ k , k = 1, 2, 3, and Γ 4 = τ 2 σ 0 . As in the previous example, see Eq. (10), the last term proportional to Γ 4 may be omitted without closing the spectral gap or violating any of the symmetries. Without this last term, the model (11) is invariant under timereversal T , with U T = τ 0 σ 2 , as well as a four-fold rotation R 4 , with U R = τ 0 e iπσ3/4 . The model (11)  In the presence of T the gapless point m = 0 is a topological phase transition between phases with and without gapless surface states, irrespective of the fourfold rotation symmetry R 4 . Schindler et al. observed that the mass term M = Γ 4 is not only antisymmetric under T , but also under the product T R 4 , so that if T R 4 is a good symmetry, the gapless point m = 0 still separates topologically different band structures, even if T and R 4 symmetries are broken individually. In this case, the topological phase is not characterized by gapless surface states (because these are gapped out on a generic surface if time-reversal symmetry is broken), but by a chiral gapless mode running along the crystal "hinges", see Fig. 4. This gapless mode is a manifestation of the topological nature of the bulk band structure and it can not be removed by changing the crystal termination. This can be understood by noticing that the relevant change of surface termination corresponds to decorating each of the four crystal faces by a two-dimensional quantized Hall insulator, which will change the number of chiral modes running along a hinge by an even number if the surface decoration is compatible with the T R 4 symmetry, see Fig. 4b. A boundary pattern shown in Fig. 4a, which has an odd number of chiral modes at each hinge, is "anomalous" -it cannot exist without a topologically nontrivial three-dimensional bulk. Because of the existence of anomalous boundary states of codimension two, the model (11) is a second-order topological phase.
The same boundary phenomenology can exist in the absence of crystalline symmetries. An early example was proposed by Sitte et al., who showed that chiral hinge modes are generic for a topological insulator in an external magnetic field 28 . Such hinge modes are not anomalous, however, since they can be removed by an appropriate surface decoration, see Fig. 5b. For this reason, the hinge states of the T R 4 -symmetric model of Eq. (11) are intrinsic, whereas hinge states that appear in the absence of a crystalline symmetry are extrinsic.
Without the term proportional to Γ 4 , the model (11) is not only invariant under time reversal and fourfold rotation, but it also satisfies three anticommuting mirror symmetries M i : Since the mass term Γ 4 is incompatible with each of these crystalline symmetries, imposing mirror or inversion symmetry while breaking T also results in a secondorder topological phase for −6t < m < 0 13,15,18-21 .

D. Higher-order topological phases
In general, a crystalline topological phase is called an nth-order topological phase if it has anomalous boundary signatures of codimension n if the boundary as a whole respects the crystalline symmetries. With this definition, a topological phase that does not rely on crystalline symmetries for its protection is a first-order topological phase. Since topological phases without crystalline symmetries always have first-order boundary signature, higher-order topological phases are necessarily crystalline topological phases. In the literature, topological phases with anomalous corner charges are sometimes also referred to as higher-order phases, although such obstructed atomic-limit phases fall outside the boundarybased classification scheme we will use in this review. (They are captured in the K-theory-based classification scheme of the bulk band structure.)

E. Domain-wall picture
To understand why a topological band structure with crystalline symmetries can give rise to higher-order boundary states, Refs. 15, 18-21 propose a "domain-wall picture". In its original form, the domain-wall picture applies if the crystalline symmetry group G admits a boundary orientation that is invariant under G. This is the case, e.g., for a mirror symmetry in two dimensions or for a rotation symmetry in three dimensions, see Fig. 6. For such an invariant boundary, the crystalline symmetry continues to act as a local symmetry at the boundary. Hence, the bulk-boundary correspondence for non-spatial (a) A two-dimensional crystal with a mirrorsymmetric edge, at which the mirror symmetry acts as a local symmetry (left) and a generic mirror-symmetric termination, at which the mirror symmetry acts as a "global" symmetry only (right). (b) A three-dimensional crystal with a rotationinvariant surface, at which the rotation symmetry acts as a local symmetry (left) and a generic fourfold rotation-symmetric termination, at which the rotation symmetry acts as a global symmetry (right). In both panels, a configuration of boundary mass terms is shown that corresponds to a second-order phase.
symmetries is applicable and guarantees the existence of an anomalous boundary state on that crystal face. The low-energy theory of that anomalous (d − 1)-dimensional boundary state has the form of a Dirac Hamiltonian, with anticommuting gamma matrices γ 2 i = 1, i = 1, . . . , d − 1. Again, we search for anticommuting mass terms µ 2 j = 1 with {µ j , γ i } = 0, i = 1, . . . , d − 1, j = 1, . . . , n. If the bulk topological phase is not a firstorder phase there must be at least one such mass term. The mass terms transform under a real representation O n of the crystalline symmetry group G, which cannot be the trivial representation, since otherwise H ∂ can be gapped out by a symmetry-preserving perturbation.
Higher-order boundary states appear when we consider deformations of the invariant boundary, so that G no longer acts locally, but continues to act on the crystal boundary as a whole. Examples of such deformations are shown in Fig. 6. Given the number n of mass terms and the representation O n of G one can derive the pattern of anomalous higher-order boundary states. Such boundary states of are of second order if d = 2. They are also of second order if d = 3 and n = 1 or if G leaves a one-dimensional subset of the deformed crystal face invariant (as is the case for, e.g., a mirror symmetry). Otherwise the boundary states are of third order in three dimensions.
A "trick" to extend this argument to symmetry groups G without invariant boundary orientation, such as inver-sion symmetry, was proposed in Ref. 18. The trick involves considering a (d + 1)-dimensional topological crystalline band structure with a symmetry group G obtained by acting with G on the first d coordinates, while leaving the (d + 1)th coordinate unchanged. For the (d + 1)-dimensional crystal the conditions of the domainwall argument are obviously fulfilled, so that one can establish the existence of higher-order boundary states using the domain-wall picture outlined above. Reference 18 then makes use of an isomorphism between (d + 1)dimensional topological band structures with symmetry group G and d-dimensional topological band structures with symmetry group G that was originally derived for non-crystalline topological phases by Fulga et al. 57 . Since this isomorphism preserves the order of the anomalous boundary states 12,18 , one can directly infer the existence of higher-order boundary states for the d-dimensional crystal with symmetry group G.
As an example, we consider the crystal described by Eq. (11). The T R 4 symmetry leaves surfaces at constant z invariant. The low-energy surface Hamiltonian has the form with γ 1 = σ 1 , γ 2 = σ 3 . The surface Hamiltonian H ∂ satisfies the product T R 4 of time-reversal and fourfold rotation symmetry, with U T R = e −iπσ2/4 . There is a single mass term µ = σ 2 , which changes sign under T R 4 . If one then deforms the invariant surface as in Fig. 6b, the faces related by fourfold rotation have opposite masses, so that there are domain walls with a sign change of the mass term at "hinges" between these faces 13 . The gapless chiral modes run along the domain walls. The same argument can be used if the model (11) is considered with a mirror symmetry M x or M y instead of with T R 4 symmetry 13,15 .

F. Boundary states from Dirac-like bulk Hamiltonian
The low-energy Dirac theory of the boundary and the transformation behavior of the boundary mass terms under the crystalline symmetry group can be obtained by direct calculation from the Dirac-like form of the bulk band structure. The starting point is the 2b-band Dirac Hamiltonian which is, e.g., the low-energy limit of the models (1), (10), or (11). Here the matrices Γ j are mutually anticommuting 2b × 2b matrices that satisfy Γ 2 j = 1, j = 0, 1, . . . , d.
The crystal boundary is modeled as the interface between regions with negative and positive m, with m negative in the interior of the crystal. Near the sample boundary, the Hamiltonian (17) has the form where x ⊥ = n · r is the coordinate transverse to the boundary, n is the outward-pointing normal, and Γ is a d-component vector containing the matrices Γ j , j = 1, . . . , d. We choose m(x ⊥ ) > 0 for x ⊥ > 0 and m(x ⊥ ) < 0 for x ⊥ < 0, so that the sample interior corresponds to negative x ⊥ . The Hamiltonian (18) admits b gapless boundary modes, the projection operator to the space of allowed 2b-component spinors being The effective b-band low-energy surface Hamiltonian is obtained using the projection operator P ( n). To illustrate this procedure, we consider a hypothetical circular or spherical "crystal" of radius R and use the polar coordinate φ or spherical coordinates (θ, ϕ) to parameterize n and the crystal boundary for d = 2 or d = 3, respectively 19 . We write the projection operator (19) as with P 2 = P ( e x ), P 3 = P ( e z ), V 2 ( n) = e φΓ2Γ1/2 , and V 3 ( n) = e (π−θ)Γ1Γ3/2 e ϕΓ2Γ1/2 . The projected Hamiltonian H at the boundary then reads, Mass terms M i , i = 1, . . . , n, of the bulk band structure (17) may be added as a position-dependent perturbation to the surface. Although such mass terms locally violate the crystalline symmetry group G, the position dependence of the surface perturbation ensures compatibility with G for the crystal as a whole. The matrices γ 2 = P 2 Γ 2 P 2 (for d = 2) or γ 1,2 = P 3 Γ 1,2 P 3 (for d = 3), combined with µ j = P d M j P d , j = 1, . . . , n, form a set of anticommuting gamma matrices and mass terms of effective dimension b. This gives the effective boundary Hamiltonian 7. A topological three-dimensional superconductor with mirror and twofold rotation symmetry admits Majorana corner modes at the rotation axis (a) or a pair of co-propagating Majorana hinge modes at the mirror plane (b). The difference between these two boundary signatures is a matter of crystal termination: The higher-order boundary modes in a crystal that has corner states as well as hinge modes are extrinsic (c).
The position dependence of the prefactors m j must be chosen such that the boundary Hamiltonian H ∂ as a whole is compatible with the crystalline symmetry group G. Hereto, we note that the induced representation u g of the crystalline symmetry operation g on the boundary Hamiltonian gives an n-dimensional real representation O n of G, so that compatibility with the crystalline symmetry group is ensured if the functions m j satisfy the requirements As a first example, we illustrate this procedure for the BBH model (4). If the BBH model is written in the form (10), a low-energy Dirac Hamiltonian of the form (17) is immediately obtained. There is only one mass term M = Γ 4 that is invariant under particle-hole conjugation P or the chiral antisymmetry C. This mass term changes sign under the fourfold rotation operation R 4 . It follows that the boundary Hamiltonian H ∂ is of the form (21) with the condition to ensure compatibility with respect to R 4 . The condition (24) implies the existence of four "domain walls" at which m(φ) changes sign. These domain walls each host an anomalous zero-energy state. The second example (which will be considered again in Sec. V) is a three-dimensional odd-parity superconductor with point-group C 2h , which is generated by commuting twofold rotation and mirror symmetries R 2,z and M z . The rotation symmetry is around the z axis; the mirror reflection is in the xy plane, see Fig. 7. Using the convention that both crystalline symmetries square to one, R 2,z /M z commute/anticommute with particle-hole conjugation P, respectively. This model is described by an eight-band Dirac Hamiltonian of the form (17) with Γ 0 = τ 2 σ 0 ρ 0 , Γ 1 = τ 1 σ 3 ρ 0 , Γ 2 = τ 1 σ 1 ρ 0 , and Γ 3 = τ 3 σ 0 ρ 0 . The relevant symmetries are represented by U P = 1, U M = τ 2 σ 2 ρ 2 , and U R = τ 0 σ 2 ρ 2 . There are two P-symmetric mass terms that break the crystalline symmetry: M 1 = τ 1 σ 2 ρ 1 and M 2 = τ 1 σ 2 ρ 3 . Both mass terms are symmetric under M z but antisymmetric under R 2,z . Hence, the effective surface theory is of the form (21) with the condition m j (θ, ϕ) = −m j (θ, ϕ + π), j = 1, 2.
Such a mass term has a singular "vortex"-like structure at the poles at θ = 0, π, resulting in the presence of protected zero modes there, see Fig. 7a. There are no protected second-order boundary states, since generically at least one of the two mass terms is nonzero away from the poles. It is interesting to point out that in this example one could also have chosen the single mass term M = τ 1 σ 2 ρ 0 . This mass term does not admit any further anticommuting mass terms. Since M is odd under M z but even under R 2,z , one would then have concluded that this model has a pair of co-propagating chiral Majorana modes at the mirror plane, see Fig. 7b. The difference between this boundary signature and the third-order boundary signature with Majorana zero modes at the twofold rotation axis is extrinsic, since it corresponds to a trivial bulk band structure. Indeed, by explicit construction, one verifies that it is possible to construct a boundary decoration that has Majorana corner modes at the rotation axis and two co-propagating chiral modes at the mirror plane, see Fig. 7c. Addition of this boundary decoration switches between the boundary signatures of Figs. 7a and b. Since the codimension-2 boundary signature of Fig. 7b can be eliminated in favor of the codimension-3 boundary of Fig. 7a by a suitable choice of termination, this model must be considered a third-order topological band structure.

G. Boundary-resolved classification
The K-theory classification classifies topological band structures without considering boundary signatures. In general, the classifying group K for a given combination of non-spatial and crystalline symmetries contains first-order topological phases, which do not rely on the presence of the crystalline symmetries for their protection, higher-order topological phases, as well as atomiclimit phases that do not have protected boundary states, but may or may not have boundary charges. To obtain a boundary-resolved classification, Ref. 21 proposes to consider a subgroup sequence where K (n) contains those elements of K that do not have intrinsic boundary signatures of order n or lower, see Fig. 8 for a schematic illustration. One verifies that K (n) is indeed a subgroup of K, since the "addition" of band structures (i.e., taking the direct sum) can not lower the order of the boundary signatures. In the language used above, this conclusion follows from the observation that under taking direct sums the number of boundary mass terms does not decrease. The quotient K (n+1) /K (n) classifies topological crystalline band structures with exactly n boundary mass terms on. Many of the examples discussed above are generators of the topological classes in these quotient groups.
To illustrate the use of the boundary-resolved classification (26), we give the subgroup sequences for one-dimensional inversion-symmetric odd-parity superconductors, of which the "Kitaev chain" (1) is an example 9-12,47 , and for two-dimensional superconductors with fourfold rotation symmetry R 4 , of which the BBH model (4) is an example, see Refs. 54-56 and the appendix. These equations summarize the discussions of the boundary resolved classifications of the last paragraphs of Secs. III A and III B, respectively.

IV. BULK-BOUNDARY CORRESPONDENCE FOR TOPOLOGICAL CRYSTALLINE PHASES
The bulk-boundary correspondence relates the topological classification of anomalous boundary states to the boundary-resolved classification (26) of the bulk band structure. It states that (i) where K (n) a is the classification group of anomalous nthorder boundary states and K (n) is the K-theory classification group of topological band structures without intrinsic boundary signature of order ≤ n, and that (ii) the topological crystalline band structures without anomalous boundary signature, which are classified by K (d) , can be smoothly deformed to atomic-limit phases. Reference 21 derives these relations using algebraic methods for order-two crystalline symmetries, crystalline symmetries that square to one. Such a general derivation is possible because of the existence of a complete K-theory classification in this case 11 . We will discuss a heuristic derivation of the bulk-boundary correspondence for general point group G at the end of this Section.

A. Anomalous boundary states
The classification group K (n) a classifies anomalous nthorder boundary states for a crystal shape that is compatible with the crystalline symmetry group G. We use the convention that the sum of an nth-order boundary state and boundary state of order larger than n is considered a boundary state of order n. The precise definition of the group K (n) a requires the notion of the G-symmetric cellular decomposition of a crystal 58 : Denoting the interior of the crystal by X, one writes where d is the spatial dimension and Ω k is the a set of disjoint "k-cells" c k -a k-cell is a k-dimensional subset of X that is homotopic to the interior or a k-dimensional sphere -which have the property that each element g ∈ G either leaves each point in c k invariant or bijectively maps the k-cell c k to a different k-dell c k . Further, for a pair of k-cells c k and c k in Ω d there is one and precisely one g ∈ G that maps these cells onto each other. Examples of G-symmetric cellular decompositions are shown in Fig. 9 for a crystal with inversion symmetry and for a crystal with mirror and twofold rotation symmetries.
To construct topological boundary states of dimension k−1, we first consider the allowed k-dimensional topological phases with support on the k-cell c k . Since the only relevant symmetries acting within c k are local -recall that each element g ∈ G either leaves each point in c k invariant or it does not act inside c k -any topological phase placed on c k satisfies the standard bulk-boundary correspondence. This establishes a one-to-one correspondence between topological phases with support on c k and boundary states at its boundary ∂c k . By placing kdimensional topological phases on Ω k in a G-compatible manner, we can generate protected k − 1-dimensional states on the crystal boundary ∂X, provided any topological boundary states that arise in the interior of the crystal mutually gap out. This procedure gives a construction of all topological boundary states on ∂Ω k ∩ ∂X, two examples for a three-dimensional crystal with inversion symmetry: Corner states at generic corners can be moved to ∂Ω1 by changing the crystal termination along two crystal hinges (a) and hinge states at a generic hinge can be moved to ∂Ω2 by changing the crystal termination at two crystal faces (b). In both cases, the hinges or faces at which the termination is changed are related by inversion.
both extrinsic and intrinsic. Since its "building blocks", topological phases defined on the k-cells, have a classification with a well-defined group structure, the result of this procedure has a group structure, too. Setting k = d + 1 − n, we refer to it as K (n) , the classifying group of all n-th order topological boundary states on ∂Ω d+1−n ∩ ∂X. Note that, in principle, codimensionn boundary states may also appear outside ∂Ω d+1−n , but such states can be moved to ∂Ω d+1−n by a suitable change of crystal termination along d + 1 − n-dimensional crystal faces, see Fig. 10.
To find the classifying group K topological states on ∂Ω d+1−n ∩ ∂X that appear as the boundary states of topological phases with support entirely within the crystal boundary ∂X. To classify such states that can be obtained by "decoration" of the crystal boundary ∂X, we make use of the induced G-symmetric cellular decomposition of the boundary ∂X, where Ω ∂ k = Ω k+1 ∩ ∂X. Figure 11 shows the induced Gsymmetric cellular decomposition of the crystal boundaries for the examples discussed above. Proceeding as before, all boundary states classified by D (n) can be obtained by "pasting" non-trivial topological phases (with the appropriate local symmetries, if applicable) onto kcells in Ω ∂ k where k ≥ d + 1 − n, with the requirement that all the states of dimension > d − n can be gapped out. The classifying group K (n) a of anomalous nth-order boundary states is then The importance of considering decorations by phases of dimension larger than d + 1 − n is that one should also consider decorations with higher-order topological phases with support on the crystal boundary. Reference 21 discusses an example for which this is relevant: A threedimensional inversion-symmetric time-reversal invariant superconductor. For this example, the classifying group K (3) = Z 2 is generated by a boundary state consisting of two Majorana Kramers pairs positioned on the boundary 0-cell Ω ∂ 0 , see Fig. 11a. Such a configuration of Majorana-Kramers pairs can also be obtained from a stand-alone two-dimensional T -symmetric superconductor with support on the crystal surface: In the G-symmetric cellular decomposition of the boundary, the two 2-cells in Ω ∂ 2 host a two-dimensional T -symmetric superconductor with helical Majorana modes. The two helical Majorana modes  Fig. 11a (a). The helical Majorana modes gap out at the interface between the two 2-cells, leaving behind two Kramers pairs of Majorana corner modes at inversion-related corners (b). Hence, for a crystal with inversion symmetry, a Kramers pair of Majorana corner modes is not an anomalous boundary signature because of the existence of a boundary decoration with the same signature.
gap out at the interface between the two 2-cells, leaving behind Kramers-Majorana zero modes at two inversionrelated corners as illustrated in Fig. 12.
To make this construction more specific, we will apply it to the example of an odd-parity topological superconductor with crystalline symmetry group C 2h in Sec. V.

B. Refined bulk classification
Close to a phase transition between two different topological phases, the low-energy description of the band structure can be chosen to take the Dirac form (17). Since we are interested in classifying strong phases that are protected by the point-group symmetry G, we may assume that the gap closing appears in the center of the Brillouin zone. For this reason, the (strong) topological classification of band structures is the same as the classification of Dirac Hamiltonians.
The key simplifying observation is that close to the phase transition, the low-wavelength description of the band structure is very "symmetric" and allows the classification problem with point-group symmetries to be mapped to a classification problem with local (i.e., onsite) symmetries or antisymmetries only 11,56,59 . With an onsite symmetry group G the Dirac Hamiltonian can be block-diagonalized, where each "block" is labeled by an irreducible representations of G. The individual blocks are no longer constrained by point-group symmetries but only by non-spatial symmetries that have same mathematical form as the fundamental symmetries P, T and/or C. Therefore, each block corresponds to one of the tenfold-way classes, the classification of which is well known 5,6 . The realization of a mapping between point-group symmetries and onsite symmetries was first provided by Shiozaki and Sato for order-two symmetries 11 , and later extended by Cornfeld and Chapman to all point-group symmetries 56 . Below we refer to this mapping as the "Cornfeld-Chapman isomorphism".
Once the topological classification K of gapped Dirac Hamiltonians is known, the next task is to refine this classification according to the boundary signatures of the different phases (26). This requires finding mass terms that are incompatible with the crystalline symmetry group G and evaluating the order of the boundary state associated with their transformation behavior under G. A complication is that there may be multiple choices for the mass terms with different numbers of mass terms and/or different order of the associated boundary signatures. If this complication occurs, the difference of boundary signatures that correspond to different choices of the mass terms is extrinsic, i.e., it is associated with a topological trivial bulk.
To find a proper correspondence between a Dirac Hamiltonian and its boundary signature, the configuration of mass terms with the maximal order of the boundary states has to be found. For simple examples this is most easily accomplished by inspection of the Dirac Hamiltonians corresponding to the generators of K, as was done in the examples discussed in Sec. III. We now discuss a systematic procedure that gives the same result.
The calculation of the subgroup K (n) of topological phases without boundary signature of order not lower than n proceeds in two steps. First one selects all ndimensional real (but not necessarily irreducible) representations O n of the point group G that correspond to (n + 1)th-order boundary signatures if n mass terms M 1 , . . . , M n were to transform under G with the representation O n . Second, the classification group K On for Dirac Hamiltonians with mass terms that transform under G in this manner is obtained. Since Dirac Hamiltonians in K On satisfy the full crystalline symmetry group G if the mass terms M 1 , . . . , M n are omitted, there is a natural inclusion K On → K. The subgroup K (n) ⊂ K is generated by the group K (n+1) and by the images K On → K, from all the representations selected in the first step.
The above procedure is simplified by the following two observations: First, the calculation of K (d) , which is needed as a starting point, is achieved using the observation made by Shiozaki 60 , that it is sufficient to consider only a single representation of the point group, the "vector representation" O vec d , in which the mass terms M 1 , . . . , M d transform in the same was as the position vector. Shiozaki proved this statement by showing that K O vec d is isomorphic to the classification of zero-dimensional phases placed onto Ω 0 , taking into account the symmetry restrictions imposed by the full crystalline symmetry group, which acts onsite on Ω 0 . Since all atomiclimit phases are obtained by placing zero-dimensional phases on different Wyckoff positions 21,46,61 , which for the case of point-group symmetries consists of single position c 0 ∈ Ω 0 , one concludes that the image of K O vec d → K is precisely K (d) . The second observation concerns the calculation of K (1) , which classifies Dirac Hamiltonians that are trivialized once the constraints posed by the point group symmetries are lifted. This group is most easily computed as the kernel of the inclusion K → K TF , where K TF is the tenfold-way classifying group without point-group constraints. This way, the bulk subgroup sequence (26) is readily obtained for d = 1, 2. In three spatial dimensions, the only additional task that needs to be accomplished is the computation of the subgroup K (2) , which requires considering all two-dimensional real representations O 2 corresponding to third-order boundary signatures.

C. Heuristic proof of bulk-boundary correspondence
The statement that the bulk classifying group K (d) of topological crystalline band structures without anomalous boundary signatures precisely describes the atomiclimit phases follows from the observation that the absence of anomalous boundary signatures implies that a crystal can be smoothly "cut" into smaller sub-units without generating gapless modes at interfaces. Performing this cutting procedure in a periodic manner gives the desired connection to an atomic-limit phase.
For the proof of the relation (29) the only nontrivial part is the surjectivity of the map K (n−1) /K (n) → K (n) a . In this regard, we note that the construction of the classifying group K (n) in Sec. IV A entails that all (anomalous) states on the boundary of a G-symmetric crystal X can be obtained by "embedding" 62 (d + 1 − n)-dimensional topological phases onto (d + 1 − n)-cells from the element Ω d+1−n of its G-symmetric cellular decomposition. It remains to be demonstrated that such "embedded" phases can be made translationally invariant. For order-two symmetries, this was proven with an algebraic method that constructs topological crystalline band structures with a Dirac low-energy description for a given element from K (n) a 21 . Alternatively, for order-two symmetries one can invoke a layer-stacking construction 63,64 . For an arbitrary symmetry group G, Refs. 38 and 65 propose the "topological crystal construction" 65 , in which X is viewed as a G-symmetric unit-cell of a periodic lattice, which is repeated periodically in space. In comparison to the construction of Sec. IV B this construction imposes one additional condition: Boundary states at the "seam" between neighboring unit-cells have to be gapped out. To the best of our knowledge, there is no proof that this condition is always met for an arbitrary symmetry group G, although we do not know of any counter examples.
Alternatively, the bulk-boundary correspondence (29) can be proven by independently calculating the bulk subgroup sequence and boundary classification using the methods reviewed in Secs. IV A and IV B. In the following Section we illustrate this procedure on one example.

V. EXAMPLE: ODD-PARITY SUPERCONDUCTOR
To make the construction of Secs. IV A and IV B more explicit, we apply it to the example of a threedimensional odd-parity topological superconductor with crystalline symmetry group C 2h (the point group generated by mutually commuting twofold rotation symmetry R 2,z and mirror symmetry M z ). Using the convention R 2 2,z = M 2 z = 1, M z anticommutes with particle-hole conjugation P, whereas and R 2,z commutes with P. An eight-band Dirac Hamiltonian for this symmetry class was discussed and analyzed in Sec. III F. An additional example in two dimensions can be found in the appendix.

A. Boundary classification
We first show that the boundary classification for this example is Calculation of the boundary classification groups requires the G-symmetric cellular decompositions of the crystal and the crystal boundary, which are shown in Figs. 9b and 11b, respectively. The result for K (1) follows immediately, as there are no first-order boundary statesthere are no non-trivial three-dimensional superconductors that can be placed onto 3-cells in Ω 3 . Second-order boundary states may be obtained by placing two-dimensional topological superconductors with chiral Majorana edge modes on the two 2-cells in the mirror plane or on the four 2-cells perpendicular to the mirror plane, see Figs. 13a and b. In the former case, the local M z symmetry imposes that the number of chiral Majorana modes of each of the two topological phases is even, since M z anticommutes with P. At the interior "seam" between the two 2-cells in the mirror plane the Majorana modes are counterpropagating and can be gapped out. Hence, placing topological phases at the 2-cells in the mirror plane results in a topological boundary state consisting of two co-propagating chiral Majorana modes in the mirror plane. The classifying group of such boundary states is 2Z, the same as the classification group of two-dimensional topological superconductors with an onsite symmetry anticommuting with P 21 . Placing two-dimensional topological superconductors with chiral Majorana edge modes on the four 2-cells perpendicular to the mirror plane does not result in a valid topological boundary state, since the chiral Majorana modes are co-propagating at the rotation axis and cannot be gapped out there, see Fig. 13b. We thus conclude that for this example K (2) = 2Z.
To construct third-order boundary states, we place one-dimensional topological superconducting phases on the two 1-cells along the rotation axis, see Fig. 13c. Because of the presence of the local R 2,z symmetry commut- 13. Construction of second-order boundary states for a crystal with Mz and R2,z symmetries by placing twodimensional topological superconductors on the two 2-cells in the mirror plane (a). Placing two-dimensional superconductors on the four 2-cells perpendicular to the mirror plane is not allowed, because the co-propagating chiral Majorana modes at the interior boundaries between cells can not be gapped out (b). Third-order boundary states with well-defined R2,z parity can be obtained by placing one-dimensional superconductors at the two 1-cells along the rotation axis (c). Extrinsic third-order boundary states, consisting of pairs of corner states with different R2,z parity, arise from one-dimensional topological phases at the four boundary 1-cells outside the mirror plane (d).
ing with P, such one-dimensional topological superconducting phases have a Z 2 2 classification, since the Majorana end states have well-defined R 2,z -parity. Since M z commutes with R 2,z , the two mirror-related Majorana states at the crystal center have the same R 2,z -parity and can mutually gap out. There are no one-dimensional topological superconductors with an onsite symmetry anticommuting with P 21 , so placing topological phases onto the two 1-cells in the mirror plane is not possible. We thus conclude that K (1) = Z 2 2 , corresponding to Majorana corner states on the rotation axis with even or odd R 2,z -parity.
To find the decoration group D (2) , we consider twodimensional topological superconductors placed on the eight faces of ∂Ω 2 , see Fig. 7c. The counter-propagating Majorana modes at the "seams" between boundary 2cells outside the mirror plane are gapped out, 66 while the pairs of co-propagating Majorana modes in the mirror plane remain. We conclude that D (2) = 2Z and, hence, K To find D (3) we add one-dimensional topological superconductors at the four boundary 1-cells outside the mirror plane, see Fig. 13d. The pairs of Majorana end states in the mirror plane can mutually gap out because M z anticommutes with P. What remains are pairs of Majorana corner modes at the two R 2,z -symmetric corners, one of each R 2,z parity. No one-dimensional topological superconductors can be pasted onto the four boundary 1cells in the mirror plane, because the existence of a local symmetry anticommuting with P rules out topological phases there. We conclude that D (3) = Z 2 and, hence, K

B. Bulk classification
We now use the general method outlined in Sec. IV B to show that from the bulk perspective the odd-parity superconductor with point group G = C 2h is classified by the subgroup sequence This subgroup sequence is consistent with the boundary classification (33). A Dirac Hamiltonian for this class was discussed in Sec. III F. There, the relevant mass terms could easily be found by inspection. To illustrate the systematic formalism of Sec. IV B, we here rederive the same results using the Cornfeld-Chapman isomorphism.
We first calculate the full classification group K of three-dimensional gapped Dirac Hamiltonians with particle-hole constraint U P with P 2 = 1, twofold rotation symmetry U R and mirror symmetry U M . At this point no explicit representation of the gamma matrices and of the symmetry representations U P , U R , and U M needs to be specified. The Dirac matrices Γ i provide a representation U Γ R = ie Γ1Γ2π/2 of twofold rotation symmetry and a representation U Γ CM = Γ 3 of a mirror antisymmetry. The superscript Γ denotes that these representations are constructed from the Dirac Hamiltonian and that they in general satisfy different algebraic relations than the point group symmetries R 2,z and M z . According to the Cornfeld-Chapman isomorphism the topological classification of Dirac Hamiltonians with the point group G is the same as the topological classification of Dirac Hamiltonians with the onsite symmetries Here the superscripts O and C indicate the representation acts as onsite symmetry or antisymmetry (i.e., chiral constraint), respectively. The local symmetries O R and C M obtained in this manner have different algebraic relations to P than the original symmetries R and M, although they still mutually commute: O R anticommutes with P, whereas C M commutes with P. By considering a basis with well defined parity ± under O R , the Dirac Hamiltonian is blockdiagonalized, H = diag (h + , h − ). Since P anticommutes with O R , h + and h − are related to each other by P, so that it is sufficient to classify the even-parity block h + . Since the only constraint on this block is the antisymmetry C M , it belongs to the tenfold way class AIII, which has a K = Z classification in three spatial dimensions 5,6 . To find the corresponding topological invariant, one has to write h + in the Dirac form similar to Eq. (35), with gamma matrices γ i , i = 0, 1, 2, 3, and find the representation u C M of the onsite antisymmetry within this block. The topological invariant N then reads 60 For the calculation of the subgroup K (3) classifying the atomic limit phases, we consider Dirac Hamiltonians with three mass terms M i , i = 1, 2, 3, and the three-dimensional vector representation O vec 3 (R 2,z ) = −O vec 3 (M z ) = diag(−1, −1, 1) of the point group. The classification group K O vec 3 , classifies an extended Dirac Hamiltonian with three "defect coordinates" x, y, and z, where H is the Dirac Hamiltonian given in Eq. (35).
Similarly as before, we construct a representationŨ Γ R = e Γ1Γ2π/2 e M1M2π/2 of twofold rotation symmetry and U Γ M = iΓ 3 M 3 of mirror symmetry using the gamma matrices and mass terms appearing in Eq. (38). Applying the Cornfeld-Chapman isomorphism, we map the problem of classifying the defect Hamiltonian (38) with the spatially non-local symmetry constraints R and M to that of the classification with the local symmetry con-straintsŨ In this case, the onsite symmetryÕ R commutes with P, whereasÕ M anticommutes with P. We use a basis with well-defined parity underÕ R to write The two blocksh ± of even/odd parity states are classified independently. Each of these blocks can again be divided into two subblocks,h ± = diag (h ±,+ ,h ±,− ), defined according to the parity underÕ M . The subblocksh ±,+ and h ±,− are not independent, since they are mapped onto each other by P. Hence, only two independent blocks remain, say h ±,+ . Each of these is in tenfold-way class A, because no further symmetry constraints apply. As shown by Teo To calculate the image in K, we first transform back to the original formulation with spatial symmetries M and R using the inverse of Eq. (39). This gives U R = ±ρ 3 σ 3 and U M = τ 3 µ 3 . Next, we use Eq. (36) to transform to the onsite constraints U O R = ±ρ 2 σ 1 and U C M = µ 1 and project onto the even-parity block h + . Calculating the topological invariant (37) then gives N = ±2. We conclude that the image of K O vec 3 → K is K (3) = 2Z. Finally, to calculate K (2) we need to consider all twodimensional representations of two mass terms that correspond to third-order boundary states for all relevant two-dimensional real representations O 2 of C 2h . We here consider the representation O 2 (M z ) = −O 2 (R 2,z ) = diag(1, 1). Other representations are possible, too, and can be treated with the same formalism, but do not affect our conclusions. The classifying group K O2 classifies defect Hamiltonians of form where H is the Dirac Hamiltonian of Eq. (35). The Cornfeld-Chapman isomorphism maps the point-group symmetries to local symmetry representations, whereŌ R andC M commute with P and mutually. Blockdiagonalizing H O2 according to theŌ R -parity gives two independent blocksh ± , which are effectively in tenfoldway class BDI because of the local constraintsC M and P.
The topological classification of three-dimensional Dirac Hamiltonians with two defect coordinates x and y is the same as classification of one-dimensional Dirac Hamiltonians 67 , which have the classifying group Z for class BDI. Since the two blocks with even and odd O R -parity are independent, we arrive at the classifying group K O2 = Z 2 .
To find the image of the inclusion K O2 → K, we start from an explicit realization of the two generators of a three-dimensional Dirac Hamiltonian in class BDI with defect dimension two, with constraintsū P = 1,ū C = µ 1 , andū O R = ±1. As before, we first map back to the formulation with the spatial symmetries U M and U R using the inverse of Eq. (42), which gives U R = ±µ 1 τ 2 σ 2 and U M = µ 2 τ 1 , and then map to a formulation with the onsite constraints U O R = ±µ 1 τ 2 and U C M = µ 1 using Eq. (36). Finally, we transform to a basis with well-defined U O R -parity and find the topological invariants N = ±1. From this, we conclude that the K (2) = Z.

VI. CONCLUSION
The discovery of higher-order topological phases 13 provided important insights for the classification of topological crystalline insulators and superconductors. In one set of classification approaches 19,26,58,65 , one first classifies anomalous higher-order boundaries and symmetryobstructed atomic limits, and then, assuming bulkboundary correspondence, finds the bulk classification group K. A different classification paradigm uses algebraic methods to classify Dirac Hamiltonians [9][10][11]56 . Both approaches provide complete classifications and a set of generating models; the latter approach provides minimal Dirac models, whereas in the former one uses the "topological crystalline construction" 65 which typically results in models with a non-minimal unit cell.
The quest for novel topological materials requires not only the knowledge of topological classification, but also an efficient method to relate the given band structure to its boundary signatures. One possibility in this regard is an algorithm that "deforms" a given band structure to a form that is close to a Dirac-like phase transition. The other possibility, which is actively pursued by many research groups, concerns the construction of easyto-compute topological invariants. If furthermore, such topological invariants are designed to "detect" only band structures with anomalous boundary states, one refers to these as symmetry-based indicators 39,46 . Symmetrybased indicators were initially constructed for insulators, and recently extended to superconductors 61,68-74 . Although symmetry-based indicators in general do not provide a full classification, their construction and subsequent application resulted in the discovery of many new topological insulator materials 75,76 . It remains to be seen if the same will be the case for topological superconductors.

Boundary classification
We first show that the boundary classification groups for this example are The R 4 symmetric cellular decomposition is shown in Fig. 14. First, we note that there can be no stable gapless points (Majorana fermions) at the 0-cell c 0 ∈ Ω 0 . To see this, note that since R 4 acts onsite at c 0 , any (Majorana) zero-energy bound states have well-defined angular momentum j = 1/2, 3/2, 5/2, or 7/2 (mod 4). Particle-hole conjugation pairwise connects these angular-momentum states, so that they can always gapped out.
To classify first-order phases, we place two-dimensional Chern superconductors onto the four 2-cells from Ω 2 , see Fig. 15a. The counter-propagating chiral Majorana modes can be gapped out on 1-cells from Ω 1 , whereas, as shown above, the 0-cell from Ω 0 does not support stable zero-energy states. Since Chern superconductors have a Z classification, we obtain K (1) a = K (1) = Z. Anomalous second-order phases are classified by placing one-dimensional Kitaev chains onto 1-cells from Ω 1 and using the fact that Ω 0 does not support topologically protected Majorana zero-states, see Fig. 15b. This gives us K (2) = Z 2 . Extrinic states are obtained by placing Kitaev chains onto bounday 1-cells from Ω ∂ 1 , see Fig. 15c; Since all boundary states obtain this way can be gapped out, it follows D (2) = 0 and, hence, K (2) a = Z 2 .

Bulk classification
The general method outlined in Sec. IV B gives the bulk subgroup sequence for two-dimensional odd-parity superconductor with point group G = C 4 15. Classification of first-order phases by pasting twodimensional Chern superconductors onto 2-cells from Ω2 (a). Second-order boundary states are classified by pasting one-dimensional superconductors onto 1-cells from Ω1 (b). Extrinsic second-order states are classified by pasting onedimensional superconductors onto boundary 1-cells from Ω ∂ 1 .
which is consistent with the boundary classification (A1).
To obtain the bulk classification group K, we first classify Dirac Hamiltonians subject to the antiunitary antisymmetry P with P 2 = 1 and to the fourfold rotation symmetry R 4 , for which with the representation U R satisfies U 4 R = −1 and commutes with P. To apply the Cornfeld-Chapman isomorphsm, we construct an additional fourfold symmetry representation U Γ R = e Γ1Γ2π/4 from the Dirac gamma matrices. The Cornfeld-Chapman isomorphism then maps the fourfold rotation symmetry R 4 to the fourfold local symmetry The local symmetry U O commutes with P and satisfies U 4 O = 1. Considering the basis with well defined angular momentum j = 0, 1, 2, 3 (mod 4), the Hamiltonian (A3) takes block-diagonal form H = diag(h 0 , h 1 , h 2 , h 3 ). The blocks h j with even angular momentum are subjected to particle-hole constraint P and belong to tenfold-way class D, so that they have a Z classification. On the other hand, the blocks h 1 and h 3 are related by particle-hole conjugation, hence for their classification we need to consider the block h 1 only. Since this block does not have any constraints, it belongs to class A and has a Z classification in two spatial dimensions. We thus conclude that K = Z 3 . The topological invariants are (Ch 0 , Ch 1 , Ch 2 ) where Ch j is the Chern number corresponding to the block with angular momentum j, where the Hamiltonian h j is written in the form (A3) with Γ i matrices γ (j) i . To find the group K (2) , we calculate the classifying group K O2 of Dirac Hamiltonians with two defect cordinates x and y, Similarly as before, we use the Gamma matrices to construct the representation of the fourfold rotation sym-metryŨ Γ R = e Γ1Γ2π/4 e M1M2π/4 . The corresponding local symmetry readsŨ which commutes with P and satisfiesŨ 4 O = −1. In the basis with well defined angular momentum j = 1/2, 3/, 5/2 ,7/2 (mod 4), the Hamiltonian (A6) takes block-diagonal form H O2 = diag(h 1/2 , h 3/2 , h 5/2 , h 7/2 ). Particle-hole symmetry relates the blocks h j and h −j to each other, thus we need to classify the blocks h 1/2 and h 3/2 only. These two blocks have no symmetry constraints and belong to class A. The classification of twodimensional Hamiltonians with two defect dimensions is isomorphic to the classification of zero-dimensional Hamiltonians 67 , i.e., K O2 = Z 2 .
It remains to find the image of the inclusion K (2) → K. Hereto, we calculate the topological invariants Ch 0 , Ch 1 and Ch 2 for the two generators of K O2 . We represesnt these generator by Hamiltonians of the form (A6), with Gamma matrices and mass terms given bỹ γ 0 = −τ 3 µ 3 , (γ 1 ,γ 2 ) = (τ 3 µ 3 σ 2 , τ 3 µ 2 ), (m 1 ,m 2 ) = (τ 3 µ 3 σ 1 , τ 3 µ 1 ), and with the representations U P = τ 1 andũ O = e iτ3π/4 , e 3iτ3π/4 for particle-hole conjugation and for the on-site fourfold symmetry for the two generators, respectively. (The only difference between the two generators is the representation ofũ O .) Using Eq. (A7), we find that the original fourfold rotation symmetry has the representation U R = e iµ2σ1π/4 e iµ2σ2π/4ũ O . The corresponding local symmetry U O then reads Forũ O = e iτ3π/4 there are four states with angular momentum j = 0 and four states with odd angular momentum. Hence, Ch 2 = 0, and explicit calculation gives Ch 0 = −2Ch 1 and Ch 1 = 1. Similarly, for the generator withũ O = e iτ33π/4 we find that Ch 1 = −2Ch 2 and Ch 2 = 1. Hence, the image of K O2 → K is generated by the elements (2, −1, 0) and (0, −1, 2). As a subgroup of K, the image of K O2 → K is isomorphic to Z × 2Z. In order to find the subgroup K (1) one can proceed in two ways: either one classifies Dirac Hamiltonians with a single mass term transforming under the one-dimensional representation O 1 (R 4 ) = −1 or one finds the kernel of the homomorphism K → K TF where the group K TF = Z classifies two-dimensional phases in class D without additional constraints. It is easy to see that this homomorphism acts as (Ch 0 , Ch 1 , Ch 2 ) → Ch 0 + 2Ch 1 + Ch 2 , hence K (1) = Z 2 .
For illustration purposes, we also calculate K (1) using the alternative approach, i.e., by considering the onedimensional representation O 1 (R 4 ) = −1. The group K O1 classifies the Hamiltonians Altough the above Hamiltonian is not in Dirac form, we find the representation of the fourfold rotation an-tisymmetryŪ Γ CR = e Γ1Γ2π/4 M . Using the Cornfeld-Chapman isomorphism we classify Hamiltonians of the form of Eq. (A11) with onsite antisymmetrȳ where (Ū C R ) 4 = 1. The twofold rotation symmetry U O R2 = (U C R ) 2 commutes with P, and states with U O R2 = ±1 define two blocks H O1 = diag(h + , h − ). The block h + has chiral constraintŪ C R that squares to one, thus it belongs to class DIII. Similarly, we find that the block h − belongs to class BDI. Therefore, K O1 = Z × Z 2 . The Z 2 part of K O1 has to be in the kernel of the inclusion K O1 → K = Z 3 . We only need to consider the generator of the free part of the group K O1 . The generator Hamiltonian reads γ 0 = τ 3 σ 2 , (γ 1 ,γ 2 ) = (τ 3 σ 1 , τ 3 σ 3 ), m = τ 2 , with symmetriesū C R = iτ 1 , U P = 1. The representation of the crystalline symmetry is U R = e iσ2π/4 τ 3 . The local symmetry representation is found to be U O = τ 3 . Thus the generator has topological invariants Ch 0 = Ch 2 = 1 and Ch 1 = 0. The product of the subgroup K (2) ⊆ K and the subgroup of the elements Ch 0 = −Ch 1 , Ch 1 = 0 gives the subgroup K (1) previously found.