Higher-order bulk-boundary correspondence for topological crystalline phases

We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order topological phases with an additional order-two symmetry or antisymmetry is contained in this work.


I. INTRODUCTION
A central paradigm in the field of topological insulators and superconductors is the bulk-boundary correspondence: A nontrivial topology of the bulk band structure uniquely manifests itself through a gapless, topologically nontrivial boundary, irrespective of the orientation of the boundary or the lattice termination. [1][2][3] On the other hand, for topological crystalline phases, which are protected by an additional non-local crystalline symmetry, 4-25 the existence of gapless boundary states for a nontrivial bulk topology is guaranteed only if the boundary is invariant under the crystalline symmetry.
Recently, it was realized that a nontrivial crystalline topology of a d-dimensional crystal may also manifest itself through protected boundary states of dimension less than d − 1. [26][27][28][29][30][31][32][33][34][35][36][37][38] A topological phase with such lowerdimensional boundary states is called a "higher-order topological phase", where the order n of the topological phase corresponds to the codimension of the boundary states. 27 [According to this definition, a topological insulator or superconductor with the conventional (d − 1)dimensional boundary states is a first-order topological phase.] The condition that guarantees the protection of such higher-order boundary states is that the orientation of the crystal faces and the lattice termination be compatible with the crystalline symmetry -i.e., the crystal faces and the corresponding lattice termination must be related to each other by the crystalline symmetry operation -, which is a much weaker condition than the condition that the crystal boundary be invariant under the symmetry operation (compare with Fig.  1). For example, whereas inversion symmetry leaves no crystal faces invariant, compatibility with inversion symmetry merely requires that crystal faces appear in inversion-related pairs (see Fig. 1c). Topological crystalline insulators with second-order boundary states were theoretically predicted for models with certain magnetic symmetries, 27 mirror symmetry, 27,29 , and rotation and inversion symmetries. 20,28,30,39 The latter two symmetries are relevant for the semimetal Bi, which shows boundary states reminiscent of that of a second-order topological insulator. 40 The presence of a crystalline symmetry is not a necessary requirement for the boundary phenomenology associated with a higher-order phase. Indeed, early examples of protected codimension-two boundary states include the superfluid 3 He-B phase 41 and a three-dimensional topological insulator with a suitable time-reversal breaking perturbation, 42,43 neither of which rely on the protection by a bulk crystalline symmetry. Instead, in these cases the appearance of higher-order protected boundary states can be solely attributed to a boundary termination that is itself topologically nontrivial, whereas the underlying bulk is essentially trivial. In Ref. 39 we called these termination-dependent higher-order topological phases extrinsic, to contrast them with the intrinsic, termination-independent higher-order boundary states of topological crystalline phases. Although for intrinsic higher-order topological phases, too, the precise form of the (d − 2)-dimensional boundary states may still depend on details of the lattice termination, their very existence is a consequence of a nontrivial bulk topology and is protected as long as the crystal termination remains compatible with the crystalline symmetry.
While a complete classification of higher-order topological phases (HOTPs) is still lacking, several authors have obtained partial classifications of higher-order topological phases, restricted to certain crystalline symmetries or for a certain Altland-Zirnbauer class. 20,29,39,44 (The Altland-Zirnbauer classes are defined with respect to the presence or absence of the fundamental non-spatial symmetry operations time-reversal T , particle-hole conjugation P and the chiral operation C = PT ). 45 ) Two approaches have been taken for the classification of intrinsic, termination-independent HOTPs: A bulk-based approach, which starts from the classification of the bulk band structure and then shows under which circumstances a nontrivial bulk topology implies a higherorder topological phase, 29,39 and a boundary-based approach, in which all topologically nontrivial boundaries of HOTPs are classified first, and a classification of intrinsic, termination-independent HOTPs is obtained upon identification of boundary states that are related by a change of termination. 20,39,44 For crystalline phases with arXiv:1805.02598v1 [cond-mat.mes-hall] 7 May 2018 a) b) c) FIG. 1. Schematic pictures of a two-dimensional crystal for which the shape is compatible with mirror symmetry (a and b) and with twofold rotation symmetry (c). The crystal in (b) has a boundary that is invariant under the mirror symmetry, whereas the boundaries of the crystals in (a) and (c) appear in symmetry-related pairs. The special situation of a crystal with mirror symmetric boundary, as shown in panel (b), is excluded from the definition of the higher-order topological phases.
an order-two crystalline symmetry, for which a complete classification of the bulk topology exists, 21 the two approaches were found to be in complete agreement for the second-order topological phases. 39,44 The boundary-based approach not only classifies the intrinsic, termination-independent HOTPs, but also the extrinsic higher-order topological phases, for which the higherorder boundary states are a manifestation of a nontrivial boundary topology rather than a nontrivial bulk topology.
In this work we provide a full classification of higherorder topological phases with an order-two crystalline symmetry or antisymmetry and in arbitrary spatial dimension d. A crystalline symmetry or antisymmetry S is called "order-two" if S 2 = ±1. Its spatial type is determined by the number d of inverted dimensions, such that d = 0 corresponds to onsite (anti)symmetry, and d = 1, 2, 3 to mirror, twofold rotation, and inversion (anti)symmetry, respectively. From a bulk perspective, we compute a subgroup series which resolves the topological crystalline phases according to their associated boundary signature. The last term in Eq. (1) K ≡ K (0) is the group that classifies the bulk band structure with an order-two symmetry (antisymmetry). The other terms K (n) ⊆ K are subgroups that exclude topological phases that are of order n and lower for any crystal shape consistent with the crystalline symmetry. The subgroup K , which classifies topological crystalline phases that are not strong, first-order topological phases, was previously studied in Ref. 39 in the context of crystals with mirror, twofold rotation or inversion symmetry, where it was called the "purely crystalline subgroup". Note that the definition of the groups K (n) excludes crystals with boundary states that can be removed by a symmetry-respecting deformation of the crystal, such as the gapless surface states on a mirror-symmetric surface of a mirror-symmetric crystal, see Fig. 1.
From the boundary perspective, we introduce classifying groups K (n) k , which classify configurations of pro-tected boundary states of codimension n, modulo continuous symmetry-preserving transformations of the Hamiltonian that preserve the bulk gap and boundary gaps of boundaries of codimension n − k and lower. The case k = 1 (for which a slightly different definition needs to be used, see Sec. IV) allows only transformations that preserve all gaps for boundaries of dimension > d − n and is referred to as the "extrinsic" classifying group K (n) e . 39 The case k = n allows all transformations that preserve the bulk gap and is referred to as the "intrinsic" classifying group K (n) i . The cases 1 < k < n are "mixed" classifications that are relevant for higher-order phases with n ≥ 3 and were not considered previously. 46 With these definitions of the bulk-based and boundarybased classification groups, the bulk-boundary correspondence takes the form K (n+1) i = K (n) /K (n+1) , n = 0, 1, 2, . . . , d. ( In case the number of inverted dimensions d < d the subgroup series (1) starts with one or more of trivial groups, so that Eq. (2) yields a complete bulk-boundary correspondence for order-two crystalline symmetries, for which a topologically nontrivial bulk is uniquely associated with a higher-order topological phase. On the other hand, if d = d (inversion symmetry), the first group in the subgroup series (1) K (d) may be nontrivial. In that case there is only a partial bulk-boundary correspondence and K (d) classifies the topological crystalline phases without topologically protected boundary states.
The above results will be expanded and made more precise in Section IV. Our calculation of the classifying groups K (n) and K (n) k strongly relies on the bulk classification of topological crystalline insulators of Shiozaki and Sato, 21 whose classifying group K equals the last group in the subgroup sequence (1). As will be shown in Sec. IV, explicit results for all groups K (n) and K (n) k can be given in terms of the bulk classifying groups of Ref. 21, the K-groups classifying topological band structures without crystalline symmetries, [47][48][49] and homomorphisms between these groups.
The remainder of this article is organized as follows: In Sec. II we review the Shiozaki-Sato classification of the topological crystalline phase stabilized by an order-two symmetry, and introduce the dimension-raising isomorphisms. In Sec. III we discuss Hamiltonians of "canonical form" and show how higher-order phases arise from the presence of crystalline-symmetry-breaking mass terms, generalizing the conclusions of Refs. 27, 29, and 39 for second-order topological phases. In Sec. IV we give the formal definitions of the groups K (n) and K (n) k , obtain explicit expressions, and establish the bulk-boundary correspondence (2). Section V discusses a few representative examples. We conclude in Sec. VI. The appendices contain results for all classifying groups introduced in C   TABLE I. The ten Altland-Zirnbauer classes are defined according to the presence or absence of time-reversal symmetry (T ), particle-hole antisymmetry (P), and chiral antisymmetry (C). The entries T ± (P ± ) denote that T 2 = ±1 (P 2 = ±1).
The chiral symmetry is assumed to square to one.
Sec. IV, as well as some derivations not presented in the main text.

II. SHIOZAKI-SATO CLASSES FOR TOPOLOGICAL PHASES WITH AN ORDER-TWO SYMMETRY
The ten Altland-Zirnbauer classes are defined according to the presence or absence of time-reversal symmetry T , particle-hole antisymmetry P, and chiral antisymmetry C, see Table I. Shiozaki and Sato 21 extend the Altland-Zirnbauer classes to include an additional crystalline unitary symmetry, 8,9,16 unitary antisymmetry, antiunitary symmetry or antiunitary antisymmetry S. The crystalline symmetry is an order-two symmetry, which means that its square is proportional to the identity operation.
It is sufficient to distinguish symmetry operations that square to one (labeled by η S = +) and to minus one (η S = −). Further, the algebraic structure of the crystalline symmetry is characterized by signs η T ,P,C indicating whether S commutes (η = +) or anticommutes (η = −) with the time-reversal operation T , particle-hole conjugation P, or the chiral symmetry operation C. Following Ref. 21, we denote the number of spatial degrees of freedom that inverted under the crystalline symmetry operation by d , so that onsite symmetries O have d = 0, reflections M have d = 1, twofold rotations R have d = 2, and inversion I has d = 3. Specifically, unitary symmetry (σ S = 1) and antisymmetry (σ S = −1) operations are represented by unitary matrices U S , with Sk = (−k , k ⊥ ), k = (k 1 , . . . , k d ), k ⊥ = (k d +1 , . . . , k d ) and Similarly, antiunitary symmetry and antisymmetry operations are  represented by unitary matrices U S , such that U S U * S = η S , U S U * T = η T U T U * S , U S U * P = η P U P U * S , and U S U * C = η C U C U S . The above characterization of unitary and antiunitary symmetry operations by the signs η S,T ,P,C and σ S may be redundant, 21 because the symmetry operations that are characterized differently may be mapped onto each other. For example, if H satisfies a crystalline unitary symmetry operation S which squares to one, then it also satisfies the unitary symmetry operation iS, which squares to minus one, or (provided T -symmetry is present) it satisfies the antiunitary symmetry T S. Using such equivalences, Shiozaki and Sato group the symmetry operations S into "equivalence classes", which, together with the Altland-Zirnbauer class of Table I, are labeled by one integer s or by two integers s and t. In this work (as in Ref. 39) we label the equivalence classes by representative (anti)symmetries that consist of a unitary crystalline symmetry S squaring to one or the product of such a crystalline symmetry and T , P, or C. These representatives are summarized in the first column of Tables II-IV for the complex Altland-Zirnbauer classes with unitary (anti)symmetries, the complex Altland-Zirnbauer classes with antiunitary (anti)symmetries, and the real Altland-Zirnbauer classes with unitary (anti)symmetries, respectively. For the complex Altland-Zirnbauer classes with antiunitary (anti)symmetries we implicitly assume that T , P commute with S. The classification of topological phases (with or without the additional crystalline symmetry or antisymmetry) has a group structure, and the symbol K (or K) is used to denote the corresponding classifying group. Formally, the group structure is obtained by the Grothendieck construction, 6 22,51 In this work, we take the latter approach and consider one-parameter family of Hamiltonians H(m), such that H(m) is in the topological class of H for −2 < m < 0 and in the topological class of H ref for 0 < m < 2, with the transition between topological classes (if any) taking place at m = 0. When considering Hamiltonian families H(m), we will often omit the parameter m and refer to it simply as the "Hamiltonian H".
The classification of strong topological crystalline phases of Ref. 21 is based on isomorphisms between the groups classifying d-dimensional Hamiltonians with the symmetries labeled by the corresponding indices, where d is the number of inverted spatial dimensions. The above mentioned isomorphisms are extensions of Teo and Kane's dimension-raising isomorphism 52 κ increasing the spatial dimension by one, to the systems with an ordertwo crystalline symmetry or antisymmetry. 21 Shiozaki and Sato introduce two isomorphisms κ and κ ⊥ , where the isomorphism κ increases both the spatial dimension d and the number of the inverted momenta d , whereas the isomorphism κ ⊥ increases only the spatial dimension d while keeping d unchanged. We review these isomorphisms in Sec. V and App. A.
For the complex and real classes with unitary (anti)symmetry the classifying groups are denoted K (s, t|d, d ) and these isomorphisms are (with d < d) with the integers s and t taken mod 2 for complex classes, and mod 8 and mod 4, respectively, for the real classes. We use the same notation for the classifying groups for the real and complex classes. When discussing specific examples we will always specify the Altland-Zirnbauer class using its Cartan symbol, so that no confusion is possible. For complex classes with antiunitary (anti)symmetry these isomorphisms are When applied repeatedly, these isomorphisms can be used to relate the classification problem of d-dimensional Hamiltonians with an order-two crystalline symmetry to a zero-dimensional classification problem with an onsite symmetry, 21,25 which can be solved with elementary methods. When no confusion is possible, we will further omit the arguments s and t in what follows, and write of K (d, d ) instead of K (s, t|d, d ) or K (s|d, d ). The classifying groups for the Altland-Zirnbauer classes (i.e., without additional crystalline symmetries) are denoted by K AZ (d).  The Shiozaki-Sato classifying groups K are the largest groups in the sequence (1), which for two-dimensional systems are listed in Tables II-IV for the complex Altland-Zirnbauer classes with unitary (anti)symmetries, the complex Altland-Zirnbauer classes with antiunitary (anti)symmetries, and the real Altland-Zirnbauer classes with unitary (anti) symmetries, respectively. The corresponding classification of three-dimensional systems is given in Tables V-VII. The Shiozaki-Sato classification of topological phases with an crystalline order-two symmetry contains only "strong" topological crystalline invariants, i.e., it addresses topological features that are unaffected by resizing of the unit cell, thus allowing the addition of perturbations that break the translation symmetry of the original (smaller) unit cell, while preserving the crystalline symmetries. Throughout this work we only consider HOTPs originating form such "strong" topology.

III. CRYSTALLINE-SYMMETRY-BREAKING MASS TERMS
In this Section we consider model Hamiltonians of a simple, "canonical" form, which are still sufficiently general that the model description can be applied to all Shiozaki-Sato classes. We count how many independent "mass terms" can be added to the Hamiltonian that satisfy the fundamental non-spatial (anti)symmetries T , P, and C defining the Altland-Zirnbauer class, but break the crystalline (anti)symmetry S that determines the Shiozaki-Sato class and show that such mass terms can be used to construct fully S-(anti)symmetric mod-els in which a "boundary mass term" appears on boundaries that are not invariant under the crystalline (anti)symmetry S. This naturally explains the phenomenology of higher-order topological phases in these models. This Section serves as the summary of the approach of the Refs. 29 and 39 and as an interlude to the subsequent, more formal Section. Explicitly, the model Hamiltonians we consider have the form with matrices Γ j that anticommute mutually and square to the identity. For the functions d j we choose although our considerations do not change if a different choice for the functions d j is made, as long as the map d/|d| : T d → S d has winding number equal to one for −2 < m < 0 and to zero for 0 < m < 2, and the vector d = (d 0 , d 1 , . . . , d d ) transforms the same as (1, k) under the crystalline (anti)symmetry S and the non-spatial (anti)symmetries T , P, and C. The nonspatial (anti)symmetries T , P, and C and the crystalline (anti)symmetry S impose restrictions on the possible choices for the matrices Γ j , j = 0, 1, . . . , d, which we do not specify explicitly here. We consider the regime −2 < m < 0, for which the Hamiltonian (8) has a band inversion near k = 0 but  not elsewhere in the Brillouin zone. In this parameter range, a Hamiltonian of the form (8) describes a nontrivial topological crystalline phase if there exists no "mass term" M -an idempotent hermitian matrix M which anticommutes with the Hamiltonian -, that satisfies the constraints imposed by S and by T , P, and/or C. The topological phase is strong -i.e., it remains nontrivial if the crystalline (anti)symmetry S is broken -if there exists no mass term M which satisfies the constraints imposed by the non-spatial (anti)symmetries T , P, and/or C alone, irrespective of the crystalline (anti)symmetry S.
On the other hand, if such an S-breaking mass terms exist, the Hamiltonian (8) describes a "purely crystalline" topological phase, which essentially relies on the crystalline (anti)symmetry S for its protection. Whereas a strong topological phase is always a first-order phase, the purely crystalline phases can be higher-order topological phase.
For a minimal canonical-form Hamiltonian in a non-trivial topological crystalline phase (i.e., for which there exist no S-preserving mass terms), we denote the number of mutually anticommuting S-breaking mass terms M l with n−1. The number of crystalline-symmetry-breaking mass terms n − 1 is uniquely defined only for minimal canonical Hamiltonians. For an arbitrary Hamiltonian H, n − 1 is defined as the maximum number, within the topological equivalence class, of mutually anticommuting S-breaking mass terms that anticommute with the Hamiltonian H.
For an order-two crystalline (anti)symmetry S with d > 0 inverted dimensions, we can add a perturbation H 1 (k) to the canonical Hamiltonian H 0 (k) that respects all the (anti)symmetries with the coefficients b (l) j numerically small. Below we show for d = 2 that the perturbation (10) gaps the boundaries not invariant under the (anti)symmetry S and that the boundary Hamiltonian has n − 1 mass terms that can be derived from the crystalline-symmetrybreaking mass terms M l . The construction is easily generalized to higher dimensions and explains the relation between the number n − 1 of the crystalline-symmetrybreaking mass terms and the order of the topological phase. Starting from the low-energy limit of the Hamiltonian H 0 of Eq. (8) with d = 2 in the vicinity of a boundary with normal n = (cos φ, sin φ), we find that the projection operator onto low-energy boundary states is 39 Projecting the bulk Hamiltonian H 0 + H 1 to the lowenergy boundary states gives where is the derivative with respect to a coordinate along the edge. We conclude that the effective boundary Hamiltonian reads where Γ 2 = P (0)Γ 2 P (0) and M l = P (0)M l P (0). Alternatively, one may arrive at the effective boundary Hamiltonian (13) by starting from the canonical-form Hamiltonian (8) and adding the perturbation M l locally at the boundary, provided the boundary is not itself invariant under S and the prefactor m l (φ) obeys the restrictions imposed by S (as it does in Eq. (13)). The boundary Hamiltonian (13) hosts zero-energy corner states between crystal edges with opposite sign of m l (φ), if all the mass terms m l (φ) go through zero at the same value of φ. For onsite order-two symmetry O with d = 0, the mass terms M l cannot be used to construct a O-preserving perturbation, which is understandable since there are no O-symmetry breaking boundaries. Accordingly a local order-two symmetry does not allow for bulk higher-order phases, consistent with the relation (3). Mirror symmetry has d = 1 flipped coordinates, which gives m l (φ) ∝ cos φ: all the mass terms m l (φ) are zero on the mirror line and one obtains a second-order phase whenever there is at least one mass term, i.e., if n ≥ 2. We conclude that for mirror symmetry the order of the bulk phase cannot be greater than two, again consistent with the relation (3). Finally, a twofold rotation symmetry has d = 2, and zero-energy corner states are obtained only if the number n − 1 of crystalline-symmetrybreaking terms is exactly one. Therefore n corresponds to the order of the phase. For n > 2, the coefficients b (l) j can be chosen to yield a fully gapped boundary, which describes the situation where the bulk is topologically nontrivial but the boundary does not host any statesin this case the group K (d) is nontrivial. Generalizing these arguments to higher dimensions, one verifies for Hamiltonians of the canonical form (8), that the presence of n − 1 crystalline-symmetry-breaking mass terms gives rise to a topological phase of order min(n, d + 1) if min(n, d + 1) ≤ d, and to a boundary without protected in-gap states if d = d and n > d.

IV. BULK AND BOUNDARY CLASSIFICATION OF HIGHER-ORDER TOPOLOGICAL PHASES
For a general topological classification, we consider a d-dimensional crystal for which the bulk Hamiltonian has an order-two crystalline symmetry or antisymmetry S, labeled by the Shiozaki-Sato parameters (s, t, d ). We further assume that the crystal shape, including the lattice termination, is compatible with the crystalline symmetry. The system is in an nth order topological phase if it has protected boundary states of codimension n, whereas the bulk and all boundaries of codimension smaller than n are gapped. In this Section we establish the formal framework for a classification of such nth order topological phases, both from a bulk perspective and from a boundary perspective, and show the extent to which they are related.
Fixed points under S.-With d inverted dimensions, the manifold of fixed points under the global crystalline symmetry form a codimension d -hyperplane, which we denote Ω f . For a mirror symmetry (d = 1) Ω f is the mirror plane, for a twofold rotation symmetry (d = 2) it is the rotation axis, and for inversion (d = d) it is the inversion center. The (d−d −1)-dimensional intersection between Ω f and the crystal boundary is denoted ∂Ω f . For inversion symmetry d = d, Ω f is a point and ∂Ω f is empty.
Classifying groups for boundary states.-To define the groups K (n) k classifying codimension n boundary states, we focus on a "proper" subset ∂Ω p of the (d − n)dimensional crystal boundary, which for n ≤ d + 1 is defined as the intersection of the crystal boundary and a (d−n+1)-dimensional manifold Ω p that is mapped to itself under the crystalline symmetry operation and chosen such that ∂Ω p is located on the (d − n)-dimensional crystal boundaries. By construction, Ω p contains the fixed point manifold Ω f if n ≤ d + 1. If n = d + 1, Ω p is unique and equal to Ω f ; if n < d + 1, Ω p is not unique. For n > d +1 Ω p is a subset of Ω f , again chosen such that its boundary ∂Ω p is located on (d − n)-dimensional crystal boundaries. We define the classifying group K (n) k as the group classifying codimension n boundary states with support entirely within ∂Ω p , where crystals that are related to each other by continuous transformations of the Hamiltonian that preserve the bulk gap and gaps of boundaries of codimension ≤ n − k are considered equivalent. To show that this definition of the groups K (n) k for k ≥ 2 agrees with that given in the introduction, we note that any configuration of boundary states of codimension n can be transformed to a configuration of boundary states localized on the proper subset ∂Ω p by locally changing the lattice termination along crystal boundaries of codimension n − 1, see Fig. 3. Equivalence of the two definitions for k ≥ 2 then follows, since such a change of lattice termination corresponds to a transformation of the Hamiltonian that preserves the bulk gap and the gaps of boundaries of codimension ≤ n − 2. For k = 1 the classifying group for the full set of boundary states of codimension n depends on the precise crystal shape, which is why it is necessary to impose the restriction to the proper boundary subset ∂Ω p for k = 1.
Bulk and boundary classifying groups for n > d .-The calculation of the bulk classifying groups K (n) and the boundary classifying groups K (n+1) k is done separately for n ≤ d and n > d . For n > d we note that a nontrivial bulk topology implies that ∂Ω f is gapless. Hence, no protected boundary states of codimension > d +1 can exist if the bulk topology is nontrivial, from where Eq. (3) follows directly. The same result follows if one notes that for d < d one may deform the crystal so that it acquires a symmetry-invariant (d−1)-dimensional boundary which, by the standard bulk-boundary correspondence, hosts a gapless boundary state. Following the construction of Refs. 27, 29, and 39, such a crystal is in a topological phase of order d + 1 or less upon returning the crystal to its original shape. Such a symmetry-invariant bound- From the boundary perspective we note that Ω p can be chosen to be entirely located inside the fixed subset ∂Ω f of the crystal boundary if n > d (see the left panels of Fig. 2). Since there are no protected boundary states outside ∂Ω p , the nth order boundary states of the bulk crystal may also be interpreted as the first-order boundary states of Ω p , which immediately gives Further, since the crystalline symmetry acts locally on Ω p , such states are classified by the Shiozaki-Sato group K(d − n, 0), where the second index refers to the number of inverted dimensions d and we have suppressed the indices s and t for brevity. This gives the "extrinsic" classifying group Finally, we note that for n > d the triviality of the intrinsic classifying groups K n+1 , combined with the triviality of the bulk classifying groups K (n) for n > d , yields the bulk-boundary correspondence (2) advertised in the introduction for n > d .
Boundary classifying groups for n ≤ d .-For n ≤ d we similarly argue that for a crystal for which all boundary states reside on the proper subset ∂Ω p , the boundary states may also be interpreted as first-order boundary states of Ω p , with an order-two crystalline symmetry with d − n inverted dimensions. Here it is essential that the crystal boundary is fully gapped away from ∂Ω p , so that the crystal away from Ω p may be considered effectively topologically trivial. This immediately gives the identification A two-dimensional second-order phase can be decorated by one-dimensional first-order phase whose boundary has classification D 2 (a). A three-dimensional third-order phase can be decorated by one-dimensional first-order phase with (extrinsic) boundary classification D 2 , or with twodimensional second-order phases with (extrinsic) boundary classification D 3 .
where K/K classifies the first-order topological phases in the corresponding Shiozaki-Sato class. [Note that K (d− n, 0) = 0 if n < d, so that formally the second line also applies to the case n = d .] The boundary classifying groups K (n+1) 1 give the broadest possible classification of codimension n + 1 boundary states on ∂Ω p , since it groups such boundary states in equivalence classes with respect to continuous transformations of the Hamitonian that preserve gaps on all boundaries of codimension ≤ n and the bulk gap. To find the remaining boundary classifying groups K (n+1) k with 1 < k ≤ n + 1, for which boundary states related by continuous transformations of the Hamitonian that preserve the bulk gap and gaps on boundaries of codimension ≤ n + 1 − k only, it is advantageous to take the complementary viewpoint, and regard K (n+1) k as the classifying group of codimension n+1 boundary states, where boundary states that differ by a "decoration", the addition or subtraction of topological phases of codimension ≥ n + 2 − k > 0 to the crystal boundary, are considered equivalent, see Fig. 4. Following the latter approach, we define the subgroup series of "decoration groups" where D is the classifying group of (possibly extrinsic) codimension n + 1 boundary states on ∂Ω p that can be obtained from (d − n + k − 2)-dimensional (k − 1)th order topological phases entirely contained within the crystal boundary and respecting the global   crystalline symmetry or antisymmetry S. The boundary classification groups K (n+1) k are then obtained as the quotients Since the crystalline symmetry S acts nonlocally for a generic position in a (d − n + k − 2)-dimensional boundary state, the (bulk) Hamiltonian of such a decoration state is "separable", i.e., it may be written as Hamiltonian without crystalline symmetries. (Note that the boundary of a decoration need not be a separable in this sense. This is illustrated schematically in Fig. 4b. Examples are given in Sec. V.) Bulk classifying groups for n ≤ d .-To formally define the bulk classifying groups K (n) for n ≤ d we construct a homomorphism which maps an equivalence class of d-dimensional Hamiltonians H in Shiozaki-Sato class (s, t, d ) to a (d + 1)dimensional Hamiltonian in Shiozaki-Sato class (s, t, d + 1) without changing the dimension of the protected boundary states. We refer to ω as the "order-raising homomorphism", because it increases the order of the topological phase by one. The repeated application of ω allows us to connect the boundary classification groups K (n+1) k , which, by the ar- guments given above, are related to the Shiozaki-Sato classifying group K(d − n, d − n), to the Shiozaki-Sato group K(d, d ), from which the bulk classifying groups K (n) are derived. In this way, the order-raising homomorphism will be key to establishing the bulk-boundary correspondence (2). Hereto, three defining properties of the order-raising homomorphism are used: • ω(H) is in the trivial class if and only if H is separable, • the homomorphism ω commutes with the dimension-raising isomorphisms κ and κ ⊥ , up to a possible sign change of the topological invariants, and • If H is a non-separable Hamiltonian with n crystalline-symmetry-breaking mass terms (bound-ary mass terms), then ω(H) is a Hamiltonian with n + 1 crystalline-symmetry-breaking mass terms (boundary mass terms).
In Sec. V, we provide a realization of the order-raising homomorphism ω and prove in App. B that it satisfies the above properties. The stacking construction previously considered in the literature 53-55 is another realization of the order-raising homomorphism -this is explicitely demonstrated in Sec. V C. The last of these properties can be used to calculate the bulk classifying groups K (n) in the subgroup series (1), since the number n − 1 of crystalline-symmetry-breaking mass terms is related to the order n of the topological phase (provided n ≤ d − 1), see Sec. III and Refs. 29 and 39. We conclude that Hamiltonians in K (n) must have at least n mass terms on a codimension-one boundary if n ≤ d , so that In particular, the "purely crystalline subgroup" K (d, d ) consists of the (classes of) Hamiltonians with at least one mass term on the boundary, Bulk-boundary correspondence.-The first property of the order-raising homomorphism ω leads to an expression for the boundary classifying groups K (n+1) k , from which the bulk-boundary correspondence is easily derived. We first consider the case n = d , for which one has K (n+1) 1 = K(d − n, 0), see Eq. (16). In this case, we find that the decoration subgroups D such that D (n+1) k includes (possibly extrinsic) codimension n + 1 boundary states from separable (k − 1)th order Hamiltonians. For the classifying group K (n+1) k this gives For n < d one finds similarly consists of products gh, with g ∈ K (d−n, d − n) and h ∈ ω k−1 . (Note that all classifying groups considered here are abelian.) This gives the compact expression only, the results can be transferred to all other Shiozaki-Sato classes using the dimension-raising and lowering isomorphisms κ and κ ⊥ .) As shown in App. C, the kernels ker ω ⊆ K(d, d ) can be obtained from the known results for K and K for most symmetry classes, whereas for a small number of symmetry classes an explicit calculation is necessary.
The remainder of the calculation of the classifying groups can be done without further explicit calculations. This relies on the key observation that the nontrivial groups in the sequence are isomorphic to Z or to Z 2 and that the succession Z → Z 2 does not occur. Since both Z and Z 2 have a single generator, it follows that any homomorphism K (d + l, d + l) → K (d + l + 1, d + l + 1) is either injective, or it maps K (d + l, d + l) to the trivial element.
Applying this observation to the order-raising homomorphism ω and denoting the first instance in which ω maps K (d+l, d +l) to the trivial element by K (d+q, d +q), we obtain the sequence where the symbol " →" denotes an injection. Since where ker ω ⊆ K (d, d ). The cut-off q can be obtained from the calculation of ker ω, see App. C. Once ker ω k and K are known, the boundary classification groups K (n+1) k follow from Eq. (25), whereas the subgroup sequence of bulk classification groups follows from the bulk-boundary correspondence (2).
The subgroup sequences for the bulk classication of two-and three-dimensional higher-order topological phases with an order-two crystalline symmetry or antisymmetry are given in Tables II-IV   phases was already obtained in Ref. 39. The classification of the bulk and boundary for all other cases can be obtained from these tables and the relations mentioned above.

V. EXAMPLES
In this Section we give various tight-binding model realizations for the minimal generators of the higher-order crystalline phases obtained by applying the order-raising homomorphism ω. Explicit examples for higher-order decoration subgroups are provided and their connection to the boundary classifying groups is clarified. We compare the realization of the order-raising homomorphism ω, derived in App. B, to that of the layer stacking procedure, 20,44,53,54 that was previously used to classify the boundaries of the topological crystalline phases. Additionally, we discuss connection to recently studied embedded topological phases. 57 In this Section we reserve the symbol ω for the concrete realization of the orderraising homomorphism given in App. B.
The models we consider can all be expressed in the canonical form where H 0 is the general canonical form of Eq. (8) and H 1 is the perturbation (10) that partially gaps out the boundaries. This Section makes use of the Shiozaki-Sato dimension-raising isomorphisms κ and κ ⊥ and the order-raising homomorphism ω, whose actions on a Hamiltonian we summarize below.
To specify the dimension-raising isomorphisms we consider a one-parameter family of Hamiltonians H(k, m). The canonical-form Hamiltonian (29) is one example of such a Hamiltonian family. The action of Teo and Kane's dimension-raising isomorphism 52 κ on a family AZ classes S symmetry κ (US ) where the pair (H κ , Γ κ ) is given in Tables XI and XII. To keep the notation consistent, if the momentum component k is flipped under the resulting crystalline symmetry then the (d + 1)-dimensional momentum is (k , k), otherwise it is (k, k ). Note that we use the convention U 2 C = 1, so that U C is hermitian. The Shiozaki-Sato isomorphisms κ and κ ⊥ are extensions of the isomorphism κ of Ref. 52; Their action on the Hamiltonian and the mapped Altland-Zirnbauer symmetries is defined in the same way as Teo and Kane's map, thus one only needs to specify the action of the Shiozaki-Sato isomorphisms κ and κ ⊥ on the additional order-two symmetry or antisymmetry; This is summarized in Table XIII for the complex Altland-Zirnbauer classes and in Table XIV for the real Altland-Zirnbauer classes.

A. Separable higher-order topological phases
In Section IV we introduced the higher-order decoration groups D (n) k which contain separable first-order as well as HOTPs that are used to "decorate" a nth order phase. For the Shiozaki-Sato classes there are in total seven classes for which the groups D   topological phases with the following representatives in two-dimensions: DIII M++ , DIII M−+ , D M+ , AII CM− and A P + M , the remaining two are separable third-order topological phases with three-dimensional representatives in classes CII R−− and AIII T + R+ . Such phases are important for understanding the bulk-boundary correspondence (2), since the higher-order separable phases need to be removed from the boundary classification 20,44 in order to obtain the bulk classification of the higher-order phases. The existence of separable HOTPs explains why in certain classes there is a maximal order of an phase, see Eq. (28). Below we give examples of higher-order separable Hamiltonians and show in which way they can be used for decorating topologically trivial bulk Hamiltonians.

Classes DIII M ++ and DIII R −+ with d = 2
Two-dimensional separable Hamiltonians can be used to decorate a three-dimensional bulk. Notice that such  operation changes mirror (twofold rotation) to twofold rotation (inversion) symmetry, see Fig. 5. Inspection of the boundary classification Table X indicates that the intrinsic boundary classification for three-dimensional superconductor in class DIII R++ is trivial while the "mixed" classification K 2 is nontrivial, although there are no topologically nontrivial bulk (crystalline) phases in this class, see Table VII. The same is true for threedimensional superconductors in class DIII I−+ , thus it follows that there has to be a separable two-dimensional superconductor phase in classes DIII M++ and DIII R−+ , that give rise to topologically protected boundary states from the group K 2 .
2d separable superconductor in class DIII M++ .-We consider a two-dimensional Hamiltonian of the form (29), a generator of Altland-Zirnbauer class DIII, with symmetries U T = σ 2 and U P = τ 1 . The above Hamiltonian has a helical Majorana mode on its boundary. We generate a separable Hamiltonian, diag [H(k), SH(k)], using U S = σ 1 , with the mirror symmetry U M = µ 1 σ 1 , which has two helical Majorana modes at its boundary, see Fig. 5a. We can deform the system into a "shell" where the mirror symmetry of the two-dimensional model acts as a twofold rotation symmetry. A perturbation of the form (10) that couples the two blocks locally at the position of the helical Majorana modes, can gap them out everywhere apart from the two points fixed under R, see Fig. 5. Parenthetically, Hamiltonians obtained after application of the order-raising homomorphism ω need not be in the manifestly separable form like the Hamiltonian constructed above. Nonetheless, if separable they can always be deformed to the manifestly separable form. 2d separable superconductor in class DIII R−+ .-We generate a separable Hamiltonian using the symmetry representation U S = σ 1 FIG. 5. Two copies of a nontrivial superconductor in class DIII that are related by mirror symmetry can be used to construct a twofold rotation symmetric, hollow third-order topological superconductor, with a Kramers pair of Majorana states at the two corners. Red and blue lines represent helical Majorana modes that are related by the order-two symmetry.
with the twofold rotation symmetry U R = µ 1 σ 1 . After deforming the system into a shell that is inversion symmetric, and applying the perturbation (10) that couples the two halves locally, two pairs of Majorana zero-energy states are obtained, that are localized on the two corners related by the inversion symmetry. This explains why the "mixed" boundary classification K 2 is nontrivial whereas the intrinsic classification is trivial in class DIII I−+ , see Table XXV. 2. Classes A P + M and A T + R with d = 2 The entry for class A P + R in Table IX indicates that there is a separable two-dimensional insulator in class A P + M , while the entry for class A T + I does not allow for separable insulators in class A T + R .
2d separable insulator in class A P + M .-A separable second-order phase can be constructed from a quantum Hall system with a single chiral mode on its boundary, to obtain a Hamiltonian from class A P + M with two counter-propagating chiral modes, with the symmetry U PM = τ 1 . The above Hamiltonian can be split into two parts and placed on the boundary of a three-dimensional system in class A P + R with the perturbation (10) acting only at the contact of the two halves, see Fig. 5. Even tough there are two mass terms, the twofold rotation symmetry acts effectively as a mirror symmetry around the contact between the two halves, thus the two mass terms need to be zero at the points fixed under R symmetry, see Sec. III. The above Hamiltonian can be continuously deformed so that it additionally has T + R symmetry. The presence of the two T + R-symmetry breaking mass terms indicates that the above insulator is not a second-order topological insulator in class A T + R . Below we demonstrate this explicitly. 2d separable insulator in class A T + R .-Using the Hamiltonian specified by Eq. (35) we obtain a twodimensional separable insulator with symmetry U T R = τ 1 . Since there are no fixed point under T R symmetry, the two mass terms generically gap out the whole boundary.

B. The order-raising homomorphism
Below we consider several examples of ordinary topological phases to which we apply the order-raising isomorphism in order to obtain the higher-order topological phase.

Higher-order phases originating from the Quantum Hall system
In this example we consider a 2d quantum Hall system with onsite symmetry U O = σ 0 , mirror antisymmetry U CM = σ 2 , and twofold rotation symmetry U R = σ 1 . We obtain a 3d second-order Chern insulator in classes A M , A CR , and A I after applying the homomorphism ω, with symmetries U M = τ 3 , U CR = σ 2 , and U I = τ 3 σ 1 . Since the Hamiltonian specified by Eq. (39) is manifestly in the image of κ , we may apply the inverse isomorphism κ −1 and obtain a two-dimensional second-order topological insulator in classes AIII M+ and AIII R− with symmetries U C = τ 3 σ 3 , U M = τ 3 and U R = τ 3 σ 1 . A subsequent application of the homomorphism ω yields a three-dimensional third-order topological insulator in classes AIII R+ and AIII I− , specified by with symmetries U C = µ 1 , U R = µ 1 σ 3 , and U I = µ 3 τ 3 σ 1 .

Higher-order phases originating from the Quantum Spin Hall system
In this example we consider a 2d quantum spin Hall systems in classes AII O+ , AII CM− and AII R− , for which the minimal Hamiltonian has the form (8) with with onsite symmetry U O = σ 0 , mirror antisymmetry U CM = σ 2 , and twofold rotation symmetry U R = τ 3 σ 1 .

C. Stacking construction
References 53, 54, and 58 construct higher-order topological phases by stacking layers of lower-dimensional ones. Like the order-raising homomorphism ω considered here, the stacking construction also involves simultaneously increasing the spatial dimension d and the number of inverted dimensions d by one, so that it, too, provides a homomorphism σ σ : K(d, d ) → K(d + 1, d + 1). (46) Further, in Ref. 58 it is argued, from the boundary perspective, that the stacking of d-dimensional "layers" that differ by a separable phase yields topologically equivalent (d+1)-dimensional crystals. This, too, is a property that is shared by the order-raising homomorphism ω. Indeed, below we show that the stacking homomorphism σ has all three defining properties of the order-raising homomorphism specified in Sec. IV. The order-raising homomorphism ω of App. B and the stacking construction are two realizations of the same homomorphism. We discuss the differences between these two realization at the end of this subsection. Specifically, the stacking procedure constructs a (d+1)dimensional crystal by alternating d-dimensional "layers" with opposite topological numbers as shown schematically in Fig. 6a. Denoting the Hamiltonians of the alternating d-dimensional layers as H d (k) andH d (k), respectively, the Hamiltonian of the (d + 1)-dimensional stack is If the d-dimensional Hamiltonians H d andH d have a crystalline (anti)symmetry with d inverted dimensions encoded by the unitary matrix U S , the (d + 1)-dimensional Hamiltonian H d+1 has two crystalline (anti)symmetries, encoded by diag (U S , U S ) and diag (e ik d+1 U S , U S ), with d and d + 1 inverted dimensions, respectively. The former (anti)symmetry yields a weak topological crystalline phase and will not be considered here. The latter (anti)symmetry has a k d+1dependent transformation matrix, which reflects the fact that it does not map the unit cell defined by the representation (47) of H d+1 to itself, see Fig. 6a. To remedy this situation we replace Eq. (47) by whereρ is a matrix that commutes with the non-spatial (anti)symmetries T , P, and C, and anticommutes with U S , and the crystalline symmetry is represented by diag (U S , U S ). (Being able to find a matrixρ with these properties may require the addition of additional, topological trivial bands.) Loosely speaking, the transformation described by Eq. (48) involves the redefinition of the unit cell as in Fig. 6b, so that the additional crystalline symmetry S maps the (d + 1)-dimensional unit cell to itself for the new choice of the unit cell.

b)
FIG. 6. Layer-stacking construction of higher-order topological phases: (a) A (d + 1)-dimensional crystal is constructed out of alternating d-dimensional "layers" with opposite topological numbers. The unit cell consisting of two such layers is not mapped to itself under the (anti)symmetry operation S, which inverts the coordinate x d+1 in the stacking direction. (b) The unit cell may be redefined, so that it is mapped to itself under S. This redefinition of the unit cell involves splitting the odd layers into two parts that are mapped onto each other under S, eventually after adding topological trivial bands.
The form of the bulk Hamiltonian (48) immediately allows us to conclude that for H d (k) separable, the Hamiltonian H d (k) can be deformed to manifestly separable form and the matrixρ can be chosen to commute with it, resulting in a k d+1 -independent, and therefore topologically trivial (aside from weak invariants) Hamiltonian σ(H d ). The reverse is also true: σ(H d ) topologically trivial implies that the upper-right block H d+1 of Eq. (48) has only weak topological invariants. Thus H d+1 can be continuously deformed to a k d+1 -independent Hamiltonian. The only possible way to remove k d+1 -dependence from e iρk d+1 /2 H d (k)e −iρk d+1 /2 is to continuously deform the Hamiltonian H d and/or the matrix ρ to mutually commute. We have therefore shown • σ(H) is in the trivial class if and only if H is separable.
The above statement is obtained from the bulk perspective, accordingly, it also holds for d-dimensional topological phases from K (d) that do not support topologically protected boundary states. The d-dimensional Hamiltonian H d (k) in Eq. (48) is to be understood as one-parameter family H d (k, m) that represents a topologically trivial Hamiltonian for m > 0. A topologically trivial Hamiltonian is separable, and we choose the parametrization where H d (k, m) is manifestly separable for m > 0. With this choice, the term e iρk d+1 /2 H d (k, m)e −iρk d+1 /2 is k d+1 -independent for m > 0, thus trivial without any additional weak invariants. Using the definitions (48) and (30), we obtain • the stacking homomorphism σ commutes with the dimension-raising isomorphisms κ and κ ⊥ .
The stacking construction has the property that if a non-separable Hamiltonian H d supports topologically protected states on its (d − 1)-dimensional boundary, σ(H) also supports topologically protected states of the same dimensionality on its d-dimensional boundary, see Refs. 53, 54, and 58, -combined with the above property it gives • If H is a non-separable Hamiltonian with n crystalline-symmetry-breaking mass terms (boundary mass terms), then σ(H) is a Hamiltonian with n + 1 crystalline-symmetry-breaking mass terms (boundary mass terms).
To see this consider a d-dimensional Hamiltonian H d with n crystalline-symmetry-breaking mass terms. By repeatedly applying the dimension-raising isomorphism κ ⊥ and κ or their inverse, we can change both the values of d and d to n + 1. The resulting inversion-symmetric (n + 1)-dimensional Hamiltonian H n+1 is guaranteed to have a zero-dimensional protected boundary states, see Sec. III. Thus σ(H n+1 ) has also zero-dimensional topologically protected boundary states, 53,54,58 and accordingly σ(H n+1 ) has n + 1 crystalline-symmetry-breaking mass terms (boundary mass terms). Since the homomorphism σ commutes with the dimension-raising isomorphism κ ⊥ and κ , the same is true for σ(H d ). We additionally checked that ω(H 0 ) ∼ = σ(H 0 ) for zerodimensional Hamiltonians H 0 . Although, as a homomorphism between classifying groups the realizations σ and ω are indistinguishable, their action on a Hamiltonian is rather different. When acting on a nearest-neighbour hopping Hamiltonian, the homomorphism ω gives a Hamiltonian of the same form. In particular, if H is a minimal canonical-form Hamiltonian, ω(H) is also a minimal canonical-form Hamiltonian. On the other hand, as evident from the definition (48), the stacking homomorphism σ generates hopping elements beyond the nearest-neighbours. Below we illustrate these differences for three examples.
The first example is one-parameter family of Hamiltonians H d (m) from Shiozaki-Sato class D O− with d = 0 with U O = σ 1 and U P = σ 3 . The stacking procedure gives a one-dimensional, one-parameter family of Hamiltonians σ(H d (m)) in class D M− . The upper-left block H d+1 of Eq. (48) takes the form where we takeρ = σ 3 in Eq. (48). The lower-right block of the Hamiltonian H d+1 of Eq. (48) is k d+1 -independent, and it does not carry any strong topological invariants. (Parenthetically, for zero-dimensional Hamiltonians, one can uniquely assign topological invariants only to oneparameter family of Hamiltonians, but not to the Hamiltonian itself.) Since the above Hamiltonian is not in the canonical form, we calculate the topological invarint N = n o (π) − n o (0) for the Hamiltonian H 1 (k 1 , m), where n o (k) is the number of the odd-parity negative-energy eigenvalues at an inversion symmetric point k in the Brillouin zone. We find that H 1 (k 1 , m) has N = 1 for m < 0 and N = −1 for m > 0, therefore the one-parameter family (50) has topological invariant N = 2 -the same is true for ω(H 0 ).
For the second example, we consider a canonical-form Hamiltonian H d from Shiozaki-Sato class D M− with d = 1, specified by with U M = σ 1 and U P = σ 3 . The above Hamiltonian describes a strong one-dimensional p-wave superconductor with a single Majorana mode localized at each end. The application of the stacking construction to the onedimensional superconductor with Hamiltonian H d specified by matrices (51) where we usedρ = σ 2 . Since this Hamiltonian is not of minimal canonical form, its topological invariant cannot simply be determined by counting the number of bands. The topological invariant N in this class takes integer values 6,21 Direct calculation gives that both σ(H d ) and ω(H d ) have N = 2 for d = 2. Finally, we apply the stacking homomorphism σ to a strong non-separable superconductor in class D R− , with two-dimensional Hamiltonian specified by with U R = σ 1 and U P = σ 3 . We chooseρ = σ 2 and obtain the upper-left block H d+1 of Eq. (48) as which has inversion symmetry with U I = σ 1 , and particle-hole antisymmetry U P = σ 3 . For class D I− in three-dimensions, similar to the previously considered classes, the topological invariant N can be evaluated via the inversion eigenvalues of the occupied bands 6,21 N = [n o (π, π, π) − n o (π, π, 0) − n o (π, 0, π) − n o (0, π, π) + n o (π, 0, 0) + n o (0, π, 0) + n o (0, 0, π) − n o (0, 0, 0)]/2, We find that both ω(H d ) and σ(H d ) have N = 1 for d = 2, accordingly they are deformable into each other.

D. Embedded topological phases
In was pointed out recently 57 that considering a lowerdimensional topological phase embedded in a higherdimensional topologically trivial bulk might give rise to interesting physical systems called "embedded topological phases". Formally, embedded topological phases in the presence of crystalline symmetries have topologically the same boundary phenomenology as the higher-order topological phases considered in this work. Can an embedded topological system with Hamiltonian H be deformed into a higher-order topological bulk system ω(H)?
The same question was recently address in the literature 59 using a slightly different approach. Figure 7a shows that the stacked-layer system σ(H) can be deformed to the corresponding embedded topological system by breaking the crystalline symmetry S locally by dimerizing the layers, while globally preserving S symmetry. Using the conclusions of the previous section we obtain that ω(H) ∼ = σ(H) is deformable to the corresponding embedded system using a deformation that breaks S locally, while preserving it globally -below we arrive at the same conclusion using different argument.
Assuming for concreteness that the Hamiltonian ω(H) is a three-dimensional inversion-symmetric, second-order Chern insulator with a single hinge mode at its boundary, Fig. 7b shows that its halves above and below the hinge mode can be trivialized as the local symmetry is broken, because ω(H) has only purely crystalline topological invariants. This construction immediately enables us to conclude that ω(H) is deformable to an embedded topological insulator.

VI. CONCLUSIONS
Topological crystalline insulators and superconductors have a more subtle boundary signatures of a nontrivial bulk topology than topological phases that do not rely on the protection by a crystalline symmetry. Whereas the latter case has bulk-boundary correspondence involving the crystal's full boundary, such that a nontrivial topol-ogy is uniquely associated by a gapless boundary state, for generic symmetry-compatible crystal shapes topological crystalline insulators or superconductors may also have protected gapless boundary states of codimension larger than one, or they may have no boundary signatures at all. In this work we provide the formal framework for a classification of topological crystalline phases that fully accounts for these different scenarios and provide such a classification for topological crystalline phases with an order-two crystalline symmetry or antisymmetry. This classification of bulk crystalline phases consists of a subgroup sequence K (d) ⊆ K (d−1) ⊆ . . . ⊆ K, where the subgroup K (n) classifies bulk phases with boundary states of codimension larger than n. The first group in the sequence, K (d) classifies those bulk phases for which no boundary signature exists. We contrast the subgroup sequence describing the bulk topology with a classification of codimension-n boundary states. After dividing out higher-codimension boundary states which can also be obtained as boundary states of lower-dimension topological phases residing on the boundary -i.e., after dividing out boundary states that can be fully attributed to the crystal's termination -, the resulting "intrinsic" boundary classifying group K (n) i = K (n−1) /K (n) . This is the bulk-boundary correspondence for topological crystalline insulators.
Our work builds on and generalizes previous works. It strongly relies on the Shiozaki and Sato's calculation of the classifying groups K , 21 the last groups in our subgroup sequence, and refines the Shiozaki-Sato theory by providing the remaining classifying groups in the subgroup sequence. Like Shiozaki and Sato, our construction follows an algebraic approach, using maps relating classifying groups for different dimensions d and crystalline symmetries with different numbers of inverted dimensions d . Our classification generalizes our own previous work with Geier and Hoskam, 39 in which the second-tolast group K in the subgroup sequence was calculated without making use of the algebraic structure of the classifying groups. Our algebraic approach shows that (up to three exceptions, which are easily considered explicitly) knowledge of K and K is sufficient to compute the full subgroup sequence.
A central role in our construction is played by an "order-raising homomorphism", which simultaneously raises the dimensionality d of the Hamiltonian, the number of inverted dimensions d of the order-two crystalline symmetry or antisymmetry, and the codimension n of the boundary states (if any). For ordertwo symmetries, we find that the layer stacking construction used in Refs. 44, 53-55 is a realization of the order-raising homomorphism. This is an important observation, since we found the explicit expression for the order-raising homomorphism ω only for order-two crystalline (anti)symmetries, whereas the layer stacking construction can be applied to arbitrary crystalline (anti)symmetry, which makes it a valuable tool in obtaining the intrinsic boundary classification of higher-order topological phases. 44 In particular, the layer stacking construction can be used to shed light on the question if there exists a bulk-boundary correspondence for other crystalline symmetries; Finding intrinsic boundary classifying groups is simpler task 20 compared to finding the bulk classifying groups, 23,24 and it would be interesting to see if the intrinsic boundary classification obtained this way agrees with the bulk classification from the other, bulk-tailored approaches. [22][23][24] The first element in the group sequence, K (d) , is zero for crystalline (anti)symmetries with d < d. These include mirror (anti)symmetry in dimensions d ≥ 2 and twofold rotation (anti)symmetry in dimensions d ≥ 3. On the other hand, for mirror symmetry with d = 1, twofold rotation symmetry with d = 2, and inversion symmetry with d = 3, K (d) may be nonzero, and a nonzero K (d) indicates that there topological phases with a nontrivial bulk topology but without topologically protected boundary states. In some cases, such topologically nontrivial phases without protected boundary states are characterized by other observable signatures, such as the presence of boundary charges (not states!), 60 or quantized electric 31,32,61 or magnetic moments. Such signatures of a nontrivial bulk topology are not part of the higher-order bulk boundary correspondence that we establish here, and it is an interesting open problem how they can be incorporated.
We hope the results of this work not only bear theoretical relevance, but will also help experimental efforts [62][63][64] to observe some of the rich boundary phenomenology of crystalline topological insulators and superconductors in solid-state systems. Currently the list of candidate materials for a second-order topological insulators consists of tin-telluride 27 and bismuth. 40 Our complete classification may facilitate the search for other material candidates. Finally, we note that in this work only strong crystalline invariants were considered. We leave it for future works the study of HOTPs originating from weak crystalline topological invariants, 65 which would further expand the list of potential solid-state material candidates.  21, the introduction of defect dimensions can be used to define the dimension-raising isomorphisms for Altland-Zirnbauer classes with an ordertwo antiunitary (anti)symmetry, making use of the fact that the complex Shiozaki-Sato classes with antiunitary (anti)symmetry are isomorphic to real Altland-Zirnbauer classes. Such an isomorphism is most easily constructed 21 by noticing that renaming the coordinates (k ⊥ , r ) →k and (r ⊥ , k ) →r gives a Hamiltonian in the corresponding Altland-Zirnbauer class, see the transformation law (A2). Such a transformation defines the isomorphism Correspondingly, for the complex Shiozaki-Sato classes with an antiunitary symmetry the dimension-raising isomorphisms are defined by first using the above isomorphism to a real Altland-Zirnbauer class, then using Teo and Kane's dimension-raising isomorphisms κ and ρ, 52 and then using the inverse of the isomorphism (A6). From this procedure it is readily seen that for complex Shiozaki-Sato classes with an antiunitary order-two symmetry one has, up to the isomorphism (A6), Appendix B: Properties of the order-raising homomorphism ω To prove the properties of the ω homomorphism, we need to find an explicit expression for it. To this end we consider the following exact sequence containing Altland-Zirnbauer and Shiozaki-Sato K-groups, which is a variant of an exact sequence considered by Turner et al. 6 and by us 16 for the classification of inversion-symmetric and mirror-symmetric topological insulators and superconductors, Here i is the natural homomorphism, in the literature 22 also called a "symmetry forgetting functor", that identifies a member of Shiozaki-Sato group as a member of the corresponding Altland-Zirnbauer group, and c t,d ,D is the homomorphism that constructs separable Hamiltonians We find that the above sequence is exact for the following choice of the order-raising homomorphism ω, The homomorphism ω acts on a Hamiltonian H from K (d, d , D, D ) first with the dimension-raising map ρ , see App. A, which maps it to a one-parameter family H(ϕ), 0 ≤ ϕ ≤ 2π, on which the symmetry S acts nonlocally, σ S U S H(ϕ)U † S = H(2π − ϕ) for ϕ ∈ [0, 2π], followed by the boundary map δ that gives a Hamiltonian with the topological numbers equal to the difference between the topological numbers of H(ϕ) at ϕ = 0, π. Lastly, the dimension-raising isomorphism κ is applied, so that the equivalence class of the obtained (d + 1)-dimensional Hamiltonian defines an element in the group K (d + 1, d + 1, D, D ). The action of the homomorphism ω for the Hamiltonians H in canonical form is summarized in Tables XVII-XVI. The maps in the exact sequence (B1) all preserve the group operations (i.e., they are homomorphisms), and the image of every map is the same as the kernel of the subsequent one. Thus exactness at K AZ (d, D) immediately gives that ω(H) is trivial if and only if H is separable, i.e., H ∈ c t,d ,D [K]. This proves the first property of the order-raising homomorphism ω listed in Sec. IV of the main text.
To prove the second property, we first notice that the natural homomorphism i commutes with the dimensionraising isomorphisms, since the latter act the same way on the Hamiltonians from the Altland-Zirnbauer and Shiozaki-Sato classes, see Sec. V, with χ = ρ, κ. The exactness of the sequence (B1) at K(d, d , D, D − 1) yields the following isomorphism thus the dimension-raising isomorphisms also preserve the subgroups img ω. We conclude that the homomorphism ω commutes with the dimension-raising isomorphisms up to an automorphism of img ω. Since the groups img ω = K are at most Z and Aut(Z) = Z 2 , the mentioned automorphism changes at most the sign of the topological invariants. Such sign change is inessential and therefore the dimension-raising isomorphisms preserve the bulk classifying groups of HOTPs K (n) . This proves the second property of the order-raising homomorphism ω.
We prove the third property using the explicit expression (B3) for the ω homomorphism. Firstly, by comparing the dimension of a nontrivial ω n (H), where H is a minimal canonical model, 9,21,67 a representative of K /K ker ω, to the minimal dimension of the representative of K (n) /K (n+1) we find that ω n (H) is also a minimal canonical model. We therefore conclude that for a minimal canonical model H, representative of either K (n) /K (n+1) or K /K ker ω, ω(H) is also a minimal canonical model.
Next we show that under the assumption that a minimal canonical model with n − 1 crystalline-symmetrybreaking mass terms H (n) (for a fixed n) is a representative of K (n−1) /K (n) for n > 1 and K /K ker ω for n = 1, ω(H) has n boundary mass terms. Since under these assumptions, ω(H) is a minimal canonical model, the number of its S-symmetry breaking mass terms does not change under the continuous Hamiltonian deformations. It is now a matter of simple algebra to show that there are no additional S-symmetry breaking mass terms beyond the ones given in Tables XVII-XVI; We illustrate how the proof works for classes BDI S++ , BDI S−− , CII S++ and CII S−− . In order to satisfy chiral symmetry, the additional mass term needs to be of the form τ 3 M n+1 which has to anticommute with M n = τ 3 U C . Thus M n+1 anticommutes with U C , which makes it a valid S-symmetry breaking mass term of the H (n) Hamiltonian, contradicting the initial assumption on the number of the crystalline symmetry breaking mass terms. This proves the third property of the ω homomorphism.
The exactness of the sequence (B1) can be proved as follows. Consider a one-parameter family H(ϕ) of a Hamiltonian H from K (d, d , D, D − 1), with the order-two symmetry (antisymmetry) U S acting locally as σ S U S H(ϕ)U † S = H(ϕ). This one-parameter family is mapped via the homomorphism c t,d ,D • i to H , that is the S symmetry now acts non-locally on the coordinate ϕ. The loop (B8) is a topologically trivial loop. Alternatively, each topologically trivial loop can be deformed to the above form with an arbitrary H(ϕ) proving that img i = ker c t,d ,D . We next show that every Hamiltonian in ker ω can be continuously deformed to the diagonal form (B2). Since κ and ρ are isomorphisms that preserve a diagonal form, it is sufficient to show that every Hamiltonian in ker δ can be deformed into the diagonal form. Hereto we note that δ(H) = 0 implies that H(0) and H(π) are both in the trivial equivalence class (nontrivial H(0) = H(π) would correspond to a weak topological phase, which we do not consider here), for which after continuous deformation, we may set H(0) = H(π) = e, e being the trivial element. Under stable equivalence we may replace H(ϕ) by H(ϕ) ⊕ e which may be smoothly deformed into H(ϕ) ≡ H(ϕ) ⊕ e for 0 ≤ ϕ < π e ⊕ H(ϕ) for π ≤ ϕ < 2π, and subsequently , into a Hamiltonian of the form (B2), since ρ Sρ −1 H(2π − ϕ) = H(ϕ). As the procedure can be run backwards we conclude ker ω = img c t,d ,D giving the exactness of the sequence (B1) at K (d, d , D, D ). Similarly, because κ is an isomorphism, to show exactness at the second stage of the sequence (B1) it is sufficient to show that any element of img δ can be smoothly deformed to the trivial element e if the crystalline symmetry ρ Sρ −1 is no longer imposed, and vice versa. Again we may assume that H(0) = e, and the continuous deformation linking H(π) H(0) to e e is H(ϕ) H(0) with 0 < ϕ < π. Similarly, if such a transformation exists, i.e., if there exists a continuous functionH(ϕ) = H(ϕ) H(0) interpolating between H(0) H(0) and H(π) H(0), then there also exists a family of ρ Sρ −1 -symmetric Hamiltonians H(ϕ) ≡ H(ϕ) for 0 ≤ ϕ < π H(2π − ϕ) for π ≤ ϕ < 2π, such thatH(ϕ) = H(ϕ) H(0). The second property of the order-raising homomorphism ω ensures that the dimension-raising isomorphism κ and κ can be applied to the subgroups ker ω ⊆ K(d, d ) and i[K ] ⊆ K AZ (d), too. With the help of the dimension-raising isomorphisms, there remain in total 44 and 10 "equivalence classes" of these subgroups for ker ω and i[K ], respectively. Since the purely crystalline subgroups K are known, 29,39 the images i[K ] K/K are readily obtained, see Tables XXII, XX and XXI. For most Shiozaki-Sato symmetry classes the kernel ker ω can be calculated uniquely from the relation (B6), which states that ker ω K AZ /i [K ]. For three cases the subgroup of the Shiozaki-Sato classifying group K (d, d ) isomorphic to K AZ /i[K ] is not unique. These classes are considered separately below for d = d = 0. Tables XXIII-XXIV list the kernels ker ω for d = d = 0. As discussed above, the results for general d and d follow upon application of the dimension-raising isomorphisms κ and κ . These tables also list the cut-off parameter q relevant for the calculation of ker ω k with k > 1, see the discussion in Sec. IV.
where n ± is the difference between the number of positive and negative energy levels of H 0 with ± parity under O symmetry. In class AII, due to Kramers degeneracy, the integers n ± need to be even. Since the local symmetry O commutes with the time-reversal symmetry (class AI), the subgroups ker ω and K are easily obtained, ker ω = {(n, n), n ∈ Z} = Z, K = {(n, −n), n ∈ Z} = Z, since Hamiltonians with n + = n − = n can be deformed into a separable Hamiltonian, whereas Hamitonians with n + +n − = 0 are trivial when the protection by the on-site symmetry O is lifted.
The subgroups ker ω ⊆ K (s|0, 0, 0, 0) for complex Shiozaki-Sato classes with an antiunitary order-two (anti)symmetry. The integer in the superscript gives the q so that ker ω k = K for k > q. Hamiltonians H from these classes are classified by with n ± = sign [Pf(H ± )], where H ± is the block of the Hamiltonian H with ± parity under O. The Hamiltonian H is taken in a basis where particle-hole antisymmetry is represented by U P = 1. In this class, the subgroups K and ker ω are identical, K = {(n, n), n ∈ Z 2 } = Z 2 , ker ω = {(n, n), n ∈ Z 2 } = Z 2 .
(C4) The Hamiltonian in class AI O− can be written as with U T = σ 0 , U O = σ 2 , and a (b) is a symmetric (antisymmetric) real matrix. This matrix has pairs of degenerate eigenvalues. The corresponding classifying group is