#  $C^{\ast}$-framework for higher-order bulk-boundary correspondences

Danilo Polo Ojito  Department of Physics and Department of Mathematical
Sciences, Yeshiva University
New York, NY 10016, USA
<EMAIL_ADDRESS>,  Emil Prodan  Department of Physics and Department
of Mathematical Sciences
Yeshiva University
New York, NY 10016, USA
<EMAIL_ADDRESS>and  Tom Stoiber  Department of Physics and Department of
Mathematical Sciences, Yeshiva University
New York, NY 10016, USA
<EMAIL_ADDRESS>
###### Abstract.

A typical crystal is a finite piece of a material which may be invariant under
some point symmetry group. If it is a so-called intrinsic higher-order
topological insulator or superconductor then it displays boundary modes at
hinges or corners protected by the crystalline symmetry and the bulk topology.
We explain the mechanism behind that using operator K-theory. Specifically, we
derive a groupoid $C^{\ast}$-algebra that 1) encodes the dynamics of the
electrons in the infinite size limit of a crystal; 2) remembers the boundary
conditions at the crystal’s boundaries, and 3) accepts a natural action by the
point symmetries of the atomic lattice. The filtrations of the groupoid’s unit
space by closed subsets that are invariant under the groupoid and point group
actions supply equivariant co-filtrations of the groupoid $C^{\ast}$-algebra.
We show that specific derivations of the induced spectral sequences in twisted
equivariant K-theories enumerate all non-trivial higher-order bulk-boundary
correspondences.

This work was supported by the U.S. National Science Foundation through the
grants DMR-1823800 and CMMI-2131760, and by U.S. Army Research Office through
contract W911NF-23-1-0127, and the German Research Foundation (DFG) Project-ID
521291358.

###### Contents

  1. 1 Introduction and Main Statements
  2. 2 Ordinary Bulk-Defect Correspondences
    1. 2.1 Point patterns and their canonical groupoids
    2. 2.2 Groupoid $C^{\ast}$-algebras and their physical significance
    3. 2.3 Automorphic actions
    4. 2.4 Mechanism of ordinary bulk-defect correspondences
  3. 3 Higher-order correspondences: Building the $C^{\ast}$-models
    1. 3.1 Technical statements
    2. 3.2 Quarter geometry
    3. 3.3 Square geometry
    4. 3.4 Cube geometry
    5. 3.5 Appending the fundamental symmetries
  4. 4 Mechanism of higher-order correspondences
    1. 4.1 Quarter geometry
    2. 4.2 Square geometry
    3. 4.3 Cube geometry
    4. 4.4 A unifying picture
    5. 4.5 Comparison with ordinary bulk boundary correspondence
  5. 5 Examples of Higher-Order Correspondences
    1. 5.1 The non-equivariant case
    2. 5.2 Inversion-symmetry
    3. 5.3 $C_{2}T$-symmetry
    4. 5.4 $C_{4}T$-symmetry
    5. 5.5 The cube geometry
  6. 6 Appendix
    1. 6.1 Twisted group representations
    2. 6.2 Twisted equivariant $K$-theory

##  1\. Introduction and Main Statements

Bulk-boundary correspondence is one of the hallmark features of topological
insulators and superconductors [61][Sec. 1.2]. In very general terms, such
correspondence supplies a prediction about the dynamics of the electrons close
to a flat boundary of a sample, based solely on input coming from bulk
properties of the material. In more precise terms, a topological material
develops propagating wave-channels along flat boundaries, which are active at
energies or frequencies where such channels are entirely inexistent in the
bulk of the material. In terms of the Hamiltonians generating the dynamics of
the electrons, this can be phrased by saying that the Hamiltonian is
spectrally gapped in a pristine infinite sample, but this gap fills with
spectrum when a flat boundary is cut into the sample. The bulk-boundary
correspondences have been the subject of intense research and, from the
mathematical point of view, the subject is in a good shape for the one-
particle sector [32, 48, 10, 11, 1, 47], though certain conjectures still lack
a proof.111For example, proving the delocalized character of the boundary
states for ${\mathbb{Z}}_{2}$-classified topological phases in space
dimensions higher than two is still an open problem.

Further innovation in the field came from the works [8, 51], where it was
observed that several flat boundaries meeting along one hinge or at a corner
can induce non-trivial electron dynamics that can be predicted entirely from
the bulk properties of the material. These works also laid down the general
principles behind this new phenomena, which were dubbed higher-order bulk-
boundary correspondences. We will try to explain these principles and their
challenges, when it comes to a rigorous mathematical formulation, using the
diagrams from Fig. 1. There, we show a regular 2-dimensional lattice that has
been cut to a finite sample with several flat boundaries. For some materials,
which are insulators, i.e. the Hamiltonian has a gapped bulk energy
spectrum,222Throughout, a spectral gap will mean an open interval which is not
contained in the spectrum of the Hamiltonian and which contains the Fermi
energy, fixed w.l.o.g. to be zero. one can witness mid-gap electron states
localized at the exposed corners. These corner states are in general unstable,
unless they exist in a spectral region made up exclusively out of corner-
supported states. In contrast, if the entire boundary hosts wave channels,
then the mentioned corner states are buried or may dissolve into the spectrum
supported by the boundaries. When the edges are also insulating, one says that
the edges are gapped. Now, even if the edges are gapped, there are other
factors of indeterminacy. Indeed, there are quasi 1-dimensional molecular
chains that are insulating yet host topological end modes, which are
completely indistinguishable from the corner modes. If one or more such chains
are deposited along the boundaries of the sample, as schematically shown by
the colored layers in Fig. 1(b), then the multiplicities of the corner states
will obviously be altered. Moreover, by coupling the additional boundary layer
with the rest of the material, one may be able to remove some or all of the
corner states. It is then clear that the corner states are, in general, very
sensitive to the physical conditions close to the boundaries.

Figure 1. (a) A crystal with a honeycomb atomic arrangement, displaying edges
and corners. The crystal is growing indefinitely while the dynamics of
electrons is observed by one experimenter at each corner. (b) The crystal seen
by one of the observers in the infinite size limit. To this observer, the
rotational symmetry of the sample is lost.

If the edges are gapped and the corner states are present, then the latter are
insensitive to perturbations as long as the edges remain gapped [25, 27] (see
also [45] for magnetic interfaces). Those type of corner states are generally
so-called extrinsic higher-order boundary states because they require
protection by both the bulk and the edge spectral gaps [23]. Calling them
extrinsic is justified because the exact number and the characteristics of the
corner-localized modes cannot be predicted entirely from the bulk properties
of the material. For example, in this approach, the bulk material can be
topologically trivial by all standards, yet a corner geometry can display
corner states depending on the details of how the boundary modifies the bulk
dynamics. In the absence of crystalline symmetries a classification of the
extrinsic-higher order correspondences was given in [26].

The higher-order bulk-boundary correspondence proposed in [8, 51], and widely
adopted by the physics community, is different. As the name suggests, the
existence and the properties of the corner modes must be determined by the
bulk, hence be insensitive to the boundary conditions. To explain how this can
be possible, let us remark that the high symmetry of the crystal shown in Fig.
1(a) is not a coincidence, but it is rather the core of the higher-order bulk-
boundary correspondence. Indeed, one of the conditions put forward in [51] is
that one considers finite samples that share the point symmetries of the
lattice. Under this setting, the boundary conditions at different parts of the
flat boundaries are related by a combination of space and possibly fundamental
symmetries333Fundamental symmetries refer to the time-reversal, particle-hole
and chiral symmetries.. In special cases, this constraint is enough to make it
impossible to remove all of the corner modes by a change of symmetry-
preserving boundary condition, even if the crystal is coated with additional
surface layers. In this case one speaks of an intrinsic higher-order
topological insulator, since topological invariants of the bulk Hamiltonian in
conjunction with the symmetry allow to predict the existence of corner modes
whenever the edges are gapped.

The same principle can be generalized to crystals cut out of 3-dimensional
materials, which can display either hinge or corner topological states. These
cases can be distinguished by introducing an order for the correspondences.
One speaks of $n$-th order bulk-boundary correspondence if $n$ is the
difference between the dimensions of the bulk and the defect. Specifically, a
3-dimensional crystal with one (2-dimensional) facet, or a 2-dimensional
crystal with one (1-dimensional) edge, or a quasi 1-dimensional crystal with
one (0-dimensional) end can host bulk-boundary correspondences of order 1 at
the mentioned boundaries. A straight prism cut out of a 3-dimensional
material, or a finite crystal cut out of a 2-dimensional material can host
bulk-boundary correspondences of order 2 along the (1-dimensional) hinges of
the prism or at the (0-dimensional) corners of the finite sample,
respectively. Lastly, a finite crystal cut out of a 3-dimensional material can
host bulk-boundary correspondences of order 3 at the (0-dimensional) corners
of the crystal. Synthetic dimensions are common in material science [46],
hence there is no actual limit on the “effective” dimension of the bulk
material and the principles of higher-order bulk-boundary correspondences work
as well for such settings.

While the above principles are now well understood, researched and explored by
the physics community [64, 65], there still remains a need for a mathematical
framework to thoroughly explain and formalize these concepts at the same level
of rigor as the ordinary bulk-boundary correspondence. What is also missing is
a rigorous device that enumerates all possible non-trivial higher-order bulk-
boundary correspondences for a specified geometry and symmetry group. For
ordinary bulk-boundary correspondence this can be done using $C^{*}$-algebras
and operator $K$-theory, however, there are several good reasons why higher-
order bulk-boundary correspondences so far resisted a similar treatment.
Firstly, we note that, rigorously speaking, such phenomena can only take place
in the infinite-size limit of a sample.444Topological phases are not separated
by sharp phase boundaries in large but finite samples, but traces of expected
topological dynamical can still be observed. Thus, we are presented with the
new challenge of building a $C^{\ast}$-algebra of observations which, although
being tailored for the infinite-size limit, continues to encode precise
information about all boundaries. While this sounds paradoxical, it can be
indeed accomplished if we think of the algebra as encoding the joint
observations of a team of several experimenters. For example, the
experimenters shown in Fig. 1(a) observe the electron dynamics as the sample
grows indefinitely, always having a corner in their field of view or reach. In
the infinite-size limit, a single observer, e.g. the one depicted in Fig.
1(b), will always see a single corner and a pattern extending infinitely
outwards. As such, the symmetry of the original sample is lost to this
experimenter.555This is most obvious if the symmetries of the crystal include
rotations or space inversion, which permute all corners with each other.
However, it is recovered when one compares the measurements of several
experimenters at symmetry-related corners. In the mathematical formulation, an
experimenter will correspond to an irreducible representation of the algebra
to be constructed and the need for a team of experimenters expresses that we
need to employ distinct irreducible representations on a priori unrelated
Hilbert spaces to get the full picture of the electron dynamics. Another
strong reason for why one needs a whole team of observers is that the corner
modes may not appear on all corners. The higher-order bulk-boundary
correspondence is a global statement about the topological modes carried
collectively by all corners. These modes can be redistributed among the
corners if coatings are deposited on the boundaries of the crystal, but they
cannot be made to disappear as long as the coatings respect the relevant
symmetries. Secondly, even if the mentioned $C^{\ast}$-algebra can be
successfully constructed, one may find that it has a rich lattice of ideals
and that there is no direct connection between the ideal corresponding to the
physical observations around the corners and the algebra of bulk observations
(as we recall in Section 2, the ordinary bulk-edge correspondence is based on
short exact sequences linking ideals of boundary observables with the bulk
algebra). Thirdly, one now has to navigate a hierarchy of boundaries and has
to determine which bulk models remain gappable up to boundaries of which
order. Conversely, the models which are not gappable at a boundary must
exhibit topologically protected boundary states and one needs to enumerate
them together with the possible manifestations of their boundary states which
may depend on the boundary condition. 666A bulk model is called gappable at
some boundary if it is possible to realize it by a gapped Hamiltonian on a
geometry which has that specified boundary. Lastly, once the correct
constructions for enumerating the non-trivial higher-order bulk-boundary
correspondences are identified, explicit computations will, in many cases,
lead to a difficult exercise which involves twisted equivariant $K$-theory.

Figure 2. The transversal of the infinite hexagonal pattern: (a) The hexagonal
pattern with different asymptotic observer positions indicated. (b) A picture
of the transversal $\Xi_{\mathcal{L}}$ in the space
${\mathcal{C}}({\mathbb{R}}^{2})$ of patterns. The discrete points are the
orbit of ${\mathcal{L}}$ under discrete translations with distances
symbolizing convergence to limit points in the Fell topology. (c) Any
neighboring observers from Fig. 1(a) will asymptotically see identical
patterns at shared edges, therefore the boundaries of the transversals match
perfectly along the indicated boundaries.

After presenting the problem and its challenges, we now describe our solutions
and how they fit into the existing mathematical landscape. The search for
model $C^{\ast}$-algebras related to locally compact spaces with boundaries
can be traced back to the works of Douglas et al [13, 14, 15] for the half-
plane, and [18, 16] for the quarter-plane. A more general and modern machinery
for building $C^{\ast}$-algebras over cones of the discrete plane was supplied
by Park in [43].777The work [25] by Hayashi on extrinsic higher-order
correspondences builds on them. At the core of these constructions sit the
Toeplitz extensions and their generalizations via pullback constructions.
These extensions have been dressed up with physical meaning in a remarkable
paper by Kellendonk, Richter and Schulz-Baldes [32], where a rigorous
explanation of the bulk-boundary correspondence observed in quantum Hall
experiments was communicated for the first time. Much of the subsequent
mathematical works on bulk-boundary correspondences use this mentioned work as
a template.

On another front, Bellissard and Kellendonk developed a groupoid formalism [6,
31], which delivers model $C^{\ast}$-algebras for generic atomic
configurations. While their work focused mainly on Delone sets that are
associated with the bulk of a material, it was observed in [47] that, when
applied to half-spaces, this formalism reproduces all $C^{\ast}$-algebras and
the associated exact sequences appearing in the standard bulk-boundary
correspondences. Briefly, the closure of the orbit of the atomic lattice
${\mathcal{L}}$ under the translation action of ${\mathbb{R}}^{d}$ on the
space ${\mathcal{C}}({\mathbb{R}}^{d})$ of closed subsets of
${\mathbb{R}}^{d}$ supplies the hull of the pattern
$\Omega_{\mathcal{L}}^{\times}$ and the transformation groupoid associated to
the dynamical system $(\Omega_{\mathcal{L}}^{\times},{\mathbb{R}}^{d})$. The
latter accepts an abstract transversal $\Xi_{\mathcal{L}}$ by letting every
point of ${\mathcal{L}}$ sit at the origin once and taking an appropriate
closure of the so generated subset in ${\mathcal{C}}({\mathbb{R}}^{d})$. The
reduction of the initial groupoid to $\Xi_{\mathcal{L}}$ supplies what we call
the canonical étale groupoid ${\mathcal{G}}_{\mathcal{L}}$ associated to
${\mathcal{L}}$. The left regular representations of
${\mathcal{G}}_{\mathcal{L}}$ and their matrix amplifications produce
translation-equivariant Hamiltonians (see subsection 2.2).

In sections 2 and 3, we demonstrate how to compute transversals in the
presence of boundaries and, as we shall see, they all share several common
features. For the pattern seen by the experimenter from Fig. 1(b), the
computation is explained in Fig. 2. As it turns out, this transversal displays
several closed subsets that are invariant under the groupoid’s action. As we
shall see in subsection 3.2, the groupoid algebras corresponding to the
restrictions of ${\mathcal{G}}_{\mathcal{L}}$ to these subsets and their
complements supply all algebras, ideals and the associated exact sequences
derived for cones of the discrete plane in [43] (see section 3.2). However, as
already explained, higher-order bulk-boundary correspondences cannot be
formalized inside a single corner algebra. Our prescription for constructing
the $C^{\ast}$-algebra that encapsulates the infinite size limit of the
crystal growing process depicted in Fig. 1 is as follows: For each finite
crystal ${\mathcal{L}}_{n}$, one constructs the transversal
$\Xi_{{\mathcal{L}}_{n}}$ as explained. This process supplies a sequence of
finite discrete subsets in ${\mathcal{C}}({\mathbb{R}}^{d})$ and we take the
(global) transversal $\Xi$ of the infinite crystal to be a limit set of this
sequence. The transversals of the patterns seen by the six observers from Fig.
1(a) are part of that limit and, as indicated in Fig. 2(c), they match
pairwise along their boundaries. This can be used to glue the corresponding
groupoids using pushouts and to construct a groupoid for the whole infinite
sample, in such a way that it includes the full algebra of boundary
conditions. The outcome is a groupoid $C^{*}$-algebra associated to $\Xi$,
which can be interpreted very concretely using our team of observers. Here is
our exact statement formulated for a general context (see subsection 3.1 for
technical details):

######  Proposition 1.1.

Let $\Lambda$ be a finite set labeling patterns
$({\mathcal{L}}^{\lambda})_{\lambda\in\Lambda}$ seen from the corners of a
crystal in the infinite-size limit. Then:

  1. i)

For each pair $\lambda,\lambda^{\prime}$, the intersection
$\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}$ in
${\mathcal{C}}({\mathbb{R}}^{d})$ is a closed subset of the unit spaces of
both ${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$ and
${\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}$, invariant under the
actions of both groupoids, and

${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\cap{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}=\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}}=\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}}.$
(1)

  2. ii)

The co-limit under the diagrams888The category of topological groupoids is co-
complete [12].

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(2)

generates the étale groupoid
${\mathcal{G}}_{\Xi}=\bigcup_{\lambda\in\Lambda}{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$
with unit space
$\Xi=\bigcup_{\lambda\in\Lambda}\Xi_{{\mathcal{L}}^{\lambda}}$.

  3. iii)

The matrix amplifications of the left-regular representations of its
$C^{\ast}$-algebra $C^{\ast}{\mathcal{G}}_{\Xi}$ supply Hamiltonians
generating the dynamics of the electrons in the infinite sample.

  4. iv)

If the point group $\Sigma\subset O(d)$ acts via permutations of $\Lambda$
then this action gives rise to an action on the $C^{\ast}$-algebra
$C^{\ast}{\mathcal{G}}_{\Xi}$.

The property (ii) ensures that any two observers, who in the infinite-size
limit perform their measurements on different patterns
${\mathcal{L}}^{\lambda}$, ${\mathcal{L}}^{\lambda^{\prime}}$, will obtain
consistent results at the shared boundaries because the dynamics of electrons
is determined by a single Hamiltonian from $C^{\ast}{\mathcal{G}}_{\Xi}$ and
the observers merely experience it from different representations of this
algebra.

The new $C^{\ast}$-algebraic framework announced above is one of the main
results of the paper. As we shall see in section 3, all the algebraic
structures seen above can be explicitly computed for the cases of interest to
us. In all instances, we found the following common characteristics:

######  Proposition 1.2.

A $d$-dimensional crystal in the infinite-size limit is described by a
transversal $\Xi\subset{\mathcal{C}}({\mathbb{R}}^{d})$, which is symmetric to
a point group $\Sigma$.

  1. i)

The space $\Xi$ of units admits filtrations

$\\{{\mathcal{L}}_{\infty}\\}=\Xi_{0}\subset\Xi_{1}\subset\cdots\subset\Xi_{d}=\Xi,$
(3)

of length $d$ by closed subspaces that are invariant to the groupoid and point
group actions. Here, ${\mathcal{L}}_{\infty}$ is the bulk lattice.

  2. ii)

In turn, this supplies a $\Sigma$-equivariant co-filtration of the groupoid
$C^{\ast}$-algebra

$C^{\ast}{\mathcal{G}}_{\Xi_{d}}\stackrel{{\scriptstyle\mathfrak{p}^{d}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{d-1}}\stackrel{{\scriptstyle\mathfrak{p}^{d-1}}}{{\twoheadrightarrow}}\cdots\stackrel{{\scriptstyle\mathfrak{p}^{2}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{1}}\stackrel{{\scriptstyle\mathfrak{p}^{1}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{0}},$
(4)

where ${\mathcal{G}}_{\Xi_{j}}$ is the restriction of ${\mathcal{G}}_{\Xi}$ to
$\Xi_{j}\subseteq\Xi$.

  3. iii)

For this co-filtration,

${\rm Ker}(C^{\ast}{\mathcal{G}}_{\Xi_{r}}\twoheadrightarrow
C^{\ast}{\mathcal{G}}_{\Xi_{r-1}})=:C^{\ast}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$
(5)

identifies the algebra of observations at selected boundaries of codimension
$r$.

The selection mentioned at point iii) is controlled by the choice of the
filtration made at point i) (see subsections 4.3 and 4.4). The transversal
$\Xi_{d}$ selects a subset of the crystalline geometry, comprised of some but
not necessarily all patterns that occur in the infinite-volume limit of a
finite crystal. To keep with the physical interpretation, there is then a
unique choice for $\Xi_{0},...,\Xi_{d-1}$ such that the ideals of Proposition
1.2 are algebras of observables localized at the boundaries of the respective
codimension. In section 3, we demonstrate how Propositions 1.1 and 1.2 play
out in the specific cases of quarter, square and cube geometries and exemplify
how the selection of the boundaries is made by the filtration. We will use
these examples in section 4 to identify the mechanism of higher-order bulk-
boundary correspondences.

For us, the final step $C^{*}{\mathcal{G}}_{\Xi_{0}}$ in the filtration will
always describe a bulk material without boundaries. To this algebra and to the
algebras $C^{\ast}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$ of boundary
observables, one can assign topological invariants using equivariant
K-functors ${\bf K}_{q}$, which are equivariant homology theories for
$C^{*}$-algebras. All gapped bulk materials and topologically protected
boundary states give rise to elements of and are classified by those K-groups.
The goal of K-theoretic bulk-boundary correspondence is to find maps between
those groups, which explain the relation between bulk and boundary topological
invariants. We construct those maps for higher-order bulk-boundary
correspondences, which have the following properties:

######  Theorem 1.3.

Choose a symmetry-adapted filtration $\\{\Xi_{n}\\}$ (3) such that the ideals
of boundary states $C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$ will be
localized to a selection of $r$-th order boundaries. Fix a subgroup
$\Gamma\subset\Sigma\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ of the point
group enhanced by Altland-Zirnbauer-type fundamental symmetries and a
$\Gamma$-equivariant K-functor together with its suspensions $({\bf
K}_{q})_{q\in{\mathbb{Z}}}$. Assume that at least for the specific value
$q=\ast$ the functor the ${\bf K}_{\ast}$ classifies stable homotopy classes
of gapped symmetric Hamiltonians in the algebras above. One has:

  1. i)

The equivariant co-filtration (4) induces a spectral sequence
$(E^{r}_{p,q},d^{r}_{p,q})$ whose terms $E^{r}_{0,q}$ are subgroups of ${\bf
K}_{q}(C^{\ast}{\mathcal{G}}_{\Xi_{0}})$ and $E^{r}_{p,q}$ for $p>1$ are
subquotients of ${\bf
K}_{-p+q}(C^{\ast}{\mathcal{G}}_{\Xi_{p}\setminus\Xi_{p-1}})$. The
differential $d^{r}_{0,q}:E^{r}_{0,q}\to E^{r}_{r,q+r-1}$ relates subquotients
of bulk and boundary K-groups.

  2. ii)

A class $x\in{\bf K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{0}})$ is in the domain
of $d^{r}_{p,\ast}$ if and only if there is a symmetric Hamiltonian $h$ in
$M_{N}({\mathbb{C}})\otimes C^{*}{\mathcal{G}}_{\Xi_{d}}$ such that dividing
out all boundaries of codimension $r$ and greater via the surjection
$(\mathfrak{p}^{r}\circ\cdots\circ\mathfrak{p}^{d})(h)$ results in a symmetric
spectrally gapped Hamiltonian in $M_{N}({\mathbb{C}})\otimes
C^{\ast}{\mathcal{G}}_{\Xi_{r-1}}$ and such that its evaluation
$(\mathfrak{p}^{1}\circ\cdots\circ\mathfrak{p}^{d})(h)$ in the bulk represents
the $K$-theory class $x$.

  3. iii)

If $d^{r}_{0,\ast}(x)$ is non-trivial for some $x\in{\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$, then any symmetric Hamiltonian in
$M_{N}({\mathbb{C}})\otimes C^{*}{\mathcal{G}}_{\Xi_{d}}$ representing this
$K$-theory class $x$ in the bulk and having a spectrally gapped image in
$M_{N}({\mathbb{C}})\otimes C^{*}{\mathcal{G}}_{\Xi_{r-1}}$ displays non-
trivial topologically stable order-$r$ order boundary states. The image
$d^{r}_{0,\ast}(x)$ as a coset in a subquotient of ${\bf
K}_{*-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$ enumerates all
possible boundary states obtainable by choice of symmetric boundary condition.

The statement (iii) is the punch line of our work. It shows that higher-order
bulk-boundary correspondence is a stable and robust phenomenon protected by
spectral gaps at suitable boundaries in combination with a specified
crystalline symmetry group and is entirely explainable by operator K-theory. A
key concept here is that of a gappable boundary: A bulk Hamiltonian will be
said to be gappable at the boundaries of codimension $r$ if its K-theory class
satisfies the equivalent conditions of Theorem 1.3(ii). To identify all
instances of order-$r$ bulk-boundary correspondence one needs to enumerate
precisely the stable homotopy classes of gapped bulk Hamiltonians which are
gappable at the boundaries of codimension $r-1$ but not at those of
codimension $r$. The domains of the differentials $d^{r}_{0,\ast}$ identify
all bulk Hamiltonians with gappable boundaries of codimension less than $r$
(see Definition 4.2 and 4.15) and to then find out if they are gappable at the
codimension $r$ boundaries one computes the differential $d^{r}_{0,\ast}$. Any
non-trivial value of $d^{r}_{0,\ast}$ identifies a topological class of bulk
Hamiltonians that delivers an order-$r$ bulk-boundary correspondence. Examples
are supplied in sections 4 and 5. In particular, we will see in subsection 5.1
that for our crystalline examples the image of $d^{r}_{0,\ast}$ for $r\geq 2$
is trivial in the absence of symmetries, therefore the presence of a symmetry
is a prerequisite to observe non-trivial higher-order bulk-boundary
correspondences.

##  2\. Ordinary Bulk-Defect Correspondences

The model $C^{\ast}$-algebras and the exact sequences relevant for the
standard bulk-defect correspondence principles can all be generated by a
mechanism described in [47], within the framework of specific (étale)
groupoids and their associated $C^{\ast}$-algebras. Our framework for higher-
order bulk-boundary correspondences builds on this formalism. The goal of this
section is to introduce a proper background and to fix the notations and
conventions.

###  2.1. Point patterns and their canonical groupoids

For the start, we will be interested in the space ${\mathcal{C}}(G)$ of closed
subsets of a locally compact second countable (lcsc) topological amenable
group $G$, which is equipped with Fell’s topology [21]. Then the space
${\mathcal{C}}(G)$ is automatically a compact Hausdorff $G$-space [17, Remark
4.4.], with the $G$-action

$g\cdot C=\\{xg^{-1},\ x\in C\\},\quad g\in G,\quad C\in{\mathcal{C}}(G).$ (6)

Throughout our exposition, a pattern in $G$ is simply an element of
${\mathcal{C}}(G)$.

######  Remark 2.1.

The examples listed in our present work all engage the abelian groups
$G={\mathbb{R}}^{d}$. However the formalisms introduced in the present section
and in section 4 are general enough to handle more general lcsc groups, such
as groups of isometries of various Riemann manifolds. To connect with
experiments, one may fix some map $G\to{\mathbb{R}}^{d}$ (or to some other
manifold) such that a pattern in $G$ amounts to a decoration of a real-space
pattern with additional on-site data. A relevant example [40] is that of the
Euclidean group $G=O(d)\ltimes{\mathbb{R}}^{d}$ with patterns made up of
artificial atoms whose macroscopic structure has a distinguishable orientation
labeled by $O(d)$. $\Diamond$

We will be interested in the following classes of patterns:

######  Definition 2.2 ([20]).

For ${\mathcal{L}}\in{\mathcal{C}}(G)$ and $S\subseteq G$, one says that
${\mathcal{L}}$ is

  1. 1)

$S$-separated if $|{\mathcal{L}}\cap g\cdot S|\leq 1$ for all $g\in G$;

  2. 1)

$S$-dense if $|{\mathcal{L}}\cap g\cdot S|\geq 1$ for all $g\in G$.

######  Definition 2.3 ([20]).

A subset ${\mathcal{L}}\subset G$ is called uniformly separated if there
exists a non-empty open set $U\subseteq G$ such that ${\mathcal{L}}$ is
$U$-separated. The set ${\mathcal{L}}$ is called uniformly dense if there
exists a compact subset $K\subseteq G$ such that ${\mathcal{L}}$ is $K$-dense.
If ${\mathcal{L}}$ is both separated and relatively dense, the it is called a
Delone set.

A Delone pattern is usually associated with the bulk of a material, while the
merely uniformly separated patterns are more closely associated with materials
displaying geometric defects. Hence, our focus is on the second type of
patterns, though Delone patterns will show up as well.

To a fixed pattern ${\mathcal{L}}\in{\mathcal{C}}(G)$ one associates its
punctured hull

$\Omega_{{\mathcal{L}}}^{\times}=\overline{\\{g\cdot{\mathcal{L}}:\ g\in
G\\}}\setminus\emptyset.$ (7)

This produces a topological dynamical system
$(G,\Omega_{{\mathcal{L}}}^{\times})$ tailored to the pattern ${\mathcal{L}}$.
To any topological dynamical system, we can attach a transformation groupoid
[63, p. 5]. For $(G,\Omega_{{\mathcal{L}}}^{\times})$, we denote this groupoid
by $\tilde{\mathcal{G}}_{{\mathcal{L}}}=\Omega_{{\mathcal{L}}}^{\times}\rtimes
G$ and its source and range maps by $\tilde{\mathfrak{s}}$ and
$\tilde{\mathfrak{r}}$, respectively. The interest, however, is not exactly in
$\tilde{\mathcal{G}}_{{\mathcal{L}}}$ but rather in a reduced form of it:

######  Definition 2.4.

If $e$ denotes the neutral element of $G$, then we define the canonical
transversal of $\tilde{\mathcal{G}}_{{\mathcal{L}}}$ as

$\Xi_{{\mathcal{L}}}=\overline{\\{{\mathcal{S}}\in\Omega_{{\mathcal{L}}}^{\times},\
e\in{\mathcal{S}}\\}}$ (8)

and the sought groupoid as the restriction

${\mathcal{G}}_{{\mathcal{L}}}:=\left.\tilde{\mathcal{G}}_{{\mathcal{L}}}\right|_{\Xi_{{\mathcal{L}}}}=\tilde{\mathfrak{s}}^{-1}(\Xi_{{\mathcal{L}}})\cap\tilde{\mathfrak{r}}^{-1}(\Xi_{{\mathcal{L}}}).$
(9)

######  Remark 2.5.

For any non-empty open set $U\subseteq G$, the set of $U$-separated sets is
closed in ${\mathcal{C}}(G)$ [20]. If ${\mathcal{L}}$ is $U$-separated, then
the limit points of its orbit are thus themselves $U$-separated. As a
consequence, all ${\mathcal{S}}\in\Xi_{\mathcal{L}}$ are uniformly separated.
Additionally, since $\Xi_{\mathcal{L}}$ is a closed subset of the compact
space ${\mathcal{C}}(G)$, it is automatically compact. $\Diamond$

######  Proposition 2.6 ([20]).

If ${\mathcal{L}}$ is uniformly separated, then the space
$\Xi_{{\mathcal{L}}}$ is an abstract transversal of
$\tilde{\mathcal{G}}_{{\mathcal{L}}}$. As a result,
$\tilde{\mathcal{G}}_{{\mathcal{L}}}$ and ${\mathcal{G}}_{{\mathcal{L}}}$ are
equivalent in the sense of [42]. Furthermore ${\mathcal{G}}_{{\mathcal{L}}}$
is a lcsc étale groupoid with compact unit space $\Xi_{{\mathcal{L}}}$ in the
Fell topology.

It will be useful to have an explicit characterization of
${\mathcal{G}}_{\mathcal{L}}$:

######  Proposition 2.7 ([40]).

The topological groupoid ${\mathcal{G}}_{{\mathcal{L}}}$ canonically
associated to the uniformly separated pattern ${\mathcal{L}}$ consists of:

  1. 1.

The set ${\mathcal{G}}_{{\mathcal{L}}}$ of tuples

$(g,{\mathcal{S}})\in G\times\Xi_{{\mathcal{L}}},\ g\in{\mathcal{S}},$ (10)

equipped with the inversion map

$(g,{\mathcal{S}})^{-1}=(g^{-1},g\cdot{\mathcal{S}})$ (11)

and with the lcsc topology inherited from $G\times{\mathcal{C}}(G)$.

  2. 2.

The subset ${\mathcal{G}}_{{\mathcal{L}}}^{2}$ of composable elements

$\big{(}(g^{\prime},{\mathcal{S}}^{\prime}),(g,{\mathcal{S}})\big{)}\in{\mathcal{G}}_{{\mathcal{L}}}\times{\mathcal{G}}_{{\mathcal{L}}},\
{\mathcal{S}}^{\prime}=g\cdot{\mathcal{S}},$ (12)

equipped with the composition

$(g^{\prime},{\mathcal{S}}^{\prime})\cdot(g,{\mathcal{S}})=(g^{\prime}g,{\mathcal{S}}).$
(13)

######  Remark 2.8.

The source and range maps of ${\mathcal{G}}_{{\mathcal{L}}}$ are

$\mathfrak{s}(g,{\mathcal{S}})=(e,{\mathcal{S}}),\quad\mathfrak{r}(g,{\mathcal{S}})=\left(e,g\cdot{\mathcal{S}}\right),$
(14)

and its space of units ${\mathcal{G}}^{0}_{{\mathcal{L}}}$ is naturally
homeomorphic to $\Xi_{{\mathcal{L}}}$. Recall that the latter is a compact
topological space. Another useful information is the action of
${\mathcal{G}}_{{\mathcal{L}}}$ on its space of units, which goes as follows:
If ${\mathcal{S}}\in\Xi_{\mathcal{L}}$ and
$\gamma\in\mathfrak{s}^{-1}({\mathcal{S}})$, then $\gamma=(g,{\mathcal{S}})$
for some $g\in{\mathcal{S}}$ and
$\gamma\cdot{\mathcal{S}}=g^{-1}\cdot{\mathcal{S}}$. $\Diamond$

###  2.2. Groupoid $C^{\ast}$-algebras and their physical significance

Any étale groupoid comes equipped with the Haar system supplied by the
counting measures. As such:

######  Proposition 2.9.

Any uniformly separated pattern ${\mathcal{L}}\in{\mathcal{C}}(G)$ carries a
canonical $C^{\ast}$-algebra, the (full) groupoid $C^{\ast}$-algebra
$C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}$ corresponding to
${\mathcal{G}}_{{\mathcal{L}}}$ and to its counting measures.

######  Remark 2.10.

All étale groupoids encountered in this work are topologically amenable when
considered with their Haar systems of counting measures, since they are
groupoid-equivalent to transformation groupoids of amenable groups $G$. As
such, there is no distinction between their reduced and full
$C^{\ast}$-algebras [2]. $\Diamond$

######  Remark 2.11.

$C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}$ accepts a family of covariant left-
regular representations indexed by $\Xi_{\mathcal{L}}$, induced by the states
$\Xi_{\mathcal{L}}\ni{\mathcal{S}}\mapsto\eta_{\mathcal{S}}(f)=f(e,{\mathcal{S}})$,
$f\in C^{\ast}{\mathcal{G}}_{\mathcal{L}}$. They are supported on the Hilbert
space
$\ell^{2}\big{(}\mathfrak{s}^{-1}({\mathcal{S}})\big{)}=\ell^{2}({\mathcal{S}})$
and act as

$\pi_{\mathcal{S}}(f)\,|g^{\prime}\rangle=\sum_{g\in{\mathcal{S}}}f(gg^{\prime-1},g^{\prime}\cdot{\mathcal{S}})\,|g\rangle$
(15)

on the canonical basis of $\ell^{2}({\mathcal{S}})$. The matrix amplifications
of the representations formalize the dynamics of electrons for the atomic
arrangement ${\mathcal{L}}$, as experienced by observers located at different
atom sites. Since the groupoid ${\mathcal{G}}_{\mathcal{L}}$ is amenable, the
direct sum $\bigoplus_{{\mathcal{S}}\in\Xi_{\mathcal{L}}}\pi_{\mathcal{L}}$ is
a faithful representation and a more detailed study readily shows that
$\pi_{\mathcal{L}}$ alone is also faithful already. $\Diamond$

On the physics side, we focus on the low energy regime in the one-electron
sector, therefore the quantum dynamics of the electrons in a material is
generated by a Hamiltonian of the type

$H_{\mathcal{S}}=\sum_{x,x^{\prime}\in{\mathcal{S}}}w_{x,x^{\prime}}({\mathcal{S}})\otimes|x\rangle\langle
x^{\prime}|,\quad w_{x,x^{\prime}}({\mathcal{S}})\in M_{N}({\mathbb{C}}),$
(16)

where the uniformly discrete ${\mathcal{S}}\subset G$ indicates the position
of the atoms, $N$ is the number of relevant atomic orbitals and
$w_{x,x^{\prime}}({\mathcal{S}})$ are the so-called coupling or hopping
matrices. Once the atomic species are fixed, the electron dynamics is fully
determined by the atom arrangements. As such, (16) can be also interpreted as
a map from the space of uniformly separated patterns to a family of coupling
matrices indexed by pairs $(x,x^{\prime})\in{\mathcal{S}}$. Each of them
should vary continuously w.r.t local displacements of a finite set of points
of ${\mathcal{S}}$. This continuity assumption is inherent to having
consistent laboratory measurements, since there are necessarily deviations
from an ideal lattice. Furthermore we assume that, in natural as well as
synthetic materials, the coupling matrices will be too small to be resolved
beyond a finite range, hence the coupling matrices
$w_{x,x^{\prime}}({\mathcal{S}})$ returned by an actual experiment may be
assumed to vanish if $x^{\prime}x^{-1}$ is outside a compact vicinity of the
origin. Lastly, if the sample is free-standing and no external fields are
present, then the coupling matrices ought to obey the equivariance relation

$w_{g\cdot x,g\cdot
x^{\prime}}(g\cdot{\mathcal{S}})=w_{x,x^{\prime}}({\mathcal{S}}),\ \forall\
g\in G.$ (17)

For $G={\mathbb{R}}^{d}$, this is a direct consequence of the Galilean
invariance of the fundamental laws of physics at low energies and the
assumptions of equivariance, continuity and finite range combine to the
statement that for any pattern ${\mathcal{L}}$ there should be a continuous
function $f\in C_{c}({\mathcal{G}}_{\mathcal{L}},M_{N}({\mathbb{C}}))$ such
that

$w_{x,x^{\prime}}({\mathcal{S}})=f(x-x^{\prime},{\mathcal{S}}-x^{\prime}),$
(18)

for all ${\mathcal{S}}\in\Xi_{\mathcal{L}}$, since the coupling matrix cannot
depend directly on $x$ and $x^{\prime}$, but only on the difference
$x^{\prime}-x\in{\mathbb{R}}^{d}$ and the relative positions on the pattern.
The relation (18) is in fact equivalent to (15) for an additively written
group action.

The physical and mathematical sides thus meet as follows:

######  Proposition 2.12 ([40]).

For a fixed atomic arrangement ${\mathcal{L}}\in{\mathcal{C}}(G)$, all
Hamiltonians of the type (16) with coupling matrices satisfying the
qualifications mentioned above can be generated from a left regular
representation of $M_{N}({\mathbb{C}})\otimes
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}$. Reciprocally, unrestricted
modifications of the coupling matrices, while keeping ${\mathcal{L}}$ fixed
and preserving the qualifications mentioned above, result in Hamiltonians (16)
that densely sample the self-adjoint sector of $M_{N}({\mathbb{C}})\otimes
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}$.

The conclusion is that a given atomic arrangement carries a canonical
$C^{\ast}$-algebra that can be equally justified by mathematical and physical
means. This fundamental principle was first discovered by Jean Bellissard [6]
and it was further refined by Johannes Kellendonk [31]. Developments that take
into account the shape of the artificial atoms and are applicable to the many-
electron sectors can be found in [39, 40]. Furthermore, the formalism can be
easily adapted to account for the presence of various symmetries (see next
subsection and subsection 3.5).

###  2.3. Automorphic actions

The structures defined in the previous subsections behave naturally under the
automorphisms of the locally compact group $G$ (which are throughout assumed
to be continuous):

######  Proposition 2.13.

Let $\sigma\in Aut(G)$. Then:

  1. i)

$\sigma$ induces a homeomorphism $\sigma:{\mathcal{C}}(G)\to{\mathcal{C}}(G)$;

  2. ii)

For a fixed uniformly separated pattern ${\mathcal{L}}$, we have natural
homeomorphisms

${\mathcal{L}}\to\sigma({\mathcal{L}}),\quad\Omega^{\times}_{\mathcal{L}}\to\Omega^{\times}_{\sigma({\mathcal{L}})},\quad\Xi_{\mathcal{L}}\to\Xi_{\sigma({\mathcal{L}})};$
(19)

  3. iii)

There is a groupoid isomorphism

$\alpha_{\sigma}:{\mathcal{G}}_{\mathcal{L}}\to{\mathcal{G}}_{\sigma({\mathcal{L}})},\quad(g,{\mathcal{S}})\mapsto(\sigma(g),\sigma({\mathcal{S}})).$
(20)

###### Proof.

i) This follows from [17, Remark 4.4.]. ii) The only challenge here is the
last homeomorphism. Note that, if $e\in{\mathcal{S}}$, then $e$ also belongs
to $\sigma({\mathcal{S}})$ because group automorphisms act trivially on the
neutral element. Thus, the map
$\Xi_{\mathcal{L}}\to\Xi_{\sigma({\mathcal{L}})}$ is obvious. iii) Let us
first check the composition law. Taking two composable elements from
${\mathcal{G}}_{\mathcal{L}}$, $(g^{\prime},g\cdot{\mathcal{S}})$ and
$(g,{\mathcal{S}})$, we have

$\alpha_{\sigma}(g^{\prime},g\cdot{\mathcal{S}})\cdot\alpha_{\sigma}(g,{\mathcal{S}})=(\sigma(g^{\prime}),\sigma(g\cdot{\mathcal{S}}))\cdot(\sigma(g),\sigma({\mathcal{S}})).$
(21)

If $\sigma$ is an automorphism, then
$\sigma(g\cdot{\mathcal{S}})=\sigma(g)\cdot\sigma({\mathcal{S}})$ and we can
see that the two elements of ${\mathcal{G}}_{\mathcal{L}}$ remain composable
after $\sigma$ is applied. Furthermore, we can complete the calculation and
conclude

$\alpha_{\sigma}((g^{\prime},g\cdot{\mathcal{S}}))\cdot\alpha_{\sigma}((g,{\mathcal{S}}))=(\sigma(g^{\prime}),\sigma(g)\cdot\sigma({\mathcal{S}})).$
(22)

On the other hand,

$\alpha_{\sigma}\big{(}(g^{\prime},g\cdot{\mathcal{S}})\cdot(g,{\mathcal{S}})\big{)}=(\sigma(g^{\prime}g),\sigma({\mathcal{S}})),$
(23)

and the two results coincide as long as $\sigma$ is an automorphism of $G$. As
for inversion, we have

$\alpha_{\sigma}\big{(}(g,{\mathcal{S}})^{-1}\big{)}=\alpha_{\sigma}(g^{-1},g\cdot{\mathcal{S}})=\big{(}\sigma(g^{-1}),\sigma(g\cdot{\mathcal{S}})\big{)},$
(24)

while

$\big{(}\alpha_{\sigma}(g,{\mathcal{S}})\big{)}^{-1}=\big{(}\sigma(g)^{-1},\sigma(g)\cdot\sigma({\mathcal{S}})\big{)}.$
(25)

The two results are identical if $\sigma$ is a group automorphism. ∎

######  Corollary 2.14.

The groupoid isomorphism induced by an automorphism of $G$ lifts to an
isomorphism of $C^{\ast}$-algebras

$C^{\ast}{\mathcal{G}}_{\mathcal{L}}\ni
f\mapsto\alpha_{\sigma}^{\ast}(f):=f\circ\alpha^{-1}_{\sigma}\in
C^{\ast}{\mathcal{G}}_{\sigma({\mathcal{L}})}$ (26)

###### Proof.

$\alpha_{\sigma}$ is bijective and preserves the Haar system. Then the
statement follows from [3, Prop. 2.7].∎

######  Example 2.15.

For our concrete physical systems, $G={\mathbb{R}}^{d}$. Then the group of
rotations, proper or otherwise, embeds in $Aut({\mathbb{R}}^{d})$. Point
groups are finite groups of rotations and they will enter in our analysis via
the actions described in Corollary 2.14. $\Diamond$

###  2.4. Mechanism of ordinary bulk-defect correspondences

A geometrical defect in a pattern, such as a local modification or boundary,
manifests itself as a feature that can be made to disappear in the limit by
translating it to infinity (which is well-described by limits in the Fell
topology). Recall that groupoids can act on topological spaces [42] (see also
[63][Def. 2.1]), in particular, ${\mathcal{G}}_{{\mathcal{L}}}$ has a
canonical action on its unit space $\Xi_{{\mathcal{L}}}$, which was spelled
out in Remark 2.8. It was observed in [47] that, in the presence of a
geometrical defect, the space of units $\Xi_{{\mathcal{L}}}$ can display one
or more closed subspaces that are invariant against the mentioned groupoid
action. Furthermore:

######  Proposition 2.16 ([63], Thm. 5.1).

If $\Xi_{{\mathcal{L}}}^{\infty}$ is a closed and invariant subspace of the
unit space $\Xi_{\mathcal{L}}$ and $\Xi_{{\mathcal{L}}}^{c}$ is its open
complement, then
$\left.C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}\right|_{\Xi_{{\mathcal{L}}}^{c}}$
is a closed ideal of the groupoid $C^{\ast}$-algebra and we have the following
short exact sequence

$0\rightarrow\left.C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}\right|_{\Xi_{{\mathcal{L}}}^{c}}\stackrel{{\scriptstyle
i}}{{\rightarrow}}C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}\stackrel{{\scriptstyle\mathfrak{p}}}{{\rightarrow}}\left.C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}\right|_{\Xi_{{\mathcal{L}}}^{\infty}}\rightarrow
0.$ (27)

Figure 3. The process of deriving $\Xi_{{\mathcal{L}}}$ for the pattern
${\mathcal{L}}={\mathbb{N}}\times{\mathbb{Z}}$ from Example 2.18.

######  Remark 2.17.

As explained in subsection 2.3, the short exact sequence becomes equivariant
under a point group if it preserves $\Xi_{\mathcal{L}}$ and
$\Xi_{\mathcal{L}}^{\infty}$. However, to simplify this brief overview of
ordinary bulk-defect correspondences, the symmetries are ignored in this
section. $\Diamond$

######  Example 2.18.

Consider the case of the pattern
${\mathcal{L}}={\mathbb{N}}\times{\mathbb{Z}}^{d-1}\subset
G={\mathbb{R}}^{d}$, hence a uniform mesh with a boundary at $x=0$. It is a
simple exercise to derive the transversal $\Xi_{{\mathcal{L}}}$ for this
pattern. First, if ${\mathcal{S}}\in\Xi_{\mathcal{L}}$, its orbit under the
groupoid action is simply ${\mathcal{O}}({\mathcal{S}})=\\{{\mathcal{S}}-x,\
x\in{\mathcal{S}}\\}$ (see Remark 2.8). Then one has

$\Xi_{{\mathcal{L}}}={\mathcal{O}}({\mathcal{L}})\cup\\{{\mathbb{Z}}^{d}\\}$
(28)

as a disjoint union, i.e. the periodic lattice
${\mathbb{Z}}^{d}\in{\mathcal{C}}({\mathbb{R}}^{d})$ is in the closure of
${\mathcal{O}}({\mathcal{L}})$. Furthermore, ${\mathbb{Z}}^{d}$ is invariant
under the groupoid action of ${\mathcal{G}}_{\mathcal{L}}$ when viewed as an
element of the unit space $\Xi_{\mathcal{L}}$. An easy way to see this is to
note that Fell’s topology on ${\mathcal{C}}({\mathbb{R}}^{d})$ coincides with
the one generated by the Hausdorff metric on the closed subsets of the one
point compactification of ${\mathbb{R}}^{d}$. The latter is popular in image
processing [5], so we can visualize ${\mathcal{O}}({\mathcal{L}})$ as the
sequence of patterns seen by an observer hopping alongside ${\mathcal{L}}$
(see Figure 3). As the edge of ${\mathcal{L}}$ distances from the observer,
the seen patterns become more and more similar to the infinite lattice.

The groupoid algebra ${\mathcal{H}}=C^{\ast}{\mathcal{G}}_{\mathcal{L}}$
generates the physical Hamiltonians supported by the half-space pattern. The
subset $\Xi_{\mathcal{L}}^{\infty}:=\\{{\mathbb{Z}}^{d}\\}$ is translation-
invariant and its groupoid algebra
$C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi^{\infty}_{\mathcal{L}}}$ reduces
to $C^{\ast}{\mathbb{Z}}^{d}$, hence it models the bulk of the system. Since
$\Xi_{\mathcal{L}}^{c}:={\mathcal{O}}({\mathbb{N}}\times{\mathbb{Z}}^{d-1})$
is open, the associated groupoid algebra
${\mathcal{E}}=C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi^{c}_{\mathcal{L}}}$
is non-unital and it only contains elements that vanish far away from the
boundary. Hence, it models the observations made around the boundary.
Furthermore, the exact sequence (27) maps into the Toeplitz extension used in
the standard bulk-boundary correspondence [32, 48] (see [47] for the explicit
mapping). $\Diamond$

######  Definition 2.19.

We say that the pattern ${\mathcal{L}}$ contains an elementary geometric
defect if the transversal $\Xi_{\mathcal{L}}$ has a unique proper closed
subset that is invariant under the ${\mathcal{G}}_{\mathcal{L}}$ action.

Example 2.18 shows that cutting a material in half results in an elementary
defect. In [47], it was shown explicitly that disclinations are also
elementary defects in the sense of Definition 2.19, and it was pointed further
that all standard material defects (see [38] for a catalog) can be
characterized in this way. Furthermore:

######  Proposition 2.20 ([47]).

Elementary defects can be classified by the isomorphism class of the extension
$ext({\Xi_{\mathcal{L}}^{\infty}})$ in (27) associated to the unique
decomposition of the transversal. As a consequence [30], each elementary
geometric defect carries the topological charge

$[ext({\Xi_{\mathcal{L}}^{\infty}})]_{1}\in
KK_{1}(C^{\ast}{\mathcal{G}}_{\mathcal{L}}|_{\Xi_{{\mathcal{L}}}^{\infty}},C^{\ast}{\mathcal{G}}_{\mathcal{L}}|_{\Xi_{{\mathcal{L}}}^{c}}),$
(29)

valued in the complex $KK$-theory.

The ordinary bulk-defect correspondences involve elementary defects and all
can be explained by the connecting maps induced in the appropriate
$K$-theories by the exact sequence (27). In fact, one can describe them in a
unifying way using a key observation from [9]. In the absence of any symmetry,
this description is as follows:

######  Proposition 2.21 ([47]).

Associated to the exact sequence (27) there is a natural connecting
homomorphism $\partial_{0}$ which makes the following sequence of complex
K-groups exact

$...\rightarrow
K_{0}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}})\stackrel{{\scriptstyle\mathfrak{p}_{*}}}{{\rightarrow}}K_{0}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi_{{\mathcal{L}}}^{\infty}})\stackrel{{\scriptstyle\partial_{0}}}{{\rightarrow}}K_{1}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi_{{\mathcal{L}}}^{c}})\stackrel{{\scriptstyle
i_{*}}}{{\rightarrow}}K_{1}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}})\rightarrow...$
(30)

Let $h_{b}\in M_{N}({\mathbb{C}})\otimes
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi_{{\mathcal{L}}}^{\infty}}$ be a
spectrally gapped bulk Hamiltonian and let $[\gamma_{h_{b}}]_{0}$ be the class
of the spectral projection onto the spectrum below the gap in the complex
$K$-group
$K_{0}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}}|_{\Xi_{{\mathcal{L}}}^{\infty}})$.
If the class

$\partial_{0}([\gamma_{h_{b}}]_{0}):=[\gamma_{h_{b}}]_{0}\times[ext({\Xi_{\mathcal{L}}^{\infty}})]_{1}\in
K_{1}(C^{\ast}{\mathcal{G}}_{\mathcal{L}}|_{\Xi_{{\mathcal{L}}}^{c}})$ (31)

is nontrivial, then any lift of $h_{b}$ to $M_{N}({\mathbb{C}})\otimes
C^{\ast}{\mathcal{G}}_{\mathcal{L}}$ under the map $\mathfrak{p}$ in (27)
fills the spectral gap of $h$. The symbol “$\times$” in (31) refers to
Kasparov’s product.

######  Remark 2.22.

The exactness of (30) means that the class $[\gamma_{h_{b}}]_{0}$ has a pre-
image in $K_{0}(C^{\ast}{\mathcal{G}}_{\mathcal{L}})$ if and only if
$\partial_{0}([\gamma_{h_{b}}]_{0})$ is trivial. The occurrence of defect-
localized states in the bulk gap protected by a K-theory class in
$K_{1}(C^{\ast}{\mathcal{G}}_{\mathcal{L}}|_{\Xi_{{\mathcal{L}}}^{c}})$ is
therefore exactly the obstruction to this lifting problem in K-theory. This
characterization will be crucial when discussing higher-order correspondences.
$\Diamond$

######  Remark 2.23.

The Kasparov product (31) is an insightful way to write the connecting map in
(30). It highlights that the connecting maps are not just plain homomorphisms
of abelian groups but have very particular properties, such as compatibility
with the associativity of the Kasparov product. Moreover, the algebraic
structure on the extension classes encodes precise and useful topological
information. It would be desirable to formalize such information also for co-
filtrations of $C^{*}$-algebras like (4), not just extensions, by putting an
algebraic structure on their equivalency classes. A tentative step in this
direction is made in Corollary 4.30. $\Diamond$

To conclude, the ordinary bulk-defect correspondences stem from filtrations
$\Xi_{\mathcal{L}}^{\infty}\subset\Xi_{\mathcal{L}}$ of length 1 of the
transversals of the patterns with elementary geometric defects. The spectral
statement spelled out in Proposition 2.21 is entirely and rigidly determined
by this filtration and by the assumed symmetries. As already explained in our
introductory remarks, the higher-order bulk-boundary correspondences are
associated with filtrations of the unit space that have lengths strictly
larger than 1.

##  3\. Higher-order correspondences: Building the $C^{\ast}$-models

In this section, we supply the technical background for Propositions 1.1 and
1.2 and demonstrate how they play out for the concrete cases of quarter,
square and cube geometries.

###  3.1. Technical statements

######  Proposition 3.1.

Let $({\mathcal{L}}^{\lambda})_{\lambda\in\Lambda}$ be a finite family of
$U$-discrete subsets of ${\mathcal{C}}(G)$ and define their global transversal
as

$\Xi:={\bigcup}\nolimits_{\lambda\in\Lambda}\,\Xi_{{\mathcal{L}}^{\lambda}}.$
(32)

Then
${\mathcal{G}}_{\Xi}:=\bigcup_{\lambda\in\Lambda}{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$
can be given the structure of a lcsc étale groupoid with unit space $\Xi$ and
the same algebraic relations and topology as in Proposition 2.7, with
$\Xi_{\mathcal{L}}$ replaced by $\Xi$. It is the co-limit under the diagrams

$\leavevmode\hbox to134.76pt{\vbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{23.13266pt}{-5.8997pt}\pgfsys@lineto{44.25279pt}{-5.8997pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad(\lambda,\lambda^{\prime})\in\Lambda\times\Lambda,$
(33)

i.e. it is the smallest topological groupoid such that each of the diagrams

${{\mathcal{G}}_{\Xi}}$${{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}}$${{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}}$${{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\cap{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}}$
(34)

commutes. Furthermore, ${\mathcal{G}}_{\Xi}$ is amenable.

###### Proof.

Consider a pair $\lambda$ and $\lambda^{\prime}$. Then
${\mathcal{O}}({\mathcal{L}}^{\lambda})$ and
${\mathcal{O}}({\mathcal{L}}^{\lambda^{\prime}})$ either coincide or are
disjoint. In the first case,
$\Xi_{{\mathcal{L}}^{\lambda}}=\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}$ while
in the second case
$\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}$ is a
closed subset invariant under the actions of both groupoids, which can be very
well the empty set. In both cases,

${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\cap{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}=\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}}=\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\cap{\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}}},$
(35)

hence
${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\cap{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}$
is a (full) subgroupoid, and

${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}^{2}=\big{(}\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\setminus(\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}})}\big{)}^{2}\cup\big{(}\left.{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\right|_{\Xi_{{\mathcal{L}}^{\lambda}}\cap\Xi_{{\mathcal{L}}^{\lambda^{\prime}}}}\big{)}^{2}.$
(36)

A statement similar to (36) holds for ${\mathcal{L}}^{\lambda^{\prime}}$. Now,
co-limits and pushouts in the category of groupoids are described in [55] and
in more detail in [37]. By following these descriptions and by taking into
account the above facts, one concludes that the co-limit under (33) in the
category of (algebraic) groupoids is just the union of the groupoids.
Similarly, in the category of topological spaces, the co-limit under (33) is
the union of topological spaces ${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$.
Therefore, the co-limit in the category of topological groupoids must coincide
with $\bigcup_{\lambda\in\Lambda}{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$, if
the latter is a topological groupoid, that is, if the algebraic structure is
compatible with the topology, and this is the case. Indeed, by construction,
the elements of the so constructed algebraic groupoid are pairs
$(g,{\mathcal{S}})$ with ${\mathcal{S}}\in\Xi$ and $g\in{\mathcal{S}}$, and
the inversion and composition of such pairs are exactly as described in
Proposition 2.7, if we replace $\Xi_{\mathcal{L}}$ by $\Xi$. Also, since each
${\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$ inherits its topology from
$G\times{\mathcal{C}}(G)$, their union also share this atribute. The
conclusion is that the topology of ${\mathcal{G}}_{\Xi}$ is also as described
in Proposition 2.7, and this topology is automatically compatible with the
algebraic structure.

We now show that ${\mathcal{G}}_{\Xi}$ is étale, i.e. that the range map
$\mathfrak{r}$ is a local homeomorphism. Since every pattern from $\Xi$ is
$U$-uniformly separated for a fixed open subset $U\in\mathcal{C}(G)$, the
statement from Lemma 3.9 from [20] continues to apply without modifications.
This statement assures us that the map $V\times\Xi\to{\mathcal{C}}(G)$ given
by $(g,{\mathcal{S}})\mapsto g\cdot{\mathcal{S}}$ is homeomorphism onto its
image, for any open subset $V\in G$ such that $V\cap V^{-1}\subset U$. From
here, we can follow the arguments from Proposition 3.10 from [20]. By
definition, the sets $U_{\gamma,V,\Gamma}=(\gamma\cdot
V\times\Gamma)\cap{\mathcal{G}}_{\Xi}$ form a basis for the topology of
${\mathcal{G}}_{\Xi}$, where $\gamma$ ranges over $G$, $V$ ranges over all
symmetric neighborhoods of the identity with $V^{2}\subseteq U$ and $\Gamma$
ranges over all open sets in $\Xi$. Note that $(\gamma\cdot
V)^{-1}(\gamma\cdot V)=V^{2}\subseteq U$, hence we are in the conditions of
[20][Lemma 3.9] and the map $\gamma\cdot V\times\Xi\to{\mathcal{C}}(G)$ given
by $(g,{\mathcal{S}})\mapsto g\cdot{\mathcal{S}}$ is a local homeomorphism. By
restricting this map to $U_{\gamma,V,\Gamma}$, we obtain a homeomorphism onto
an image contained in $\Xi$, which coincides with the restriction of the range
map on $U_{\gamma,V,\Gamma}$. Hence, ${\mathcal{G}}_{\Xi}$ is étale.

For the remaining statement, we will use an equivalent characterization of
${\mathcal{G}}_{\Xi}$. Defining a hull

$\Omega_{\Xi}^{\times}=\overline{\\{g\cdot{\mathcal{L}}:\ g\in
G,{\mathcal{L}}\in\Xi\\}}\setminus\emptyset,$ (37)

one again obtains a locally compact $G$-invariant subset of ${\mathcal{C}}(G)$
and can therefore define a transformation groupoid
$\tilde{{\mathcal{G}}}_{\Xi}:=\Omega_{\Xi}^{\times}\rtimes G$. Restricting it
to the invariant subset
$\tilde{\mathfrak{s}}^{-1}(\Xi)\cap\tilde{\mathfrak{r}}^{-1}(\Xi)$, one
obtains precisely ${\mathcal{G}}_{\Xi}$. Now, [20, Prop. 3.8] can be used
without alteration to prove that $\Xi$ is an abstract transversal of
$\Omega_{\Xi}^{\times}\rtimes G$, therefore the amenability of
${\mathcal{G}}_{\Xi}$ follows from the amenability of $G$, which is assumed
throughout. ∎

Since we are dealing with a lcsc étale groupoid, there is a natural
$C^{\ast}$-algebra:

######  Corollary 3.2.

To any finite set of patterns $({\mathcal{L}}^{\lambda})_{\lambda\in\Lambda}$
with transversal
$\Xi=\bigcup_{\lambda\in\Lambda}\Xi_{{\mathcal{L}}^{\lambda}}$, we can
associate the groupoid $C^{\ast}$-algebra $C^{\ast}{\mathcal{G}}_{\Xi}$
corresponding to the groupoid ${\mathcal{G}}_{\Xi}$ constructed in Proposition
3.1 and to its Haar system of counting measures. The $C^{\ast}$-algebras
$C^{\ast}{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}$ fit into commutative
diagrams dual to (34):

${C^{*}({\mathcal{G}}_{{\mathcal{L}}^{\lambda}}\cap{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}})}$${C^{*}{\mathcal{G}}_{{\mathcal{L}}^{\lambda^{\prime}}}}$${C^{*}{\mathcal{G}}_{{\mathcal{L}}^{\lambda}}}$${C^{*}{\mathcal{G}}_{\Xi}}$
(38)

######  Remark 3.3.

The left regular representations of the $C^{\ast}{\mathcal{G}}_{\Xi}$ supply
representations of $C^{*}{\mathcal{G}}_{{\mathcal{L}}}$ on the Hilbert spaces
$\ell^{2}({\mathcal{L}}^{\lambda})$. As explained in the introduction, if we
interpret $C^{*}{\mathcal{G}}_{\Xi}$ as an algebra which encodes observations
made by multiple observers placed on different locations of a crystal in the
infinite-size limit, then the sole consistency condition necessary is that all
their observations come from representations of the same element of
$C^{\ast}{\mathcal{G}}_{\Xi}$. $\Diamond$

The construction can be also used to recover the group actions. Indeed, as a
consequence of Corollary 2.14, we have:

######  Corollary 3.4.

If $\sigma\in Aut(G)$ maps $\Xi$ into itself, then it gives rise to an
automorphism of $C^{\ast}{\mathcal{G}}_{\Xi}$.

Let us also comment briefly on ideals in those algebras:

######  Proposition 3.5 ([63], Thm. 5.1).

If $\tilde{\Xi}\subset\Xi$ is a closed and ${\mathcal{G}}_{\Xi}$-invariant
subset then restriction of the unit space

$C^{*}{\mathcal{G}}_{\Xi\setminus\tilde{\Xi}}:=C^{*}{\mathcal{G}}_{\Xi}\rvert_{\Xi\setminus\tilde{\Xi}}$

gives a closed ideal in $C^{*}{\mathcal{G}}_{\Xi}$ such that
$C^{*}{\mathcal{G}}_{\Xi}/C^{*}{\mathcal{G}}_{\Xi\setminus\tilde{\Xi}}\simeq
C^{*}{\mathcal{G}}_{\tilde{\Xi}}$.

Since the unit space $\Xi\setminus\tilde{\Xi}$ is open,
$C^{*}{\mathcal{G}}_{\Xi\setminus\tilde{\Xi}}$ will, by definition of the
groupoid algebra as $C^{*}$-completion of the convolution algebra
$C_{c}({\mathcal{G}}_{\Xi\setminus\tilde{\Xi}})$, only contain those elements
of $C^{*}{\mathcal{G}}_{\Xi}$ that asymptotically vanish in regions that look
more and more like patterns contained in $\tilde{\Xi}$, hence by choosing
appropriate subsets of $\Xi$ one can isolate elements localized to certain
boundaries or other geometric defects.

In the remainder of this section, we will consider examples how those
constructions play out to demonstrate how the unit spaces glue naturally for
typical crystals and also determine filtrations of the unit spaces by closed
invariant subsets.

###  3.2. Quarter geometry

Figure 4. (a) The pattern
${\mathcal{L}}={\mathbb{N}}\times{\mathbb{N}}\times{\mathbb{Z}}\subset{\mathbb{R}}^{3}$
considered in subsection 3.2. The edges of each layer are indicated with
fictitious dashed lines and the hinge of the pattern by a solid line. (b) A
top view of the pattern indicating directions in which an observer will find
different asymptotic patterns. (c) Representation of the unit space
$\Xi_{\llcorner}$ inside the space ${\mathcal{C}}({\mathbb{R}}^{3})$. The
decreasing size of the shaded squares symbolizes convergence to a limit point.

We analyze here the pattern
${\mathcal{L}}={\mathbb{N}}\times{\mathbb{N}}\times{\mathbb{Z}}^{d-2}\subset
G={\mathbb{R}}^{d}$, shown in Fig. 4(a) for the case $d=3$, with the point
group $\Sigma$ of symmetries containing only the reflection against the
diagonal which exchanges the two factors of ${\mathbb{N}}$ (for simplicity we
ignore possible symmetries that only engage the last $d-2$ coordinates). This
case is special because we only need one observer, hence its analysis can be
completed with just the tools presented in the previous section. We will
denote the canonical groupoid for this patter by ${\mathcal{G}}_{\llcorner}$.

######  Proposition 3.6.

The transversal of the quarter pattern is the disjoint union

$\Xi_{\llcorner}={\mathcal{O}}({\mathcal{L}})\cup{\mathcal{O}}({\mathcal{L}}^{1}_{\infty})\cup{\mathcal{O}}({\mathcal{L}}^{2}_{\infty})\cup\\{{\mathcal{L}}_{\infty}\\}$
(39)

where, ${\mathcal{L}}_{\infty}={\mathbb{Z}}^{d}$ and
${\mathcal{L}}^{1}_{\infty}={\mathbb{N}}\times{\mathbb{Z}}\times{\mathbb{Z}}^{d-2}$
and
${\mathcal{L}}^{2}_{\infty}={\mathbb{Z}}\times{\mathbb{N}}\times{\mathbb{Z}}^{d-2}$
are the limits in the Fell topology of the translates ${\mathcal{L}}-ne_{i}$
for $n\to\infty$, with $e_{i}$ being the unit vectors for the first two
directions.

###### Proof.

First, it should be clear that the pattern remains unchanged if the observer
moves in any direction except for the first two. Thus, the last $d-2$
directions can be ignored and, as such, it is enough to look at the pattern
from the top, as in Figure 4(b). Now, it is standard to prove that the limits
${\mathcal{L}}_{\infty}^{i}$ exist and that they coincide with the patterns
studied in Example 2.18. Then the transversal
$\Xi_{{\mathcal{L}}_{\infty}^{i}}$ of ${\mathcal{L}}_{\infty}^{i}$ are limit
sets of ${\mathcal{O}}({\mathcal{L}})$ (see Fig. 4(c)). Lastly, from Example
2.18, we have
$\Xi_{{\mathcal{L}}_{\infty}^{i}}={\mathcal{O}}({\mathcal{L}}^{i}_{\infty})\cup\\{{\mathcal{L}}_{\infty}\\}$
and an observer starting at the corner and traveling into different directions
will eventually see a pattern from one of the four components listed in Eq.
(39) (see Figure 4(b)). ∎

As opposed to the case of elementary defects, this transversal has several
closed and invariant proper subsets, which are $\\{{\mathcal{L}}_{\infty}\\}$,
$\Xi_{i}^{\infty}={\mathcal{O}}({\mathcal{L}}^{i}_{\infty})\cup\\{{\mathcal{L}}_{\infty}\\}$,
$i=1,2$, and more importantly,

$\Xi_{\llcorner}^{\infty}={\mathcal{O}}({\mathcal{L}}^{1}_{\infty})\cup{\mathcal{O}}({\mathcal{L}}^{2}_{\infty})\cup\\{{\mathcal{L}}_{\infty}\\}=\Xi_{{\mathcal{L}}_{\infty}^{1}}\cup\Xi_{{\mathcal{L}}_{\infty}^{2}}.$
(40)

For the reader’s convenience, we illustrate these subsets in Fig. 5. As a
result, we can generate several filtrations of $\Xi_{\llcorner}$ by closed
invariant subsets, and these filtrations have maximal length 3. However:

######  Proposition 3.7.

With the assumed group of point symmetries, the space of units has the unique
filtration

$\\{{\mathcal{L}}_{\infty}\\}\subset\Xi_{\llcorner}^{\infty}\subset\Xi_{\llcorner}$
(41)

by closed proper subsets that are invariant to the actions of both the
groupoid and group of symmetries. In the dual picture, this supplies the
equivariant co-filtration

$\bar{\mathcal{Q}}:=C^{\ast}{\mathcal{G}}_{\llcorner}\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{2}}}{{\twoheadrightarrow}}\bar{\mathcal{P}}:=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}}\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{1}}}{{\twoheadrightarrow}}{\mathcal{B}}:=C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}$
(42)

mentioned in Proposition 1.2.

To prepare the ground for section 4, we introduce and characterize useful
ideals in the $C^{\ast}$-algebras listed above. The kernel of the epimorphism
$\bar{\mathfrak{p}}^{1}$ is the $C^{\ast}$-algebra

$\bar{\mathcal{C}}:=\left.C^{\ast}{\mathcal{G}}_{\llcorner}\right|_{\Xi_{\llcorner}\setminus\Xi_{\llcorner}^{\infty}}=C^{\ast}{\mathcal{G}}_{\Xi_{\llcorner}\setminus\Xi_{\llcorner}^{\infty}},$
(43)

which can be rightfully referred to as the corner algebra, since it relates to
physical observations made around the corner of the quarter-space (for
simplicity we will also call this a corner for $d>2$, even though it
corresponds to a hinge in $d=3$ and is more generally called a ridge in
polyhedral geometry). The kernel of the epimorphism $\bar{\mathfrak{p}}^{0}$
is the $C^{\ast}$-algebra

$\bar{\mathcal{F}}:=\left.C^{\ast}{\mathcal{G}}_{\llcorner}\right|_{\Xi^{\infty}_{\llcorner}\setminus\\{{\mathcal{L}}_{\infty}\\}}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}\setminus\\{{\mathcal{L}}_{\infty}\\}},$
(44)

which can be rightfully called the face algebra because it relates to the
physical observations around the facets and away from the corner.
Additionally, as in Example 2.18, we can define
${\mathcal{H}}_{i}:=\left.C^{\ast}{\mathcal{G}}_{\llcorner}\right|_{\Xi^{\infty}_{i}}$
and
${\mathcal{F}}_{i}:=\left.C^{\ast}{\mathcal{G}}_{\llcorner}\right|_{\Xi^{\infty}_{i}\setminus\\{{\mathcal{L}}_{\infty}\\}}$,
which are the standard half-space and face $C^{\ast}$-algebras corresponding
to the two surfaces $x=0$ and $y=0$, respectively. It is easy to see that
$\bar{\mathcal{F}}={\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2}$.

Figure 5. Illustrations in the space ${\mathcal{C}}({\mathbb{R}}^{d})$ of
patterns, showing the relevant closed and invariant subspaces of the unit
space $\Xi_{\llcorner}$ for the quarter geometry described in subsection 3.2.

######  Remark 3.8.

For $d=2$, the $C^{\ast}$-algebras encountered above can be identified with
$C^{\ast}$-algebras from the theory of quarter-space Toeplitz operators. Note
that, instead of a simple quarter-space at right-angles, we could have more
generally defined ${\mathcal{L}}$ to have angles $\alpha$, $\beta$ w.r.t. the
horizontal, which would merely modify the asymptotic half-space patterns
${\mathcal{L}}^{1}_{\infty},{\mathcal{L}}^{2}_{\infty}$. In the notation of
[43], $\bar{\mathcal{Q}}$ then coincides with the algebra
$\mathfrak{T}^{\alpha,\beta}$, $\bar{\mathcal{P}}$ coincides with the algebra
$\mathcal{S}^{\alpha,\beta}$ and ${\mathcal{H}}_{i}$’s coincide with
$\mathfrak{T}^{\alpha}$ and $\mathfrak{T}^{\beta}$. Furthermore, one has the
short exact sequences

${\mathcal{F}}_{2}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}\setminus\Xi^{\infty}_{1}}\rightarrowtail\bar{\mathcal{P}}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}}\twoheadrightarrow{\mathcal{H}}_{1}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{1}};$
(45)
${\mathcal{F}}_{1}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}\setminus\Xi^{\infty}_{2}}\rightarrowtail\bar{\mathcal{P}}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}}\twoheadrightarrow{\mathcal{H}}_{2}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{2}}.$
(46)

While the transversal of the quarter-space is generated by a single pattern,
the same is not true for all of its closed subgroupoids. In particular,
$\bar{\mathcal{P}}=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\llcorner}}$ is the
groupoid algebra associated to the transversal
$\Xi^{\infty}_{\llcorner}=\Xi_{{\mathcal{L}}^{1}_{\infty}}\cup\Xi_{{\mathcal{L}}^{2}_{\infty}}$
and thus by Corollary 3.2 it is the pull-back of groupoid algebras

${{\mathcal{B}}}$${{\mathcal{H}}_{1}}$${{\mathcal{H}}_{2}}$${{\bar{\mathcal{P}}}}$
(47)

which is how $\bar{\mathcal{P}}$ was defined in [43].

Figure 6. (a) A wire with a square cross section is growing laterally while
being observed by four experimenters with a corner in their field of view; (b)
The transversals $\Xi_{{\mathcal{L}}^{\lambda}}$ reported by the four
observers; (c) The transversal $\Xi_{\square}$ of the whole wire in the
infinite-size limit.

###  3.3. Square geometry

We consider here the scaling limit of infinitely long wires cut out of a
${\mathbb{Z}}^{d}$ mesh, which are infinite in $d-2$ directions and in the
remaining two directions have square cross sections growing laterally as
indicated Fig. 6(a). As for the symmetry group $\Sigma$, we take the full
point symmetry of such shape (isomorphic to the dihedral group $D_{4}$). The
sample is inspected by four observers who are fixed somewhere at the indicated
corners (in a top view). They can be naturally labeled by
$\lambda\in{\mathbb{Z}}_{4}$ and the point group $\Sigma$ acts on these labels
by bijections. In the extreme limit, the observers sitting at the four corners
see different patterns ${\mathcal{L}}^{\lambda}$ and report the transversals
$\Xi_{{\mathcal{L}}^{\lambda}}$ seen in Fig. 6(b). There is an action of
$\Sigma$ on ${\mathbb{R}}^{d}$ such that
$\Xi_{{\mathcal{L}}^{\sigma\cdot\lambda}}=\sigma(\Xi_{{\mathcal{L}}^{\lambda}})$.
We single out the closed invariant subsets $\Xi_{\lambda,\\#}^{\infty}$,
$\\#\in\\{1,2,\llcorner\\}$ analogous to the previous section fixing the
labels as in Figure 6(a).

The global transversal of the square geometry in the infinite-size limit is

$\Xi_{\square}:={\bigcup}\nolimits_{\lambda\in{\mathbb{Z}}_{4}}\,\Xi_{{\mathcal{L}}^{\lambda}},$
(48)

which is depicted in Fig. 6(c). A remarkable fact here is that the
transversals $\Xi_{{\mathcal{L}}^{\lambda}}$ match along their boundaries as
indicated in Fig. 6(b). Indeed, for example, as observer 1 walks to the right
and observer 2 walks to the left, they will eventually see the same half-space
pattern, indicated as
${\mathcal{L}}_{\infty}^{1,1}={\mathcal{L}}_{\infty}^{2,2}$ in Fig. 6(a).

Figure 7. The largest closed and invariant proper subset of the space of units
$\Xi_{\square}$ studied in section 3.3.

We denote by ${\mathcal{G}}_{\square}$ the topological groupoid for
transversal $\Xi_{\square}$. The latter is invariant to the group of point
symmetries and it displays a large number of closed and invariant proper
subsets. As such, there are many filtrations which have lengths smaller or
equal to 3. However:

######  Proposition 3.9.

The space of units $\Xi_{\square}$ accepts a unique filtration of length 2,

$\\{{\mathcal{L}}_{\infty}\\}\subset\Xi_{\square}^{\infty}\subset\Xi_{\square},$
(49)

by closed subspaces that are invariant to ${\mathcal{G}}_{\square}$ and
$\Sigma$ actions. $\Xi_{\square}^{\infty}$, which is the largest closed and
invariant subspace of $\Xi_{\square}$, is displayed in Fig. 7. In the dual
picture, this supplies the equivariant co-filtration

${\mathcal{Q}}:=C^{\ast}{\mathcal{G}}_{\square}\stackrel{{\scriptstyle\mathfrak{p}^{2}}}{{\twoheadrightarrow}}{\mathcal{P}}:=C^{\ast}{\mathcal{G}}_{\Xi^{\infty}_{\square}}\stackrel{{\scriptstyle\mathfrak{p}^{1}}}{{\twoheadrightarrow}}{\mathcal{B}}:=C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}$
(50)

mentioned in Proposition 1.2.

${\mathcal{Q}}$ is the $C^{\ast}$-algebra encoding the experimental
observations on a wire with a large square cross section. The groupoid
$C^{\ast}$-algebra corresponding to the complement
$\Xi_{\square}\setminus\Xi_{\square}^{\infty}$ coincides with ${\rm
Ker}\,\mathfrak{p}^{1}$ and relates to the physical observations made around
the corners of the sample, hence, it can be rightfully called the algebra of
the corners. We denote it by ${\mathcal{C}}$ and
${\mathcal{C}}=\bigoplus_{\lambda\in{\mathbb{Z}}_{4}}\bar{\mathcal{C}}_{\lambda}$
for four isomorphic copies of the corner algebra $\bar{\mathcal{C}}$ of the
previous section. The groupoid $C^{\ast}$-algebra corresponding to the
complement $\Xi_{\square}^{\infty}\setminus\\{{\mathcal{L}}_{\infty}\\}$
coincides with ${\rm Ker}\,\mathfrak{p}^{0}$ and will be denoted by
${\mathcal{F}}$. Clearly,
${\mathcal{F}}=\oplus_{\lambda\in\Lambda}{\mathcal{F}}_{\lambda}$,
${\mathcal{F}}_{\lambda}:=C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}^{\lambda,1}}$,
and, as such, it can be rightfully called the face algebra.

###  3.4. Cube geometry

We consider here a crystal of cubic shape cut out of ${\mathbb{Z}}^{3}$ mesh,
as shown in Fig. 8, and we take as the group $\Sigma$ of symmetries the full
point symmetry group of the cubic lattice. The analysis may feel repetitive,
but this is exactly the purpose of this exercises, to help us reveal the
hierarchical structure of the space of units described in Propositions 1.1 and
1.2.

Figure 8. (a) Schematic of a growing process of a crystal with cubic symmetry.
Also indicated are three asymptotic directions in which observer $\lambda$
experiences quarter geometries. (b) The transversal
$\Xi_{{\mathcal{L}}^{\lambda}}$ consists of the union of the open subset
${\mathcal{O}}({\mathcal{L}}^{\lambda})$ and the transversals (shown in
matching colors) of the quarter patterns experienced by the experimenter from
the points indicated in panel (a). (c) The union of
$\Xi_{{\mathcal{L}}^{\lambda}}\setminus{\mathcal{O}}({\mathcal{L}}^{\lambda})$
over $\lambda$ supplies the set $\Xi_{2}$, which is the largest closed and
invariant proper subset of $\Xi_{\mbox{\small{\char 28\relax}}}$. The
characterization from Fig. 5 supplies the additional closed proper subset
$\Xi_{1}$ of $\Xi_{2}$, which is also invariant against
${\mathcal{G}}_{\mbox{\small{\char 28\relax}}}$ and $\Sigma$ actions.

Following our general strategy, we imagine such sample growing indefinitely
while experimenters located at each of the corners (of finite crystals)
observe and report their findings. They are labeled by the set $\Lambda$ and,
given our assumptions, $\Lambda$ accepts a natural action by the point
symmetry group $\Sigma$. Now, in the limit all but one corner vanishes from
the field of view of the observer $\lambda$, who will therefore report a
transversal $\Xi_{{\mathcal{L}}^{\lambda}}$. As seen in Fig. 8(a), the
observer $\lambda$ will in different asymptotic limits encounter three
different quarter patterns ${\mathcal{L}}_{\infty}^{\lambda,i}$ which are
familiar to us from section 3.2. The transversals of these quarter patterns
are limit sets for ${\mathcal{O}}({\mathcal{L}}^{\lambda})$ and, regardless of
the walking direction, the observer will eventually see a pattern from one of
these transversals as he/she walks away from the corner. Therefore,

$\Xi_{{\mathcal{L}}^{\lambda}}={\mathcal{O}}({\mathcal{L}}^{\lambda})\cup\Xi_{{\mathcal{L}}_{\infty}^{\lambda,1}}\cup\Xi_{{\mathcal{L}}_{\infty}^{\lambda,2}}\cup\Xi_{{\mathcal{L}}_{\infty}^{\lambda,3}}.$
(51)

This is schematically shown in Fig. 8(b). Since the point group of the lattice
acts naturally on the reported transversals,
$\sigma(\Xi_{{\mathcal{L}}^{\lambda}})=\Xi_{{\mathcal{L}}^{\sigma\cdot\lambda}}$,
(51) together with the analysis of the quarter patterns from subsection 3.2
give the complete picture of the reported transversals.

The global transversal of the cube geometry is

$\Xi_{\mbox{\small{\char
28\relax}}}:={\bigcup}\nolimits_{\lambda\in\Lambda}\,\Xi_{{\mathcal{L}}^{\lambda}},$
(52)

which is illustrated in Fig. 8(c). It reflects the remarkable fact that, if
the corners $\lambda$ and $\lambda^{\prime}$ are connected by a hinge, then
their corresponding transversals have a common boundary
$\Xi_{\lambda}\cap\Xi_{\lambda^{\prime}}$ that coincides with one of quarter-
space transversals $\Xi_{{\mathcal{L}}_{\infty}^{\lambda,j}}$. By
construction, $\Xi_{\mbox{\small{\char 28\relax}}}$ is invariant against the
action of the point group symmetry of the lattice.

######  Proposition 3.10.

For the assumed group of point symmetries, the transversal
$\Xi_{\mbox{\small{\char 28\relax}}}$ accepts a unique filtration of length 3

$\\{{\mathcal{L}}_{\infty}\\}=\Xi_{0}\subset\Xi_{1}\subset\Xi_{2}\subset\Xi_{3}=\Xi_{\mbox{\small{\char
28\relax}}},$ (53)

by closed proper subsets that are invariant against the groupoid
${\mathcal{G}}_{\mbox{\small{\char 28\relax}}}$ and group $\Sigma$ actions.
The notation in Eq. (53) is the same as in Fig. 8(c). In the dual picture, the
cube geometry carries a canonical co-filtration of the algebra of physical
observations

$C^{\ast}{\mathcal{G}}_{\mbox{\small{\char 28\relax}}}\twoheadrightarrow
C^{\ast}{\mathcal{G}}_{\Xi_{2}}\twoheadrightarrow
C^{\ast}{\mathcal{G}}_{\Xi_{1}}\twoheadrightarrow
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}},$ (54)

which is the equivariant co-filtration listed in Proposition 1.2.

######  Remark 3.11.

The uniqueness of the filtrations presented so far spurs from the transitive
character of the action of symmetry group on the set of observers.
Transitivity, however, is lost for simpler symmetry groups, such as those
containing just two elements, in which case there are more than one option for
proper symmetry-adapted filtrations. However, there is always a unique
filtration by the codimension of the respective boundaries. This issue will be
taken up again in section 4. $\Diamond$

###  3.5. Appending the fundamental symmetries

As we shall see from the examples supplied in section 5, the higher-order
bulk-boundary correspondences are aided by fundamental symmetries, such as
time-reversal, particle-hole or chiral symmetries. We briefly describe here
how they are appended to the groupoid formalism we already described, by
following the standard procedure devised in [22]. More details can be found in
our Appendix 6. If all fundamental symmetries are present, then $\Sigma$ is
enhanced to
$\bar{\Sigma}:=\Sigma\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$. As
explained in [22], physics constrains time reversal $T$ and particle-hole
exchange $P$ operations, which are the generators of the ${\mathbb{Z}}_{2}$
groups just introduced, to act anti-linearly on the algebras of observables
and $P$ to change the sign of the Hamiltonians. These complications can be
dealt with by passing to an extension of $\bar{\Sigma}$ by ${\mathbb{T}}$. Up
to isomorphisms, such extensions are enumerated by group morphisms
$\phi:\bar{\Sigma}\to{\rm Out}({\mathbb{T}})={\rm
Aut}({\mathbb{T}})\simeq{\mathbb{Z}}_{2}$ and by two-cocycles $\tau\in
H^{2}_{\phi}(\bar{\Sigma},{\mathbb{T}})$ and, as such, it is natural to denote
the mentioned group extension by $\bar{\Sigma}^{\phi}_{\tau}$. As a set,
$\bar{\Sigma}^{\phi}_{\tau}={\mathbb{T}}\times\bar{\Sigma}$ with
multiplication determined by $\phi$ and $\tau$. The twist $\phi$ fixes which
group elements will act anti-linearly in representations and the twisting
cocycle $\tau$ specifies whether the (matrix) representatives of $T$ or $P$
should square to $1$ or $-1$. In addition, the elements of $\bar{\Sigma}$ that
will eventually be required to reverse the signs of the Hamiltonians are fixed
by the non-trivial value of the grading homomorphism
$c:\bar{\Sigma}\to{\mathbb{Z}}_{2}$, which assigns $c(P)=1$, $c(T)=0$ and
$c(\sigma)=0$ for all $\sigma\in\Sigma$.

Now, let $C^{\ast}{\mathcal{G}}_{\Xi}$ be one of the groupoid
$C^{\ast}$-algebras equipped with a $\Sigma$-action $\alpha$, introduced in
the previous subsections. It is extended uniquely to a complex linear
$\bar{\Sigma}$-action $\alpha^{*}$ by letting
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ act trivially. Any element of
$C^{\ast}{\mathcal{G}}_{\Xi}$ can be written uniquely as a continuous function
in $C_{0}({\mathcal{G}}_{\Xi},{\mathbb{C}})$, since ${\mathcal{G}}_{\Xi}$ is
étale and amenable. One lets $\bar{\Sigma}_{\tau}^{\phi}$ act on $f\in
C_{0}({\mathcal{G}}_{\Xi},{\mathbb{C}})$ by setting

$(t,\bar{\sigma})\cdot
f=\begin{cases}t\,\overline{\alpha^{*}_{\bar{\sigma}}(f)}&\text{if
}\phi(\bar{\sigma})=1\\\ t\,\alpha^{*}_{\bar{\sigma}}(f)&\text{if
}\phi(\bar{\sigma})=0\end{cases}$ (55)

with the complex conjugation on ${\mathbb{C}}$, which is an
${\mathbb{R}}$-linear automorphism of $C^{\ast}{\mathcal{G}}_{\Xi}$. As
announced above, the morphism $\phi$ indicates when a group element
$\bar{\sigma}$ acts linearly or anti-linearly on
$C^{\ast}{\mathcal{G}}_{\Xi}$.

Given a set of data $(\phi,c,\tau)$, we are now ready to specify the algebras
from where the Hamiltonians are drawn. In appendix 6.1, we recall the
definition of $(\phi,c,\tau)$-twisted/graded $\bar{\Sigma}$-representations.
The spin and atomic orbital degrees of freedom carried by each atom are
encoded in a finite dimensional vector space ${\mathcal{V}}$. The algebra
$B({\mathcal{V}})$ of linear maps is equipped with a grading automorphism $C$
and inherits $\phi$-twisted $c$-graded $\bar{\Sigma}$-action stemming from a
graded algebra morphism

$U:{\mathbb{R}}\bar{\Sigma}_{\tau}^{\phi}\to
B_{\mathbb{R}}({\mathcal{V}}),\quad U\circ c=C\circ U$ (56)

which represents ${\mathbb{T}}$ as scalar multiplication. The Hamiltonians
generating the low energy dynamics of the electrons are produced from the
graded $\bar{\Sigma}_{\tau}^{\phi}$-$C^{\ast}$-algebra
$B({\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}_{\Xi}$.999Since
$C^{\ast}{\mathcal{G}}_{\Xi}$ is trivially graded, the graded and un-graded
tensor products coincide. A specific Hamiltonian will be a self-adjoint
element $h$ of this algebra which may or may not (anti-)commute with the
representatives of $\bar{\Sigma}$, but if it does satisfy

$[U_{\bar{\sigma}},h]=0,\quad\forall\bar{\sigma}\in\bar{\Sigma}_{\tau}^{\phi},$
(57)

where $[\cdot,\cdot]$ is the graded commutator, then it is called $c$-twisted
invariant under $\bar{\Sigma}$ (see Appendix 6.2). In most symmetry classes
those relations will only be required to hold for a specific subgroup of
$\bar{\Sigma}$.

######  Example 3.12.

In systems with chiral symmetry, the time-reversal and particle-hole
symmetries are broken, but the combination $S=TP$ persists. As a combination
of two anti-unitary transformations, $S$ is unitary, hence the data
$(\phi,\tau)$ is trivial, but $c(S)=1$. The minimal example is when
$B({\mathcal{V}}_{S})=M_{2}({\mathbb{C}})$, which we view as being generated
by Pauli matrices $\\{\alpha_{i}\\}_{i={1,2,3}}$. It is equipped with the
outer grading that changes the signs of the Pauli matrices, hence the standard
outer grading on the odd Clifford algebra, and also with the graded
representation of the sub-group $\\{I,S\\}$, $S\mapsto Ad_{\alpha_{3}}$. The
symmetric Hamiltonians anti-commute with the representative $\alpha_{3}$ of
$S$ and therefore take the off-diagonal form

$h=\begin{pmatrix}0&w^{\ast}\\\
w&0\end{pmatrix}=\tfrac{1}{2}\alpha_{1}\otimes(w+w^{\ast})+\tfrac{\imath}{2}\alpha_{2}\otimes(w-w^{\ast}),\quad
w\in C^{\ast}{\mathcal{G}}_{\Xi}.$ (58)

If there are additional (crystalline) symmetries then they should commute with
$S$ and the blocks $w,w^{*}$ of a symmetric $h$ should be separately
invariant. $\Diamond$

##  4\. Mechanism of higher-order correspondences

In this section, we first analyze higher-order bulk-boundary correspondences
for the geometries studied in section 3. In the process, we formalize what it
means for a model to be gapped at specific boundaries and then we reveal a
pattern in the formulation of the higher-order bulk-boundary principle. Based
on these findings, we provide a general picture of the principle and
demonstrate how it fits into the spectral sequences induced by the co-
filtrations from Proposition 1.2.

Before we start, we need to fix a relation between gapped Hamiltonians and
$K$-theory classes, which is essential in the field of topological condensed
matter systems. For this, let $C^{\ast}{\mathcal{G}}$ be any of the groupoid
$\Sigma$-$C^{\ast}$-algebras introduced so far. For some subgroup $\Gamma$ of
the extended symmetry group
$\bar{\Sigma}=\Sigma\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$, we will
want to use an $\Gamma$-equivariant K-functor together with its suspensions
$({\bf K}_{q})_{q\in{\mathbb{Z}}}$ to assign K-theory classes to gapped
Hamiltonians. To stay concrete, one may just want to fix the twisted
equivariant K-functor ${\bf K}_{q}={}^{\phi}K^{\Gamma}_{0+q,c,\tau}$ or ${\bf
K}_{q}={}^{\phi}K^{\Gamma}_{-1+q,c,\tau}$ with the two versions of the twisted
equivariant K-functor as defined in appendix 6.2, and twisting data
$(\phi,c,\tau)$ obtained from restriction of respective data on
$\bar{\Sigma}$. Other similarly defined functors will also work or are already
implicitly included in this generality, see Remark 6.4. Depending on the type
of K-functor and precise picture of K-theory not all ${\bf K}_{q}$ are
naturally represented in terms of physically relevant Hamiltonians and are
instead drawn from suspensions (though this can usually be remedied by going
to K-theory for graded algebras where suspensions can be replaced by graded
tensor products with Clifford algebras in exchange for other technical
complications). However, as seen in remark 6.4, there are indeed many relevant
cases where each ${\bf K}_{q}$ can be represented in terms of Hamiltonians
symmetric under some subgroup $\Gamma_{q}\subset\bar{\Sigma}$ that depends on
$q$ with either $2$\- or $8$-fold periodicity.

In any case, we focus on one particular value $q=\ast$, for which Hamiltonians
can in principle define classes in ${\bf K}_{\ast}$ without explicit
suspensions. Throughout, a Hamiltonian will be a self-adjoint element drawn
from one of the algebras $B({\mathcal{V}})\otimes C^{*}{\mathcal{G}}$ where
${\mathcal{V}}$ is a finite-dimensional graded vector space furnished with a
twisted representation of the symmetry group $\Gamma\subset\bar{\Sigma}$
carrying twisting data $(\phi,c,\tau)$. We will call a Hamiltonian $h\in
B({\mathcal{V}})\otimes C^{*}{\mathcal{G}}$ symmetric if it satisfies the
symmetry requirements to canonically define a class in ${\bf K}_{\ast}$, but
possibly does not have a spectral gap. For the functor ${\bf
K}_{\ast}={}^{\phi}K^{\Gamma}_{0,c,\tau}$ this means explicitly that $h$ is
$c$-twisted invariant under $\Gamma$ for the fixed $(\phi,c,\tau)$-twisted
representation on ${\mathcal{V}}$. If $h$ has a spectral gap then
$[\gamma_{h}]_{\ast}=[{\rm sgn}(h)]_{0}$ with the functional calculus in
$B({\mathcal{V}})\otimes C^{*}{\mathcal{G}}$ (we assume throughout that the
Fermi energy is fixed at zero and thus when we speak of spectral gaps we will
therefore always mean open intervals in the resolvent set which contain $0$).
In a unitary picture such as ${\bf K}_{\ast}={}^{\phi}K^{\Gamma}_{-1,c,\tau}$
or complex equivariant $K$-theory ${\bf K}_{\ast}=K^{\Gamma}_{1}$ one will
additionally assume that ${\mathcal{V}}$ has a balanced grading, i.e.
${\mathcal{V}}={\mathcal{W}}\oplus{\mathcal{W}}$ and $\gamma={\bf
1}_{\mathcal{W}}\oplus(-{\bf 1}_{\mathcal{W}})$, and symmetric Hamiltonians
are not only ($c$-twisted) invariant under $\Gamma$ but also odd w.r.t. the
grading. Then $h\mapsto[\gamma_{h}]_{\ast}$ for invertible $h$ maps to the
class defined by either of the two off-diagonal parts of ${\rm sgn}(h)$ (which
of the two components one picks is a matter of convention). Comparing to (58),
this corresponds precisely to Hamiltonians which have a chiral symmetry.

In any equivariant picture of K-theory one can, while preserving the ${\bf
K}_{\ast}$-theoretic class, amplify gapped Hamiltonians $h\in
B({\mathcal{V}})\otimes C^{*}{\mathcal{G}}$ to ones in
$B({\mathcal{V}}\oplus{\mathcal{W}})\otimes C^{*}{\mathcal{G}}$ with
${\mathcal{W}}$ any admissible finite-dimensional (graded) representation
space of the symmetry group by adding to $h$ in a direct sum a gapped
symmetric Hamiltonian in $B({\mathcal{W}})$ which represents the trivial class
in ${\bf K}_{\ast}$.

######  Definition 4.1.

Two gapped symmetric Hamiltonians $h_{i}\in B({\mathcal{V}}_{i})\otimes
C^{\ast}{\mathcal{G}}$, $i=1,2$ are stably symmetric-preserving homotopic if
there are amplifications ${\mathcal{W}}_{1}$ and ${\mathcal{W}}_{2}$ so that
${\mathcal{V}}_{1}\oplus{\mathcal{W}}_{1}\simeq{\mathcal{V}}_{2}\oplus{\mathcal{W}}_{2}\simeq\bar{\mathcal{V}}$
(as representation spaces) and if the corresponding amplifications of $h_{1}$,
$h_{2}$ in $B(\bar{\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}$ can be norm-
continuously deformed into each other within the self-adjoint invertible
symmetric operators.

The standing assumption (which is true for the explicitly given K-functors
above) is that $[h]\mapsto[\gamma_{h}]_{\ast}\in{\bf K}_{\ast}$ is a one-to-
one correspondence with the stable homotopy equivalence classes of gapped
symmetric Hamiltonians.

###  4.1. Quarter geometry

Using the notations from subsection 3.2, we will consider model Hamiltonians
$h=h^{\ast}\in B({\mathcal{V}})\otimes\bar{\mathcal{Q}}$, symmetric under a
subgroup of a ${\bar{\Sigma}}$, generating the dynamics of electrons in a
sample filling a quarter of the space. For now we will leave the K-functor and
symmetry group unspecified. The Hamiltonian can be projected to a symmetric
bulk Hamiltonian $h_{b}\in B({\mathcal{V}})\otimes{\mathcal{B}}$ by using the
amplification of the composition
$\bar{\mathfrak{p}}^{1}\circ\bar{\mathfrak{p}}^{2}$ of the maps introduced in
equation (42). Since the bulk is assumed to be insulating, the latter is
assumed to have a gap in its spectrum, which we refer to as the bulk gap. We
first address the question of what it means for $h$ to be gapped at the facets
of the sample. As the words suggest, if one probes the quarter sample near the
faces and moves farther and farther away from the corners, one will find that
the Hamiltonian increasingly resembles a gapped operator. Using the physical
interpretation of the left-regular representations of
$\bar{\mathcal{Q}}=C^{\ast}{\mathcal{G}}_{\llcorner}$ (see Remark 2.11), we
can express this in precise mathematical terms by stating that
$\pi_{\mathcal{S}}(h)$ is a spectrally gapped operator for all
${\mathcal{S}}\in\Xi_{{\mathcal{L}}_{\infty}^{1}}\cup\Xi_{{\mathcal{L}}_{\infty}^{2}}$.
Given the definition of $\bar{\mathcal{P}}$, we arrive at the following
definition:

######  Definition 4.2.

A symmetric Hamiltonian $h\in B({\mathcal{V}})\otimes\bar{\mathcal{Q}}$ is
gapped at the codimension 1 boundaries (hence faces) if its projection
$\bar{\mathfrak{p}}^{2}(h)\in B({\mathcal{V}})\otimes\bar{\mathcal{P}}$ has a
spectral gap contained inside the spectral gap of $h_{b}$.

######  Definition 4.3.

Let $h_{b}=h_{b}^{\ast}\in B({\mathcal{W}})\otimes{\mathcal{B}}$ be a
symmetric gapped bulk Hamiltonian. We say that $h_{b}$ is gappable at the
codimension 1 boundaries if there exists a symmetric Hamiltonian
$h^{\prime}_{b}\in B({\mathcal{V}})\otimes{\mathcal{B}}$ which is in the same
${\bf K}_{\ast}$-theoretic class
$[\gamma_{h_{b}}]_{\ast}=[\gamma_{h_{b}^{\prime}}]_{\ast}$ and which lifts
under $\bar{\mathfrak{p}}^{1}\circ\bar{\mathfrak{p}}^{2}$ to a symmetric
Hamiltonian $h\in B({\mathcal{V}})\otimes\bar{\mathcal{Q}}$ that is gapped at
the codimension 1 boundaries.

In the following all Hamiltonians are assumed to be symmetric unless stated
otherwise. Being gappable at some boundary means precisely that K-theory does
not provide an obstruction to the existence of a spectral gap at that
boundary. Conversely, if a bulk Hamiltonian is not gappable at some boundary
then any lift must within the bulk gap host boundary modes that are localized
at that respective boundary and which are protected by K-theoretic invariants.

######  Remark 4.4.

The goal of this paper is to find all topological obstructions that are the
result of the bulk K-theory class, which naturally means that we classify
Hamiltonians up to stable equivariant homotopy. Accordingly, to construct a
gapped lift of a bulk Hamiltonian one is by the definition above allowed to
stabilize by adding as direct summands topologically trivial gapped
Hamiltonians of the respective symmetry class.

The following equivalent characterization highlights that one does not need to
think in terms of concrete Hamiltonians at all:

######  Proposition 4.5.

$h_{b}\in B({\mathcal{V}})\otimes{\mathcal{B}}$ is gappable at the codimension
1 boundaries if and only if its class $[\gamma_{h_{b}}]_{\ast}\in{\bf
K}_{\ast}({\mathcal{B}})$ accepts a lift in ${\bf
K}_{\ast}(\bar{\mathcal{P}})$ under the map $\bar{\mathfrak{p}}^{1}_{\ast}$.

###### Proof.

If $h_{b}$ is gappable then by assumption there exists some
$h_{b}^{\prime}\in{\mathcal{B}}({\mathcal{V}})\otimes\bar{\mathcal{B}}$ with
$[\gamma_{h_{b}}]_{\ast}=[\gamma_{h_{b}^{\prime}}]_{\ast}$ with a lift
$h\in{\mathcal{B}}({\mathcal{V}})\otimes\bar{\mathcal{Q}}$ that is gapped at
the codimension 1 boundaries. Therefore,
$\bar{\mathfrak{p}}^{1}_{*}([\gamma_{\bar{\mathfrak{p}}^{2}(h)}]_{\ast})=[\gamma_{h_{b}^{\prime}}]_{\ast}$
provides a pre-image of $[\gamma_{h_{b}}]_{\ast}$. Conversely, any class in
$(\bar{\mathfrak{p}}^{1}_{*})^{-1}([\gamma_{h_{b}}]_{\ast})\subset{\bf
K}_{\ast}(\bar{\mathcal{P}})$ can be represented by a gapped symmetric
Hamiltonian $\tilde{h}\in B({\mathcal{V}})\otimes\bar{\mathcal{P}}$ and one
can always lift that to a symmetric Hamiltonian $h\in
B({\mathcal{V}})\otimes\bar{\mathcal{Q}}$ by picking any self-adjoint lift and
averaging it over a twisted representation of the subgroup
$\Gamma\subset\bar{\Sigma}$ which implements ${\bf K}_{*}$. One can then set
$h_{b}^{\prime}=(\bar{\mathfrak{p}}^{1}\circ\bar{\mathfrak{p}}^{2})(h)$. ∎

The formulation of the higher-order bulk-boundary correspondence engages only
(symmetric) bulk models that are gappable at the edges. The long exact
sequence in K-theory induced by the equivariant epimorphism
$\bar{\mathfrak{p}}^{1}:\bar{\mathcal{P}}\twoheadrightarrow{\mathcal{B}}$,

${\bf
K}_{\ast}(\bar{\mathcal{F}})\stackrel{{\scriptstyle\bar{i}^{1}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast}(\bar{\mathcal{P}})\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{1}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast}({\mathcal{B}})\stackrel{{\scriptstyle\bar{\partial}^{1}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast-1}(\bar{\mathcal{F}})\stackrel{{\scriptstyle\bar{i}^{1}_{\ast-1}}}{{\longrightarrow}}{\bf
K}_{\ast-1}(\bar{\mathcal{P}})\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{1}_{\ast-1}}}{{\longrightarrow}}{\bf
K}_{\ast-1}({\mathcal{B}}),$ (59)

gives an equivalent characterization, since exactness at ${\bf
K}_{\ast}({\mathcal{B}})$ in conjunction with Proposition 4.5 literally means:

######  Corollary 4.6.

Up to stable symmetry-preserving homotopies, the symmetric bulk Hamiltonians
that are gappable at the codimension 1 boundaries correspond precisely to the
classes in

${\rm Im}\,\bar{\mathfrak{p}}^{1}_{\ast}={\rm
Ker}\,\bar{\partial}^{1}_{\ast}\subset{\bf K}_{\ast}({\mathcal{B}}).$ (60)

We now turn our attention to the exact sequence derived from the
$\bar{\mathfrak{p}}^{2}:\bar{\mathcal{Q}}\to\bar{\mathcal{P}}$ epimorphism:

${\bf
K}_{\ast}(\bar{\mathcal{C}})\stackrel{{\scriptstyle\bar{i}^{2}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast}(\bar{\mathcal{Q}})\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{2}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast}(\bar{\mathcal{P}})\stackrel{{\scriptstyle\bar{\partial}^{2}_{\ast}}}{{\longrightarrow}}{\bf
K}_{\ast-1}(\bar{\mathcal{C}})\stackrel{{\scriptstyle\bar{i}^{2}_{\ast-1}}}{{\longrightarrow}}{\bf
K}_{\ast-1}(\bar{\mathcal{Q}})\stackrel{{\scriptstyle\bar{\mathfrak{p}}^{2}_{\ast-1}}}{{\longrightarrow}}{\bf
K}_{\ast-1}(\bar{\mathcal{P}}).$ (61)

Due to the exact nature of the sequence any symmetric lift of a gapped
Hamiltonian $h\in B({\mathcal{V}})\otimes\bar{\mathcal{P}}$ to
${\mathcal{B}}({\mathcal{V}})\otimes\bar{\mathcal{Q}}$ must have corner modes
protected by K-theoretic obstructions if and only if
$\bar{\partial}_{*}^{2}([\gamma_{h}]_{\ast})\in{\bf
K}_{*-1}(\bar{\mathcal{C}})$ is non-trivial. Taking the analogue of
Proposition 4.5 as a definition, a gapped bulk Hamiltonian shall be called
gappable at the corner if and only if its K-theory class lifts from ${\bf
K}_{\ast}({\mathcal{B}})$ to ${\bf K}_{\ast}(\bar{\mathcal{P}})$ and then
further to ${\bf K}_{\ast}(\bar{\mathcal{Q}})$. Conversely, if a bulk model is
gappable at the edges, but not gappable at the corner, then every lift that is
gapped at the edges will display protected corner modes inside the bulk gap.
In this case we will speak of an order-2 bulk-boundary correspondence, since
the existence of some corner modes is enforced by the K-theory class of the
bulk material, no matter which lift to
${\mathcal{B}}({\mathcal{V}})\otimes\bar{\mathcal{P}}$ one chooses, as long as
it is gapped and symmetric. In particular, this means the fact that corner
modes occur is independent of such a choice of boundary condition, but not
their number and type. We can enumerate all such instances as follows:

######  Proposition 4.7.

There exists a well defined bulk-corner map

$\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{\ast}:{\rm
Im}\,\bar{\mathfrak{p}}^{1}_{\ast}={\rm
Ker}\,\bar{\partial}^{1}_{\ast}\subseteq{\bf
K}_{\ast}({\mathcal{B}})\to\frac{\mathrm{Im}\,\bar{\partial}^{2}_{\ast}}{\bar{\partial}^{2}_{\ast}({\rm
Ker}\,\bar{\mathfrak{p}}^{1}_{\ast})}=\frac{\mathrm{Im}\,\bar{\partial}^{2}_{\ast}}{\mathrm{Im}\,(\bar{\partial}^{2}_{\ast}\circ\bar{i}^{1}_{\ast})}\subseteq\frac{{\bf
K}_{\ast-1}(\bar{\mathcal{C}})}{\bar{\partial}^{2}_{\ast}({\rm
Ker}\,\bar{\mathfrak{p}}^{1}_{\ast})},$ (62)

and whose support identifies all symmetric bulk Hamiltonians that host non-
trivial order-2 bulk-boundary correspondences.

###### Proof.

By definition, a class $x\in{\rm Im}\,\bar{\mathfrak{p}}^{1}_{\ast}\subset{\bf
K}_{\ast}({\mathcal{B}})$ has a pre-image $\tilde{x}\in{\bf
K}_{\ast}(\bar{\mathcal{P}})$, but it need not be unique. However, any two
pre-images $\tilde{x},\tilde{x}^{\prime}$ differ only by an element
$\tilde{x}-\tilde{x}^{\prime}\in{\rm
Ker}\,\bar{\mathfrak{p}}^{1}=i^{1}_{\ast}({\bf K}_{\ast}(\bar{\mathcal{F}}))$,
with the latter following from equality being the exactness of (59).
Therefore, $\bar{\partial}_{\ast}^{2}(\tilde{x})$ and
$\bar{\partial}_{\ast}^{2}(\tilde{x}^{\prime})$ differ only by an element of
$\bar{\partial}_{\ast}^{2}({\rm Ker}\,\bar{\mathfrak{p}}^{1}_{\ast})$, which
makes

$\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{\ast}:x\mapsto\bar{\partial}_{\ast}^{2}(\tilde{x})+\bar{\partial}_{\ast}^{2}({\rm
Ker}\,\bar{\mathfrak{p}}^{1}_{\ast})$

for arbitrary choice of lift $\tilde{x}$ a well-defined homomorphism of
abelian groups. One can lift $x$ to ${\bf K}_{\ast}(\bar{\mathcal{Q}})$ if and
only if there is a pre-image $\tilde{x}\in{\bf K}_{\ast}(\bar{\mathcal{P}})$
with $\bar{\partial}_{\ast}^{2}(\tilde{x})=0$, hence if and only if
$\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{\ast}(x)=0$. ∎

One takes the quotient by $\bar{\partial}^{2}_{\ast}({\rm
Ker}\,\bar{\mathfrak{p}}^{1}_{\ast})$ which by definition enumerates precisely
the corner modes which can be carried by models that are trivial in the bulk,
i.e. pure surface layers. Generally it would be difficult to directly
enumerate the classes in ${\bf K}_{\ast}({\mathcal{B}})$ that lift to ${\bf
K}_{\ast}(\bar{\mathcal{P}})$ but not to ${\bf K}_{\ast}(\bar{\mathcal{Q}})$,
since the latter two groups are in practice not known explicitly. That is
precisely the point of rewriting all components of (62) in terms of more
computable expressions using exactness. For example, it is usually quite
feasible to compute ${\bf K}_{\ast}(\bar{\mathcal{F}})$ and then
$\mathrm{Im}\,(\bar{\partial}^{2}_{\ast}\circ\bar{i}^{1}_{\ast})$. The domain
and range of $\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{\ast}$ can then be
determined by computing the boundary maps $\bar{\partial}_{*}^{1}$ and
$\bar{\partial}_{*}^{2}$ for sufficiently many Hamiltonians.

######  Example 4.8.

Take $d=2$. Then the diagonal mirror symmetry together with chiral symmetry
are enough to produce a non-trivial bulk-boundary correspondence of order 2.
In this case, $\Sigma=\\{e,\sigma_{m}\\}\simeq{\mathbb{Z}}_{2}$ and we only
need to engage the ordinary ${\mathbb{Z}}_{2}$-equivariant $K$-theories, i.e.
the K-functor above becomes ${\bf K}_{q}=K_{q}^{{\mathbb{Z}}_{2}}$, which is
to be distinguished from the non-equivariant complex K-functor $K_{q}$ which
also plays a role in the following. Classes in $K^{{\mathbb{Z}}_{2}}_{0}$ are
defined by symmetric matrix-valued projections, in particular the spectral
projections of gapped symmetric Hamiltonians, and classes in
$K^{{\mathbb{Z}}_{2}}_{1}$ by symmetric unitary matrices, in particular the
polar decompositions of the off-diagonal parts $w$ of oddly graded gapped
symmetric Hamiltonians as in (58).

For $S_{1},\ S_{2}$ the unitary generators of the group $C^{*}$-algebra
${\mathcal{B}}=C^{\ast}{\mathbb{Z}}^{2}$, the ${\mathbb{Z}}_{2}$-action
$\sigma_{m}$ interchanges $S_{1}$ and $S_{2}$. We fix the directions of the
shifts by imposing that projecting them to
$\ell^{2}({\mathbb{N}}\times{\mathbb{N}})$ makes them isometries. We can see
from the start that
$\bar{\mathcal{C}}\simeq\mathbb{K}({\mathbb{N}}\times{\mathbb{N}})$, the
compact operators on $\ell^{2}({\mathbb{N}}\times{\mathbb{N}})$, which yields
${K}_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})=0$ and

$K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})=K_{0}(C^{\ast}{\mathbb{Z}}_{2})\otimes
K_{0}(\bar{\mathcal{C}})={\mathbb{Z}}[\chi_{+}\otimes
E_{0}]_{0}\oplus{\mathbb{Z}}[\chi_{-}\otimes E_{0}]_{0}$ (63)

with representatives in
${\mathcal{B}}({\mathcal{V}})\otimes\bar{{\mathcal{C}}}$. Here
${\mathcal{V}}={\mathbb{C}}e\oplus{\mathbb{C}}\sigma_{m}$ is a two-dimensional
vector space, $\chi_{\pm}=\frac{1}{2}(e\pm\sigma_{m})$ are the central
projections supporting the trivial/non-trivial irreducible representations of
${\mathbb{Z}}_{2}$, and
$E_{0}=|\delta_{0}\otimes\delta_{0}\rangle\langle\delta_{0}\otimes\delta_{0}|$
is the rank-one projection onto the site located exactly at the corner. The
equivariant $K$-theories of the face algebra are also straightforward. Indeed,
we have $\bar{\mathcal{F}}={\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2}$,
${\mathcal{F}}_{i}\simeq{\mathbb{K}}({\mathbb{N}})\otimes
C^{\ast}{\mathbb{Z}}$, and ${\mathbb{Z}}_{2}$ acts by an isomorphism between
${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$. Then Proposition 6.5 gives

$\begin{split}K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{F}})&={\mathbb{Z}}[\chi_{+}\otimes(P_{1}\oplus
P_{2})]_{0},\\\
K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{F}})&={\mathbb{Z}}[\chi_{+}\otimes\left((S_{1}P_{2}+P_{2}^{\perp})\oplus(S_{2}P_{1}+P_{1}^{\perp})\right)]_{1}\end{split}$
(64)

where $P_{i}$ are the images of $|\delta_{0}\rangle\langle\delta_{0}|\otimes
1\in{\mathbb{K}}({\mathbb{N}})\otimes C^{\ast}{\mathbb{Z}}$ through the stated
isomorphisms. Since $K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})$ is trivial,
the only option left for us to observe a non-trivial bulk-corner
correspondence is to consider $\ast=1$ in 4.7. As such, we are forced to
engage the $K_{1}$-group of the bulk algebra and this calls for a chiral
symmetry on the level of self-adjoint Hamiltonians.

######  Proposition 4.9.

We have

$K^{{\mathbb{Z}}_{2}}_{0}(\mathcal{B})={\mathbb{Z}}[\chi_{+}\otimes
1_{\mathcal{B}}]_{0}\oplus{\mathbb{Z}}[\chi_{-}\otimes
1_{\mathcal{B}}]_{0},\quad
K^{{\mathbb{Z}}_{2}}_{1}(\mathcal{B})={\mathbb{Z}}[u_{\mathcal{F}}]_{1}\oplus{\mathbb{Z}}[u_{\mathcal{C}}]_{1},$
(65)

where

$u_{\mathcal{F}}=\chi_{+}\otimes S_{1}S_{2}+\chi_{-}\otimes
1_{\mathcal{B}},\quad
u_{\mathcal{C}}=\frac{1}{2}\begin{pmatrix}-S_{1}^{\ast}-S_{2}^{\ast}&S_{1}^{\ast}-S_{2}^{\ast}\\\
S_{1}-S_{2}&S_{1}+S_{2}\end{pmatrix},$ (66)

with the latter written as an element of
$M_{2}({\mathbb{C}})\otimes{\mathcal{B}}$ with the ${\mathbb{Z}}_{2}$ action
on ${\mathbb{C}}^{2}$ given by $M={\scriptsize\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}}$, such that
$(Ad_{M}\otimes\sigma_{m})(u_{\mathcal{C}})=u_{\mathcal{C}}$.

###### Proof.

The equivariant $K^{\ast}_{{\mathbb{Z}}_{2}}$-groups of the 2-torus can be
easily computed from the symmetry-adapted CW-filtration $X_{0}\subset
X_{1}\subset{\mathbb{T}}^{2}$, defined as follows: If ${\mathbb{T}}^{2}$ is
seen as a square with identified opposite edges, then $X_{0}$ is any of the
corners and $X_{1}$ is the union of left and down edges, as well as the
mirror-invariant diagonal. An elementary computation with the Atiyah-
Hirzebruch spectral sequence shows that
$K^{{\mathbb{Z}}_{2}}_{0}({\mathcal{B}})\simeq K_{0}(C^{*}{\mathbb{Z}}_{2})$
and $K^{{\mathbb{Z}}_{2}}_{1}({\mathcal{B}})\simeq{\mathbb{Z}}^{2}$, with the
two generators distinguished by the following properties: As unitary functions
on the torus, the first, $u_{\mathcal{F}}$, has winding number $1$ on both
edges of the square and the second, $u_{\mathcal{C}}$, becomes a diagonal
matrix when restricted to the diagonal of the square and there has winding
numbers $1$ and $-1$ respectively in the two eigenspaces of $M$. ∎

######  Proposition 4.10.

The domain of $\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}$ is ${\rm
Im}\,\bar{\mathfrak{p}}^{1}_{1}={\mathbb{Z}}[u_{\mathcal{C}}]_{1}$.

###### Proof.

As a self-adjoint invertible element with chiral symmetry, $u_{\mathcal{C}}$
is represented by

$h_{b}=\begin{pmatrix}0&u_{\mathcal{C}}\\\
u_{\mathcal{C}}^{\ast}&0\end{pmatrix}\in B({\mathcal{V}}_{S})\otimes
M_{2}({\mathbb{C}})\otimes{\mathcal{B}},$ (67)

where $B({\mathcal{V}}_{S})$ and its structure are described in Example 3.12.
By separating the generators $S_{i}$, we have the decomposition
$h_{b}=h_{1}(S_{1})+h_{2}(S_{2})$. A useful feature to notice is that
$h_{1}(X)$ anti-commutes with $h_{2}(Y)$ whenever $X$ and $X^{\ast}$ both
commute with $Y$ and $Y^{\ast}$, but not necessarily with themselves. Now, the
pair
$\hat{h}=(h_{1}(\hat{S}_{1})+h_{2}(S_{2}),h_{1}(S_{1})+h_{2}(\hat{S}_{2}))$
supplies a symmetric lift of $h_{b}$ to
$M({\mathbb{C}}_{4})\otimes\bar{\mathcal{P}}$, where the hat indicates the
standard Toeplitz extension and $\bar{\mathcal{P}}$ is viewed as the pullback
explained in Remark 3.8. Since
$h_{1}(S_{1})^{2}=h_{2}(S_{2})^{2}=\frac{1}{2}$, we have

$\hat{h}^{2}=(h_{1}(\hat{S}_{1})^{2}+h_{2}(S_{2})^{2},h_{1}(S_{1})^{2}+h_{2}(\hat{S}_{2})^{2})=\tfrac{1}{2}\,1_{\bar{\mathcal{P}}}+(h_{1}(\hat{S}_{1})^{2},h_{2}(\hat{S}_{2})^{2}),$
(68)

hence $\hat{h}$ is invertible. As such, $\hat{h}$ supplies a class in
$K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{P}})$, which proves that
$[u_{\mathcal{C}}]_{1}$ belongs to the range of $\bar{\mathfrak{p}}^{1}_{1}$.
The other generator $u_{\mathcal{F}}$ has non-zero weak odd Chern numbers in
both directions, hence it leads to non-trivial classes under
$\bar{\partial}_{1}^{1}$ and, as such, to un-gappable models (see Section 5.1
below for the more general case). ∎

######  Proposition 4.11.

The image of $\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}$ is isomorphic to
${\mathbb{Z}}_{2}$ and the generator $[u_{\mathcal{C}}]_{1}$ of the domain of
$\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}$ is mapped into the generator
${\mathbb{Z}}_{2}$.

###### Proof.

Our first task is to evaluate $\bar{\partial}_{1}^{2}([\hat{u}_{c}]_{1})$,
where $\hat{u}_{c}$ is the unitary operator in
$M_{2}({\mathbb{C}})\otimes\bar{\mathcal{P}}$ obtained from the spectral
flattening of $\hat{h}$. A concrete expression of the connecting map for the
non-equivariant case was given in [48, Prop. 4.3.2], the class in
$K_{0}(\bar{\mathcal{C}})$ of $\bar{\partial}_{1}^{2}([\hat{u}_{c}]_{1})$ is
represented by the projection

$\bar{P}=e^{-\imath\frac{\pi}{2}\varphi(\bar{h})}{\rm
diag}(1_{M_{2}({\mathbb{C}})\otimes\bar{\mathcal{Q}}},0)e^{\imath\frac{\pi}{2}\varphi(\bar{h})}\in
M_{4}({\mathbb{C}})\otimes\bar{\mathcal{C}},$ (69)

where $\bar{h}$ is any chirally symmetric lift of $\hat{h}$ to
$M_{4}({\mathbb{C}})\otimes\bar{\mathcal{Q}}$ and
$\varphi:{\mathbb{R}}\to[-1,1]$ a continuous non-decreasing odd function with
variation inside the spectral gap of $\hat{h}$. This expression will also
represent the element in the equivariant
$K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})$, if we choose a
${\mathbb{Z}}_{2}$-invariant lift
$\bar{h}=h_{1}(\bar{S}_{1})+h_{2}(\bar{S}_{2})$, where $\bar{S}_{i}$’s are the
lifts of $S_{i}$ supplied by truncating to the quarter geometry. Furthermore,
we are in the conditions of Proposition 4.3.3 in [48], hence the projection in
(69) is equivalent to $J\bar{P}_{0}+{\rm
diag}(0_{2},1_{{M_{2}({\mathbb{C}})}})$, where $\bar{P}_{0}$ is the spectral
projection of $\bar{h}$ onto $\\{0\\}$ and
$J=1_{M_{2}({\mathbb{C}})}\oplus(-1_{M_{2}({\mathbb{C}})})$ is the operator
implementing chiral symmetry. Using the anti-commuting structure mentioned
above, it is easily verifiable that the kernel of $\bar{h}$ has dimension one
and that the corresponding projector coincides with $\chi_{+}\otimes{\rm
diag}(E_{0},0)$. Following a similar procedure for the generator of
$K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{F}})$, one finds

$\bar{\partial}_{1}^{2}\Big{(}[\chi_{+}\otimes\left((S_{1}P_{2}+P_{2}^{\perp})\oplus(S_{2}P_{1}+P_{1}^{\perp})\right)]_{1}\Big{)}=-2[\chi_{+}\otimes
E_{0}]_{0},$ (70)

and the statement follows. ∎

These calculations demonstrate that our framework not only enable us to
identify a non-trivial bulk model $h_{b}$, but also to conclude that, for a
quarter geometry and diagonal mirror symmetry, there are no other
(topologically distinct) models that can produce bulk-boundary correspondences
of order-2. $\Diamond$

######  Remark 4.12.

Had we used in the previous example complex non-equivariant $K$-theory, we
would have found that the natural map $K_{1}(\bar{\mathcal{F}})\to
K_{1}(\bar{\mathcal{P}})$ is an isomorphism, which readily implies
$\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}=0$. This absence of second-order
bulk-boundary correspondence means that without the crystalline symmetry
fixing any particular $K$-theory class in the bulk never constrains the corner
states in any way.$\Diamond$

######  Remark 4.13.

After using the map $\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}$ in
equivariant $K$-theory to assert the existence of a corner mode it is actually
stable even if one breaks the symmetry at the corner (but not if the
asymptotic symmetry between the two half-spaces adjacent to the corner is
broken). This is because the possible corner states in
$\mathrm{Im}\,\delta^{{\mathcal{B}}\bar{\mathcal{C}}}_{1}$ all correspond to
classes $K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})$ which get mapped
injectively when one forgets the symmetry via the homomorphism
$K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})\to K_{0}(\bar{\mathcal{C}})$. The
commutative diagram

${K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{P}})}$${K_{0}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{C}})}$${K_{1}(\bar{\mathcal{P}})}$${K_{0}(\bar{\mathcal{C}})}$$\scriptstyle{{\bar{\partial}}^{2}_{1}}$$\scriptstyle{{\bar{\partial}}^{2}_{1}}$
(71)

then tells us that if we start with a class in
$K_{1}^{{\mathbb{Z}}_{2}}(\bar{\mathcal{P}})$ we can still detect the corner
states using the boundary map in non-equivariant $K$-theory where no symmetry
is enforced at the corner. In other scenarios, however, a symmetry-protected
corner mode can be trivial in non-equivariant $K$-theory and then one can
remove it by breaking the symmetry at the corner.$\Diamond$

###  4.2. Square geometry

Once a filtration of the unit space is fixed, all statements from the previous
subsection remain valid for the wire geometry, and they can be formulated in
exactly the same form but with the bar removed from above the symbols and the
$K$-theoretic functor properly adjusted. Examples of non-trivial higher-order
bulk-boundary correspondences for this geometry will be supplied in section 5.

###  4.3. Cube geometry

This geometry displays faces, hinges and corners, which we refer to as
boundaries of codimension 1, 2 and 3, respectively. As such, the crystal
geometry can support non-trivial correspondences of order 1, 2 and 3. An
order-3 bulk boundary correspondence is carried by the corners of the cube and
one will want to use a filtration of appropriate length:

######  Proposition 4.14.

There exists a unique filtration

$\Xi_{0}\subset\Xi_{1}\subset\Xi_{2}\subset\Xi_{3},$ (72)

starting at $\Xi_{0}=\\{{\mathcal{L}}_{\infty}\\}$ and ending at
$\Xi_{3}={\bigcup}_{\lambda\in\Lambda}\ \Xi_{{\mathcal{L}}^{\lambda}}$ such
that each $C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$, $1\leq r\leq 3$,
consists out of those observables localized to the boundaries of codimension
$r$.

###### Proof.

The beginning $\Xi_{0}$ and the end $\Xi_{3}$ of the filtration are fixed. One
can assign to each pattern in $\Xi_{3}$ uniquely a codimension, which is $2$
for all quarter-space patterns and $1$ for all half-space patterns. One needs
to take for $\Xi_{r}$ precisely the set of all patterns with codimension $r$
or less to achieve the mentioned localization property for
$C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$ (see the comment below
Definition 3.5). ∎

The cube is invariant under a finite symmetry group $\Sigma$, w.l.o.g. the
point symmetry group of the cube, and we again fix a subgroup
$\Gamma\subset\bar{\Sigma}$ with twisting data $(\phi,c,\tau)$ together with
some twisted $\Gamma$-equivariant K-functor ${\bf K}_{\ast}$. The filtration
(72) is by construction term-wise $\Gamma$-invariant and a symmetric model
Hamiltonian $h\in B({\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}_{\Xi_{3}}$ can
therefore again be projected to a symmetric bulk Hamiltonian $h_{b}\in
B({\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}$ using
the amplifications of the maps in the equivariant co-filtration

$C^{\ast}{\mathcal{G}}_{\Xi_{3}}\stackrel{{\scriptstyle\mathfrak{p}^{3}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{2}}\stackrel{{\scriptstyle\mathfrak{p}^{2}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{1}}\stackrel{{\scriptstyle\mathfrak{p}^{1}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}.$
(73)

To observe protected topological face, hinge or corner modes, the bulk
Hamiltonian $h$ should in the first place be an insulator, i.e. have a
spectral gap. To classify protected corner modes one should further not
already have face or hinge modes inside the bulk gap. Given the definition and
physical interpretation of the left regular representations, this is the same
as saying that $\pi_{\mathcal{S}}(h)$ has a spectral gap for all
${\mathcal{S}}\in\Xi_{1}$ respectively ${\mathcal{S}}\in\Xi_{2}$. These
arguments leads to:

######  Definition 4.15.

The symmetric Hamiltonian $h\in B({\mathcal{V}})\otimes
C^{\ast}{\mathcal{G}}_{\Xi_{3}}$ is gapped at the boundaries of codimension
$r$ if the projection $(\mathfrak{p}^{r+1}\circ...\circ\mathfrak{p}^{3})(h)$
in $B({\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}_{\Xi_{r}}$ has a spectral
gap.

A symmetric gapped bulk Hamiltonian $h_{b}\in B({\mathcal{W}})\otimes
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}$ is said to be gappable at the
boundaries of codimension $r$ if there is a symmetric Hamiltonian
$h_{b}^{\prime}\in B({\mathcal{V}})\otimes
C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}}$ with the same bulk K-theory
class $[\gamma_{h_{b}}]_{\ast}=[\gamma_{h_{b}^{\prime}}]_{\ast}\in{\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}})$ which admits a