symmetric lift $h\in B({\mathcal{V}})\otimes C^{\ast}{\mathcal{G}}_{\Xi_{3}}$
that is gapped at the boundaries of codimension $r$.

######  Remark 4.16.

The spectrum can only become smaller under unital morphisms, hence a model
which is gapped or gappable at codimension $r$ boundaries is automatically
also gapped or gappable at all boundaries with smaller codimension. $\Diamond$

######  Proposition 4.17.

The symmetric gapped bulk Hamiltonian $h_{b}$ is gappable at the boundaries of
codimension $r$ if and only if the class $[\gamma_{h_{b}}]_{\ast}\in{\bf
K}_{\ast}({\mathcal{B}})$ can be lifted to an element of ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{r}})$.

###### Proof.

The argument is practically identical with the one for Proposition 4.5. ∎

######  Corollary 4.18.

Up to stable symmetry-preserving deformations, the symmetric bulk Hamiltonians
that are gappable at the boundaries of codimension 1 and 2 are respectively
listed by

${\rm Im}\,\mathfrak{p}^{1}_{\ast}\subseteq{\bf K}_{\ast}({\mathcal{B}}).$
(74)

and

${\rm Im}\,\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast}\subseteq{\bf
K}_{\ast}({\mathcal{B}}).$ (75)

The ideal ${\mathcal{C}}={\rm Ker}\,\mathfrak{p}^{3}$ supplies the algebra of
observations around the corners and the connecting map $\partial^{3}_{\ast}$
induced by $\mathfrak{p}^{3}$ sends the classes in ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{2}})$ of symmetric lifts of bulk
Hamiltonians to classes in ${\bf K}_{\ast-1}({\mathcal{C}})$. A symmetric bulk
Hamiltonian that is gappable at the codimension-2 boundaries (hinges) can be
(up to stable equivalence) lifted to $C^{\ast}{\mathcal{G}}_{\Xi_{2}}$ under
$\mathfrak{p}^{1}\circ\mathfrak{p}^{2}$, but such a lift to a gapped symmetric
Hamiltonian from $C^{\ast}{\mathcal{G}}_{\Xi_{3}}$ under
$\mathfrak{p}^{1}\circ\mathfrak{p}^{2}\circ\mathfrak{p}^{3}$ will exist if and
only if there is a pre-image of $[\gamma_{h_{b}}]_{\ast}$ in ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{2}})$ that is mapped by
$\partial^{3}_{\ast}$ to the trivial class of ${\bf
K}_{\ast-1}({\mathcal{C}})$, equivalently if and only if there exists a choice
of symmetric boundary condition for which there are no protected corner modes.
If conversely there are always protected corner modes, then we say that we
have a non-trivial order-3 bulk-boundary correspondence. The ambiguity in the
choice of lift to ${\bf K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{2}})$ is enumerated
by ${\rm Ker}(\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast})\subset{\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{2}})$ therefore these instances can be
detected using a group homomorphism like in Proposition 4.7:

######  Proposition 4.19.

There exists a well defined bulk-corner map

$\delta^{{\mathcal{B}}{\mathcal{C}}}_{\ast}:{\rm
Im}\,(\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast})\subseteq{\bf
K}_{\ast}({\mathcal{B}})\to\frac{\mathrm{Im}\,\partial^{3}_{\ast}}{\partial^{3}_{\ast}({\rm
Ker}\,\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast})}\subseteq\frac{{\bf
K}_{\ast-1}({\mathcal{C}})}{\partial^{3}_{\ast}({\rm
Ker}\,\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast})},$ (76)

obtained by computing $\partial^{3}_{\ast}$ for any lift from ${\rm
Im}\,\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast}$ to ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{2}})$. Its support identifies all symmetric
bulk models that generate non-trivial order-3 bulk-boundary correspondences
supported by $\Xi_{3}$. These correspondences cannot be destroyed by symmetry
preserving boundary conditions.

######  Remark 4.20.

The support of $\delta^{{\mathcal{B}}{\mathcal{C}}}_{\ast}$ identifies all
bulk models that are gappable at the faces and hinges of the cube, but not at
the corners. Conversely, any symmetric lift of such a bulk model to the cube
which is gapped at the faces and hinges must display corner modes in the bulk
gap protected by a non-trivial class in ${\bf K}_{*-1}({\mathcal{C}})$.
However, the latter may depend on the specific lift (i.e. boundary condition)
up to an element of ${\partial^{3}_{\ast}({\rm
Ker}\,\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast})}\subset{\bf
K}_{*-1}({\mathcal{C}})$, which enumerates again the corner modes that can be
carried by surface layers that are trivial in the bulk. $\Diamond$

######  Remark 4.21.

While Proposition 4.19 is formally very similar to Proposition 4.7,
determining the codomain can be significantly more difficult. The kernel of
$\mathfrak{p}^{1}_{\ast}\circ\mathfrak{p}^{2}_{\ast}$ is not easily computed
using the definitions alone. In the next section we will see how one can go
about this systematically. $\Diamond$

It happens frequently that a bulk Hamiltonian will display bulk-boundary
correspondences of mixed order, for example, there can be protected surface
modes at some faces while other faces can still be gapped and then there may
be additional second-order hinge modes or third-order corner modes at the
hinges respectively corners that do not border the gapless faces. We will see
examples of this in section 5.5. To resolve all phenomena of mixed higher
bulk-boundary correspondence one must potentially investigate all possible
filtrations adapted to the chosen symmetry group $\Gamma$, not just (72).
Generally, to detect a mixed order-$r$ bulk-boundary correspondence induced by
a specific class in ${\bf K}_{\ast}({\mathcal{B}})$ one should adapt the
filtration to exclude so many boundaries as to make it no longer be subject to
bulk-boundary correspondences of order $(r-1)$ and lower, starting with
removing all faces that potentially carry first-order boundary states. There
are some subtleties involved; a change of boundary condition can move the
higher boundary modes between different parts of the boundary and one can
further have both too many or too few boundaries left to stabilize higher-bulk
boundary correspondences, which means that one may need to sample several
filtrations until one finds the correct combinations of gappable boundaries
that stabilize interesting higher-order boundary modes. Since the relevant
boundary states should still be localized to faces, hinges or corners one can
restrict oneself to filtrations of the form

$(\tilde{\Xi}\cap\Xi_{0})\subset(\tilde{\Xi}\cap\Xi_{1})\subset(\tilde{\Xi}\cap\Xi_{2})\subset(\tilde{\Xi}\cap\Xi_{3})$
(77)

enumerated by closed invariant transversals
$\tilde{\Xi}\subset\Xi_{\mbox{\small{\char 28\relax}}}$ that select parts of
the cube geometry. This choice ensures that the boundary ideals
$C^{*}{\mathcal{G}}_{(\tilde{\Xi}\cap\Xi_{r})\setminus(\tilde{\Xi}\cap\Xi_{r-1})}$
still localize precisely to the boundaries of codimension $r$ contained in
$\tilde{\Xi}$. A particular example of this form is the transversal
$\Xi_{\square}$ of section 3.3 in three dimensions, which can be seen as a
closed invariant subset of $\Xi_{\mbox{\small{\char 28\relax}}}$ that excludes
all corners and all but four of the hinges and faces each, thereby isolating
mixed second-order bulk-boundary correspondences which localize to the
selected hinges. Other than using a different transversal respectively
filtration, the mathematical formalism remains unchanged.

###  4.4. A unifying picture

We now consider a crystal geometry in $d$ dimensions displaying boundaries of
codimension between $1$ and $d$ and with a global transversal $\Xi$ (which may
be only a subset of all possible patterns constructed in the scaling limit of
an actual crystal such as the wire geometry of section 3.3).

In accordance with the standing assumptions of this section, we fix again a
finite group of the form
$\Gamma\subset\bar{\Sigma}=\Sigma\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
together with a twist $(\phi,c,\tau)$ such that the twisted
$\Gamma$-equivariant K-functor ${\bf K}_{\ast}$ classifies the corresponding
stable homotopy classes of gapped symmetric Hamiltonians. We assume that
$\Gamma$ acts on ${\mathcal{G}}_{\Xi}$ via (anti-)linear automorphisms, thus
in particular it acts on the unit space $\Xi$ via homemorphisms. Suppose we
aim to discover the order-$r$ bulk-boundary correspondences supported by the
selected boundaries of codimension $r$, protected by spectral gaps at the
boundaries of codimension $r-1$, the symmetry $\Gamma$ and possibly some
fundamental symmetries. We assume that we have a filtration

$\\{{\mathcal{L}}_{\infty}\\}=\Xi_{0}\subset\ldots\subset\Xi_{d-1}\subset\Xi_{d}=\Xi,$
(78)

by $\Gamma$-symmetric closed invariant subsets, which mathematically defines
which patterns $\Xi_{r}\subset\Xi$ we consider to have boundaries of
codimension $r$ or less. Then the algebra of physical observables
$C^{\ast}{\mathcal{G}}_{\Xi}$ accepts a $\Gamma$-equivariant co-filtration

$C^{\ast}{\mathcal{G}}_{\Xi_{d}}\stackrel{{\scriptstyle\mathfrak{p}^{d}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{d-1}}\stackrel{{\scriptstyle\mathfrak{p}^{d-1}}}{{\twoheadrightarrow}}\cdots\stackrel{{\scriptstyle\mathfrak{p}^{2}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{1}}\stackrel{{\scriptstyle\mathfrak{p}^{1}}}{{\twoheadrightarrow}}C^{\ast}{\mathcal{G}}_{\Xi_{0}}=C^{\ast}{\mathcal{G}}_{{\mathcal{L}}_{\infty}},$
(79)

and each ${\rm
Ker}\,\mathfrak{p}^{r}=C^{\ast}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}}$ has
a sensible interpretation as an algebra of observations near the boundaries of
codimension precisely $r$.

Regarding the correspondence itself, the arguments are very similar to the
ones from the previous subsections:

######  Definition 4.22.

The gapped symmetric Hamiltonian $h_{b}\in B({\mathcal{V}})\otimes
C^{*}{\mathcal{G}}_{\Xi_{0}}$ is called gappable at the codimension $r$
boundaries if and only if $[\gamma_{h_{b}}]_{\ast}\in{\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$ admits a pre-image under
$s^{r}:=\mathfrak{p}^{1}\circ\cdots\circ\mathfrak{p}^{r}:C^{*}{\mathcal{G}}_{\Xi_{r}}\to
C^{*}{\mathcal{G}}_{\Xi_{0}}$ in ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r}})$. We also say that $h_{b}$ exhibits an
order-$r$ bulk boundary correspondence if $h_{b}$ is gappable at the
codimension $r-1$ boundaries but not at the codimension $r$ boundaries.

Let $\partial^{r}_{\ast}:{\bf K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r-1}})\to{\bf
K}_{*-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$ be the connecting
map induced by the epimorphism $\mathfrak{p}^{r}$. Any particular lift to
${\bf K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{r-1}})$ can be further lifted to
${\bf K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{r}})$ if and only if it is in the
kernel of $\partial^{r}_{\ast}$. If the image under $\partial^{r}_{\ast}$ is
conversely non-trivial for all lifts of a fixed class in ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{0}})$ then we have found an order-$r$
bulk-boundary correspondence. The ambiguities for the lifts from ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{0}})$ to ${\bf
K}_{\ast}(C^{\ast}{\mathcal{G}}_{\Xi_{r-1}})$, induced by choosing different
symmetric boundary conditions, are enumerated by ${\rm
Ker}\,(s^{r-1}_{*})\subset{\bf K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r-1}})$.
Thus, after these are quotiented out, we obtain an enumeration of the
order-$r$ correspondences supported by the boundaries selected by $\Xi_{r}$:

######  Proposition 4.23.

The stable equivariant homotopy classes of symmetric bulk Hamiltonians which
are gappable at the boundaries of codimensions less than $r$ are listed by
${\rm Im}\,s^{r-1}_{\ast}\subset{\bf K}_{\ast}({\mathcal{B}})$. One obtains a
well-defined homomorphism

$\delta^{r}_{\ast}:{\rm Im}\,s^{r-1}_{\ast}\subseteq{\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})\to\frac{\mathrm{Im}\,\partial^{r}_{\ast}}{\partial^{r}_{\ast}({\rm
Ker}\,s^{r-1}_{\ast})}$ (80)

by computing the connecting map for any lift from ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$ to ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r-1}})$ and then taking the stated quotient
to make it independent of the lift. If a bulk Hamiltonian $h_{b}$ is gappable
at the codimension $r-1$ boundaries then the range of $\partial^{r}_{\ast}$
evaluated on lifts of $[\gamma_{h_{b}}]_{\ast}$ to ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r-1}})$ is precisely given by the coset
$\delta^{r}_{\ast}([\gamma_{h_{b}}]_{\ast})$ as a subset of ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$. In particular,
if $\delta^{r}_{\ast}([\gamma_{h_{b}}]_{\ast})$ evaluates to a non-trivial
value (i.e. a coset that does not contain the neutral element) then $h_{b}$ is
not gappable at the codimension $r$ boundaries and therefore exhibits an
order-$r$ bulk-boundary correspondence.

In this way we can enumerate both the Hamiltonians which exhibit order-$r$
bulk-boundary correspondences as well as their possible boundary states at the
order $r$ boundaries under the assumption that the order $r-1$ boundaries are
gapped.

######  Remark 4.24.

The higher boundary maps $(\delta^{s}_{\ast})_{s\leq r}$ together identify the
bulk classes in ${\bf K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$ which cannot be
lifted to ${\bf K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r}})$ and assign an order
to the obstruction, which is the lowest order boundary which must be un-
gapped. For bulk models which are already not gappable at the codimension
$r-1$ boundaries one does not obtain any information about possible
codimension $r$ boundary states which may coexist with the lower order
boundary states. As sketched in the previous section, it is largely a modeling
problem to choose the filtrations (respectively geometries) in such a way that
the constructed boundary maps help one to identify and enumerate those
phenomena of higher-order bulk-boundary correspondence one is actually
interested in and one may have to repeat the computations for different
filtrations until one obtains a satisfactory picture of the entirety of the
possible boundary states.

We now prepare to place the higher-order bulk-boundary maps
$\delta^{r}_{\ast}$ in their rightful framework:

######  Theorem 4.25 ([49]).

For a cofiltration of $C^{\ast}$-algebras,

${\mathcal{A}}_{d}\twoheadrightarrow{\mathcal{A}}_{d-1}\twoheadrightarrow...\twoheadrightarrow{\mathcal{A}}_{0},$
(81)

there exists a spectral sequence $(E^{r}_{p,q},d^{r}_{p,q})$ converging to
${\bf K}({\mathcal{A}}_{d})$.

Although standard (see e.g. [36]), we need to describe this spectral sequence
in some detail. First of all one extends the co-filtration to
$({\mathcal{A}}_{n})_{n\in{\mathbb{Z}}}$ by setting
${\mathcal{A}}_{n}={\mathcal{A}}_{d}$ for $n\geq d$ and ${\mathcal{A}}_{n}=0$
for $n<0$. The first page of the spectral sequence is then the bigraded
complex

$E^{1}=\bigoplus_{p,q}E^{1}_{p,q},\quad E^{1}_{p,q}:={\bf
K}_{-p+q}({\mathcal{E}}_{p}),\quad{\mathcal{E}}_{p}={\rm
Ker}({\mathcal{A}}_{p}\twoheadrightarrow{\mathcal{A}}_{p-1}),$ (82)

and one considers the auxiliary bigraded complex
$D^{1}=\bigoplus_{p,q}D^{1}_{p,q}$ with $D^{1}_{p,q}:={\bf
K}_{-p+q}({\mathcal{A}}_{p})$. They form an exact couple, i.e. a diagram

${D^{1}}$${D^{1}}$${E^{1}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$
(83)

where every map has as its kernel the image of the previous one. The morphism
$\alpha$ maps each $D^{1}_{p,q}$ to $D^{1}_{p-1,q-1}$ and is given by ${\bf
K}_{-p+q}({\mathcal{A}}_{p})\to{\bf K}_{-p+q}({\mathcal{A}}_{p-1})$, induced
from ${\mathcal{A}}_{p}\twoheadrightarrow{\mathcal{A}}_{p-1}$. The morphism
$\beta$ maps each $D^{1}_{p,q}$ to $E^{1}_{p+1,q}$ and is given by the
connecting map induced by
${\mathcal{A}}_{p+1}\twoheadrightarrow{\mathcal{A}}_{p}$. The morphism
$\gamma$ maps each $E^{1}_{p,q}$ to $D^{1}_{p,q}$ and is given by ${\bf
K}_{-p+q}({\mathcal{E}}_{p})\rightarrow{\bf K}_{-p+q}({\mathcal{A}}_{p})$.
This first page is just a reformulation of the long exact sequence of
$K$-theory.

To any exact couple, one can canonically associate a new derived exact couple
$(E^{2},D^{2})$ and iteration results in a spectral sequence $E^{r}_{p,q}$
with differentials

$d^{r}_{p,q}:E^{r}_{p,q}\to E^{r}_{p+r,q+r-1},$ (84)

defined in terms of combinations of $\alpha$, $\beta$ and $\gamma$. Concretely
(e.g. [36, Proposition 2.8])

$E^{r}=\frac{\gamma^{-1}\alpha^{\circ(r-1)}(D^{1})}{\beta({\rm
Ker}\,\alpha^{\circ(r-1)})},$ (85)

which can be identified with the homology $H(E^{r-1},d^{r-1})$ w.r.t. the
differentials

$d^{r}=\beta(\alpha^{-1})^{\circ(r-1)}\gamma,$ (86)

where the inverse of $\alpha$ is well-defined since it does not depend on the
choice of lift. The construction is iterative and uses that
$D^{r}=\alpha^{r-1}(D^{1})$ forms another exact couple with $E^{r}$ and
$\alpha^{(r)}=\alpha$, $\beta{(r)}=\beta(\alpha^{-1})^{\circ(r-1)}$ and
$\gamma^{(r)}=\gamma$.

######  Remark 4.26.

There is a similar spectral sequence due to Schochet [52] also constructed
from an exact couple: Let
${\mathcal{I}}_{-1}={\mathcal{I}}_{0}\subset{\mathcal{I}}_{1}\subset...\subset{\mathcal{I}}_{d}={\mathcal{A}}$
be a filtration of a $C^{*}$-algebra ${\mathcal{A}}$ by closed ideals. Then
there is an exact couple with

$\bar{E}^{1}_{p,q}=K_{p+q}({\mathcal{I}}_{p}/{\mathcal{I}}_{p-1}),\quad\bar{D}^{1}_{p,q}=K_{p+q}({\mathcal{I}}_{p}).$
(87)

One can write any filtration of a $C^{*}$-algebra equivalently as a
cofiltration and then both approaches yield equivalent spectral sequences that
converge to $K_{\ast}({\mathcal{A}})$ [49], hence the higher-order bulk-
boundary maps can be constructed using either version. $\Diamond$

The following statement provides the connection between the boundary maps
constructed above with the spectral sequence and therefore completes the proof
of our main Theorem 1.3:

######  Proposition 4.27.

Let $(E^{r}_{p,q},d^{r}_{p,q})$ be the spectral sequence corresponding to co-
filtration (79). Then the corestriction of the $r$-th differential

$d^{r}_{0,q}:{\rm Ker}(d^{r-1}_{0,q})\to\mathrm{Im}(d^{r}_{0,q})$ (88)

for $q=\ast$ coincides with the corestriction of $\delta^{r}_{\ast}:{\rm
Ker}(\delta^{r-1}_{\ast})\to\mathrm{Im}(\delta^{r}_{\ast})$.

###### Proof.

One can write (85) as

$E^{r}_{p,q}=\frac{\gamma^{-1}\mathrm{Im}({\bf
K}_{-p+q}({\mathcal{A}}_{p+r-1})\to{\bf
K}_{-p+q}({\mathcal{A}}_{p}))}{\beta{\rm Ker}({\bf
K}_{-p+q+1}({\mathcal{A}}_{p-1})\to{\bf K}_{-p+q+1}({\mathcal{A}}_{p-r}))},$
(89)

which, for $p=0$, specializes to

$E^{r}_{0,q}=\mathrm{Im}({\bf K}_{q}({\mathcal{A}}_{r-1})\to{\bf
K}_{q}({\mathcal{A}}_{0}))={\rm Im}\,s_{\ast}^{r-1}.$ (90)

The codomain of $d^{r}_{0,q}$ is $E^{r}_{r,q+r-1}$, hence one takes the
quotient by

$\beta{\rm Ker}({\bf K}_{q}({\mathcal{A}}_{p-1})\to{\bf
K}_{q}({\mathcal{A}}_{p-r}))=\partial^{r}_{q}({\rm Ker}s_{*}^{r-1}).$

Since $d_{0,\ast}^{r}$ and $\delta_{\ast}^{r}$ have the same domain, are both
defined in the same way (by choosing a lift from ${\bf
K}_{\ast}({\mathcal{A}}_{0})$ to ${\bf K}_{\ast}({\mathcal{A}}_{p-1})$,
computing the connecting map to ${\bf K}_{\ast-1}({\mathcal{E}}_{p})$ and then
taking the quotient by the same group) they corestrict to the same map. ∎

One main advantage of using the spectral sequence is that it breaks the
computation into smaller pieces; the iterative construction shows that the
subquotients can be expanded as follows:

######  Corollary 4.28.

The codomain of $d^{r}_{0,\ast}$ is equivalently given by

$\frac{\partial^{r}_{\ast}((s_{*}^{r-1})^{-1}{\bf
K}_{\ast}({\mathcal{A}}_{0}))}{\partial^{r}_{\ast}({\rm
Ker}\,s^{r-1}_{\ast})}=\frac{\partial^{r}_{q}((s_{*}^{r-1})^{-1}{\bf
K}_{\ast}({\mathcal{A}}_{0}))}{\mathrm{Im}(d^{r-1}_{1,1+\ast})+\mathrm{Im}(d^{r-2}_{2,2+\ast})...+\mathrm{Im}(d^{1}_{r-1,r-1+\ast})}.$

The image $\mathrm{Im}(d^{s}_{r-s,r-s+\ast})$ enumerates the possible boundary
states at the codimension $r$ boundaries which are protected by surface
topological insulators at the codimension $(r-s)$-boundaries through an
order-$s$ correspondence. Hence, to compute the codomain of $d^{r}_{0,\ast}$
one should first classify those lower-dimensional higher-order topological
insulators of order less than $r$. This can be done through an iterative
process, in fact it happens automatically if one computes one page after
another of the spectral sequence. Upon completion, those groups enumerate
exhaustively the possible dependence of the boundary states of any fixed bulk
Hamiltonian on the choice of boundary condition, which turns out to be in
K-theory completely equivalent to investigating the effects of decorating the
surface with additional layers.

The framework of spectral sequences supplies other powerful tools. For
example, we can recognize when two higher-order bulk-boundary correspondences
are identical from a topological point of view:

######  Proposition 4.29.

Let there be two cofiltrations

$...\twoheadrightarrow{\mathcal{A}}_{n}\twoheadrightarrow{\mathcal{A}}_{n-1}\twoheadrightarrow...\twoheadrightarrow{\mathcal{A}}_{0}\twoheadrightarrow
0,\quad\tilde{{\mathcal{A}}}_{n}\twoheadrightarrow\tilde{{\mathcal{A}}}_{n-1}\twoheadrightarrow...\twoheadrightarrow\tilde{{\mathcal{A}}}_{0}\to
0$ (91)

and homomorphisms $\psi_{p}:{\bf K}_{q}({\mathcal{A}}_{p})\to{\bf
K}_{q}(\tilde{{\mathcal{A}}}_{p})$, $\varphi_{n}:{\bf
K}_{q}({\mathcal{E}}_{p})\to{\bf K}_{q}(\tilde{{\mathcal{E}}}_{p})$ such that
the diagram

${{\bf K}_{q}({\mathcal{A}}_{p})}$${{\bf K}_{q}({\mathcal{A}}_{p-1})}$${{\bf
K}_{q-1}({\mathcal{E}}_{p})}$${{\bf K}_{q-1}({\mathcal{A}}_{p-1})}$ ${{\bf
K}_{q}(\tilde{{\mathcal{A}}}_{p})}$${{\bf
K}_{q}(\tilde{{\mathcal{A}}}_{p-1})}$${{\bf
K}_{q-1}(\tilde{{\mathcal{E}}}_{p})}$${{\bf
K}_{q-1}(\tilde{{\mathcal{A}}}_{p-1})}$
$\scriptstyle{\psi_{p}}$$\scriptstyle{\psi_{p-1}}$$\scriptstyle{\partial}$$\scriptstyle{\varphi_{p}}$$\scriptstyle{\psi_{p-1}}$$\scriptstyle{\partial}$
(92)

commutes, then the higher order boundary maps satisfy
$\tilde{d}^{p}_{0,q}\circ\varphi_{p}=\varphi_{p}\circ d^{p}_{0,q}$. If all
homomorphisms $\varphi_{p}$ are isomorphisms then the induced spectral
sequences are term-wise isomorphic.

###### Proof.

For a commutative diagram of exact couples (in unraveled form)

${D^{1}}$${D^{1}}$${E^{1}}$${D^{1}}$
${\tilde{D}^{1}}$${\tilde{D}^{1}}$${\tilde{E}^{1}}$${\tilde{D}^{1}}$
$\scriptstyle{\gamma}$$\scriptstyle{\psi}$$\scriptstyle{\alpha}$$\scriptstyle{\psi}$$\scriptstyle{\beta}$$\scriptstyle{\varphi}$$\scriptstyle{\gamma}$$\scriptstyle{\psi}$$\scriptstyle{\alpha}$$\scriptstyle{\tilde{\gamma}}$$\scriptstyle{\tilde{\alpha}}$$\scriptstyle{\tilde{\beta}}$$\scriptstyle{\tilde{\gamma}}$$\scriptstyle{\tilde{\alpha}}$
(93)

one naturally obtains induced homomorphisms relating the derived pages
$D^{r}$, $E^{r}$ with $\tilde{D}^{r}$, $\tilde{E}^{r}$, again in a commutative
diagram of the same shape [35, I.5]. For the last statement, note that if the
maps $\varphi_{p}:{\bf K}_{q}({\mathcal{E}}_{p})\to{\bf
K}_{q}(\tilde{{\mathcal{E}}}_{p})$ are isomorphisms, then the maps
$\psi_{p}:{\bf K}_{q}({\mathcal{A}}_{p})\to{\bf
K}_{q}(\tilde{{\mathcal{A}}}_{p})$ are also isomorphisms since
${\mathcal{A}}_{0}={\mathcal{E}}_{0}$ and one can iteratively apply the five
lemma from this base case. ∎

######  Corollary 4.30.

Let there be a commutative diagram of equivariant morphisms

${C^{*}{\mathcal{G}}_{\Xi_{d}}}$${C^{*}{\mathcal{G}}_{\Xi_{d-1}}}$${...}$${C^{*}{\mathcal{G}}_{\Xi_{0}}}$${C^{*}{\mathcal{G}}_{\tilde{\Xi}_{d}}}$${C^{*}{\mathcal{G}}_{\tilde{\Xi}_{d-1}}}$${...}$${C^{*}{\mathcal{G}}_{\tilde{\Xi}_{0}}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\psi_{d-1}}$$\scriptstyle{\psi_{0}}$
(94)

between cofiltrations corresponding to two different crystals. If all of the
homomorphisms $\psi_{n}$ induce isomorphisms in $K$-theory, then the induced
spectral sequences are term-wise isomorphic and the two bulk-boundary
correspondences can be declared identical.

######  Example 4.31.

Consider a situation where the lattice is deformed in compact neighborhoods of
the corners of the cube, such that the point symmetry $\Sigma$ is preserved.
Then Corollary 4.30 assures us that the two systems support the same higher-
order bulk-boundary correspondences. $\Diamond$

######  Remark 4.32.

Corollary 4.30 can be also used to obtain partial (complete) information about
the bulk-boundary maps by relating to simpler (equivalent) groupoids for which
it may be more feasible to compute the boundary maps. $\Diamond$

######  Remark 4.33.

If the pattern at infinity ${\mathcal{L}}_{\infty}$ is simply the lattice
${\mathbb{Z}}^{d}$ one has $C^{*}{\mathcal{G}}_{\Xi_{0}}\simeq
C({\mathbb{T}}^{d})$. It may then be interesting to extend the cofiltration by
merging it with a skeleton filtration of the Brillouin torus
${\mathbb{T}}^{d}=X_{d}\supset X_{d-1}\supset...\supset X_{0}$ as a CW-complex

$C^{\ast}{\mathcal{G}}_{\Xi_{d}}\to C^{\ast}{\mathcal{G}}_{\Xi_{d-1}}\to...\to
C^{\ast}{\mathcal{G}}_{\Xi_{0}}\to C(X_{d-1})\to...\to C(X_{0}).$ (95)

The higher-order differentials of the associated spectral sequence then relate
specific parts of the skeleton to topologically forced band-touching points in
the Brillouin zone as well as to higher-order boundary invariants localized at
edges, hinges, corners etc, therefore allowing in theory to pinpoint exactly
which bulk topological invariants cause which (higher order) boundary states.
For some space-groups in three dimensions, one already has non-trivial third-
order differentials in the part which comes from the skeleton filtration of
the torus alone [57]. $\Diamond$

We end the section with a few words about the challenges of concrete
calculations. For any polyhedron representing a $d$-dimensional crystal, one
can construct a groupoid algebra ${\mathcal{G}}_{\Xi}$ as in Section 3 by
gluing together patterns which model its corners. A natural filtration is then
given by grouping together boundaries of the same linear dimension. Such a
filtration makes the groupoid solvable in the sense that there is an
equivariant isomorphism (or at least Morita equivalence)

${\mathcal{E}}_{r}=C^{*}({\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})\simeq\bigoplus_{m=1}^{N_{r}}C({\mathbb{T}}^{d-r})\otimes{\mathbb{K}}(\ell^{2}({\mathcal{L}}_{r,m}))$
(96)

with compact operators on some discrete $r$-dimensional pattern. Each direct
summand represents one disjoint piece of a boundary, e.g. a single facet,
hinge or corner. The equivalence arises since one can choose a large enough
unit cell compatible with the boundaries and then Fourier transform the
remaining translation-invariant directions. Any element of the symmetry group
acts either as an automorphism of one or an isomorphism between two of the
direct summands. Thanks to Proposition 6.5 one can compute the $K$-theory of
${\mathcal{E}}_{n}$ by choosing orbit representatives and then computing the
$H_{r,m}$-equivariant $K$-theories of each summand
$C({\mathbb{T}}^{d-r})\otimes{\mathbb{K}}(\ell^{2}({\mathcal{L}}_{r,m})$ where
$H_{r,m}$ is its stabilizer group. In many cases the individual pieces are
just groups $K^{H_{r,m}}_{q}(C({\mathbb{T}}^{d-r}))$, i.e. (possibly twisted)
equivariant $K$-groups describing $d-r$-dimensional topological crystalline
insulators with a point group appropriate for $d-r$ dimensions. In conclusion,
the first page of the spectral sequence is very often feasible to compute and
therefore provides a good starting point for further investigations. The
higher derivations on the other hand elude systematic computation; spectral
sequences are bookkeeping tools to systematically compile information about
the connecting maps but one usually cannot compute those easily using their
definition alone. In practice one needs to construct a sufficiently large (but
finite) collection of Hamiltonians for which one can exactly determine on
which boundaries their gaps close and what the associated $K$-theoretic
invariants are. Any such Hamiltonian supplies some information on domain and
image of some of the differentials. Here standards will differ between the
fields: for the requirements of theoretical physics numerical solutions of the
Hamiltonians on large enough finite volumes are perfectly acceptable and in
practice even fairly small numerical simulations will reproduce the correct
classification. For rigorous computation of the boundary maps more
sophisticated methods or at least construction of exactly solvable
Hamiltonians will be necessary.

######  Remark 4.34.

While we consider only crystals in this work, quasi-crystals can be described
using the literally same formalism, simply by using different patterns
${\mathcal{L}}^{\lambda}$ to make the global transversal. This is of interest
in particular since those allow point symmetries that are not possible in
crystalline systems, such as five-fold rotational symmetry. Likewise,
continuous models for topological phases can be realized by similar groupoid
algebras. As the formalism of this section applies to general
$C^{\ast}$-algebras, the only thing that changes is that the basic building
blocks in the direct sum decompositions (96) will be different algebras.

###  4.5. Comparison with ordinary bulk boundary correspondence

In our formalism we have a co-filtration

$C^{*}{\mathcal{G}}_{\Xi_{d}}\to C^{*}{\mathcal{G}}_{\Xi_{d-1}}\to...\to
C^{*}{\mathcal{G}}_{\Xi_{0}}$

and we say that a gapped bulk Hamiltonian exhibits higher-bulk boundary
correspondence of order $r$ or lower if and only if its class in ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$ cannot be lifted to ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{r}})$. In $K$-theory this obstruction is
precisely the boundary map of the exact sequence

$0\to C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}}\to
C^{*}{\mathcal{G}}_{\Xi_{r}}\to C^{*}{\mathcal{G}}_{\Xi_{0}}\to 0,$

thus, it seems that an alternative approach to classification would be to
compute the associated boundary map

$\mathring{\partial}^{r}_{\ast}:{\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})\to{\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$ (97)

with all the higher-order boundary states corresponding to non-trivial classes
in ${\bf K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$. From this
point of view the higher-order bulk boundary correspondence is almost the same
thing as an ordinary bulk-boundary correspondence. The crucial difference is
that in higher-order bulk boundary correspondence we impose an additional gap
condition at the codimension $(r-1)$-boundary which allows us to localize the
obstruction to a subquotient of ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$. Let us sketch
how the boundary map above relates to our higher boundary maps and why the
latter are more practical:

  1. i)

Without first understanding the K-groups of
$C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}}$ one cannot feasibly compute the
boundary map since one will be unable to pinpoint classes in ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$. Unfortunately,
this can be complicated. We know from examples that many different classes in
${\bf K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$ are related
by a change of boundary conditions, which means that they must be one and the
same class when included in ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$. Furthermore, the
relations to the boundary states ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{p}\setminus\Xi_{p-1}})$ for $p\leq r$ are
encoded in a spectral sequence which converges to the groups ${\bf
K}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$, namely the one associated
to the co-filtration

$C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}}\to
C^{*}{\mathcal{G}}_{\Xi_{r-1}\setminus\Xi_{0}}\to...\to
C^{*}{\mathcal{G}}_{\Xi_{1}\setminus\Xi_{0}}\to 0.$

The quotients in this co-filtration are precisely the same ideals
${\mathcal{E}}_{p}=C^{*}{\mathcal{G}}_{\Xi_{p}\setminus\Xi_{p-1}}$ as before,
except for the quotient at $p=0$ which is now trivial, and therefore the first
page is almost the same. Thus our spectral sequence approach can also be used
to compute ${\bf K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$ up
to a finite number of group extension problems involving subquotients of ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$.

  2. ii)

Our higher-order boundary maps give us partial information: By construction,
$\mathring{\partial}^{r}_{\ast}$ maps a bulk class in ${\bf
K}_{\ast}(C^{*}{\mathcal{G}}_{\Xi_{0}})$ to a non-trivial class in ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$ if and only if one
of the boundary maps $\delta^{p}_{\ast}$ of order $p\leq r$ maps it to a non-
trivial value.

  3. iii)

When we impose the gap condition at the codimension $(r-1)$-boundaries this
narrows us down to classes in ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$ which are in the
image of the natural map ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})\to{\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$. Recalling that
$\delta^{r}_{\ast}$ takes values in a subquotient of ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{r-1}})$, every
representative of the same coset in $\mathrm{Im}(\delta^{r}_{\ast})$ gives
rise to one and the same class in ${\bf
K}_{\ast-1}(C^{*}{\mathcal{G}}_{\Xi_{r}\setminus\Xi_{0}})$.

The conclusion is that the boundary map (97) is intimately related to the
higher-order bulk-boundary correspondences and the tools required to
understand it are essentially the same as we have developed to map the higher-
order bulk-boundary correspondences. While $\mathring{\partial}^{r}_{\ast}$ is
certainly non-trivial in the cases of interest, in our setup of higher-order
bulk boundary correspondence where the codimension $(r-1)$ boundaries are
assumed to be gapped it cannot provide any additional information over the
boundary maps $\delta^{p}_{\ast}$, which are at this point both easier to
compute and more directly allow us to derive the possible manifestations of
the boundary states. The classification by (97) would, however, be the
relevant one if one imposes only a bulk gap assumption.

######  Example 4.35.

In the setting of example 4.8, $\mathring{\partial}^{2}_{1}$ corresponds to a
boundary map of the exact sequence $0\to{\rm
Ker}(\bar{{\mathcal{Q}}}\to{\mathcal{B}})\to\bar{{\mathcal{Q}}}\to{\mathcal{B}}\to
0$. The computations given there and the spectral sequence mentioned above
yield $K_{0}^{{\mathbb{Z}}_{2}}({\rm
Ker}(\bar{{\mathcal{Q}}}\to{\mathcal{B}}))\simeq{\mathbb{Z}}\oplus{\mathbb{Z}}_{2}$
as the unique solution to a group extension problem. Under
$\mathring{\partial}^{2}_{1}$ the bulk classes $[u_{\mathcal{F}}]_{1}$ and
$[u_{\mathcal{C}}]_{1}$ map to the first respectively second of those
generators. This gives us information on possible experimental signatures of
the corner states which may remain if the edges are not gapped. In contrast,
for non-equivariant K-theory one finds that naturally $K_{0}({\rm
Ker}(\bar{{\mathcal{Q}}}\to{\mathcal{B}})))\simeq K_{0}(\bar{\mathcal{F}})$
which reiterates that the bulk K-theory can only enforce first-order boundary
states at the edges but not corner states.

##  5\. Examples of Higher-Order Correspondences

In this section, we supply additional example of higher-order bulk-boundary
correspondences. While working in three dimensions we will use the infinite
square $C^{*}{\mathcal{G}}_{\square}$ algebra with $d-2$ infinite translation-
invariant direction. Thus, in real space, our crystal models a straight prism
with infinite cross-section. In three dimensions it represents the infinite-
volume limit of infinitely long wires as one increases the (square) cross-
section.

From Section 3.3, we have a co-filtration of $C^{\ast}$-algebras (50)

${\mathcal{Q}}\stackrel{{\scriptstyle\mathfrak{p}^{2}}}{{\twoheadrightarrow}}{\mathcal{P}}\stackrel{{\scriptstyle\mathfrak{p}^{1}}}{{\twoheadrightarrow}}{\mathcal{B}}\twoheadrightarrow
0$ (98)

with ${\mathcal{B}}=C^{*}{\mathbb{Z}}^{d}\simeq C({\mathbb{T}}^{d})$ and
kernels

$\displaystyle{\mathcal{F}}$ $\displaystyle={\rm
Ker}({\mathcal{P}}\twoheadrightarrow{\mathcal{B}})\simeq\bigoplus\nolimits_{\lambda\in{\mathbb{Z}}_{4}}{\mathcal{F}}_{\lambda},\quad$
$\displaystyle{\mathcal{F}}_{\lambda}$ $\displaystyle\simeq
C^{*}{\mathbb{Z}}^{d-1}\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}}))$ (99)
$\displaystyle{\mathcal{C}}$ $\displaystyle={\rm
Ker}({\mathcal{Q}}\twoheadrightarrow{\mathcal{P}})\simeq\bigoplus\nolimits_{\lambda\in{\mathbb{Z}}_{4}}{\mathcal{C}}_{\lambda},\quad$
$\displaystyle{\mathcal{C}}_{\lambda}$ $\displaystyle\simeq
C^{*}{\mathbb{Z}}^{d-2}\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}}))$

We need to actually describe the $K$-theoretic invariants and boundary maps in
some detail to finally exhibit non-trivial examples of higher-order bulk-
boundary correspondence. As before, the equivariant $K$-functors will be
specified by ${\bf K}_{\ast}$, while the non-equivariant ones by $K_{\ast}$.

###  5.1. The non-equivariant case

We will begin with the non-equivariant case, i.e. we discuss the complex
$K$-groups. Each of the algebras ${\mathcal{B}}$, ${\mathcal{F}}_{\lambda}$,
${\mathcal{C}}_{\lambda}$ is Morita-equivalent to the continuous functions on
a torus of some dimension. It is well-known that

$K_{\ast}(C^{*}{\mathbb{Z}}^{n})=K_{\ast}({\mathbb{T}}^{n})\simeq{\mathbb{Z}}^{2^{n-1}},\quad
i=0,1$ (100)

but we need to find a way to consistently label the elements of those
$K$-groups. Since the groups are torsion-free one, it is convenient to label
all generators using the numerical pairings with cyclic cohomology. It is
known that the cyclic cohomology of $C({\mathbb{T}}^{n})$ is spanned by the
so-called Chern cocycles which we describe now. For
${\mathcal{A}}\in\\{{\mathcal{B}},{\mathcal{F}}_{\lambda},{\mathcal{C}}_{\lambda}\\}$
one has a natural (densely defined) trace ${\mathcal{T}}_{\mathcal{A}}$
induced from the Haar measure on $C({\mathbb{T}}^{n})$ and the trace on the
compact operators. There is an ${\mathbb{R}}^{d}$-action on
$C^{\ast}{\mathcal{G}}_{\square}$ acting via

$(\Theta_{x}f)(g,S)=e^{2\pi\imath(x\cdot g)}f(g,S),\quad
x\in{\mathbb{R}}^{d},\ g\in{\mathcal{S}},\ {\mathcal{S}}\in\Xi_{\square}$
(101)

on elements of the convolution algebra $C_{c}{\mathcal{G}}_{\Xi_{\square}}$,
which also gives rise to an action on the quotients ${\mathcal{A}}$. This
allows us to consistently label spatial directions between the different
algebras:

For any tuple $v=(v_{1},...,v_{n})$ made up of unit vectors in
${\mathbb{R}}^{d}$ and $(n+1)$-tuple $(f_{0},\ldots,f_{n})$ made up of
elements of a suitable dense subalgebra of ${\mathcal{A}}$ one can define the
Chern cocycle

${\rm
Ch}_{{\mathcal{A}},v}(f_{0},...,f_{n})\;=\;c_{{\mathcal{A}},n}\,\sum_{\rho\in
S_{n}}(-1)^{\rho}\,{\mathcal{T}}_{\mathcal{A}}\big{(}f_{0}\nabla_{v_{\rho(1)}}f_{1}\ldots\nabla_{v_{\rho(n)}}f_{n}\big{)}\;,$
(102)

with some normalization constants and the densely defined derivation
$\nabla_{v}$ in the direction $v$ of the action $\Theta$. Using a picture of
$K$-theory where $K_{0}$ is represented by projections and $K_{1}$ by
unitaries, one has well-defined pairings between
$K_{q}({\mathcal{A}})=K_{q\,\mathrm{mod}\,2}({\mathcal{A}})$ and the cyclic
cocycles of parity $n\,\mathrm{mod}\,2=q\,\mathrm{mod}\,2$ via

$\langle[x]_{q},[{\rm Ch}_{{\mathcal{A}},v}]\rangle=\begin{cases}{\rm
Ch}_{{\mathcal{A}},v}(x,...,x)&q\text{ even}\\\ {\rm
Ch}_{{\mathcal{A}},v}(x^{-1},x,x^{-1},...,x,x^{-1})&q\text{ odd}\end{cases}$
(103)

We assume the choice of normalization constants is made as in [48] such that
all non-trivial pairings will exactly have ${\mathbb{Z}}$ as their range
whenever $v$ made up of unit vectors (see [48, Section 5.7]). We will now
identify tuples of unit vectors $v_{I}=(e_{i_{1}},...,e_{i_{n}})$,
$i_{1}<i_{2}<...<i_{n}$ with subsets $I=\\{i_{1},...,i_{n}\\}$ of elements in
$\\{1,...,d\\}$ and write ${\rm Ch}_{{\mathcal{A}},I}={\rm
Ch}_{{\mathcal{A}},v_{I}}$. Let us then say a class $x_{{\mathcal{A}},I}\in
K_{q}({\mathcal{A}})$ is dual to ${\rm Ch}_{{\mathcal{A}},I}$ if

$\langle[x_{{\mathcal{A}},I}]_{q},[{\rm
Ch}_{{\mathcal{A}},J}]\rangle=\delta_{I,J}$ (104)

for all subsets $I,J\subset\\{1,...,d\\}$ whose length has the same parity as
$q$. For the torus such dual classes exist and are unique, hence we can label
(assuming $d\geq 2$)

  1. (i)

A basis of $K_{q}({\mathcal{B}})$ by the $2^{d-1}$ even respectively odd
subsets of $\\{1,...,d\\}$.

  2. (ii)

A basis of $K_{q}({\mathcal{F}}_{\lambda})$ by the $2^{d-2}$ even respectively
odd subsets of $\\{1,...,d\\}$ which do not contain the direction $1$
respectively $2$ which is parallel to the normal vector $n_{\lambda}$ of the
face $\lambda$ (see Fig. 9).

  3. (iii)

A basis of $K_{q}({\mathcal{C}}_{\lambda})$ by the $2^{d-3}$ even respectively
odd subsets of $\\{3,...,d\\}$ if $d\geq 3$, for $d=2$ there is only the even
subset $I=\emptyset$ and no odd subsets.

We can now express the first-order bulk-boundary map
$\partial^{{\mathcal{B}}{\mathcal{F}}}_{q}:K_{q}({\mathcal{B}})\to
K_{q-1}({\mathcal{F}})$. Since ${\mathcal{F}}$ is a direct sum of four
algebras one can treat them individually writing
$\partial^{{\mathcal{B}}{\mathcal{F}}}_{q}=\oplus_{\lambda}\partial^{{\mathcal{B}}{\mathcal{F}}_{\lambda}}_{q}$
and one has:

######  Proposition 5.1.

For any tuple $v$ of directions in ${\mathbb{R}}^{d}$ with the opposite parity
as $q\in\\{0,1\\}$, the Chern numbers relate under the boundary map by

$(-1)^{q}\langle\partial^{{\mathcal{B}}{\mathcal{F}}_{\lambda}}_{q}[x]_{q},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},v}]\rangle=\langle[x]_{q-1},[{\rm
Ch}_{{\mathcal{B}},v\times n_{\lambda}}]\rangle,$ (105)

which uniquely determines
$\partial^{{\mathcal{B}}{\mathcal{F}}_{\lambda}}_{i}$.

###### Proof.

The exact sequence connecting ${\mathcal{B}}$ and ${\mathcal{F}}_{\lambda}$
has the form

$0\to{\mathcal{F}}_{\lambda}\to{\mathcal{H}}_{\lambda}\simeq
C^{*}{\mathbb{Z}}^{d-1}\otimes\mathfrak{T}\to{\mathcal{B}}\to 0,$ (106)

where $\mathfrak{T}$ is the Toeplitz algebra. Under this isomorphism a
translation in the direction $n_{\lambda}$ corresponds to the co-isometry
which generates the Toeplitz extension. The identity for the Chern cocycles is
then a well-known duality for cyclic cocycles under under this exact sequence
(see [32, 48, 54] for treatments at various level of detail). ∎

The boundary maps
$\partial^{{\mathcal{F}}{\mathcal{C}}}_{i}=\oplus_{\lambda}\partial^{{\mathcal{F}}{\mathcal{C}}_{\lambda}}_{\ast}$
are only slightly more complicated:

######  Proposition 5.2.

With the same notation as Proposition 5.1 one has

$\begin{split}(-1)^{q}\langle\partial^{{\mathcal{F}}{\mathcal{C}}_{\lambda}}[(x_{1},x_{2},x_{3},x_{4})]_{q},[&{\rm
Ch}_{{{\mathcal{C}}_{\lambda}},v}]\rangle\\\ =\langle[x_{\lambda}]_{q},[&{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},v\times
n_{\lambda-1}}]\rangle+\langle[x_{\lambda-1}]_{q},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda-1}},v\times n_{\lambda}}]\rangle.\end{split}$
(107)

###### Proof.

The exact sequence of groupoid algebras relating each face algebra
${\mathcal{F}}_{\lambda}$ to the adjacent corner algebra
${\mathcal{C}}_{\lambda}$ is isomorphic to an exact sequence

$0\to{\mathcal{C}}_{\lambda}\to
C^{\ast}{\mathbb{Z}}^{d-2}\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}}))\otimes\mathfrak{T}\to{\mathcal{F}}_{\lambda}\to
0.$ (108)

In this extension the translation in direction $n_{\lambda-1}$ plays the role
of the co-isometry generating the Toeplitz extension which results analogously
to Proposition 5.1 in the expression

$(-1)^{q}\langle\partial([x_{\lambda}]_{q}),[{\rm
Ch}_{{\mathcal{C}}_{\lambda},v}]\rangle=\langle[x_{\lambda}]_{q},[{\rm
Ch}_{{\mathcal{F}}_{\lambda},v\times n_{\lambda-1}}]\rangle$ (109)

for its boundary map. Denoting $\mathring{{\mathcal{Q}}}={\rm
Ker}({\mathcal{Q}}\to{\mathcal{B}})$ one has a commutative diagram

${0}$${{\mathcal{C}}_{\lambda}}$${C^{*}{\mathbb{Z}}^{d-2}\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}}))\otimes\mathfrak{T}}$${{\mathcal{F}}_{\lambda}}$${0}$${0}$${{\mathcal{C}}}$${\mathring{{\mathcal{Q}}}}$${{\mathcal{F}}}$${0}$
(110)

for each $\lambda$. Similarly one relates each face algebra
${\mathcal{F}}_{\lambda}$ with the other adjacent corner
${\mathcal{C}}_{\lambda+1}$ and then naturalness of the connecting
homomorphisms implies that the contributions of ${\mathcal{F}}_{\lambda}$ and
${\mathcal{F}}_{\lambda-1}$ add up in such a way that the Chern numbers
satisfy (107). ∎

One should note that if one wants to label the Chern cocycles in terms of sets
of standard directions $I$, as we did above, some of the contributions would
acquire minus signs due to the algebraic property ${\rm
Ch}_{{\mathcal{A}},v\times(-n)}=-{\rm Ch}_{{\mathcal{A}},v\times n}$ of the
Chern cocycles

$n_{1}$$n_{2}$$n_{3}$$n_{4}$$1$$2$$3$$4$$a)$$w$$x$$y$$z$$w-z$$x-w$$y-x$$z-y$$b)$
Figure 9. a) Labeling of the corners and half-spaces with their outward
normals. b) Pictorial description of the face-corner correspondence: At the
faces we mark the Chern numbers ${\rm Ch}_{{{\mathcal{F}}_{\lambda}},v\times
n_{\lambda+1}}$ of some class in $K_{i}({\mathcal{F}})$ and at the corners the
Chern numbers $(-1)^{q}{\rm Ch}_{{{\mathcal{C}}_{\lambda}},v}$ of its image
under $\partial^{{\mathcal{F}}{\mathcal{C}}}$.

It is now easy to compute the range of the map
$\partial^{{\mathcal{F}}{\mathcal{C}}}$. Any class in $K_{i}({\mathcal{C}})$
is determined by $4$ sets of corner Chern numbers ${\rm
Ch}_{{\mathcal{C}}_{\lambda},I}$, $\lambda\in{\mathbb{Z}}_{4}$,
$I\subset\\{3,...,d\\}$. Any assignment of corner Chern numbers to the corners
$1,2,3$ can be re-produced from a pre-image under
$\partial^{{\mathcal{F}}{\mathcal{C}}}$ by choosing appropriate face Chern
numbers. However, the sums

$\sum_{\lambda=1}^{4}\langle\partial^{{\mathcal{F}}{\mathcal{C}}_{\lambda}}[(x_{1},x_{2},x_{3},x_{4})]_{q},[{\rm
Ch}_{{{\mathcal{C}}_{\lambda}},I}]\rangle$ (111)

are constrained to be zero for each $I\subset\\{3,...,d\\}$ (see Figure 9)
which determines the Chern numbers of the remaining corner. We can write this
as follows:

######  Proposition 5.3.

In complex (non-equivariant) $K$-theory the map
$\partial^{{\mathcal{F}}{\mathcal{C}}}_{q}:K_{q}({\mathcal{F}})\to
K_{q-1}({\mathcal{C}})$ has the range

$\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{q}\simeq
K_{q-1}({\mathcal{C}}_{1})\oplus K_{q-1}({\mathcal{C}}_{2})\oplus
K_{q-1}({\mathcal{C}}_{3})$ (112)

and
$K_{q}({\mathcal{C}})/\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{q}\simeq
K_{q-1}({\mathcal{C}}_{4})$.

In principle the sums of the corner Chern numbers can therefore be independent
of the lift of any fixed bulk class $[x]_{q}\in K_{q}({\mathcal{B}})\cap{\rm
Ker}\,\partial^{{\mathcal{B}}{\mathcal{F}}}_{q}$. Nevertheless, we have the
negative result:

######  Proposition 5.4.

The bulk-corner map $\delta^{{\mathcal{B}}{\mathcal{C}}}_{q}$ is the zero-map.

###### Proof.

The kernel of $\partial^{{\mathcal{B}}{\mathcal{F}}}_{q}$ is spanned by
precisely those basis elements which are dual to the Chern cocycles ${\rm
Ch}_{{\mathcal{B}},I}$, where $I$ contains neither $1$ nor $2$. This span is
nothing but the image of the inclusion $K_{q}(C({\mathbb{T}}^{d-2}))\to
K_{q}({\mathcal{B}})$. Those generators can be represented by
projections/unitaries that act trivially on the first factor of the
decomposition
$\ell^{2}({\mathbb{Z}}^{d})=\ell^{2}({\mathbb{Z}}^{2})\otimes\ell^{2}({\mathbb{Z}}^{d-2})$
and therefore their restrictions to any half- or quarterspace will still be a
projections/unitaries. Hence they are in the kernel of
$\delta^{{\mathcal{B}}{\mathcal{C}}}_{q}$. ∎

This is the reason why one needs to enhance the $K$-theory by spatial
symmetries to obtain non-trivial higher-order bulk-hinge correspondence. We
used here complex $K$-theory but the same is true for the real Altland-
Zirnbauer classes. Those can be labeled by real K-theory groups
$KO^{i}({\mathbb{T}}^{d})$ and due to the Künneth formula the kernel of the
first boundary map is spanned by the image of $KO^{i}({\mathbb{T}}^{d-2})$. We
omit the details.

The same applies to the cube geometry (see below) and it seems to us that for
all polyhedral geometries the higher-order bulk-boundary maps are all zero
unless one imposes an additional spatial symmetry. However, there are some
subtleties which make it difficult to prove or disprove this conjecture,
especially if not all boundaries are at $90$ degree angles. We will leave a
more precise analysis to future work.

###  5.2. Inversion-symmetry

We now want to impose inversion-symmetry, i.e. the order two symmetry
generated by the involutive transformation

$\sigma:x\in{\mathbb{R}}^{3}\mapsto-x\in{\mathbb{R}}^{3}.$ (113)

Here we consider only the three-dimensional case, since spatial symmetries are
difficult to treat uniformly across dimensions and we only know generators for
the equivariant $K$-groups in low dimensions. The order $2$-symmetry induces
an involutive automorphism $\sigma$ on $C^{\ast}{\mathcal{G}}_{\square}$,
which we also consider as a ${\mathbb{Z}}_{2}$-action denoted by the same
letter. To distinguish from the other symmetries considered in this work, we
denote the ${\mathbb{Z}}_{2}$-equivariant $K$-groups w.r.t. inversion symmetry
by
$K_{\ast}^{I}(C^{\ast}{\mathcal{G}}_{\square})=K_{\ast}^{{\mathbb{Z}}_{2}}(C^{\ast}{\mathcal{G}}_{\square})$.

We compute first
$K^{I}_{\ast}({\mathcal{C}})/\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{\ast}$,
which embeds the codomain of the second order boundary map. The face algebra
${\mathcal{F}}=\oplus_{\lambda\in{\mathbb{Z}}_{4}}{\mathcal{F}}_{\lambda}$ can
be decomposed into two orbits under inversion-symmetry by writing

${\mathcal{F}}={\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2}\oplus\sigma({\mathcal{F}}_{1})\oplus\sigma({\mathcal{F}}_{2})\simeq({\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2})\otimes
C({\mathbb{Z}}_{2}),$ (114)

with $C({\mathbb{Z}}_{2})$ the functions ${\mathbb{Z}}_{2}\to{\mathbb{C}}$.
The isomorphism is equivariant when $C({\mathbb{Z}}_{2})$ carries the regular
representation. Hence, one has by Proposition 6.5

$K_{\ast}^{I}({\mathcal{F}})\simeq
K_{\ast}({\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2}).$ (115)

Similarly one can decompose ${\mathcal{C}}$ into orbits and obtain

$K_{\ast}^{I}({\mathcal{C}})\simeq
K_{\ast}({\mathcal{C}}_{1}\oplus{\mathcal{C}}_{2}).$ (116)

Here we do not distinguish between $K^{I}_{-1}\simeq K^{I}_{1}$ and represent
both by invariant unitaries (see Remark 6.4(i)). The boundary maps in
equivariant $K$-theory are now easy to compute by forgetting the equivariance

${K_{\ast}^{I}({\mathcal{F}})}$${K_{\ast-1}^{I}({\mathcal{C}})}$${K_{\ast}({\mathcal{F}})}$${K_{\ast-1}({\mathcal{C}})}$$\scriptstyle{{\partial}^{{\mathcal{F}}{\mathcal{C}}}_{\ast}}$$\scriptstyle{{\partial}^{{\mathcal{F}}{\mathcal{C}}}_{\ast}}$
(117)

but keeping in mind that the image of $K_{\ast}^{I}({\mathcal{F}})\to
K_{\ast}({\mathcal{F}})$ is generated by $\sigma$-invariant representatives.
Therefore, it is enough to compute the boundary map for representatives
$[x_{1},x_{2},x_{3},x_{4}]_{\ast}\in\bigoplus_{\lambda=1}^{4}K_{\ast}({\mathcal{F}}_{i})$
of the form $[x_{1},x_{2},\sigma(x_{1}),\sigma(x_{2})]_{\ast}$. Here,
$x_{\lambda}\in M_{N}({\mathbb{C}})\otimes{\mathcal{F}}_{\lambda}$ and
$\sigma$ acts on the first factor by some unitary representation of
${\mathbb{Z}}_{2}$. This implies relations for the Chern numbers of the faces

$\displaystyle\langle[x_{\lambda}]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},e_{3}\times n_{\lambda+1}}]\rangle$
$\displaystyle=\langle\sigma([x_{\lambda}]_{0}),[{\rm
Ch}_{{{\mathcal{F}}_{\lambda+2}},(-e_{3})\times(-n_{\lambda+1})}]\rangle$
(118) $\displaystyle=-\langle\sigma([x_{\lambda}]_{0}),[{\rm
Ch}_{{{\mathcal{F}}_{\lambda+2}},e_{3}\times n_{\lambda+3}}]\rangle,$

since $\sigma$ maps ${\mathcal{T}}_{{\mathcal{F}}_{\lambda}}$ to
${\mathcal{T}}_{{\mathcal{F}}_{\lambda+2}}$ but flips the signs of all
derivations $\sigma\circ\nabla_{i}=-\nabla_{i}\circ\sigma$ due to
$\Theta_{x}\circ\sigma=\sigma\circ\Theta_{-x}$. Those Chern numbers are the
only relevant ones for the computation of
$\partial^{{\mathcal{F}}{\mathcal{C}}}$ via (107) since
$K_{1}({\mathcal{C}})\simeq{\mathbb{Z}}^{4}$ is labeled precisely by the four
Chern cocycles ${\rm Ch}_{{{\mathcal{C}}_{\lambda}},e_{3}}$. Due to
$\sigma$-invariance, classes in
$K^{I}_{1}({\mathcal{C}})\simeq{\mathbb{Z}}^{2}$ are then already determined
precisely by the pairings with ${\rm Ch}_{{{\mathcal{C}}_{\lambda}},e_{3}}$
for $\lambda\in\\{1,2\\}$.

One can now read off the possible values for the corner Chern numbers from
Figure 9: The inversion symmetry implies the constraints $y=-w$ and $z=-x$ and
therefore the difference of the two corner Chern numbers $w-z$ and $x-w=-z-w$
lies in $2{\mathbb{Z}}$. This can be written as

######  Proposition 5.5.

$K_{1}^{I}({\mathcal{C}})\supset\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq{\mathbb{Z}}\oplus(2{\mathbb{Z}})$
(119)

and

$K_{1}^{I}({\mathcal{C}})/\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq{\mathbb{Z}}_{2}.$
(120)

Above, ${\mathbb{Z}}_{2}$ encodes the parity of the difference of the corner
Chern numbers of the two not symmetry-related corners.

######  Remark 5.6.

Interestingly, it would have been impossible to obtain such a constraint for a
single corner or even two symmetry-related corners. Here we get the payoff for
constructing the novel groupoid algebra in Section 3: Only by taking the
geometric relations between the hinges into account and gluing them together
to provide an algebra for an infinite crystal with boundary we managed to
obtain those non-trivial maps in equivariant $K$-theory. $\Diamond$

What this does not show immediately is whether there is in fact a generator of
$K_{0}^{I}({\mathcal{B}})$ that maps non-trivially under
$\delta^{{\mathcal{B}}{\mathcal{C}}}_{0}$. This requires a more detailed
analysis, in particular of the bulk $K$-group $K_{0}^{I}({\mathcal{B}})$. The
bulk algebra is isomorphic to ${\mathcal{B}}\simeq C({\mathbb{T}}^{3})$ with
the same action ${\mathbb{T}}^{3}\ni t\mapsto-t$, if one thinks of the torus
as ${\mathbb{R}}^{3}/{\mathbb{Z}}^{3}$. It is standard to compute its
equivariant $K$-groups using the Atiyah-Hirzebruch spectral sequence (AHSS)
and a decomposition of the torus into an equivariant CW-complex:

######  Proposition 5.7 ([57, Page 22]).

One has

$K_{0}^{I}(C({\mathbb{T}}^{3}))\simeq{\mathbb{Z}}^{12},\quad
K_{1}^{I}(C({\mathbb{T}}^{3}))=0.$ (121)

On the converged pages of the spectral sequence one has an extension of
abelian groups

$0\to E^{\infty}_{2,0}\to K_{0}^{I}(C({\mathbb{T}}^{3}))\to
E^{\infty}_{0,0}\to 0$ (122)

as well as $K_{1}^{I}(C({\mathbb{T}}^{3}))\simeq 0$. Here
$E_{0,0}^{\infty}\simeq{\mathbb{Z}}^{9}$ and
$E_{2,0}^{\infty}\simeq{\mathbb{Z}}^{3}$ are subquotients of the $K$-groups of
the $0$\- and $2$-skeletons respectively. The group extension problem
trivially has the unique solution
$K_{0}^{{\mathbb{Z}}_{2}}(C({\mathbb{T}}^{3}))\simeq{\mathbb{Z}}^{12}$. Let us
examine those integers in more detail.

There are $8$ points of the torus invariant under inversion, called time-
reversal-invariant momenta (TRIM), since time-reversal is just inversion
composed with complex conjugation. Parametrizing the torus as $[0,1]^{3}$ they
are $(s_{1},s_{2},s_{3})$ with $s_{1},s_{2},s_{3}\in\\{0,1\\}$ and the zero-
skeleton $X_{0}$ is made up of precisely those $8$ points. To any invariant
(virtual) vector bundle on the torus one can assign $8$ integers $n_{a}$,
$a=1,...,8$, which are the multiplicities of the sign representation of
${\mathbb{Z}}_{2}$ at the TRIM. For any invariant vector bundle which is
everywhere defined on the torus one must have

$\sum\nolimits_{a}n_{a}=0\mod 2,$ (123)

a constraint which results from a non-vanishing third differential in the
AHSS. In physics, it is understood that conversely an odd parity results in
gap-closing Weyl points and therefore one would have a semimetal instead of an
insulator. The three integers from $E^{\infty}_{2,0}$ correspond to vector
bundles which transform trivially under inversion at the TRIM but have even
weak two-dimensional Chern numbers, i.e. Chern numbers on the two-dimensional
slices of the torus.

To go from elements of $E_{0,0}^{\infty}$ to
$K^{{\mathbb{Z}}_{2}}_{0}(C({\mathbb{T}}^{3}))$ one needs to find pre-images,
i.e. construct for any admissible set of symmetry eigenvalues $n_{a}$ at the
TRIM a symmetric projection in $M_{N}({\mathbb{C}})\otimes
C({\mathbb{T}}^{3})$ which reproduces those multiplicities. This is not easy,
since those projections are far from non-trivial: Those pre-images contain
projections which have any possible combination of weak $2$-dimensional Chern
numbers and can also have non-trivial Chern-Simons invariant. More precisely,
the Chern-Simons invariant

$\theta_{\mathrm{CS}}=\frac{1}{4\pi}\int\mathrm{tr}(dA\wedge
A+\frac{2}{3}A\wedge A\wedge A)$

where $A$ is the Berry connection on the vector bundle, is quantized to $0$ or
$\pi$ in the presence of inversion symmetry. If all weak Chern numbers are
trivial it is related to the symmetry eigenvalues via

$2\frac{\theta_{\mathrm{CS}}}{\pi}=\sum_{a}n_{a}\mod 4,$ (124)

where the right-hand side can only take values of $0$ or $2$.

Another relation is between the symmetry eigenvalues and the parity of the
weak Chern numbers: For directions $\\{i,j,k\\}=\\{1,2,3\\}$ one has

$\sum_{\begin{subarray}{c}a=1,...,8\\\
(\xi_{a})_{k}=\pi\end{subarray}}n_{a}\mod 2=\mathrm{Ch}_{k}\mod 2,$ (125)

where $\mathrm{Ch}_{k}$ is the Chern number on a two-dimensional slice of the
torus. In particular, the $E^{0,0}_{\infty}$-group does include the generators
with odd-valued weak Chern numbers which were conspicuously absent from
$E^{2,0}_{\infty}$.

######  Remark 5.8.

The two formulas (124) and (125) are discussed and derived by elementary
methods at various places in the physics literature, e.g. [29], but they still
lack a $K$-theoretic interpretation. Nevertheless, they give us hint for what
the pre-images should look like. $\Diamond$

Figure 10. Spectral analysis of Hamiltonian (126) for the indicated atomic
geometries and for $\gamma=0$ as well as for $\gamma=0.5$. Both slabs contain
30 atomic layers, while the wire has a square cross section of size $30\times
30$. In panel (a), the eigenvalues are rendered in their increasing order. In
all remaining panels, the eigenvalues are resolve by the momenta parallel to
the slab or momentum parallel to the wire. Panels b) and c) demonstrate that
the model is gappable at the boundaries of codimension 1. In panel c), near
the origin of the vertical axis, we see a locus (parametrized by $k_{z}$) of
non-degenerate eigenvalues. They must be and they are indeed localized at
hinges.

Non-trivial secondary characteristic classes such as the Chern-Simons
invariant have been associated with higher-order boundary states in the
literature [50, 62], however, there appears to be as of yet no structural
argument why that might be the case.

From this heuristic it seems that a generator which realizes the non-trivial
parity in (124) would be a good candidate for non-trivial second order bulk-
edge correspondence. A simple lattice model for that is given below as a
slight modification of the bulk Hamiltonian listed in [29][Eq. 145]

$h=\tfrac{1}{2\imath}\sum_{i=1}^{3}\Gamma_{i}\otimes(S_{i}-S_{i}^{\ast})+\Gamma_{0}\otimes\Big{(}2+\tfrac{1}{2}\sum_{i=1}^{3}(S_{i}+S^{\ast}_{i})\Big{)}+\gamma\Gamma_{B}\otimes
1,$ (126)

where $S_{i}$’s are the generators of $C^{\ast}{\mathbb{Z}}^{3}$, and
$\Gamma_{1}=\sigma_{3}\otimes\sigma_{1}$, $\Gamma_{2}={\bf
1}\otimes\sigma_{2}$, $\Gamma_{3}=\sigma_{2}\otimes\sigma_{1}$,
$\Gamma_{0}={\bf 1}\otimes\sigma_{3}$ and, finally,
$\Gamma_{B}=\frac{1}{2}(\sigma_{1}+\sigma_{2})\otimes({\bf 1}+\sigma_{3})$,
with the Pauli matrices $\sigma_{i}$. On the atomic orbitals space
$M_{4}({\mathbb{C}})$, we assume that the action of the time reversal is
implemented by conjugation with $(\sigma_{2}\otimes 1){\mathcal{K}}$,
${\mathcal{K}}$ being complex conjugation on $\ell^{2}({\mathbb{Z}}^{3})$, and
inversion by conjugation with ${\bf 1}\otimes\sigma_{3}$. If the parameter
$\gamma$ is set to $0$, the time reversal and the space inversion become
separate symmetries of the model and the Hamiltonian (126) becomes a strong
topological insulator from the class $AII$ of the classification table [61].
This means that any surface cut into a bulk sample will fill the bulk spectral
gap seen in Fig. 10(a) with surface spectrum. For a perfectly flat surface,
the latter can be resolved by the quasi-momenta $k_{j}\in{\mathbb{T}}$ along
the surfaces, by effectively using the isomorphism
$C^{\ast}{\mathbb{Z}}^{3}\simeq C^{\ast}{\mathbb{Z}}\otimes
C({\mathbb{T}}^{2})$, $S_{j}\mapsto u_{j}(k_{j})=e^{\imath k_{j}}$, where $j$
samples the cartesian directions parallel to surface. If done so, one can see
in the momentum-resolved spectrum as in Fig. 10(b,c) the hallmark Dirac
spectral singularities originating from protected surfaces states. 101010The
$xy$-slab, for example, results from a confinement of the material in the
$x$-direction. Thus, a slab displays two infinite surfaces, but these surfaces
are separated by a distance large enough to reduce the effects of their
interference below the resolution of our figures. When $\gamma$ is set to a
non-zero value, the time-reversal symmetry is broken but the space inversion
symmetry persists. As seen in Fig. 10(b,c), the Dirac singularities are lifted
and the surfaces of the slabs become spectrally gapped, regardless of the
orientations of the cuts. Yet, when we cut a wire with a large square section
out of the bulk sample, the spectrum becomes again un-gapped due to two
infinitely thin bands crossing the slabs’ spectral gaps in Fig. 10(d). This
spectrum is supported by 1-dimensional wave channels that develop along the
hinges of the wires.

The numerical evidence for the existence of the hinge modes seems unambiguous
and if they exist then the given bulk $K$-theory class does map non-trivially
under the second order boundary map. We will not attempt a rigorous proof of
the non-triviality of the hinge modes here; while there are methods to do this
in principle, they either require non-trivial major adaptations to apply to
the present case or lengthy computations. Instead of doing cumbersome spectral
analysis for specific operators on quarter-spaces, one will want to resort to
index- or $K$-theoretic methods which allow determination of the hinge-
invariants without resolving the full spectral decomposition. Let us highlight
in this context the recent result [28] which allows to compute the invariants
of parametrized families of quarter-space Toeplitz operators in terms of an
extended symbol space, which also applies to this case. A different approach
is to start from the time-reversal symmetric model and track the change of its
topological invariants as the surface Dirac cones are gapped by the symmetry-
breaking mass terms. Indeed, such arguments can be justified mathematically
[7, 56, 48] but are not understood in the generality required here. Another
promising idea is an adiabatic approach where one replaces the sharp
boundaries with slowly varying domain walls, which allows one to construct
Hamiltonians with computable topological invariants from topologically non-
trivial symbols, i.e. matrix-valued functions which depend on space and
momentum. Such ideas are popular in physics [59, 53] and one can make them
rigorous, in particular for models on continuous space via pseudodifferential
methods (see [4] for a relevant example). Such adiabatic quantizations can
also be given meaning in $K$-theory using continuous fields or asymptotic
morphisms [19]. We leave exploration of those approaches to systematically
compute higher boundary maps to future work.

###  5.3. $C_{2}T$-symmetry

Here we consider an anti-linear symmetry. In three dimensions $C_{2}$ shall be
rotation by $\pi$ in the $x_{1}$-$x_{2}$-plane, hence implemented by the
operation

$C_{2}:(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}\mapsto(-x_{1},-x_{2},x_{3}).$
(127)

As for inversion this is an order two symmetry, whose action on
$C^{\ast}{\mathcal{G}}_{\square}$ we denote by the same symbol $C_{2}$. We
compose it with time-reversal $T$ which acts trivially on ${\mathbb{R}}^{3}$
but acts by complex conjugation on the convolution algebra
$C^{\ast}{\mathcal{G}}_{\square}$, i.e. it is an involutive anti-automorphism.
The same is true for the composition $C_{2}T$. One can therefore consider the
twisted equivariant $K$-groups

$K_{\ast}^{C_{2}T}(C^{\ast}{\mathcal{G}}_{\square}):={}^{\phi}K_{\ast,c,\tau}^{{\mathbb{Z}}_{2}}(C^{\ast}{\mathcal{G}}_{\square}),$
(128)

for $\ast\in\\{0,-1\\}$ and where $c=0$ and $\tau$, $\phi$ just express that
the generator shall be represented anti-unitarily in any
$(\phi,c,\tau)$-twisted representation of ${\mathbb{Z}}_{2}$.

As for inversion-symmetry opposite sides and corners of the square are
conjugate under the symmetry:

######  Proposition 5.9.

$K_{\ast}^{C_{2}T}({\mathcal{F}})\simeq
K_{\ast}({\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2})$ (129)

and

$K_{\ast}^{C_{2}T}({\mathcal{C}})\simeq
K_{\ast}({\mathcal{C}}_{1}\oplus{\mathcal{C}}_{2}).$ (130)

Figure 11. Same as Fig. 10 but for the Hamiltonian (134). The more refined
scale shown in the inset of panel (c) is needed to see the topological hinge
modes.

Here one has $K_{1}=K_{-1}$ due to Bott periodicity of complex K-theory. One
can again think of $K_{\ast}^{C_{2}T}({\mathcal{F}})$ as being represented by
elements
$[x_{1},x_{2},x_{3},x_{4}]_{\ast}\in\bigoplus_{\lambda=1}^{4}K_{\ast}({\mathcal{F}}_{i})$
of the form $[x_{1},x_{2},C_{2}T(x_{1}),C_{2}T(x_{2})]_{\ast}$. The Chern
numbers of the are related by

$\displaystyle\langle[x_{\lambda}]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},e_{3}\times n_{\lambda+1}}]\rangle$
$\displaystyle=\langle C_{2}T([x_{\lambda}]_{0}),[{\rm
Ch}_{{{\mathcal{F}}_{\lambda+2}},e_{3}\times(-n_{\lambda+1})}\circ T]\rangle$
(131) $\displaystyle=-\langle C_{2}T([x_{\lambda}]_{0}),[{\rm
Ch}_{{{\mathcal{F}}_{\lambda+2}},e_{3}\times n_{\lambda+3}}]\rangle$

since $C_{2}$ flips the signs of $\nabla_{1},\nabla_{2}$ but leaves the other
derivations unchanged. Complex conjugation $T$ also flips the sign of the
Chern numbers since they would be purely imaginary if not for the
normalization constants. We conclude that one has precisely the same
constraints as for inversion symmetry:

######  Proposition 5.10.

$K_{-1}^{C_{2}T}({\mathcal{C}})\supset\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq{\mathbb{Z}}\oplus(2{\mathbb{Z}})$
(132)

and

$K_{-1}^{C_{2}T}({\mathcal{C}})/\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq{\mathbb{Z}}_{2}.$
(133)

We now present evidence that there actually is a gapped Hamiltonian that
realizes the non-trivial values of those parities. Below is a Hamiltonian
inspired by [51][Eq. 1]

$h=\tfrac{1}{2\imath}\sum_{i=1}^{3}\Gamma_{i}\otimes(S_{i}-S_{i}^{\ast})+\Gamma_{0}\otimes\Big{(}2+\tfrac{1}{2}\sum_{i=1}^{3}(S_{i}+S^{\ast}_{i})\Big{)}+\gamma\Gamma_{B}\otimes
1,$ (134)

where $S_{i}$ are the generators of $C^{\ast}{\mathbb{Z}}^{3}$, and
$\Gamma_{i}=\sigma_{1}\otimes\sigma_{i}$, $\Gamma_{0}=\sigma_{z}\otimes 1$ and
finally $\Gamma_{B}=1\otimes(\sigma_{1}+\sigma_{2})$. On the atomic orbitals
space $M_{4}({\mathbb{C}})$, the action of the time reversal is implemented by
conjugation with $(1\otimes\sigma_{2})K$, while that of the 2-fold rotation by
conjugation with ${\bf 1}\otimes e^{\imath\frac{\pi}{2}\sigma_{3}}$. The
spectral characteristics of Hamiltonian (134) under various underlying atomic
configurations are summarized in Fig. 11. The outcome of these results is a
strong evidence that the Hamiltonian (134) belongs to the support of
$\delta^{2}_{0}$.

######  Remark 5.11.

The $C_{2}T$-invariant $K$-theory of the torus has been computed recently [62]
and the only generator which can potentially exhibit non-trivial second-order
boundary maps corresponds to a vector bundle that is invariant under both
$C_{2}T$\- and inversion symmetry and has a non-trivial Chern-Simons parity.
The Hamiltonian above is not invariant under inversion but must represent that
same class. $\Diamond$

###  5.4. $C_{4}T$-symmetry

We consider the fourfold rotation

$C_{4}:(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}\mapsto(-x_{2},x_{1},x_{3})$
(135)

composed with time-reversal $T$, i.e. complex conjugation, which defines an
order four anti-linear automorphism $C_{4}T$ on
$C^{\ast}{\mathcal{G}}_{\square}$. Moreover, the rotation shall be of so-
called fermionic type, by which we mean that the representations of $C_{4}T$
shall be twisted by

$U^{4}=-{\bf 1}$ (136)

where $U$ is the anti-unitary generator of the projective ${\mathbb{Z}}_{4}$
representation. The relevant $K$-groups are

$K_{\ast}^{C_{4}T}(C^{\ast}{\mathcal{G}}_{\square}):={}^{\phi}K^{{\mathbb{Z}}_{4}}_{\ast,c,\tau}(C^{\ast}{\mathcal{G}}_{\square})$
(137)

for $\ast\in\\{0,-1\\}$ and where $c=0$, $\phi$ is determined by the fact that
$C_{4}T$ and $(C_{4}T)^{2}$ are anti-linear automorphisms. The twist $\tau$ is
the unique $2$-cocycle such that (136) is imposed for the generators of the
$(\phi,c,\tau)$-twisted representations of ${\mathbb{Z}}_{4}$.

In this case all four sides and corners form a single orbit each under
rotations, which leads to:

######  Proposition 5.12.

$K_{\ast}^{C_{4}T}({\mathcal{F}})\simeq K_{\ast}({\mathcal{F}}_{1})$ (138)

and

$K_{\ast}^{C_{4}T}({\mathcal{C}})\simeq K_{\ast}({\mathcal{C}}_{1}).$ (139)

Therefore $K_{\ast}^{C_{4}T}({\mathcal{F}})$ is represented by elements
$[x_{1},x_{2},x_{3},x_{4}]_{\ast}\in\bigoplus_{\lambda=1}^{4}K_{\ast}({\mathcal{F}}_{\lambda})$
of the form
$[x_{1},C_{4}T(x_{1}),(C_{4}T)^{2}(x_{1}),(C_{2}T)^{3}(x_{1})]_{\ast}$, where
$x_{1}\in M_{N}({\mathbb{C}})\otimes{\mathcal{F}}_{1}$ and $C_{4}T$ acts on
$M_{N}({\mathbb{C}})$ by some $(\phi,c,\tau)$-twisted representation. This
implies the following relation for the relevant Chern numbers

$\displaystyle\langle[C_{4}T(x_{\lambda})]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda+1}},e_{3}\times n_{\lambda+2}}]\rangle$
$\displaystyle=\langle[x_{\lambda}]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},e_{3}\times n_{\lambda+1}}\circ T]\rangle$
(140) $\displaystyle=-\langle[x_{\lambda}]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{\lambda}},e_{3}\times n_{\lambda+1}}]\rangle.$

For the Chern cocycle which determines
$K_{-1}^{C_{4}T}({\mathcal{C}})\simeq{\mathbb{Z}}$ one therefore has by (107)

$\langle\partial^{{\mathcal{F}}{\mathcal{C}}_{1}}([x_{1},C_{4}T(x_{1}),(C_{4}T)^{2}(x_{1}),(C_{4}T)^{3}(x_{1})],[{\rm
Ch}_{{{\mathcal{C}}_{1}},e_{3}}]\rangle=2\langle[x_{1}]_{0},[{\rm
Ch}_{{{\mathcal{F}}_{1}},e_{3}\times n_{4}}]\rangle.$

Since this is any even integer we conclude

######  Proposition 5.13.

$K_{-1}^{C_{4}T}({\mathcal{C}})\supset\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq
2{\mathbb{Z}}$ (141)

and

$K_{-1}^{C_{4}T}({\mathcal{C}})/\mathrm{Im}\,\partial^{{\mathcal{F}}{\mathcal{C}}}_{0}\simeq{\mathbb{Z}}_{2}.$
(142)

Let us highlight something remarkable about this fact: The
${\mathbb{Z}}_{2}$-invariant is the parity of any of the four corner Chern
numbers. Hence if a bulk Hamiltonian maps to the odd parity sector then the
corner Chern number of any single corner is non-trivial and has odd parity,
even though it would not be possible to detect the global $C_{4}T$-symmetry of
the crystal. Indeed, to derive the maps in equivariant $K$-theory which
guarantee the topological protection of the hinge mode we had to adjoin three
additional hinges. The only remnant of the symmetry at a single hinge is that
the two asymptotic half-spaces adjacent to the hinge are related by a
$C_{4}T$-transformation.

We now present evidence that there actually are gapped Hamiltonians which
realize the non-trivial values of those parities. Below is a slight
modification of the bulk Hamiltonian listed in [51][Eq. 1]

$h=\tfrac{1}{2\imath}\sum_{i=1}^{3}\Gamma_{i}\otimes(S_{i}-S_{i}^{\ast})+\Gamma_{0}\otimes\Big{(}2+\tfrac{1}{2}\sum_{i=1}^{3}(S_{i}+S^{\ast}_{i})\Big{)}+\gamma\Gamma_{B}\otimes(S_{1}+S^{\ast}_{1}-S_{2}-S^{\ast}_{2}),$
(143)

where $S_{i}$ are the generators of $C^{\ast}{\mathbb{Z}}^{3}$, and
$\Gamma_{i}=\sigma_{1}\otimes\sigma_{i}$, $\Gamma_{B}=\sigma_{2}\otimes 1$ and
finally $\Gamma_{0}=\sigma_{3}\otimes 1$. On the atomic orbitals space
$M_{4}({\mathbb{C}})$, the action of the time reversal is implemented by
conjugation with $(1\otimes\sigma_{2})K$, while that of the 4-fold rotation by
conjugation with ${\bf 1}\otimes e^{\imath\frac{\pi}{4}\sigma_{3}}$. The
spectral characteristics of Hamiltonian (143) under various underlying atomic
configurations are summarized in Fig. 12. Again, the outcome of these results
is a strong evidence that the Hamiltonian (134) belongs to the support of
$\delta^{2}_{0}$.

Figure 12. Same as Fig. 10 but for the Hamiltonian (143).

###  5.5. The cube geometry

Let us now consider the cube groupoid ${\mathcal{G}}_{\mbox{\small{\char
28\relax}}}$ from Section 3.4. There is a natural filtration of the unit space
by faces, hinges, corners such that the co-filtration (53), written
schematically as
${\mathcal{A}}_{3}\to{\mathcal{A}}_{2}\to{\mathcal{A}}_{1}\to{\mathcal{A}}_{0}\to
0$, has ideals

$\displaystyle{\mathcal{E}}_{0}$ $\displaystyle={\mathcal{B}}\simeq
C({\mathbb{T}}^{3})$ (144) $\displaystyle{\mathcal{E}}_{1}$
$\displaystyle\simeq(C({\mathbb{T}}^{2})\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}})))^{\otimes
6}$ $\displaystyle{\mathcal{E}}_{2}$
$\displaystyle\simeq(C({\mathbb{T}})\otimes{\mathbb{K}}(\ell^{2}({\mathbb{N}}\times{\mathbb{N}})))^{\otimes
12}$ $\displaystyle{\mathcal{E}}_{3}$
$\displaystyle\simeq({\mathbb{K}}(\ell^{2}({\mathbb{N}}\times{\mathbb{N}}\times{\mathbb{N}})))^{\otimes
8}.$

The first-order differentials can again be computed easily in terms of Chern
numbers and Toeplitz extensions and one finds that there are no non-trivial
higher-order bulk-boundary correspondences in complex K-theory. In fact, the
kernel of the first differential $\partial^{1}_{q}$ is precisely the image of
$K_{q}({\mathbb{C}})\to K_{\ast}({\mathbb{T}}^{3})$, i.e. only trivial
elements remain. Enhancing the K-groups by crystalline symmetries, one can
stabilize second-order hinge modes or third-order corner modes. Since the
mechanism and formalism should be clear by now we do not need to give an
explicit example of a third-order bulk-boundary correspondence.

Figure 13. Surface states for the inversion-, $C_{2}T$\- and
$C_{4}T$-symmetric examples

Instead, we want to elaborate on the importance of scanning different possible
symmetry adapted filtrations (77). If one places the bulk Hamiltonians given
in section 5.2 to 5.4 on a cube geometry, one will find hinge and surface
states as depicted in Figure 13. The hinge Chern numbers have a concrete
interpretation as current flows, whose direction and magnitude is determined
by sign and magnitude of the Chern number (this connection to charge transport
is completely analogous to the case of edge currents [32][48, Proposition
7.1.2]). For inversion-symmetry one obtains a closed loop of hinge currents
which flow around the surface of the cube. Only the existence of the current
flow is topologically protected; the exact path it takes depends on the choice
of boundary condition. We can now explore different closed invariant
transversals $\tilde{\Xi}\subset\Xi_{\mbox{\small{\char 28\relax}}}$ and see
for which we have second-order bulk boundary correspondences. The smallest
non-trivial transversal $\tilde{\Xi}$ contains just two opposite hinges. From
the analysis of subsection 5.2 we know that this choice will return only
trivial correspondences since, given gapped faces, one can gap any two fixed
opposite hinges by adding a surface layer. If $\tilde{\Xi}$ contains exactly
the four vertical hinges we get back to the situation of subsection 5.2 and
have possible hinge-modes protected by a non-trivial parity. If one adds more
hinges to the filtration one will still have a non-trivial correspondence and
the possible patterns of current flows can be enumerated by appropriate sum
rules.

On the other hand, in the $C_{2}T$\- and $C_{4}T$-symmetric case one finds
that the top bottom surface are ungapped and host Dirac-type surface states.
One can see this as a result of current conservation: If one fixes the
currents on the vertical hinges to be as in the computations above then it is
impossible to complete them to a $C_{2}T$\- respectively $C_{4}T$-invariant
configuration of hinge currents which satisfies Kirchhoffs current law at each
corner (as we did for the inversion symmetric case). The only possible
conclusion is that currents must be able to flow across the top and bottom
face. This constraint does exist naturally in $K$-theory: Any possible surface
state is characterized by a K-theory class in the image of a boundary map
$\partial^{p}_{*}$. As a consequence of the long exact sequences of K-theory
it is therefore in the kernel of the consecutive differential
$\partial^{p+1}_{*-1}$. In the case of the boundary map $\partial_{0}^{2}$ the
kernel of $\partial_{-1}^{3}$ consists precisely of those classes whose hinge
Chern numbers satisfy the Kirchhoff current law. Thus, the second-order bulk-
boundary map must be trivial for those combinations of filtration and symmetry
classes.

This indicates that the non-trivial order-2 correspondences can be only
resolved if we choose a smaller transversal $\tilde{\Xi}$ such that
$\tilde{\Xi}\cap\Xi_{1}$ contains all faces except the top and bottom ones.
Then $\tilde{\Xi}\cap\Xi_{2}$ should add to that the four vertical hinges of
the cube which brings us back to the co-filtrations used in subsections 5.3
and 5.4. Lastly, the topological modes supported by the faces, seen in Fig. 13
for the $C_{2}T$\- and $C_{4}T$-symmetric cases, can be discovered via order-1
bulk-boundary correspondences supported by any $\tilde{\Xi}$ that includes the
top and the bottom faces.

##  6\. Appendix

###  6.1. Twisted group representations

As is well-known since Wigner’s analysis, the symmetries of quantum systems
are given by groups acting on rays in a Hilbert space. They lift to projective
representations which act as unitary or anti-unitary operators, sometimes
called PUA (projective unitary/anti-unitary) representations. In a more modern
form that also includes the possibility of introducing a grading on the groups
they have been studied in the remarkable article [22] and for actions on
$C^{*}$-algebras we refer to [33, 34, 60, 24].

######  Definition 6.1.

A twist $(\phi,c,\tau)$ of a group $\Gamma$ consists of

  * •

homomorphisms $\phi,c:\Gamma\to{\mathbb{Z}}_{2}$.

  * •

A $2$-cocycle $\tau\in H^{2}_{\phi}(\Gamma,{\mathbb{T}})$ where $\Gamma$ acts
on ${\mathbb{T}}=\\{t\in{\mathbb{C}}:\,\left\lvert t\right\rvert=1\\}$ via
$g\cdot t={}^{\phi(g)}t$ where ${}^{\phi(g)}t=t$ if $\phi(g)=0$ and
${}^{\phi(g)}t=\overline{t}$ if $\phi(g)=1$.

Those cocycles classify the $\phi$-twisted group extensions, i.e. exact
sequences

$1\to{\mathbb{T}}\to\Gamma^{\phi}_{\tau}\to\Gamma\to 1,$

where $\Gamma$ acts on the abelian subgroup ${\mathbb{T}}$ via
$gtg^{-1}={}^{\phi(g)}t$ for all $g\in\Gamma$.

For simplicity we will assume that all groups are finite, as is sufficient for
the purposes of this work. Those groups can also act on operator algebras, in
particular complex $C^{*}$-algebras, in a way that incorporates those twists.
While those do not necessarily need to possess a (distinguished) Real
structure, the actions may be anti-linear:

######  Definition 6.2.

A $\phi$-twisted $\Gamma$-action on a $C^{*}$-algebra ${\mathcal{A}}$ is an
${\mathbb{R}}$-linear homomorphism
$\alpha:\Gamma\to\mathrm{Aut}_{\mathbb{R}}({\mathcal{A}})$ such that
$\alpha_{g}$ is complex linear for $\phi(g)=0$ and conjugate linear for
$\phi(g)=1$.

A $(\phi,\tau)$-twisted $\Gamma$-action on a $C^{*}$-algebra ${\mathcal{A}}$
is a $\phi$-twisted $\Gamma^{\tau}$-action such that
$\alpha\rvert_{{\mathbb{T}}}=\textup{id}.$

The grading and twist is furthermore incorporated by representations on graded
Hilbert modules:

######  Definition 6.3.

Let $({\mathcal{A}},\alpha)$ be a $\phi$-twisted $\Gamma$-$C^{*}$-algebra. A
$(\phi,c,\tau)$-twisted representation on a graded Hilbert
${\mathcal{A}}$-module $E$ is an $\phi$-linear homomorphism
$U:\Gamma^{\phi}_{\tau}\to{\mathcal{L}}_{\mathbb{R}}(E)$
(${\mathbb{R}}$-linear bounded operators that are not necessarily adjointable)
where $U\rvert_{\mathbb{T}}$ is realized by scalar multiplication, and one has

$\langle U(g)\xi,U(g)\eta\rangle_{E}=\alpha_{g}(\langle\xi,\eta\rangle_{E})$

and

$\gamma_{E}(U(g)\xi)=(-1)^{c(g)}U(g)\gamma_{E}(\xi)$

for $\gamma_{E}$ the grading of $E$. In particular, $U(g)$ is odd if $c(g)=1$
and even otherwise, $U(g)$ is anti-linear if $\phi(g)=1$ and linear otherwise.

For representations on graded Hilbert spaces, i.e. the case
${\mathcal{A}}={\mathbb{C}}$, the action $\alpha_{g}$ can only be trivial or
complex conjugation, thus $\phi(g)$ decides if $U(g)$ is unitary or anti-
unitary.

###  6.2. Twisted equivariant $K$-theory

The notion of twisted equivariant $K$-theory was developed for spaces in [22]
and for operator algebras in [33, 34, 24]. Of course, equivariant $K$-theory
itself even including twisted actions is significantly older, the difference
is that these works incorporate anti-linear actions and also those of graded
groups in a very natural way that closely aligns with the needs of solid state
physics.

There is a natural notion of $(\phi,c,\tau)$-twisted equivariant $KK$-theory
based on Hilbert ${\mathcal{A}}$-${\mathcal{B}}$-modules of $\phi$-twisted
$\Gamma$-$C^{*}$-algebras which carry $(\phi,c,\tau)$-twisted representations,
from which one can define

${}^{\phi}K^{\Gamma}_{c,\tau}({\mathcal{A}}):={}^{\phi}KK^{\Gamma}_{c,\tau}({\mathbb{C}},{\mathcal{A}})$

where we think of ${\mathbb{C}}$ as a $\phi$-twisted $\Gamma$-algebra on which
elements $g\in\Gamma$ act trivially or by complex conjugation depending on
$\phi(g)$.

For practical computations the definition as $KK$-groups are fairly
inconvenient since they are closer to the Fredholm picture of $K$-theory than
the standard picture. Kubota [33] also defines a more useful van-Daele-like
picture:

Let $({\mathcal{A}},\alpha)$ be a $\phi$-twisted ungraded
$\Gamma$-$C^{*}$-algebra and assume for now it is unital. For any
$(\phi,c,\tau)$-twisted finite-dimensional representation of $\Gamma$ on a
finite-dimensional vector space ${\mathcal{V}}$ the algebra
${\mathcal{A}}\otimes B({\mathcal{V}})$ carries a $(\phi,\tau)$-twisted
$\Gamma$-action which we denote by the same letter $\alpha$.

An element $a\in{\mathcal{A}}\otimes B({\mathcal{V}})$ is called $c$-twisted
invariant if $\alpha_{g}(a)=(-1)^{c(g)}a$ for all $g\in\Gamma$ and the space
of $c$-twisted invariant self-adjoint unitaries is denoted
${}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}})$.

  1. (i)

One can take the inductive limit over all $(\phi,c,\tau)$-twisted
representations

${}^{\phi}K^{\Gamma}_{0,c,\tau}({\mathcal{A}}):=\lim_{{\mathcal{V}}}\pi_{0}({}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}}))$

with respect to the inclusion

${}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}})\hookrightarrow{}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}\oplus{\mathcal{W}}}({\mathcal{A}}),\quad
a\mapsto a\oplus\gamma_{\mathcal{W}}$

where $\gamma_{\mathcal{W}}$ is the grading operator on ${\mathcal{W}}$
(recall that we are using $c$-graded group representations) and $\pi_{0}$ are
equivalence classes w.r.t. norm-continuous homotopy. With the direct sum
$[a_{1}]+[a_{2}]:=[a_{1}\oplus a_{2}]$ this becomes an abelian group with the
inverse $-[a]=[-\gamma_{\mathcal{V}}a\gamma_{{\mathcal{V}}}]$ where the
representative is in ${\mathcal{A}}\otimes B({\mathcal{V}}^{\mathrm{op}})$
with the opposite grading
$\gamma_{{\mathcal{V}}^{\mathrm{op}}}=-\gamma_{{\mathcal{V}}}$ since
$a\oplus(-\gamma_{\mathcal{V}}a\gamma_{{\mathcal{V}}})$ is homotopic to
$\gamma_{\mathcal{V}}\oplus(-\gamma_{\mathcal{V}})$ in
${}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}\oplus{\mathcal{V}}^{\mathrm{op}}}({\mathcal{A}})$.

  2. (ii)

Denote by
${}^{\phi}{\mathcal{U}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}})$ the
space of unitaries $u\in{\mathcal{A}}\otimes B({\mathcal{V}})$ such that
$\alpha_{g}(u)=u$ if $\phi(g)+c(g)=0$ and $\alpha_{g}(u)=u^{*}$ if
$\phi(g)+c(g)=1$. Then set

${}^{\phi}K^{\Gamma}_{-1,c,\tau}({\mathcal{A}}):=\lim_{{\mathcal{V}}}\pi_{0}({}^{\phi}{\mathcal{U}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}}))$

with respect to the inclusion

${}^{\phi}{\mathcal{U}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}})\hookrightarrow{}^{\phi}{\mathcal{U}}^{\Gamma}_{c,\tau,{\mathcal{V}}\oplus{\mathcal{W}}}({\mathcal{A}}),\quad
a\mapsto a\oplus{\bf 1}_{\mathcal{W}}.$

With the direct sum $[u_{1}]+[u_{2}]:=[u_{1}\oplus u_{2}]$ this becomes an
abelian group with inverse $-[u]=[u^{*}]$.

In the non-unital case one defines

${}^{\phi}K^{\Gamma}_{i,c,\tau}({\mathcal{A}})={\rm
Ker}({}^{\phi}K^{\Gamma}_{i,c,\tau}({\mathcal{A}}^{+})\to{}^{\phi}K^{\Gamma}_{i,c,\tau}({\mathbb{C}})).$

Classes in ${}^{\phi}K^{\Gamma}_{0,c,\tau}({\mathcal{A}})$ are defined by
band-flattenings ${\rm sgn}(h)$ of $c$-twisted invariant self-adjoint
invertibles $h$, while in some sense $K_{-1}$ is the natural range of the
boundary map as one may see below.

In general one can define higher $K$-groups by suspension

${}^{\phi}K^{\Gamma}_{p-q,c,\tau}({\mathcal{A}})={}^{\phi}K^{\Gamma}_{0,c,\tau}(S^{p,q}{\mathcal{A}})\simeq{}^{\phi}K^{\Gamma}_{-1,c,\tau}(S^{p+1,q}{\mathcal{A}})$

where $S^{p,q}$ is the tensor product of ${\mathcal{A}}$ with the algebra
$C_{0}({\mathbb{R}}^{p+q})$ where the Real structure

$\overline{f}(x_{1},...,x_{p},y_{1},...,y_{q})=\overline{f(x_{1},...,x_{p},-y_{1},...,-y_{q})}$

is used to extend the $\phi$-linear action from ${\mathcal{A}}$ to
$S^{p,q}{\mathcal{A}}$ via

$\alpha_{g}(f)(x,y)=\begin{cases}\alpha_{g}(f(x,-y))&\text{if }\phi(g)=1\\\
\alpha_{g}(f(x,y))&\text{if }\phi(g)=0\end{cases}.$

These groups only depend on $p-q$ up to isomorphism.

Twisted equivariant $K$-theory is a homology theory and for every equivariant
short exact sequence

$0\to{\mathcal{J}}\to{\mathcal{A}}\to{\mathcal{A}}/{\mathcal{J}}\to 0$

of $\phi$-twisted $\Gamma$-$C^{*}$-algebras there exists a boundary map which
fits into a long exact sequence

$...\to{}^{\phi}K^{\Gamma}_{n,c,\tau}({\mathcal{J}})\to{}^{\phi}K^{\Gamma}_{n,c,\tau}({\mathcal{A}}/{\mathcal{J}})\stackrel{{\scriptstyle\partial}}{{\to}}{}^{\phi}K^{\Gamma}_{n-1,c,\tau}({\mathcal{J}})\to{}^{\phi}K^{\Gamma}_{n-1,c,\tau}({\mathcal{A}})\to...$

For a class $[x]_{0}\in K^{\Gamma}_{0,c,\tau}({\mathcal{A}}/{\mathcal{J}})$
represented by
$x\in{}^{\phi}{\mathcal{F}}_{c,\tau,{\mathcal{V}}}^{\Gamma}({\mathcal{A}}/{\mathcal{J}})$
it is defined by choosing any self-adjoint lift
$\tilde{x}\in{}^{\phi}{\mathcal{F}}_{c,\tau,{\mathcal{V}}}^{\Gamma}({\mathcal{A}})$
and setting

$\partial[x]_{0}=[-\exp(-\imath\pi\tilde{x})]_{-1}.$

This class can be seen as a
${}^{\phi}K^{\Gamma}_{-1,c,\tau}({\mathcal{J}})$-valued index for the self-
adjoint lift $\tilde{x}$ in the sense that it is precisely the $K$-theoretic
obstruction to its invertibility.

######  Remark 6.4.

The definition of the boundary map given above uses both pictures of Kubota’s
$K$-theory and for various constructions one needs to pass between
${}^{\phi}K^{\Gamma}_{0,c,\tau}(S^{p,q}\cdot)\simeq{}^{\phi}K^{\Gamma}_{-1,c,\tau}(S^{p+1,q}\cdot)$
which can be inconvenient. In general one prefers “unsuspended” versions in
which higher $K$-classes can be represented in terms of projections
(equivalently self-adjoint unitaries) or unitary elements with certain
symmetries. This is indeed possible at least in special cases:

  1. (i)

For trivial $\phi,c,\tau=0$ one has
${}^{\phi}K^{\Gamma}_{n,c,\tau}({\mathcal{A}})=K_{n}^{\Gamma}({\mathcal{A}})$,
the usual complex equivariant $K$-groups defined by [44]. In particular the
$K$-groups are $2$-periodic.

  2. (ii)

For $\Gamma=\Gamma_{0}\times{\mathbb{Z}}_{2}$, $c=0$ and $(\phi,\tau)$ such
that $\Gamma_{0}$ acts by linear and ${\mathbb{Z}}_{2}$ by anti-linear
automorphisms one has

${}^{\phi}K^{\Gamma}_{n,c,\tau}({\mathcal{A}})=KR_{n}^{\Gamma_{0}}({\mathcal{A}}),$

where the anti-linear ${\mathbb{Z}}_{2}$-action provides the Real structure on
${\mathcal{A}}$ which commutes with the action of $\Gamma_{0}$. Those
$KR$-groups are $8$-periodic.

  3. (iii)

In case (ii) one further can express the $KR$-groups in terms of twisted
equivariant $K$-theory by adjoining $CT$-type symmetries

$KR_{n}^{\Gamma_{0}}({\mathcal{A}})={}^{\phi_{n}}K^{\Gamma_{0}\times\Gamma_{n}}_{0,c_{n},\tau_{n}}({\mathcal{A}})$

where either $n\in\\{0,...,7\\}$ stands for the anti-unitary symmetry classes
(enumerated in order as $\mathrm{AI}$, $\mathrm{BDI}$, $\mathrm{D}$,
$\mathrm{DIII}$, $\mathrm{AII}$, $\mathrm{CII}$, $\mathrm{C}$, $\mathrm{CI}$),
$\Gamma_{n}$ is one of the abelian groups $0$, ${\mathbb{Z}}_{2}$ or
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ and the grading and twist
$(\phi_{n},c_{n},\tau_{n})$ are used to implement the usual commuting or anti-
commuting symmetries, one can write the complex $\mathrm{AIII}$-class as

$K_{1}^{\Gamma}(A)={}^{\phi}K^{{\mathbb{Z}}_{2}}_{0,c,\tau}(A)$

by adjoining a single oddly graded generator of a trivial
${\mathbb{Z}}_{2}$-action and trivial twists $\phi$,$\tau$. In this way all
symmetry classes of the tenfold way fit naturally into Real or complex
$K$-theory.

$\Diamond$

One helpful isomorphism that we need in the main text is that one can
sometimes reduce equivariant to non-equivariant $K$-theory:

######  Proposition 6.5.

For $H$ a subgroup of the finite group $\Gamma$ let
$C(\Gamma/H)=C(\Gamma/H,{\mathbb{C}})$ be the Real $C^{*}$-algebra of
functions on $\Gamma/H$ with pointwise complex conjugation we define the
$\phi$-twisted left translation

$\alpha_{\tilde{g}}(f)(gH)=\begin{cases}\overline{f(\tilde{g}^{-1}gH)}&\text{if
}\phi(\tilde{g})=1\\\ f(\tilde{g}^{-1}gH)&\text{if
}\phi(\tilde{g})=0\end{cases}.$

For any $\phi$-twisted trivially graded $\Gamma$-$C^{*}$-algebra
${\mathcal{A}}$, ${\mathcal{A}}\otimes C(\Gamma/H)$ is a $\phi$-twisted
$\Gamma$-$C^{*}$-algebra and one has

${}^{\phi}K^{\Gamma}_{n,c,\tau}({\mathcal{A}}\otimes
C(\Gamma/H))={}^{\phi\rvert_{H}}K^{H}_{n,c\rvert_{H},\tau\rvert_{H}}({\mathcal{A}}).$

###### Proof.

It is enough to discuss the case $n=0$ since one can just suspend
${\mathcal{A}}$. Let ${\mathcal{V}}$ be a $(\phi,c,\tau)$-twisted
representation of $\Gamma$. Denote the action on ${\mathcal{A}}\otimes
B({\mathcal{V}})$ by $\alpha$ and that on $({\mathcal{A}}\otimes
C(\Gamma/H))\otimes B({\mathcal{V}})$ by $\beta$.

The $c$-twisted $\beta$-invariant functions $f\in
C(\Gamma/H,{\mathcal{A}}\otimes B({\mathcal{V}}))=({\mathcal{A}}\otimes
C(\Gamma/H))\otimes B({\mathcal{V}})$ are determined by their value at the
identity coset $f(eH)$ via

$(\beta_{g}(f))(gH)=(-1)^{c(g)}\alpha_{g}(f(eH))$ (145)

since $\Gamma$ acts transitively on $\Gamma/H$.

Classes in ${}^{\phi}K^{\Gamma}_{0,c,\tau}({\mathcal{A}}\otimes C(\Gamma/H))$
are determined by couples of $(\phi,c,\tau)$-twisted $\Gamma$-representations
and $c$-twisted invariant elements
$f\in{}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}}\otimes
C(\Gamma/H))$. Any finite-dimensional $(\phi,c,\tau)$-twisted
$\Gamma$-representations restricts to a representation of $H$ and the map
$[f]_{0}\mapsto[f(eH)]_{0}$ results in a well-defined homomorphism
${}^{\phi}K^{\Gamma}_{0,c,\tau}({\mathcal{A}}\otimes C(\Gamma/H))\to
K^{H}_{0,c,\tau}({\mathcal{A}})$. It depends only on the class of $f$ since
evaluating a $\Gamma$-equivariant homotopy in
${}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}}\otimes
C(\Gamma/H))$ on $eH$ gives an $H$-equivariant homotopy in
${}^{\phi}{\mathcal{F}}^{H}_{c,\tau,{\mathcal{V}}}({\mathcal{A}})$. The
element $f(eH)$ is $c$-twisted $H$-invariant since $H$ acts trivially on
$C(\Gamma/H)$.

To prove that this map is an isomorphism it is enough to construct an inverse:
W.l.o.g. we can assume that any
$(\phi\rvert_{H},c\rvert_{H},\tau\rvert_{H})$-twisted representation of $H$ is
the restriction of a $(\phi,c,\tau)$-twisted representation of $\Gamma$ on
some graded vector space ${\mathcal{V}}$, since one can always induce a
$\Gamma$-representation by enlarging ${\mathcal{V}}$. One then has for each
$a\in{}^{\phi\rvert_{H}}{\mathcal{F}}^{H}_{c\rvert_{H},\tau\rvert_{H},{\mathcal{V}}}({\mathcal{A}})$
a unique function $f$ with (145) and $f(eH)=a$.

The homomorphism well-defined and injective: Any path in
${}^{\phi\rvert_{H}}{\mathcal{F}}^{H}_{c\rvert_{H},\tau\rvert_{H},{\mathcal{V}}}({\mathcal{A}})$
lifts uniquely to one in
${}^{\phi}{\mathcal{F}}^{\Gamma}_{c,\tau,{\mathcal{V}}}({\mathcal{A}}\otimes
C(\Gamma/H))$. If the lift $f$ represents the neutral element of
${}^{\phi}K^{\Gamma}_{0,c,\tau}({\mathcal{A}}\otimes C(\Gamma/H))$ then there
exists (possibly after stabilization) a path connecting $f$ to
$\gamma_{\mathcal{V}}$ and evaluating that path at the identity coset gives a
path in
${}^{\phi\rvert_{H}}{\mathcal{F}}^{H}_{c\rvert_{H},\tau\rvert_{H},{\mathcal{V}}\oplus{\mathcal{W}}}({\mathcal{A}})$
connecting $f(eH)$ to $\gamma_{\mathcal{V}}$, hence it represents the neutral
element of
${}^{\phi\rvert_{H}}K^{H}_{0,c\rvert_{H},\tau\rvert_{H}}({\mathcal{A}})$. ∎

In particular, in the special case where $H=\\{e\\}$ the equivariant
$K$-groups always reduce to the usual complex $K$-groups $K_{0}$ and $K_{1}$.

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