#  $\mathbbm{Z}_{2}$ Spin Hopf Insulator: Helical Hinge States and Returning
Thouless Pump

Penghao Zhu  Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
Taylor L. Hughes  Department of Physics and Institute for Condensed Matter
Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,
USA

###### Abstract

We introduce a time-reversal symmetric analog of the Hopf insulator that we
call a spin Hopf insulator. The spin Hopf insulator harbors nontrivial Kane-
Mele $\mathbb{Z}_{2}$ invariants on its surfaces, and is the first example of
a class AII delicate topological insulator. We show that the Kane-Mele
$\mathbb{Z}_{2}$ topology on the surface is generically unstable, but can be
stabilized by the addition of a composite of the particle hole and inversion
symmetry. Such a symmetry not only protects the surface $\mathbb{Z}_{2}$
invariant, but immediately manifests gapless helical hinge states on the spin
Hopf insulator. Furthermore, we show that in the presence of $C_{4}$ rotation
symmetry the spin Hopf insulator exhibits a returning Thouless pump, and
associated surface states on sharp boundary terminations.

††preprint: APS/123-QED

##  I Introduction

The Hopf insulator is a three-dimensional bulk insulator that has delicate
topology, i.e., the topology is unstable to the addition of trivial valence or
_conduction_ bands [1, 2]. The bulk topology of the Hopf insulator can be
diagnosed by the _Hopf invariant_ , which is a Brillouin Zone(BZ) integral of
the Abelian Chern-Simons three-form [1]:

$\chi=-\frac{1}{4\pi^{2}}\int_{BZ}d^{3}k\bm{A}(\bm{k})\cdot\bm{\mathcal{F}}(\bm{k}),$
(1)

where $\bm{A}(\bm{k})=i\langle u|\nabla_{\bm{k}}u\rangle$ is the Berry
connection, $\bm{\mathcal{F}}(\bm{k})=\nabla\times\bm{A}(\bm{k})$ is the Berry
curvature, and $|u\rangle$ are the occupied Bloch functions. We note that the
conventional model for a Hopf insulator has two-bands [1], and thus the Berry
connection only includes a single occupied band and is hence Abelian.
Additionally, for Eq. 1 to be gauge invariant, the the Chern number of any
two-dimensional momentum slice in the three-dimensional BZ must vanish.

It has been recently demonstrated that the Hopf insulator has an unusual bulk-
boundary correspondence [3] that relates the bulk Hopf invariant $\chi$ to the
Chern number on surface facets having normal vector $\hat{\bm{n}}$ via:

$\displaystyle
C(\hat{\bm{n}})\equiv\int_{rBZ}d^{2}k\operatorname{Tr}\left[\bm{\mathcal{F}}_{\hat{\bm{n}}}(\bm{k})\cdot\hat{\bm{n}}\right]=\chi,$
(2)

where $\bm{\mathcal{F}}_{\hat{\bm{n}}}(\bm{k})$ is the Berry curvature of
boundary-localized bands, “$\operatorname{Tr}$” is over _all_ boundary-
localized bands 111We will explicitly describe what we mean by boundary-
localized bands below. (including both occupied and unoccupied bands), and
$rBZ$ indicates the reduced Brillouin zone (rBZ) of the surface facet. We dub
$C(\hat{\bm{n}})$ the _surface Chern number_ , and it is equal to the Hopf
invariant $\chi$ for any orientation of $\hat{\bm{n}}$. Unlike conventional
topological insulators, the Hopf bulk-boundary correspondence indicates that
the sum over _all_ boundary-localized bands has non-trivial first Chern
number, instead of just the _occupied_ boundary-localized bands. Furthermore,
we recently showed that the correspondence between the surface Chern number
and the Hopf invariant also implies the existence of chiral hinge states [5].
Thus, the topology diagnosed by the Hopf invariant has a higher order
character [6, 7].

In addition to the novel bulk-boundary correspondence, rotation symmetric Hopf
insulators have been shown to generate a returning Thouless pump (RTP) [3, 2,
8]. A RTP is a newly discovered symmetry protected property of delicate
topological insulators. In general, an RTP describes charge adiabatically
pumped by a multiple of the lattice period during the first half of the pump,
and during the second half of the pump the charge returns to its original
position [3, 2, 8]. Interestingly, the phase transition between a delicate TI
with a RTP and a trivial insulator is through a novel topological band
degeneracy called a Berry-dipole band degeneracy [2, 8].

Inspired by these properties of the Hopf insulator, we aim to find a spin-1/2,
time-reversal (TR) symmetric analog of the Hopf insulator that will describe
previously unknown class-AII delicate topological insulators. Since non-
magnetic materials are more abundant than magnetic materials in nature, the TR
symmetric analogs of the Hopf insulator could be easier to realize in
experiments. For these reasons, in this paper, we explore a variety of
phenomena of the TR symmetric analog of the Hopf insulator. For convenience,
we will denote the TR symmetric analog of the Hopf insulator as a spin Hopf
insulator. In analogy to previous work on Hopf insulators we expect the spin
Hopf insulator to have a nontrivial $\mathbb{Z}_{2}$ Kane-Mele invariant [9,
10, 11, 12, 13, 14, 15, 16] on the surface, and to exhibit a TR symmetric RTP
protected by rotation symmetries.

The remainder of the article is organized as follows. In Sec. II we introduce
a model for the spin Hopf insulator and its surface
$\mathbb{Z}_{2}$-invariant. Then in Sec. III we show that through an
adiabatic, symmetry-preserving process one can generically eliminate the
surface $\mathbb{Z}_{2}$-invariant by pumping the surface Kane-Mele
$\mathbb{Z}_{2}$-invariant across the bulk via a gapless, Bloch-Wannier band
transition. However, in Sec. IV we show that with the addition of a particle
hole symmetry combined with inversion symmetry the surface $\mathbb{Z}_{2}$
invariant can be stabilized. Indeed, we go on to show that this symmetry and
the nontrivial surface $\mathbb{Z}_{2}$ invariant can lead to stable helical
hinge states. Finally, in Sec. V we discuss the RTP protected by four-fold
rotation symmetry, and the special band-degeneracy that appears at the
topological phase transition point between the rotation invariant spin Hopf
insulator and a trivial insulator.

##  II Model Hamiltonian of spin Hopf insulator and surface
$\mathbb{Z}_{2}$-invariant

In order to construct a model Hamiltonian of the spin Hopf insulator let us
first briefly review the model Hamiltonian of the Hopf insulator constructed
by Moore, Ran and Wen (MRW) [1]. The MRW Bloch Hamiltonian is given by:

$\displaystyle z=(z_{1}+iz_{2},z_{3}+iz_{4})^{T},$ (3)
$\displaystyle\mathbf{d}=z^{{\dagger}}\bm{\sigma}z,\
\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z}),$ $\displaystyle
H_{MRW}(\mathbf{k})=\mathbf{d}\cdot\bm{\sigma},$

where

$\displaystyle z_{1}=\sin k_{x},z_{2}=\sin k_{y},z_{3}=\sin k_{z},$ (4)
$\displaystyle z_{4}=u-\cos k_{x}-\cos k_{y}-\cos k_{z}.$

If we take $1<u<3$, then the above model is a Hopf insulator with Hopf
invariant $\chi=1$. Note that in the MRW model, any two-dimensional slice of
the BZ has trivial Chern number, which indeed is required for $\chi$ to be
gauge invariant under gauge transformations of the Bloch wavefunctions.

Being an integral of the Chern-Simons three-form [cf. Eq. (1)], $\chi$
transforms like a pseudoscalar under crystallographic spatial transformations
and is odd under TR. Thus, TR symmetry is incompatible with a nontrivial Hopf
invariant $\chi$. To make progress on a spin Hopf insulator, we consider an
enlarged Hilbert space that is the tensor product of the original Hilbert
space with a spin-half Hilbert space. In this case, TR symmetry interchanges
spin with the representation $T=-i\tau_{y}\mathbbm{1}K,$ and squares to minus
the identity. To construct a model that is invariant under $T$, we note that
the MRW model Hamiltonian $H_{MRW}$ ($1<u<3$) is a Hamiltonian with $\chi=1$,
and replacing $z_{2}$ by $-z_{2}$ in $H_{MRW}$ leads to a new Hamiltonian
$H_{MRW}^{\prime}$ with $\chi=-1$. Hence, we can use $H_{MRW}$ for the spin-up
sector and $H_{MRW}^{\prime}$ for the spin-down sector to form the direct sum

$\displaystyle H_{sh}(\mathbf{k})$ $\displaystyle=H_{MRW}\oplus
H_{MRW}^{\prime}$ (5)
$\displaystyle=2z_{1}z_{3}\mathbbm{1}\sigma_{x}+2z_{2}z_{4}\tau_{z}\sigma_{x}-2z_{2}z_{3}\tau_{z}\sigma_{y}$
$\displaystyle+2z_{1}z_{4}\mathbbm{1}\sigma_{y}+\left(z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2}\right)\mathbbm{1}\sigma_{z},$

where $z_{i}$ for $i=1,2,3,4$ is defined in Eq. (4), and $\tau_{x,y,z}$
($\sigma_{x,y,z}$) are Pauli matrices for spin (orbital), and we have left the
tensor product between Pauli matrices implicit. We see that
$H_{sh}(\mathbf{k})$ satisfies $TH_{sh}(\mathbf{k})T^{-1}=H_{sh}(-\mathbf{k})$
as expected.

Since the opposite Hopf invariants of each spin sector generate opposite
surface Chern numbers [cf. Eq.(2)], the spin Hopf insulator inherits a
nontrivial surface $\mathbb{Z}_{2}$ invariant – defined as the
$\mathbb{Z}_{2}$ Kane-Mele invariant [10] of all boundary-localized Bloch-
Wannier bands. To better understand this point, let us clarify what we mean by
Bloch-Wannier bands here. Without loss of generality, we consider a slab
geometry with $z$-direction open and $x,y$-directions periodic. We assume that
a gap exists in the energy spectrum over the rBZ of the surface facet, and
separates two orthogonal subspaces with projection operators $P$ (occupied
subspace) and $Q$ (unoccupied subspace). By diagonalizing the projected
position operators, $P\hat{Z}P$ and $Q\hat{Z}Q$, where $\hat{Z}$ is the
position operator in the $z$-direction, we can obtain the eigenstates which
form a set of Bloch-Wannier bands [17, 18]. These bands are extended (in the
$xy$-plane) as Bloch functions with crystal wavevector
$\bm{k}_{\perp}=(k_{x},k_{y})$, but exponentially localized as Wannier
functions in the $z$-direction. The eigenvalues of $P\hat{Z}P$ and $Q\hat{Z}Q$
represent the Wannier center location in the $z$-direction (see Fig. 1 (d) and
(e) for two examples).

Now let us consider the topological properties of the surface Bloch-Wannier
bands. For simplicity, we take a semi-infinite slab with only one surface
facet, which has normal vector $+\hat{z}$, and we label the layers along the
$z$-direction by $z=1,2,3\ldots$ with $z=1$ being closest to the surface,
$z=2$ the next closest, and so on. In each layer, there are two Bloch Wannier
bands related by the TR symmetry (see Fig. 1 (d) and (e)). Far away from the
surface the Bloch-Wannier bands are indistinguishable (up to exponentially
small corrections) from Bloch-Wannier bands obtained with periodic boundary
conditions in all three spatial coordinates since the projectors $P,Q$ decay
exponentially in the coordinate representation [19]. Hence we dub these bands
_bulk_ Bloch-Wannier bands, and all other Bloch-Wannier bands are _boundary-
localized_ bands. We will denote the Kane-Mele $\mathbb{Z}_{2}$ invariant of
the pair of Bloch-Wannier bands in the _occupied_ subspace and localized in
the layer labelled by $z$ as $\nu_{v}(+\hat{z},z)$, and define the surface
valence $\mathbb{Z}_{2}$ invariant as

$\nu_{v}(+\hat{z})=\sum_{z=1}^{bulk}\nu_{v}(+\hat{z},z),$ (6)

where $\sum_{z=1}^{bulk}$ means that the sum extends deep enough into the slab
to the region containing bulk Bloch-Wannier bands. The sum in Eq. (6) is
uniquely defined because bulk bands have trivial $\mathbb{Z}_{2}$ Kane-Mele
invariant, which is inherited from the vanishing Chern number of the bulk
bands of the Hopf insulator in each spin sector. Similarly, we can define
$\nu_{c}(+\hat{z},z)$ and the surface conduction $\mathbb{Z}_{2}$ invariant
$\nu_{c}(+\hat{z})$ for the unoccupied subspace.

All the definitions discussed above can be extended to any insulating surface
facets by replacing $+\hat{z}$ with a generic $\hat{\bm{n}}$ representing the
normal vector of the surface facet of interest. Then, as defined above, the
_surface $\mathbb{Z}_{2}$ invariant_ for a surface facet with normal vector
$\hat{\bm{n}}$ is the $\mathbb{Z}_{2}$ Kane-Mele invariant of _all_ boundary-
localized bands, i.e.,

$\nu({\hat{\bm{n}}})=\nu_{v}({\hat{\bm{n}}})+\nu_{c}({\hat{\bm{n}}}).$ (7)

Figure 1: (a) A section of the bulk Bloch-Wannier bands for the spin-Hopf
model at the $\mathbb{Z}_{2}$ phase transition point. The Kramers pair of
Bloch-Wannier bands indicated by the blue color has touched its top and bottom
neighbors at Dirac-like points (indicated by red dots) related by TR symmetry
on the $\rm{X}-\Gamma-\rm{X}$ line in the BZ. (b) and (c) show the spectra of
Wilson loops built from projectors of the occupied and unoccupied surface
Bloch-Wannier bands (i.e., the layer closest to $z=10$). The Wilson loops are
parallel transported along the $k_{y}$-direction and then plotted vs $k_{x}.$
(b) and (c) show the results _before_ and _after_ the $\mathbb{Z}_{2}$ phase
transition point. The bands that contribute to the Wilson loop projectors are
highlighted in red in (d) and (e) for the calculations done before and after
the transition respectively. In (b) and (c) we show the intersections between
the Wilson loop spectra and an arbitrary $\Gamma X$ line (colored blue), which
we have indicated by blue triangles. The parity of the number of intersections
tells us the $\mathbb{Z}_{2}$ Kane-Mele invariant. We find an odd number in
(b) and an even number in (c) indicating their respective non-triviality and
triviality. Note that we have used 10 unit cells in the $z$-direction for
these calculations.

##  III Trivialization from Pump of $\mathbb{Z}_{2}$ Kane-Mele invariant

While the explicit model we wrote in Eq. (5) has a nontrivial surface
$\mathbb{Z}_{2}$ invariant inherited from the surface Chern numbers of the
Hopf insulators for each spin sector, we must be careful to understand the
stability of this nontrivial surface $\mathbb{Z}_{2}$ invariant under gap and
symmetry-preserving deformations. Indeed, we find that with no additional
symmetries other than TR, the spin Hopf insulator can be continuously
connected to a trivial band insulator while preserving the bulk energy gap. We
provide an explicit symmetry and gap-preserving interpolation from the model
Hamiltonian in Eq. (5) to a trivial insulator in the Appendix A. This is
consistent with an earlier homotopic classification in Ref. 20 which asserts
there are no delicate topological insulators in Altland-Zirnbauer class AII
(in the first order topology sense).

Interestingly, even though the bulk _energy_ gap is maintained during our
interpolation, we expect that the spectral gap of the bulk Bloch-Wannier bands
_must_ close at some point to effect a change in the surface $\mathbb{Z}_{2}$
invariant. Indeed, this gap closing realizes a real-space pump of the
$\mathbb{Z}_{2}$ Kane-Mele invariant, i.e., one quantum of the
$\mathbb{Z}_{2}$ Kane-Mele invariant leaves the surface and propagates through
the bulk. If this propagation occurs in the occupied (resp. unoccupied) bulk
states, it is the bulk spectral gap of $P\hat{Z}P$ (resp. $Q\hat{Z}Q$) that
closes. For a finite slab of the spin Hopf insulator, this indicates that the
$\mathbb{Z}_{2}$ Kane-Mele invariant will be pumped from each surface and
annihilate in the bulk, leading to a net cancellation of the surface
$\mathbb{Z}_{2}$ invariant on both surfaces – this is the TR-symmetric analog
of the Berry-curvature teleportation that occurs at the critical transition
between trivial and Hopf insulators [3].

With this phenomenology in mind, let us illustrate this pump of the
$\mathbb{Z}_{2}$ Kane-Mele invariant on a slab geometry that is open in the
$z$-direction and periodic in the $x,y$-directions. In our interpolation we
couple the two spin sectors of Eq. (5) and modify the surface Hamiltonian to
create an energy gap between the occupied and unoccupied subspaces. This
surface energy gap will remain throughout the part of our interpolation in
which the $\mathbb{Z}_{2}$ pump occurs. For our interpolated Hamiltonian [see
Appendix A], the topologically nontrivial Bloch-Wannier bands on the top and
bottom surface that participate in the $\mathbb{Z}_{2}$ pump lie in the
occupied subspace. Therefore, it is sufficient to focus our attention on
$P\hat{Z}P$ alone. At the critical point of the interpolation we plot the
spectrum of $P\hat{Z}P$ in Fig. 1 (a). As we can see, each bulk Bloch-Wannier
band in the occupied space touches its top and bottom neighbors at two TR
related points on the $\rm{X}-\Gamma-\rm{X}$ high symmetry line (thus each
forms four Dirac-like points as indicated by red dots in Fig. 1 (a)).

The transition where the $\mathbb{Z}_{2}$ pump occurs is analogous to the
phase transition of the 2D quantum spin Hall phase [21, 22], i.e., the
critical point where the $\mathbb{Z}_{2}$ pump occurs is a $\mathbbm{Z}_{2}$
phase transition point for the two-dimensional bulk Bloch-Wannier bands.
Because of the translation symmetry in the bulk, each bulk Bloch-Wannier band
(indicated by blue in Fig. 1 (a)) touches with both its top and bottom
neighbors simultaneously, and thus the $\mathbbm{Z}_{2}$ phase transition
happens twice for a bulk Bloch-Wannier band – once from the top and once from
the bottom. Then the $\mathbbm{Z}_{2}$ invariant of the bulk Bloch-Wannier
bands will not change during this process. However, the boundary-localized
Bloch-Wannier bands on both top and bottom surface facets touch with the bulk
Bloch-Wannier bands from only one side, i.e., the boundary-localized Bloch-
Wannier bands on the top (bottom) touch with only the bulk Bloch-Wannier bands
below (above) it. Thus, there will be a quantum of $\mathbbm{Z}_{2}$
invariant, $\nu=1$, pumped from one surface to another through the bulk
occupied bands during this process and all of the Bloch-Wannier bands will
become trivial. After this critical point, the gaps in the bulk Bloch-Wannier
bands reopen and the system will be a trivial insulator. We note that breaking
translation symmetry does not change our conclusions since the full
$\mathbb{Z}_{2}$ pump can still occur, but in perhaps a non-uniform fashion
where some of the bulk Bloch-Wannier bands become $\mathbb{Z}_{2}$ non-trivial
during an intermediate stage. We also point out that the $\mathbb{Z}_{2}$ pump
occurs in the $P$ subspace but not in the $Q$ subspace. This should be
contrasted with previously-studied pumps associated to a first or second Chern
number, where a Berry-curvature quantum is simultaneously pumped in both $P$
and $Q$ subspaces [23, 24].

To confirm our results we use the fact that the $\mathbb{Z}_{2}$ invariant can
be determined from spectrum of the Wilson loop [25, 26, 27]. Let
$p(\mathbf{k})$ be the projector onto a chosen group of bands at $\mathbf{k}$.
We can define the Wilson loop operator over this group of bands as

$\widehat{\mathcal{W}}=\prod_{\mathbf{k}}^{\mathscr{C}}p(\mathbf{k}),$ (8)

where $\prod^{\mathscr{C}}_{\mathbf{k}}$ is a path-ordered product over the
loop $\mathscr{C}$ in momentum space. The exponents of all unit modulus
complex eigenvalues of the Wilson loop operator are defined to be the spectrum
of the Wilson loop. Hereafter, we denote the spectrum of the Wilson loop to be
$\lambda_{W}$. To derive the Kane-Mele $\mathbb{Z}_{2}$ invariant of a given
group of bands, we consider $\mathscr{C}$ to be a non-contractible loop along
the $k_{y}$-direction, e.g., from $k_{y}=0$ to $k_{y}=2\pi$ at fixed values of
$k_{x}$. To determine the invariant we can use the parity of the number of
intersections between a line parallel to $\Gamma$-$X$ and the spectrum of the
Wilson loop: if the number of intersections is odd (even), then
$\mathbb{Z}_{2}$ invariant is one (zero) and thus nontrivial (trivial). Note
that the line parallel to $\Gamma$-$X$ can be arbitrarily chosen.

In Fig. 1 (b) and (c), we show numerical plots of the Wilson loop spectra
along $k_{y}$ (parallel transport along $k_{y}$-direction) for a projector
that includes both the occupied and unoccupied Bloch-Wannier bands localized
at the top surface (i.e., the layer closest to $z=10$) before and after the
phase transition point of the bulk Bloch-Wannier bands respectively.
Specifically, before (after) the phase transition point, the parallel
transport is performed on the surface-localized occupied and unoccupied Bloch-
Wannier bands indicated by the red colored states in Fig. 1 (d) (Fig. 1 (e)).
As shown in Fig. 1 (b), the blue reference line parallel to $\Gamma$-$X$
crosses the Wilson loop spectrum at one point indicated by a blue triangle.
Thus the surface $\mathbb{Z}_{2}$ invariant is non-trivial on the top surface
before the critical point. By comparison, in Fig. 1 (c), the blue reference
line parallel to $\Gamma$-$X$ crosses the Wilson loop spectrum at two points
indicated by the two blue triangles. Thus, in this case we have a trivial
surface $\mathbb{Z}_{2}$ invariant for the top surface after the critical
point. These results are consistent with our analysis above. More details
about the evolution of the surface $\mathbbm{Z}_{2}$ invariant are given in
the Appendix B.

While our symmetry-preserving interpolation trivializes our model, we note
that the topology can remain robust as long as the Bloch-Wannier band gap
remains open in addition to the bulk energy gap. Indeed, if the Bloch-Wannier
gap is kept open the $\mathbb{Z}_{2}$ pump will not occur, and the surface
$\mathbb{Z}_{2}$-invariant will be a robust indicator of the nontrivial
topology. This is similar to the boundary obstructed topology of some electric
multipole insulators [28, 29, 30, 31], in the sense that the topology is
protected by the bulk energy gap together with bulk Bloch-Wannier gap. In the
next subsection we will show that instead of requiring that the spectral gap
of the Bloch-Wannier bands is preserved, we can impose a symmetry that can
generically prevent the pump and stabilize the surface $\mathbb{Z}_{2}$
invariant. We also demonstrate the existence of stable helical hinge modes in
the spin Hopf insulator under this symmetry.

##  IV Symmetry protected surface $\mathbb{Z}_{2}$ topology and helical hinge
states

###  IV.1 Particle-hole inversion symmetry

In this section we will show that the surface $\mathbb{Z}_{2}$ invariant of
the TR-symmetric spin-Hopf insulator can be stabilized if we impose an
additional particle-hole times spatial inversion symmetry in each spin sector:
$\mathcal{C}^{\prime}c_{\bm{R},i,s}\mathcal{C}^{\prime-1}=\epsilon_{ij}c_{-\bm{R},j,s}^{{\dagger}}$,
where $i,j$ label the orbital degrees of freedom, $s$ labels the spin,
$c_{\bm{R},i,s}^{{\dagger}}$ creates an electron at $\bm{R}$, and
$\epsilon_{ij}$ is the two-dimensional Levi-Civita tensor. Note that this
symmetry is a spin-doubled version of that discussed in Ref. 5, and is related
to a symmetry used in Ref. 32 to extend the Hopf insulator to multiple bands.
For simplicity we shorten the name of the symmetry to $\mathcal{C}^{\prime}$
symmetry in the following. The $\mathcal{C}^{\prime}$ symmetry of the second-
quantized Hamiltonian constrains the first-quantized Bloch Hamiltonian to obey
an anti-symmetry:

$\displaystyle{\mathbbm{1}\sigma_{y}H^{\star}_{sh}(\bm{k})\mathbbm{1}\sigma_{y}=-H_{sh}(\bm{k}),}$
(9)

where ⋆ is the complex conjugate.

With this definition, let us demonstrate that the $\mathcal{C}^{\prime}$
symmetry can stabilize the surface $\mathbb{Z}_{2}$ invariant, i.e., any
continuous deformations that preserve the symmetries and bulk energy gap
_cannot_ change the surface $\mathbb{Z}_{2}$ invariant 222Note that the
particle-hole symmetry is broken by the adiabatic interpolation (see the
Appendix A) from the model Hamiltonian in Eq. (5) to a trivial Hamiltonian,
owing to a Hamiltonian term $tz_{1}\tau_{y}\sigma_{x}$ that is introduced in
the interpolation. . We note that the explicit interpolation we used in the
previous section violates $\mathcal{C}^{\prime}$-symmetry so it is no longer
valid. Instead of proving directly that no such interpolation can exist in the
presence of $\mathcal{C}^{\prime}$ symmetry, we will take a different approach
below to prove the stability of the spin Hopf phase protected by the addition
of $\mathcal{C}^{\prime}$.

Our intuition about the nature of the stability comes from the fact that if a
$\mathbb{Z}_{2}$ Kane-Mele quantum is pumped from the bottom facet to the top
through the bulk occupied Bloch-Wannier bands, the $\mathcal{C}^{\prime}$
symmetry enforces that a $\mathbb{Z}_{2}$ Kane-Mele quantum will also be
pumped from top to bottom through the bulk unoccupied Bloch-Wannier bands, and
hence the processes would cancel each other. Without loss of generality, let
us illustrate this idea in a spin-Hopf insulator with $z$-direction open, and
both $x,y$-directions periodic.

The $\mathcal{C}^{\prime}$ symmetry will enforce

$\nu_{v}(+\hat{z})=\nu_{c}(-\hat{z}),\ \nu_{c}(+\hat{z})=\nu_{v}(-\hat{z}),$
(10)

where we recall that $\nu_{v}(\hat{\bm{n}})$ ($\nu_{c}(\hat{\bm{n}})$) is
defined to be the Kane-Mele $\mathbbm{Z}_{2}$ invariant of boundary-localized
occupied (unoccupied) Bloch Wannier bands. Eq. (10) can be easily understood
from the Bloch-Wannier bands. If we have $\mathcal{C}^{\prime}$ symmetry then
$P\hat{Z}P=-\mathcal{C}^{\prime}Q\hat{Z}Q\mathcal{C}^{\prime-1}$. Thus, if
$|w_{z}^{v}(k_{x},k_{y})\rangle$ is an eigenstate of $P\hat{Z}P$ with
eigenvalue $z(k_{x},k_{y})$, then
$\mathcal{C}^{\prime-1}|w_{z}^{v}(k_{x},k_{y})\rangle=|w_{-z}^{c}(k_{x},k_{y})\rangle$
is an eigenstate of $Q\hat{Z}Q$ with eigenvalue $-z(k_{x},k_{y})$. Note that
the subscript $z$ and $-z$ indicates the corresponding eigenvalues of the
eigenstates. Since the Bloch-Wannier wave functions at $\pm z$ are related by
an antiunitary operation that maps $(k_{x},k_{y})\rightarrow(k_{x},k_{y})$,
they must have the same Kane-Mele index, meaning
$\nu_{v}(\pm\hat{z})=\nu_{c}(\mp\hat{z})$. A detailed proof following this
symmetry argument can be found in the Appendix C.

In terms of the quantities $\nu_{v}$ and $\nu_{c}$ discussed above, the total
surface $\mathbbm{Z}_{2}$ invariant of a surface facet with the normal vector,
e.g., $+\hat{z}$, is $\nu(+\hat{z})=\nu_{v}(+\hat{z})+\nu_{c}(+\hat{z})$.
Since the bulk Bloch-Wannier bands have trivial $\mathbb{Z}_{2}$ invariant,
the full slab $\mathbb{Z}_{2}$ invariant of the occupied (unoccupied)
subspace, $\nu_{v/c}$, is contributed by only the top and bottom surfaces,
i.e., $\nu_{v/c}=\nu_{v/c}(+\hat{z})+\nu_{v/c}(-\hat{z})\ \mathrm{mod}\ 2$.
Now, we are ready to prove our target conclusion: any continuous deformations
that preserve the symmetries and bulk gap _cannot_ change the surface
$\mathbb{Z}_{2}$ invariant. We separate our proof into two steps. First, if we
preserve both the bulk and surface gaps in the energy spectrum, $\nu_{v/c}$
cannot change. From Eq. (10), we can derive

$\displaystyle\nu_{v/c}=\nu_{v/c}(+\hat{z})+\nu_{v/c}(-\hat{z})$ (11)
$\displaystyle=\nu_{v/c}(+\hat{z})+\nu_{c/v}(+\hat{z})=\nu(+\hat{z})$
$\displaystyle=\nu_{c/v}(-\hat{z})+\nu_{v/c}(-\hat{z})=\nu(-\hat{z}).$

Thus, if $\nu_{v/c}$ remains invariant, so do both $\nu(+\hat{z})$ and the
$\nu(-\hat{z})$. Hence, $\mathcal{C}^{\prime}$ symmetry along with bulk and
surface gaps rule out a change of $\nu(+\hat{z})$ and $\nu(-\hat{z})$ due to a
$\mathbb{Z}_{2}$ pump that occurs at the gap closing point in the spectra of
$P\hat{Z}P$ and $Q\hat{Z}Q$. Second, we relax the requirement for a surface
energy gap. Since the surface $\mathbb{Z}_{2}$ invariant is the
$\mathbb{Z}_{2}$ Kane-Mele invariant over the rBZ of _all_ boundary-localized
Bloch-Wannier bands, it will not change even if the surface gap in the energy
spectrum closes and reopens since such a scenario, at worst, simply exchanges
the topology between the occupied and unoccupied subspaces.

In conclusion, even if the surface gap is not preserved, the surface
$\mathbbm{Z}_{2}$ invariants $\nu(\hat{z})$ and $\nu(-\hat{z})$, will never
change as long as the bulk gap, TR symmetry, and $\mathcal{C}^{\prime}$
symmetry are preserved. In other words, in the spin-Hopf insulator with the
particle-hole inversion symmetry and TR symmetry, the surface
$\mathbbm{Z}_{2}$ invariant is stable against any adiabatic deformations, and
can be a well-defined topological invariant. Our proof also indicates that the
surface $\mathbb{Z}_{2}$ topology is stable to the simultaneous addition of
momentum-independent Kramers pairs of bands (which would have trivial
$\mathbb{Z}_{2}$ topology everywhere) in the valence and conduction subspaces
such that the particle-hole inversion symmetry is preserved. This generalizes
the spin-Hopf insulator to multi-band cases. We note that the above proof
still works if we change $\hat{z}$ to any other surface normal vector
$\hat{\bm{n}}$.

Figure 2: (a) An illustration of the helical hinge modes for a TR-symmetric
spin-Hopf insulator with nontrivial surface $\mathbbm{Z}_{2}$ invariant which
has $\mathcal{C}^{\prime}$ symmetry. (b) is the energy spectrum for the spin-
Hopf insulator in the same geometric configuration as (a) which has open
boundaries in $y,z$ and periodic in $x$. The red-colored (blue-colored) in-gap
helical modes is (are) localized on the red-colored (blue-colored) hinge(s) in
(a). Green-colored hinges also host hinge modes which are, however, not shown
in this energy-momentum plot. We used a lattice having 10 unit cells along the
$z$-direction and 40 unit cells along the $y$-direction.

###  IV.2 Stable helical hinge modes

We have now shown that imposing $\mathcal{C}^{\prime}$ symmetry on a TR-
symmetric spin Hopf insulator, or just requiring the gap of the bulk Bloch-
Wannier bands to be preserved, stabilizes the surface $\mathbb{Z}_{2}$
invariant. An important implication of the stable surface $\mathbb{Z}_{2}$
invariant is that there could be gapless helical modes on the hinges where two
facets with distinct surface valence $\mathbb{Z}_{2}$ invariants meet. It is
then natural to ask if these helical modes are stable in the presence of
surface perturbations.

If we only have TR symmetry, even if we require the bulk Bloch Wannier gap to
be preserved, we find that the helical hinge modes are not stable. That is,
they can be removed by adding certain surface perturbations to make all
surface facets have the same surface valence $\mathbb{Z}_{2}$ invariant (while
preserving the energy gap and the gap of the bulk Bloch-Wannier bands). This
is because the surface valence $\mathbb{Z}_{2}$ invariant can be changed by
perturbations on the surface facets that close and reopen the _surface_ energy
gap. It is only the surface $\mathbb{Z}_{2}$ invariant of both the occupied
and unoccupied bands combined that is fixed.

In contrast, the particle-hole inversion symmetry can stabilize gapless
helical hinge modes by ensuring the existence of domain walls on the closed
two-dimensional surface of our 3D system. Recall from Eq. (10) that the
symmetry enforces $\nu_{v}(\hat{\bm{n}})=\nu_{c}(-\hat{\bm{n}})$ for all
$\hat{\bm{n}}$. Then, for spin Hopf insulators with $\nu(\hat{\bm{n}})=1$ for
all $\hat{\bm{n}}$ [cf. Eq. (5)], we have

$\nu_{v}(\hat{\bm{n}})-\nu_{v}(-\hat{\bm{n}})=1\ \rm{mod}\ 2.$ (12)

Thus, there must be a domain wall between distinct Kane-Mele $\mathbbm{Z}_{2}$
invariants on the two-dimensional surface of the spin-Hopf insulator, and this
domain wall will harbor gapless helical hinge modes. As a representative
illustration, Fig. 2 (a) shows a pattern of helical hinge modes for a
particular, symmetric surface termination of our model (whose bulk Hamiltonian
is given in Eq. (5)). In Fig. 2 (b) we show the corresponding energy spectrum
for our model where periodic boundary conditions have been imposed in the
$x$-direction, and open boundaries in both the $y,z$ directions such that
there are four surface facets with normals in the $\pm\hat{y}$ and
$\pm\hat{z}$ directions. Perturbations can change the spatial pattern of
helical hinge states, but cannot remove them entirely.

##  V Returning Thouless pump protected by a four-fold rotation symmetry

###  V.1 Theory for the Returning Thouless Pump

After discussing the nontrivial surface $\mathbb{Z}_{2}$ topology and helical
hinge states in the spin Hopf insulator, in this Section we discuss the
symmetry protected returning Thouless pump (RTP), of the spin Hopf insulator
that is stabilized by a four-fold rotation symmetry.

Let us first review the general theory of the RTP in $n$-fold rotation
symmetric insulators [2]. Without loss of generality, let us focus on the
$n$-fold rotation symmetry along the $z$-direction ($C_{nz}$), and denote the
fixed points of the $C_{nz}$ rotation in the $k_{x}-k_{y}$ plane as
$\bm{\Lambda}$, i.e., $C_{nz}\bm{\Lambda}=\bm{\Lambda}$ up to a reciprocal
lattice vector. At each $\bm{\Lambda}$, the Bloch Hamiltonian represents a
gapped 1D Hamiltonian along the $z$-direction $H_{\bm{\Lambda}}(k_{z})$. For
convenience, we always choose a basis in which $C_{nz}$ is diagonal so that at
each $\bm{\Lambda}$, $H_{\bm{\Lambda}}(k_{z})$ is always block diagonal, and
the blocks are labelled by the eigenvalues of $C_{nz}$. Given that $C_{nz}$ is
a unitary operator, the eigenvalues of $C_{nz}$ generally take the form of
$e^{i2\pi l/n}$, where $l$ is the angular momentum.

Now we can calculate the charge polarization along the $z$-direction
contributed by all eigenstates having angular momentum $l$ of the 1D Bloch
Hamiltonian at momentum $\bm{\Lambda}$, and denote it as
$\mathscr{P}_{l}(\bm{\Lambda})$. Since we are considering the polarization of
_all_ of the energy bands (both occupied and unoccupied) the Hilbert space
spanned by eigenstates of $H_{\bm{\Lambda}}(k_{z})$ with angular momentum $l$
is independent of $\bm{\Lambda}$, because it is always the full Hilbert space
spanned by basis orbitals with angular momentum $l$ in each unit cell.
Therefore, the difference of $\mathscr{P}_{l}$ between a pair of
$\bm{\Lambda}$’s is always quantized to be an integer:

$\Delta_{12}\mathscr{P}_{l}\equiv\mathscr{P}_{l}(\bm{\Lambda}_{1})-\mathscr{P}_{l}(\bm{\Lambda}_{2})=m,\
m\in\mathbb{Z},$ (13)

where $\bm{\Lambda}_{1}$ and $\bm{\Lambda}_{2}$ are two different rotation
invariant points. A quantized difference $\Delta_{12}\mathcal{P}$ indicates
that if we go from $\bm{\Lambda}_{1}$ to $\bm{\Lambda}_{2}$, the polarization
along the $z$-direction continuously changes by an integer. Then if we go from
$\bm{\Lambda}_{2}$ to $\bm{\Lambda}_{1}$ by completing a noncontractible loop
across the BZ, the polarization returns back to its original value. Hence, in
analogy with the Thouless pump where the polarization changes by an integer
over a loop, we say that in this case there is a RTP with value $m$ in sector
$l$ along the loop $\bm{\Lambda}_{1}\to\bm{\Lambda}_{2}\to\bm{\Lambda}_{1}$ if
$\Delta_{12}\mathscr{P}_{l}=m$.

Let us consider a special case where the eigenstates at rotation invariant
points with a given angular momentum $l$ are either all in the occupied
subspace or all in the unoccupied subspace (this is the mutually disjoint
condition defined in Ref. 2), then the polarization of all the occupied bands
can be expressed as

$\mathscr{P}_{v}(\bm{\Lambda})=\sum_{l\in
l_{v}}\mathscr{P}_{l}(\bm{\Lambda}),$ (14)

where $l_{v}$ indicates the set of all possible angular momenta in the
occupied subspace. Using Eqs. (13) and (14), it is straightforward to see that
the difference of $\mathcal{P}_{v}$ between a pair of rotation invariant
points is also quantized to be an integer. This quantized difference cannot be
changed as long as the bulk energy gap and the $C_{nz}$ symmetry are
preserved. Similarly, under these conditions we say there is an RTP with value
$m$ in the occupied valence subspace along the loop
$\bm{\Lambda}_{1}\to\bm{\Lambda}_{2}\to\bm{\Lambda}_{1}$ if
$\Delta_{12}\mathscr{P}_{v}=m.$

Now that we have finished reviewing the general theory of the RTP, we are
ready to discuss the RTP for the spin-Hopf insulator. Although we did not
emphasize it, Eq. (5) already has a four-fold rotation symmetry along the
$z$-direction, for which the operator is
$C_{4z}=\exp\left(i\frac{\pi}{2}\tau_{z}(\frac{\sigma_{z}}{2}+\mathbbm{1})\right)$.
The two valence (conduction) eigenbands have $C_{4z}$ eigenvalues $\exp(\pm
i3\pi/4)$ ($\exp(\pm i\pi/4)$) respectively, at both $C_{4z}$ symmetric points
$\Gamma$ and $M$. This corresponds to $l_{v}=\\{3/2,-3/2\\}$
($l_{c}=\\{1/2,-1/2\\}$). Clearly, the mutually disjoint condition is
satisfied because $l_{v}\cap l_{c}=\emptyset$. As shown in Fig. 3 (a), the RTP
data in different angular momentum sectors are $\Delta_{\Gamma
M}\mathscr{P}_{\pm 3/2}=-1$ and $\Delta_{\Gamma M}\mathscr{P}_{\pm 1/2}=+1$.
Thus, the RTP in the valence subspace has a value $\Delta_{\Gamma
M}\mathscr{P}_{v}=-2$. Hence, the nontrivial polarization difference at
$\Gamma$ and $M$ leads to a RTP, which is robust to perturbations and
deformations (e.g., introducing couplings between two spin sectors) as long as
we preserve the $C_{4z}$ symmetry and the bulk energy gap.

Figure 3: (a) An RTP along $M-\Gamma-M$ for valence bands (left) with $l=\pm
3/2$ and conduction bands (right) with $l=\pm 1/2$. (b) schematically
illustration of Bloch-Wannier states protruding from the bulk to the surface
(colored red for $l=\pm 3/2$ and blue for $l=\pm 1/2$ ). Since the Hilbert
space in each layer is fixed there is are compensating surface states (SS)
(represented by a green line and an orange line) with the compensating angular
momenta at $\Gamma$ and $M$. (c) shows the spectrum of the model with a sharp
termination where in-gap SS localized on the top surface, i.e., $z=N$
(highlighted by the green color), and on the bottom surface, i.e., $z=1$
(highlighted by the orange color) are shown.

###  V.2 Implications of the RTP for Gapless surface states on sharp
terminations

Delicate TIs with a nonzero RTP exhibit an unusual bulk-boundary
correspondence as discussed in Ref. 8. As an example of this, in this
subsection, we will discuss gapless surface states (SS) on sharp terminations
of the spin-Hopf model. By a sharp termination we mean a Hamiltonian that is
nearly identical to the corresponding Hamiltonian under periodic boundary
conditions, except that all hopping matrix elements across the terminations
are turned off.

In a $C_{nz}$ symmetric system with sharp terminations along the
$z$-direction, each diagonal block $H_{l}(\bm{\Lambda})$ of the effective 1D
tight-binding Hamiltonian at a $C_{nz}$ invariant $\bm{\Lambda}$ is a block
Toeplitz matrix, i.e., the hopping matrix elements between unit cells at
$\mathbf{R}$ and $\mathbf{R}^{\prime}$ depend on only
$\mathbf{R}-\mathbf{R}^{\prime}$ and satisfy
$H_{l,\mathbf{R}\mathbf{R}^{\prime}}\equiv
H_{l,\mathbf{R}-\mathbf{R}^{\prime}}=H^{{\dagger}}_{l,\mathbf{R}^{\prime}-\mathbf{R}}$.
The spectrum theorem of block Toeplitz matrices tells us that in the
thermodynamic limit, the spectrum of $H_{l}(\bm{\Lambda})$ is bounded by the
spectrum of its corresponding Bloch Hamiltonian $H_{l}(k_{z},\bm{\Lambda})$
[8].

For our spin-Hopf insulator model (which obeys the mutual disjoint condition)
with sharp terminations, the above conclusion means that at each $C_{nz}$
invariant $\bm{\Lambda}$, any SS with angular momentum $l\in l_{v}$ ($l\in
l_{c}$) must lie in the bulk valence (conduction) bands. Let us now focus on
the top surface. As shown in Fig. 3 (a), the bulk RTP indicates that occupied
(unoccupied) states at $M$ ($\Gamma$) will localize one layer higher along the
$z$-direction than the occupied (unoccupied) states at $\Gamma$ ($M$). Thus,
on the top surface, i.e., the layer with $z=N$ as shown in Fig. 3 (b), there
will be an extra pair of occupied (unoccupied) states with angular momenta
$+3/2,-3/2$ ($+1/2,-1/2$) at $M$ ($\Gamma$) due to the bulk RTP. In each layer
along the $z$-direction, there should always be four states in total having
angular momenta $\pm 1/2,\pm 3/2$ at both $\Gamma$ and $M$ since the Hilbert
space in each layer is spanned by the four basis orbitals 333 As we can see
from Fig. 3 (b), this condition is satisfied at each bulk layer (i.e., layers
with $z=2,3,\ldots,N-1$) . Thus, the extra states from the bulk RTP at $M$
($\Gamma$) with angular momenta $\pm 3/2$ ($\pm 1/2$) should be compensated by
a pair of SS with angular momenta $\pm 1/2\in l_{c}$ ($\pm 3/2\in l_{v}$), as
illustrated in Fig. 3 (b). This means that the SS at $\Gamma$ ($M$) must lie
in the bulk valence (conduction) bands in the energy spectrum, which
guarantees the energy of SS must cross the bulk gap as one traverses from
$\Gamma$ to $M$ as shown in Fig. 3 (c). The same arguments can also be made
for the bottom surface to imply the existence of SS on sharp terminations.

###  V.3 Dirac dipole at topological phase transition point

Finally, in this subsection we will consider the nature of the critical point
separating a topological spin Hopf insulator and a trivial insulator. We will
identify the topological phase transition point of the spin Hopf model to be a
novel band degeneracy that we call a Dirac dipole in analogy with previous
work on the conventional Hopf insulator. Indeed, as discussed in Refs. 3, 8,
at the topological phase transition point between the trivial phase and the
Hopf/RTP insulator in the MRW model [c.f. Eq. (3)], the low-energy
$\mathbf{k}\cdot\mathbf{p}$ expansion around $\mathbf{k}=0$ yields a Berry
dipole. To understand why this is called a Berry dipole we can consider a
sphere in momentum space that surrounds $\mathbf{k}=0$ where such a band
degeneracy appears at the topological transition point $u=3$. Then, over the
upper (lower) hemisphere the integral of the Berry curvature defined over the
occupied subspace is quantized to be $2\pi$ ($-2\pi$). This has the structure
of a dipole with opposite charges in opposite hemispheres.

For convenience one can explicitly correlate the Berry curvature calculations
with the winding of the Wilson loop spectrum over the hemisphere. For example,
as shown in Fig. 4 (a), at each latitude on the upper hemisphere, we construct
the Wilson loop operator along the warp $\mathscr{C}_{w}$, i.e., we take the
path-ordered product of the projectors $p_{v}(\mathbf{k})$ onto the occupied
subspace as shown in Eq. (8). Explicitly, let
$p_{v}(\bm{k})=\sum_{j=1}^{n_{occ}}|u_{j}(\mathbf{k})\rangle\langle
u_{j}\mathbf{k})|$ where $|u_{j}(\mathbf{k})\rangle$ is the $j$-th eigenstate
of the Bloch Hamiltonian, and $n_{occ}$ is the number of occupied bands. In
Fig. 4 (b), we plot the Wilson loop spectra of the MRW model (with $u=3$)
versus the latitude from the north pole to the equator. We observe that the
argument of the unit-modulus eigenvalue of $\hat{\mathcal{W}}_{w}$ changes by
$+2\pi$, and thus there is a positive winding of Wilson loop spectrum over the
hemisphere, which implies a $+2\pi$-quantized Berry flux over the hemisphere.
This winding flips sign if we repeat this process over the southern hemisphere
moving from the south pole to the equator. Note that since the MRW model is a
two-band model with a single occupied band, each $\hat{\mathcal{W}}_{w}$ has
only one unit-modulus eigenvalue. Since a Weyl point would have a $2\pi$
quantized Berry flux if we integrate over whole sphere surrounding it, the
degeneracy of the Hopf insulator critical point is called a Berry dipole.
Interestingly, the Berry dipole in the MRW model is protected by an emergent
mirror symmetry along the $z$-direction [8, 35].

Since the spin Hopf model is a spin-doubled version of the MRW model, at the
topological transition point $u=3$ there is a 4-fold band degeneracy at
$\mathbf{k}=0$. For this degneracy we expect to see a positive winding of the
upper hemisphere Wilson loop spectrum in one spin sector, but a negative
winding in the other spin sector [see Fig. 4 (c)]. For the spin Hopf model,
even though the net winding (i.e., the sum of the windings of the two unit-
modulus eigenvalues) is zero, there is a nontrivial relative winding in the
Wilson loop spectrum, i.e., the difference of the windings of the two unit-
modulus eigenvalues is nonzero. As a TR invariant analog of a Berry dipole, we
call this novel band degeneracy a Dirac dipole because a single Dirac point
degeneracy has a relative winding in the Wilson loop spectrum over the whole
sphere surrounding it. In Appendix D, we prove that this Dirac dipole, or
equivalently, the relative winding in the Wilson loop spectrum over a
hemisphere of the sphere surrounding the band degeneracy point 444This means
that we always assume the band degeneracy is preserved when discussing the
stableness of the relative winding, i.e., we do not consider perturbations
that lift the band degeneracy, is stabilized by TR, four-fold rotation, and
emergent inversion symmetries even when there are couplings between the two
spin sectors.

Figure 4: (a) A sphere in momentum space surrounding the critical point at
$\mathbf{k}=0$. Wilson loop spectra are calculated on the lattitude warps
$\mathscr{C}_{w}.$ For MRW and spin-Hopf Hamiltonians with $u=3$, we plotted
the spectra of Wilson loops along warps over the upper hemisphere versus the
latitude (direction from north pole to equator is indicated by the black
arrow) in (b) and (c). These spectra will be flipped when repeating the
calculation from the equator to the south pole.

##  VI Conclusion and remark

In conclusion, our work shows that the spin-Hopf insulator is a novel phase
with a surface $\mathbb{Z}_{2}$ Kane Mele order. This topology has a higher-
order character that is manifested by the helical hinge states in the presence
of particle-hole inversion symmetry. With four-fold rotation symmetry, the
spin Hopf insulator is the first concrete model for delicate class-AII
topological insulators that have nonzero RTP and associated gapless states on
sharp boundaries. In addition, we have also explored the novel Dirac dipole
band degeneracy at the phase transition point between a spin-Hopf insulator
and a trivial insulator.

Besides the phenomena of the spin-Hopf model discussed in this work, there are
also some open questions for future research. (i) Though the topology of spin
Hopf insulator with TR symmetry and particle-hole inversion symmetry can be
diagnosed by the surface $\mathbb{Z}_{2}$ invariant, it is unclear whether
there is a corresponding bulk topological invariant, i.e., whether the spin
Hopf insulator is bulk obstructed or boundary obstructed is still unclear.
(ii) We have mentioned that adding momentum-independent bands into the system
while preserving the TR and particle-hole inversion symmetry will not destroy
the surface $\mathbb{Z}_{2}$ invariant, and thus generalizes the spin-Hopf
model to larger numbers of bands. However, it is unclear if there is a
homotopic invariant for larger band spin-Hopf insulators like that for the
larger band Hopf insulator protected by particle-hole inversion symmetry
discussed in Ref. 32. (iii) Recently, Ref. 37 has discussed the potential for
a large transverse photovoltaic effect in class A delicate TIs with nonzero
RTP. The photovoltaic effect (both transverse and longitudinal) and related
phenomena in rotation symmetric class AII delicate TIs is still an open
question. We leave these questions for future research.

## Acknowledgements

P.Z and T.L.H. thank A. Alexandradinata for launching this project, and thank
A. Alexandradinata and Tomáš Bzdušek for insightful discussions. P.Z and
T.L.H. thank the US Office of Naval Research (ONR) Multidisciplinary
University Research Initiative (MURI) grant N00014-20- 1-2325 on Robust
Photonic Materials with High-Order Topological Protection for support.

##  Appendix A Adiabatic path connecting the spin Hopf insulator to a trivial
band insulator without breaking the time-reversal symmetry

Here, we explicitly construct the adiabatic path via three steps, and each of
them can be described by a Bloch Hamiltonian with an adiabatic parameter. The
first step is

$\displaystyle
H_{1}(\mathbf{k},t)=2(1-t)z_{1}z_{3}\mathbbm{1}\sigma_{x}+2z_{2}z_{4}\tau_{z}\sigma_{x}-2(1-t)z_{2}z_{3}\tau_{z}\sigma_{y}+2z_{1}z_{4}\mathbbm{1}\sigma_{y}$
(S1)
$\displaystyle+\left(z_{1}^{2}+z_{2}^{2}-(1-t)^{2}z_{3}^{2}-z_{4}^{2}+t^{2}z_{3}^{2}\right)\mathbbm{1}\sigma_{z}+2t(1-t)z_{3}^{2}\tau_{x}\sigma_{y}-2tz_{4}z_{3}\tau_{x}\sigma_{x},$

where $t\in\left[0,1\right]$ and $H_{1}(\mathbf{k},0)=H_{sh}(\mathbf{k})$. The
energy spectrum varies with $t$ as
$E_{1}(\mathbf{k},t)=\pm\big{(}z_{1}^{2}+z_{2}^{2}+(1-t)^{2}z_{3}^{2}+z_{4}^{2}+t^{2}z_{3}^{2}\big{)}$,
and we note that the bands with negative and positive energies are both two-
fold degenerate. When $1<u<3$, $z_{1}$, $z_{2}$, $z_{3}$ and $z_{4}$ cannot be
zero simultaneously. This is because if we want $z_{1}$, $z_{2}$ and $z_{3}$
to be zero, $k_{x}$, $k_{y}$ and $k_{z}$ must be $0$ or $\pi$, then only when
$u=3,1,-3,-1$ can we have $z_{4}=0$. Thus, the bulk gap will never close for
$t\in\left[0,1\right]$ if $1<u<3$. This step tunes the Bloch Hamiltonian so
that at $t=1$, all the four by four matrices in the Bloch Hamiltonian
$H_{1}(\bm{k},1)$ anti-commute with each other.

The second step is

$\displaystyle
H_{2}(\mathbf{k},t)=\left(2(1-t)z_{1}z_{4}+tz_{2}\right)\mathbbm{1}\sigma_{y}+tz_{1}\tau_{y}\sigma_{x}+\left(2(1-t)z_{2}z_{4}+tz_{3}\right)\tau_{z}\sigma_{x}-2(1-t)z_{4}z_{3}\tau_{x}\sigma_{x}$
(S2)
$\displaystyle+\left((1-t)^{2}z_{1}^{2}+(1-t)^{2}z_{2}^{2}-z_{4}^{2}+(1-t)^{2}z_{3}^{2}\right)\mathbbm{1}\sigma_{z},$

where $t\in\left[0,1\right]$ and $H_{2}(\mathbf{k},0)=H_{1}(\mathbf{k},1)$.
Since all five four by four matrices in the above Hamiltonian anti-commute
with each other, the energy gap closes if and only if all five coefficients of
the five matrices are simultaneously zero. Let us first consider $t=0,1$. At
$t=0$, the energy spectrum is given by
$E_{2}(\mathbf{k},0)=\pm(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2})$, and hence
the gap cannot close because $z_{i}$ for $i=1,2,3,4$ cannot be zero
simultaneously. At $t=1$,
$E_{2}(\mathbf{k},1)=\pm(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{4})$, so the gap
cannot close for the same reason. Then, let us focus on $0<t<1$. If we want
the second term and the fourth term to be zero, then we either have
$z_{1}=z_{4}=0$ or $z_{1}=z_{3}=0$. If $z_{1}=z_{4}=0$, then the factor in the
fifth term becomes $(1-t)^{2}(z_{2}^{2}+z_{3}^{2})$ which equals zero only
when $z_{2}=z_{3}=0$. This is forbidden because $z_{i}$ for $i=1,2,3,4$ cannot
be zero simultaneously when $1<u<3$. If $z_{1}=z_{3}=0$, then the factor in
the first term becomes $tz_{2}$ which equals zero only when $z_{2}=0$. By
substituting $z_{1}=z_{2}=z_{3}=0$ into the fifth term, the factor becomes
$-z_{4}^{2}$ which equals zero only when $z_{4}=0$. Again, this cannot happen
because $z_{i}$ for $i=1,2,3,4$ cannot be zero simultaneously when $1<u<3$.
Thus, we have proved that the energy gap remains open during this step.

Finally, the last step is

$\displaystyle
H_{3}(\mathbf{k},t)=(-z_{4}^{2}-tz_{3}^{2}-tz_{2}^{2}-tz_{1}^{2})\mathbbm{1}\sigma_{z}+(1-t)z_{1}\tau_{y}\sigma_{x}+(1-t)z_{2}\mathbbm{1}\sigma_{y}+(1-t)z_{3}\tau_{z}\sigma_{x},$
(S3)

where again $t\in\left[0,1\right]$ and
$H_{3}(\mathbf{k},0)=H_{2}(\mathbf{k},1)$. For the same reason, the gap will
never close in this step. As we can see,
$H_{3}(\mathbf{k},1)=M\mathbbm{1}\sigma_{z}$, with mass
$M=-\sum_{i=1}^{4}z_{i}^{2}<0$ for all $\mathbf{k}$ in the BZ. Thus, it is
just a trivial band insulator. One can check explicitly that all these time-
dependent Hamiltonians are symmetric under the TR symmetry
$T=-i\tau_{y}\mathbbm{1}K$.

In conclusion, this specific path indeed adiabatically connects the spin Hopf
insulator with nontrivial surface $\mathbb{Z}_{2}$ number, to a trivial band
insulator with trivial surface $\mathbb{Z}_{2}$ number without breaking the TR
symmetry $T=-i\tau_{y}\mathbbm{1}K$.

##  Appendix B Evolution of surface $\mathbbm{Z}_{2}$ invariant during our
adiabatic path

We numerically studied the evolution of the energy spectrum and the surface
$\mathbbm{Z}_{2}$ invariant for insulators described by $H_{1}$, $H_{2}$, and
$H_{3}$ with the $z$-direction open and the $x$, $y$-directions periodic.
During the first step, i.e., $H_{1}$, the nontrivial $\mathbbm{Z}_{2}$
invariant on the top surface is in boundary-localized occupied bands, and the
nontrivial $\mathbbm{Z}_{2}$ invariant on the bottom surface is in boundary-
localized unoccupied bands bands as shown in Fig. S1. At
$H_{1}(t=1)=H_{2}(t=0)$, the surface gap closes on the top surface, and a
phase transition occurs between the boundary-localized occupied bands and
unoccupied bands on the top surface. Then, for $H_{2}(t)$ with $0<t<0.33$, the
nontrivial $\mathbbm{Z}_{2}$ invariants are in boundary-localized unoccupied
bands for both top and bottom surfaces. At $H_{2}(t=0.33)$ and
$H_{2}(t=0.41)$, the surface closes along $\rm{X}-\Gamma-\rm{X}$ on the top
and bottom surface subsequently. Again, phase transition happens at each of
these two points, and we have the nontrivial $\mathbbm{Z}_{2}$ invariants in
boundary-localized occupied bands for both top and bottom surfaces in
$H_{2}(t)$ with $0.41<t<0.43$. Next, as discussed in the main text, the Bloch-
Wannier bands touch with their neighbors at $H_{2}(t\approx 0.43)$, and pump a
quantum of $\mathbbm{Z}_{2}$ invariant $\nu=1$ from one surface to another.
Since this $\mathbbm{Z}_{2}$ pump only occurs in the occupied subspace in our
adiabatic path, it leads to the removal of the nontrivial surface
$\mathbb{Z}_{2}$ invariant.

Figure S1: (a) and (c) are Bloch-Wannier eigenbands of $Q\hat{z}Q$ and
$P\hat{z}P$ for $H_{1}(t=0.5)$ with 10 unit cells along the $z$-direction,
respectively. (b) shows the spectrum of Wilson loop along the
$k_{y}$-direction (and plotted vs $k_{x}$) for Bloch-Wannier eigenbands
localized at $z=1$ (marked as red in (a)) of $Q\hat{z}Q$; (d) shows the
spectrum of Wilson loop along the $k_{y}$-direction (and plotted vs $k_{x}$)
for Bloch-Wannier eigenbands localized at $z=10$ (marked as blue in (c)) of
$P\hat{z}P$.

##  Appendix C Proof of Eq. (10) using the Wilson loop of the Bloch-Wannier
bands

According to Eq. (8), we can define and numerically calculate the Wilson loop
operator for the Bloch-Wannier bands. We denote the Wilson loop operator along
the $k_{y}$-direction (parallel transport in the $k_{y}$-direction, i.e.,
$\mathscr{C}$: $(k_{x},k_{y}=0)\rightarrow(k_{x},k_{y}=2\pi)$) of the occupied
Bloch-Wannier bands in the top half of the slab (derived by diagonalizing
$P\hat{Z}P$) as $\hat{\mathcal{W}}_{v,t,y}(k_{x})$, such that

$\hat{\mathcal{W}}_{v,t,y}(k_{x})=\prod_{k_{y}}^{2\pi\leftarrow
0}\sum_{z>0}|w_{z}^{v}\rangle\langle w_{z}^{v}|,$ (S4)

where $z>0$ characterizes all Bloch-Wannier bands in the top half of the slab,
and $\prod_{k_{y}}^{2\pi\leftarrow 0}$ denotes a path-ordered product along
$k_{y}:0\rightarrow 2\pi$. Similarly, we denote the Wilson loop operator along
the $k_{y}$-direction of the unoccupied Bloch-Wannier bands in the bottom half
of the slab (derived by diagonalizing $Q\hat{Z}Q$) as
$\hat{\mathcal{W}}_{c,b,y}(k_{x})$, such that

$\hat{\mathcal{W}}_{c,b,y}(k_{x})=\prod_{k_{y}}^{2\pi\leftarrow
0}\sum_{z<0}|w_{z}^{c}\rangle\langle w_{z}^{c}|,$ (S5)

where $z<0$ characterizes all Bloch-Wannier bands in the bottom half of the
slab.

Then, the relation
$\mathcal{C}^{\prime-1}|w_{z}^{v}(k_{x},k_{y})\rangle=|w_{z}^{c}(k_{x},k_{y})\rangle$
discussed in the main text implies $\hat{\mathcal{W}}_{v,t,y}(k_{x})$ is
unitarily equivalent to $\hat{\mathcal{W}}_{c,b,y}^{\star}(k_{x})$, where ⋆ is
the complex conjugate due to the complex conjugate operator $K$ in the
particle-hole inversion operator. As mentioned in the main text, it is known
that the $\mathbb{Z}_{2}$ Kane-Mele invariant can be derived by counting the
parity of the number of intersections between a line parallel to the
$k_{x}$-axis and the spectrum of the Wilson loop $\lambda_{W}(k_{x})$ [25, 26,
27]. Let the spectrum of $\hat{\mathcal{W}}_{v,t,y}(k_{x})$,
$\lambda_{W_{vt}}(k_{x})$, cross an arbitrary line parallel to the
$k_{x}$-axis in the region $\Gamma$-$X$ an odd number of times and thus have
$\nu_{v}(\hat{z})=1\ \rm{mod}\ 2$. Then, given the unitary equivalence between
$\hat{\mathcal{W}}_{v,t,y}(k_{x})$ and
$\hat{\mathcal{W}}^{\star}_{c,b,y}(k_{x})$, the spectrum of
$\hat{\mathcal{W}}_{c,b,y}(k_{x})$,
$\lambda_{W_{cb}}(k_{x})=\lambda^{\star}_{W_{vt}}(k_{x})$, must also cross
such a line an odd number of times and thus have $\nu_{c}(-\hat{z})=1\
\rm{mod}\ 2=\nu_{v}(+\hat{z})$. By a similar procedure, we can also get
$\nu_{v}(-\hat{z})=\nu_{c}(+\hat{z})$.

##  Appendix D Symmetry protected Dirac dipole

To discuss the Dirac dipole protected by TR, four-fold rotation, and emergent
inversion symmetry, we first write down the $\mathbf{k}\cdot\mathbf{p}$
expansion of the spin Hopf model at transition point $u=3$ around the
$\mathbf{k}=0$ point:

$\displaystyle
H_{\Gamma}(\mathbf{k})=2k_{x}k_{z}\mathbbm{1}\sigma_{x}-2k_{y}k_{z}\tau_{z}\sigma_{y}+\left(k_{x}^{2}+k_{y}^{2}-k_{z}^{2}\right)\mathbbm{1}\sigma_{z},$
(S6)

which has an emergent inversion symmetry
$\mathcal{I}H_{\Gamma}(\mathbf{k})\mathcal{I}^{-1}=H_{\Gamma}(-\mathbf{k})$,
where $\mathcal{I}=\mathbbm{1}_{4\times 4}$.

To show that the Dirac dipole in the spin-Hopf model is robust, we first
analyze the symmetry constraints on the Wilson loop operator along a warp on
the sphere surrounding $\mathbf{k}=0$ point [see Fig. 4 (a) in the main text].
Without loss of generality, let us consider a warp at a given $k_{z}$, and set
the starting point for the Wilson loop operator to be $\mathbf{k}_{0}$. Then,
the Wilson loop operator is

$\hat{\mathcal{W}}_{w}=P(\mathbf{k_{0}})\ldots
P(\mathbf{k_{0}}+2\delta\mathbf{k})P(\mathbf{k_{0}}+\delta\mathbf{k})P(\mathbf{k_{0}}).$
(S7)

A time reversal ($T=\tau K$, $T^{2}=-1$) times inversion symmetry
($\mathcal{I}$) acts on this Wilson loop operator as

$T\mathcal{I}\hat{\mathcal{W}}_{w}(T\mathcal{I})^{-1}=\tau\mathcal{I}^{\star}\hat{\mathcal{W}}^{\star}_{w}(\tau\mathcal{I}^{\star})^{-1}=\hat{\mathcal{W}}_{w},$
(S8)

because $T\mathcal{I}P(\mathbf{k})(T\mathcal{I})^{-1}=P(\mathbf{k})$. This
means that $\hat{\mathcal{W}}^{\star}_{w}$ and $\hat{\mathcal{W}}_{w}$ are
similar to each other, and thus they have the same spectrum. Then, for each
eigenvalue of $\hat{\mathcal{W}}_{w}$, its complex conjugate must also be an
eigenvalue of $\hat{\mathcal{W}}_{w}$. Therefore, the unit-modulus eigenvalues
of $\hat{\mathcal{W}}_{w}$ can only be $\pm 1$ or appear as a pair of complex
conjugate numbers. This constraint guarantees that the spectrum of Wilson loop
along a warp is always symmetric with respect to $\lambda_{w}=0,\pi$, and thus
the degeneracy in the spectrum can only happen at $\lambda_{w}=0,\pi$ . If we
furthermore show that the degeneracy at $\lambda_{w}=0,\pi$ in Fig. 4 (c)
cannot be lifted through hybridizing, then we can prove the relative winding
is robust.

Next, we show that the four-fold rotation symmetry can protect the degeneracy
at $\lambda_{w}=0,\pi$. From the four-fold rotation symmetry $C_{4z}$, we can
construct an operator $O$ such that $O^{4}=-\hat{\mathcal{W}}_{w}^{{\dagger}}$
and thus $[O,\hat{\mathcal{W}}^{{\dagger}}_{w}]=0$. Explicitly, the operator
$O$ takes the form

$O(\mathbf{k}_{0})=\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z},$
(S9)

where $\mathbf{k}_{0}$ is the starting point of the Wilson loop operator on a
given warp, and $R_{4}$ rotates the momentum by $\pi/2$, and
$O^{4}=-\hat{\mathcal{W}}_{w}^{{\dagger}}$ can be proved through the basic
relationship $C_{4z}P(\mathbf{k})C_{4z}=P(R_{4}\mathbf{k})$:

$\displaystyle O^{4}(\mathbf{k_{0}})$
$\displaystyle=\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}$
(S10)
$\displaystyle=\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)\left(\prod_{\mathbf{k}=R^{2}_{4}\mathbf{k}_{0}}^{R_{4}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C^{2}_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}$
$\displaystyle=\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)\left(\prod_{\mathbf{k}=R^{2}_{4}\mathbf{k}_{0}}^{R_{4}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)\left(\prod_{\mathbf{k}=R^{3}_{4}\mathbf{k}_{0}}^{R^{2}_{4}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C^{3}_{4z}\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C_{4z}$
$\displaystyle=\left(\prod_{\mathbf{k}=R_{4}\mathbf{k}_{0}}^{\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)\left(\prod_{\mathbf{k}=R^{2}_{4}\mathbf{k}_{0}}^{R_{4}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)\left(\prod_{\mathbf{k}=R^{3}_{4}\mathbf{k}_{0}}^{R^{2}_{4}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C^{3}_{4z}\left(\prod_{\mathbf{k}=\mathbf{k}_{0}}^{R_{4}^{3}\mathbf{k}_{0}}P(\mathbf{k}^{\prime})\right)C^{4}_{4z}=-\hat{\mathcal{W}}^{{\dagger}}_{w},$

where the factor $-1$ is because of $C^{4}_{4z}=-1$ in our case. It is
straightforward to see that $O^{4}=-\hat{\mathcal{W}}_{w}^{{\dagger}}$
directly indicates $[O,\hat{\mathcal{W}}_{w}]=0$. Thus, if two degenerate
eigenstates of $\hat{\mathcal{W}}^{{\dagger}}_{w}$ (and thus of
$\hat{\mathcal{W}}_{w}$) have different eigenvalues of $O$, their degeneracy
cannot be lifted by any $C_{4z}$ symmetry-preserving perturbations [38].

With these symmetry constraints, we are now ready to show that in the spin-
Hopf model, the relative winding in the spetrum of Wilson loop over a
hemisphere is robust as long as the symmetries and the band degeneracy are
preserved. First, at the north pole of the sphere, the Hamiltonian simplifies
and the Wilson loop operator at the fixed momentum $(0,0,k_{z0})$ has two
unit-modulus eigenvalues that are degenerate and equal to $1$ (i.e.,
$\lambda_{W}=0$).

Second, for the crossing points at $\lambda_{W}=\pi$ [see Fig. 4 (c)], because
$O^{4}=-\hat{\mathcal{W}}^{{\dagger}}_{w}$ the eigenvalues of $O$ can take
only discrete values $\exp(in\pi/2)$ with $n=0,1,2,3$. For the spin Hopf
model, the two eigenstates of $\hat{\mathcal{W}}_{w}$ with unit-modulus
eigenvalues are eigenstates of $O$ with eigenvalues $\exp(i\pi/2)$ and
$\exp(i3\pi/2)$. Continuous deformations preserving symmetries and the band
degeneracy can never change the eigenvalues of $O$ given that they only take
discrete values. Since $\exp(i\pi/2)\neq\exp(i3\pi/2)$, this crossing at
$\lambda_{W}=\pi$ is then robust against any symmetry preserving
perturbations.

Finally, for $\hat{\mathcal{W}}_{w}$ along the equator of the spin Hopf model
with $u=3$, the two unit-modulus eigenvalues are both $1$ (i.e.,
$\lambda_{W}=0$). These two eigenstates of $\hat{\mathcal{W}}_{w}$ are
eigenstates of $O$ with eigenvalues $\exp(i\pi/4)$ and $\exp(i7\pi/4)$.
Following the same argument in the last paragraph, the degeneracy at
$\lambda_{W}=0$ in the spectrum of $\hat{\mathcal{W}}_{w}$ along the equator
is also robust against any symmetry preserved perturbations. Thus, starting
from the spin Hopf model with spin-$U(1)$ symmetry, the relative winding in
the spectrum of the Wilson loops is robust against any perturbations that
preserve the symmetries.

We note that by assumption we consider only band degeneracy preserving
perturbations when discussing the Dirac dipole. However, symmetry preserved
perturbation terms do not automatically preserve the band degeneracy. For
clarification, let us now show all possible perturbation terms for the 4-band
spin-Hopf model preserving both the band degeneracy and the symmetries up to
second order in $k_{x},k_{y},k_{z}$, which are perturbation terms we focused
on.

The inversion symmetry precludes all terms linear in $k_{x},k_{y},k_{z}$.
Under TR transformation, five out of fifteen $4\times 4$ Hermitian matrices
are invariant, and all others gain an extra minus sign. The five TR invariant
matrices are $\tau_{x}\sigma_{y}$, $\tau_{y}\sigma_{y}$, $\tau_{z}\sigma_{y}$,
$\mathbbm{1}\sigma_{x}$, and $\mathbbm{1}\sigma_{z}$. Under
$C_{4z}=\exp\left(i\frac{\pi}{2}\tau_{z}(\frac{\sigma_{z}}{2}+\mathbbm{1})\right)$,
the transformations of the fifteen $4\times 4$ Hermitian matrices are
summarized in Table S1. Then, all possible terms that preserve the TR, four-
fold rotation, inversion symmetries and the band degeneracy at $\mathbf{k}=0$
are:

$k_{x}k_{y}\tau_{x}\sigma_{y},\quad k_{x}k_{y}\tau_{y}\sigma_{y},\quad
k_{x}k_{z}\tau_{z}\sigma_{y}+k_{y}k_{z}\mathbbm{1}\sigma_{x},\quad
k_{x}k_{z}\mathbbm{1}\sigma_{x}-k_{y}k_{z}\tau_{z}\sigma_{y},\quad[a(k_{x}^{2}+k_{y}^{2})+bk_{z}^{2}]\mathbbm{1}\sigma_{z},$
(S11)

where only the first two terms couple different spin sectors. We have
numerically confirmed that the relative winding is indeed robust under these
perturbation terms (especially the first two terms that break the spin-$U(1)$
symmetry). Thus, even if we move away from the simple spin-$U(1)$ symmetric
case, we can still have the Dirac dipole band degeneracy.

Table S1: Transformation of fifteen $4\times 4$ Hermitian matrices under $C_{4z}$ $M$ |  $\tau_{x}\sigma_{x}$ |  $\tau_{x}\sigma_{y}$ |  $\tau_{x}\sigma_{z}$ |  $\tau_{x}\mathbbm{1}$
---|---|---|---|---
$C_{4z}MC_{4z}^{-1}$ |  $-\tau_{x}\sigma_{x}$ |  $-\tau_{x}\sigma_{y}$ |  $\tau_{y}\mathbbm{1}$ |  $\tau_{y}\sigma_{z}$
$M$ |  $\tau_{y}\sigma_{x}$ |  $\tau_{y}\sigma_{y}$ |  $\tau_{y}\sigma_{z}$ |  $\tau_{y}\mathbbm{1}$
$C_{4z}MC_{4z}^{-1}$ |  $-\tau_{y}\sigma_{x}$ |  $-\tau_{y}\sigma_{y}$ |  $-\tau_{x}\mathbbm{1}$ |  $-\tau_{x}\sigma_{z}$
$M$ |  $\tau_{z}\sigma_{x}$ |  $\tau_{z}\sigma_{y}$ |  $\tau_{z}\sigma_{z}$ |  $\tau_{z}\mathbbm{1}$
$C_{4z}MC_{4z}^{-1}$ |  $-\mathbbm{1}\sigma_{y}$ |  $\mathbbm{1}\sigma_{x}$ |  $\tau_{z}\sigma_{z}$ |  $\tau_{z}\mathbbm{1}$
$M$ |  $\mathbbm{1}\sigma_{x}$ |  $\mathbbm{1}\sigma_{y}$ |  $\mathbbm{1}\sigma_{z}$ |
$C_{4z}MC_{4z}^{-1}$ |  $-\tau_{z}\sigma_{y}$ |  $\tau_{z}\sigma_{x}$ |  $\mathbbm{1}\sigma_{z}$ |

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# The photography method: solving pentagon equation

Vassily Olegovich Manturov  Moscow Institute of Physics and Technology, Moscow
141700, Russia
Nosov Magnitogorsk State Technical University, Zhilyaev Laboratory of
mechanics of gradient nanomaterials, 38 Lenin prospect, Magnitogorsk, 455000,
Russian Federation
<EMAIL_ADDRESS>and  Zheyan Wan  Yanqi Lake Beijing Institute of
Mathematical Sciences and Applications, Beijing 101408, China
<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we consider two applications of the pentagon equation.
The first deals with actions of flips on edges of triangulations labelled by
rational functions in some variables. The second can be formulated as a system
of linear equations with variables corresponding to triangles of a
triangulation. The general method says that if there is some general data
(say, edge lengths or areas) associated with states (say, triangulations) and
a general data transformation rule (say, how lengths or areas are changed
under flips) then after returning to the initial state we recover the initial
data.

Keywords: Braid groups, cluster algebras, photography method, pentagon
equation

MSC 2020: 20F36, 13F60, 57K20, 57K31

##  1\. Introduction

Many structures in mathematics are of extreme importance not because they just
solve a problem or provide nice invariants (which may also be the case) but
because they uncover close ties between various branches of mathematics. This
allows one to construct a “universal translator” between different languages
different mathematicians speak.

Figure 1. Cluster algebras’ connections and applications [1], see also [2, 3,
4, 5, 6, 9, 21, 23].

Among such famous objects, we have cluster algebras (see Fig. 1). We also
mention Coxeter groups and Associahedra (Stasheff polytopes). Among other
branches of mathematics, the groups $\Gamma$ are related at least to these
three ones.

The groups $\Gamma_{n}^{k}$ were first introduced in [16] and intensively
studied in [15]. In [20], the groups $\Gamma_{n}^{k}$ motivated the authors to
look for representations and actions of the braid group. This leads to the
photography method. By the photography method (in a broad sense) we mean two
very general methods for solving equations and constructing invariants of
objects.

For invariants, we consider objects (say, manifolds) represented by states
(say, triangulations) modulo local moves (say, Pachner moves). With each state
we associate some data (say, lengths, areas, angles, cocylces) and write down
some equations so that the set of solutions (maybe with some structure on it)
before applying a move is in bijection with the set of solutions (with the
same structure) after the move. The equations are taken from some natural data
transmission rules which are local, so that the invariance evidently follows
from geometrical reasons.

For equations, we take some data (say, edge-lengths) as variables, and we
define data transmission law in such a way that the composition of local moves
corresponding to flips between states (say, Pachner moves) leads to the
initial data. So, the composition of data transmission operators applied to
any data is identical. For further generalisations of the photography method,
see [8].

The Ptolemy transformation (1) satisfies the pentagon equation, i.e., the
composition of five Ptolemy transformations shown in Fig. 3 is the identity.
Namely, in order to show that Ptolemy yields pentagon, we need not calculate
anything (see [7, 15, 16, 17]): we just say that if two quadrilaterals sharing
three vertices are both inscribed then the whole pentagon is inscribed, and
the data concerning any of its quadrilaterals can be restored from the data of
the two quadrilaterals.

What do we mean by data? In geometry, those can be edge lengths, angles,
dihedral angles, areas, volumes, etc. But probably other data can come from
interesting number theory and hence yield new invariants of (at least) braids
and 3-manifolds.

We expect that many representations of the groups $\Gamma$ and other groups
(known or still to be discovered) can be explained by using this method.
Probably, many statements and relations in the cluster algebra theory also
follow from this method.

The general idea of the photography method belongs to the first named author,
as well as the possible strategy what to do in further directions (see the
last section). Most of the calculations are due to the second named author.

The paper is structured as follows. In Sec. 2, we give a solution to the
pentagon equation using the Ptolemy identity and photography method. In Sec.
3, we give another solution to the pentagon equation using the areas and
photography method. In Sec. 4, we give further directions on photography
method.

###  1.1. Acknowledgements

On the way towards the present state of the art, the first named author did a
lot of research jointly with I.M. Nikonov; the paper and the book coauthored
with Nikonov are just a tiny bit of it. The present paper could not be written
without Nikonov’s collaboration.

The authors are extremely grateful to L.H. Kauffman for sharing their joy in
working on this project, and to I.G. Korepanov who indicated to the first
author the idea which led to the understanding of how much can be done with
this “photography method”. The authors are also grateful to Joshua Abraham,
Zichang Han, and Seongjeong Kim for helpful discussion and comments.

The study was supported by the grant of Russian Science Foundation (No.
22-19-20073 dated March 25, 2022 “Comprehensive study of the possibility of
using self-locking structures to increase the rigidity of materials and
structures”).

##  2\. Two faces of the pentagon equation and solutions to them

The goal of the present section and the next section is to give two forms of
the pentagon equation (as an action on lengths and as a matrix-form action on
areas) and give a solution to them. Let

(1) $\displaystyle f(a,b,c,d,x)=\frac{ac+bd}{x}.$

This is the well known Ptolemy formula for the other diagonal of the
circumscribed quadrilateral with edge-lengths $a,b,c,d$ and one diagonal of
length $x$.

Now, we follow the discussion in [1]. Consider the hyperbolic plane
$\mathbb{H}^{2}=\\{z\in{\mathbb{C}}|Im(z)>0\\}$ with metric
$ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}$ and curvature $-1$. In this model,
geodesics are half-lines and half-circles perpendicular to the real axis. The
map $f(z)=\frac{z-i}{z+i}$ transforms the upper half-plane model to the
Poincaré unit disk model.

Let $A,B\in\partial\mathbb{H}^{2}$ be two points at the boundary of the
hyperbolic plane. The hyperbolic distance between them is infinite, but the
infinity is concentrated around $\partial\mathbb{H}^{2}$ and can be dealt with
by using horocycles as follows (in the upper half-plane model of
$\mathbb{H}^{2}$, a horocycle centered at $\infty$ is a horizontal line,
horocycles centered at other points are circles tangent to $\partial H^{2}$).
Choose horocycles $h_{A}$ and $h_{B}$ centered at $A$ and $B$ (see Fig. 2 for
the pictures in the upper half-plane and unit disk models). Let $l_{AB}$ be
the signed distance between the horocycles $h_{A}$ and $h_{B}$ ($l_{AB}$ is
zero if the horocycles are tangent and negative if the horocycles intersect
each other). Denote $\lambda_{AB}=e^{l_{AB}/2}$, the lambda length of $AB$.

Given an ideal quadrilateral $ABCD$ in $\mathbb{H}^{2}$ (a quadrilateral with
all vertices at $\partial\mathbb{H}^{2}$) and a choice of horocycles around
each of the vertices, one can prove that the lambda lengths for $ABCD$ satisfy
Ptolemy identity [22]:

(2)
$\displaystyle\lambda_{AB}\cdot\lambda_{CD}+\lambda_{BC}\cdot\lambda_{DA}=\lambda_{AC}\cdot\lambda_{BD}.$

$\lambda_{AB}\cdot\lambda_{CD}+\lambda_{BC}\cdot\lambda_{DA}=\lambda_{AC}\cdot\lambda_{BD}$$\lambda_{AB}$$=e^{l_{AB}/2}$$A$$B$$C$$D$$A$$B=\infty$$C$$D$$l_{AB}$$l_{CD}$

Figure 2. Lambda-lengths: removing infinity by horocycles

It is easy to show that given $a,b,c>0$, one can construct (in a unique way up
to isometry) an ideal hyperbolic triangle with a choice of horocycles at its
vertices such that $a=\lambda_{BC}$, $b=\lambda_{AC}$, and $c=\lambda_{AB}$.
Also, triangles can be attached to each other along the edges with the same
lambda lengths. Therefore, given a triangulated polygon with positive numbers
assigned to its edges and diagonals in the triangulation, one can construct an
ideal hyperbolic polygon with horocycles assigned to its vertices such that
for every edge and diagonal the assigned number will coincide with the
corresponding lambda length. In particular, arbitrary positive numbers
satisfying Ptolemy identity are realisable as the lambda lengths of the edges
and diagonals of an ideal hyperbolic quadrilateral.

Consider two ideal hyperbolic triangles with lambda lengths of the edges
$a,b,x$ and $c,d,x$ and glue them together along the edge $x$ to get an ideal
hyperbolic quadrilateral with lambda lengths of the edges $a,b,c,d$ so that
the edge of lambda length $b$ is adjacent to the edge of lambda length $c$. By
(2), $f(a,b,c,d,x)=\frac{ac+bd}{x}$ is the lambda length of the other diagonal
of the quadrilateral.

Given a pentagon with vertices $A_{1},A_{2},A_{3},A_{4},A_{5}$. Associate
variables $a,b,c,d,e$ to the edges $A_{1}A_{2},\cdots,A_{5}A_{1}$. Associate
variables $x,y$ to the diagonals $A_{1}A_{3}$ and $A_{1}A_{4}$.

Start doing the following flips. First, we leave the diagonal $A_{1}A_{3}$
untouched with label $x$; replace $A_{1}A_{4}$ with with $A_{3}A_{5}$ and put
label $z=f(x,c,d,e,y)$ on the new diagonal.

We get two diagonals with labels on them; these labels are $x$ and some
function on it. Perform this operation five times (each time to a different
quadrilateral) until we get back to the diagonals $A_{1}A_{3}$ and
$A_{1}A_{4}$, see Fig. 3. At each step we get functions of $a,b,c,d,e,x,y$.

Figure 3. Flip the diagonals of a pentagon five times.

######  Lemma 2.1.

The new labels will be $x$ for $A_{1}A_{3}$ and $y$ for $A_{1}A_{4}$.

###### Proof.

We just use the following obvious fact. If seven lengths are given
arbitrarily, then one can construct an ideal hyperbolic pentagon with a choice
of horocycles at its vertices such that the lambda lengths of the five edges
and two diagonals are exactly the given seven lengths. This can be done by
attaching three triangles to each other along the edges with the same lambda
lengths. Then we apply the Ptolemy identity (2) five times (each time to a
different quadrilateral) and complete the proof.

∎

By the pentagon equation we mean that the composition of five transformation
shown in Fig. 3 is the identical transformation. This equation can be written
in various forms depending on what these operators act on.

From this, we can get solutions to the pentagon equation and actions of the
braid group by using the function $f$. The function $f$ gives an action of the
pure braid group [17] on labelled triangulations where labels of edges remain
rational functions. The beauty of the above observation is that for the action
of the braid group, we need not know what the formula really is, we only need
to have Lemma 2.1.

The action of the pure braid group on labelled triangulations is an instance
of the photography method. The data are the lengths and the data transmission
rule is the Ptolemy identity. We start from the data and data transmission
rule. They are natural because they correspond to some geometry. Finally we
end up with an action of the pure braid group.

We can do other things with other data, say, with angles of hyperbolic
triangles (the sum of angles is not $\pi$) or with triangles on the round
sphere. For example, one can take angles of triangles and sets of axioms
coming from Euclidean geometry (opposite angles of an inscribed quadrilateral
sum to $\pi$) or Hyperbolic geometry (sums of opposite angles of an inscribed
quadrilateral are equal) to write down the data transmission rules coming from
flips.

##  3\. Towards actions of the braid group inspired by $\Gamma_{n}^{4}$

In this section, we mention a way to get an action of the braid group inspired
by the group $\Gamma_{n}^{4}$ which is related to a solution of the pentagon
equation, which was communicated to the first named author by I.G. Korepanov.
After several conversations, the authors realized that this solution can be
interpreted as a partial case of the “photography method”. First, we recall
the definition of the group $\Gamma_{n}^{4}$ given in [16, 15].

######  Definition 3.1.

The group $\Gamma_{n}^{4}$ is the group generated by

$\\{d_{(ijkl)}\mid\\{i,j,k,l\\}\subset\\{1,\dots,n\\},|\\{i,j,k,l\\}|=4\\}$

subject to the following relations:

  1. (1)

$d_{(ijkl)}^{2}=1$ for $\\{i,j,k,l\\}\subset\\{1,\dots,n\\}$,

  2. (2)

$d_{(ijkl)}d_{(stuv)}=d_{(stuv)}d_{(ijkl)}$ for
$|\\{i,j,k,l\\}\cap\\{s,t,u,v\\}|<3$,

  3. (3)

$d_{(ijkl)}d_{(ijlm)}d_{(jklm)}d_{(ijkm)}d_{(iklm)}=1$ for distinct
$i,j,k,l,m$,

  4. (4)

$d_{(ijkl)}=d_{(kjil)}=d_{(ilkj)}=d_{(klij)}=d_{(jkli)}=d_{(jilk)}=d_{(lkji)}=d_{(lijk)}$
for distinct $i,j,k,l$.

Here $d_{(ijkl)}$ corresponds to the flip of diagonals in the quadrilateral
$ijkl$. The third relation is the pentagon relation.

We will associate a $2\times 2$ matrix to each generator of $\Gamma_{n}^{4}$.
Below we construct the matrix for $d_{(1234)}$, the matrices for other
$d_{(ijkl)}$ are constructed similarly. The two diagonals divide a
quadrilateral $1234$ into 4 parts, the areas of the 4 parts are denoted by
$A,B,C,D$, see Fig. 4.

Figure 4. Flip the diagonals of a quadrilateral.

Then the generator $d_{(1234)}$ flips the diagonal 13 to 24 of the
quadrilateral 1234. It maps $\left(\begin{array}[]{cc}A+C\\\
B+D\end{array}\right)$ to $\left(\begin{array}[]{cc}A+B\\\
C+D\end{array}\right)$. We try to associate a matrix
$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ to $d_{(1234)}$ such
that

$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{cc}A+C\\\
B+D\end{array}\right)=\left(\begin{array}[]{cc}A+B\\\ C+D\end{array}\right).$

Then

$(1,1)\left(\begin{array}[]{cc}A+C\\\
B+D\end{array}\right)=(1,1)\left(\begin{array}[]{cc}A+B\\\
C+D\end{array}\right)$

for all $A,B,C,D$ implies that $(a+c,b+d)=(1,1)$.

The inverse of the generator $d_{(1234)}$ flips the diagonal 24 to 13 of the
quadrilateral 1234. The matrix associated to $d_{(1234)}^{-1}$ is

$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)^{-1}=\left(\begin{array}[]{cc}a&b\\\
1-a&1-b\end{array}\right)^{-1}=\left(\begin{array}[]{cc}\frac{1-b}{a-b}&\frac{-b}{a-b}\\\
\frac{a-1}{a-b}&\frac{a}{a-b}\end{array}\right)=:\left(\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{array}\right).$

We find that $(a^{\prime}+c^{\prime},b^{\prime}+d^{\prime})=(1,1)$. Note that
the matrices associated to $d_{(1234)}$ and $d_{(1234)}^{-1}$ are not equal in
general.

Let $\zeta_{i}$ ($i=1,2,3,4$) be variables associated to the vertices of the
quadrilateral, then we can construct the matrix of $d_{(1234)}$ explicitly as

(5)
$\displaystyle\left(\begin{array}[]{cc}\frac{\zeta_{3}-\zeta_{2}}{\zeta_{4}-\zeta_{2}}&\frac{\zeta_{1}-\zeta_{2}}{\zeta_{4}-\zeta_{2}}\\\
\frac{\zeta_{3}-\zeta_{4}}{\zeta_{2}-\zeta_{4}}&\frac{\zeta_{1}-\zeta_{4}}{\zeta_{2}-\zeta_{4}}\end{array}\right).$

Note that the sum of the elements in each column of this matrix is 1.

We verify that the matrices associated to the flips of diagonals satisfy the
pentagon relation (see Fig. 3). Namely,

(6) $\displaystyle
a_{1345}a_{1235}a_{2345}^{-1}a_{1245}^{-1}a_{1234}^{-1}=I_{3}$

where $a_{ijkl}$ is the $3\times 3$ matrix associated to the flip of diagonals
$d_{(ijkl)}$ in the quadrilateral $ijkl$ which fixes a triangle in the
pentagon $12345$. Note that each triangulation of the pentagon consists of
three triangles, each flip of the diagonals changes two of the three triangles
and fixes the other one. The $3\times 3$ matrix $a_{ijkl}$ is obtained from
the $2\times 2$ matrix constructed for $d_{(ijkl)}$ by adding a diagonal entry
1 corresponding to the fixed triangle. By direct computation, we verify that

(16) $\displaystyle\left(\begin{array}[]{ccc}1&&\\\
&\frac{\zeta_{4}-\zeta_{3}}{\zeta_{5}-\zeta_{3}}&\frac{\zeta_{1}-\zeta_{3}}{\zeta_{5}-\zeta_{3}}\\\
&\frac{\zeta_{4}-\zeta_{5}}{\zeta_{3}-\zeta_{5}}&\frac{\zeta_{1}-\zeta_{5}}{\zeta_{3}-\zeta_{5}}\end{array}\right)\left(\begin{array}[]{ccc}\frac{\zeta_{3}-\zeta_{2}}{\zeta_{5}-\zeta_{2}}&\frac{\zeta_{1}-\zeta_{2}}{\zeta_{5}-\zeta_{2}}&\\\
\frac{\zeta_{3}-\zeta_{5}}{\zeta_{2}-\zeta_{5}}&\frac{\zeta_{1}-\zeta_{5}}{\zeta_{2}-\zeta_{5}}&\\\
&&1\end{array}\right)\left(\begin{array}[]{ccc}1&&\\\
&\frac{\zeta_{5}-\zeta_{2}}{\zeta_{4}-\zeta_{2}}&\frac{\zeta_{3}-\zeta_{2}}{\zeta_{4}-\zeta_{2}}\\\
&\frac{\zeta_{4}-\zeta_{5}}{\zeta_{4}-\zeta_{2}}&\frac{\zeta_{4}-\zeta_{3}}{\zeta_{4}-\zeta_{2}}\end{array}\right)\cdot{}$
(23)
$\displaystyle\cdot\left(\begin{array}[]{ccc}\frac{\zeta_{5}-\zeta_{1}}{\zeta_{4}-\zeta_{1}}&&\frac{\zeta_{2}-\zeta_{1}}{\zeta_{4}-\zeta_{1}}\\\
&1&\\\
\frac{\zeta_{4}-\zeta_{5}}{\zeta_{4}-\zeta_{1}}&&\frac{\zeta_{4}-\zeta_{2}}{\zeta_{4}-\zeta_{1}}\end{array}\right)\left(\begin{array}[]{ccc}\frac{\zeta_{4}-\zeta_{1}}{\zeta_{3}-\zeta_{1}}&\frac{\zeta_{2}-\zeta_{1}}{\zeta_{3}-\zeta_{1}}&\\\
\frac{\zeta_{3}-\zeta_{4}}{\zeta_{3}-\zeta_{1}}&\frac{\zeta_{3}-\zeta_{2}}{\zeta_{3}-\zeta_{1}}&\\\
&&1\end{array}\right)=I_{3}.$

Note that all generators of $\Gamma_{n}^{4}$ are involutions while these
matrices are not. So, formally, this is not a representation of
$\Gamma_{n}^{4}$, though it does give rise to an action of the pure braid
group on labelled triangulations (see [15]).

The Korepanov’s solution is another instance of the photography method. The
data are areas and the data transmission rule is (5). We start from the data
and data transmission rule. They are natural because they correspond to some
geometry. Finally we end up with an action of the pure braid group.

##  4\. Towards an algorithm: How it works and what to do further

In this section we give a general strategy for how to solve various equations
(like the pentagon) and give representations of various groups (like $\Gamma$)
by using very mild data. We consider states of an object in question (e.g.,
triangulations of a manifold). We consider moves between states (say, Pachner
moves, flips). We consider some data together with a data transformation law
which should be natural and sufficient.

For data we can take, say, edge lengths (or $\lambda$-lengths), by a natural
data transformation law we consider, say, the Ptolemy transformation. It is
natural because it originates from some geometry, and the data is sufficient
because the edge-lengths ($\lambda$-lenghs) of any triangulation (state)
uniquely define the data of any other state. See Sec. 2.

This leads to a formula (in our case, the Ptolemy formula) which can be
treated as a function (the way of rewriting lengths). Geometric naturality
guarantees that the function will satisfy certain equation (in our case,
Ptolemy yields pentagon). Hence, Lemma 2.1 can be checked by a brute force
algorithm [15]. After such a purely algebraic proof, one can completely forget
about what the initial objects come from (lengths are natural numbers
satisfying some mild conditions), and take abstract variables.

We also describe some possible caveats which may appear in constructing these
solutions, representations, and invariants, and give possible ways to overcome
it. Also, we indicate possible applications of this method in various branches
of mathematics.

1\. Space with data. Given a homogeneous space $S$ with data which is
invariant under the group of isometries. Examples: Euclidean space,
Lobachevsky space, Spheres, other homogeneous spaces.

2\. Nice family: for an integer $k$, on $S$ we have a family of “nice
$k$-objects” so that through each $k$ points there exists a unique nice
$k$-object. Examples: $k=3$, Euclidean space, $2$-dimensional sphere,
hyperbolic space. Caveats: For ${\mathbb{R}}^{2}$ we need not only circles,
but also lines, for the hyperbolic plane we also need horocycles and
equidistants. Moreover (see further) sometimes we have to deal with coincident
points; in this case we have to justify our methods.

3\. A formula for data (existence). By putting together two $k$-tuples of
points ( two vertices of a $(k-1)$-simplex, we get a bypiramid where the whole
data can be restored from the data for the two $k$-tuples. Examples: Two
inscribed triangles give rise to an inscribed quadrilateral, where the second
diagonal is expressed in terms of the first.

4\. With or without family. Sometimes, the photography method works without
any “nice family”.

5\. From formula to function. Functions are sometimes algebraic; variables can
be abstract. The pentagon equation deals with sides of a pentagon on the
Euclidean space, hence, with non-negative real numbers. However, the
implication “Ptolemy yields Pentagon” for the function $x\to\frac{ac+bd}{x}$
can be proved by purely algebraic methods.

Hence, we can use it not for numbers, but rather, for variables or other
abstract objects. This led to the action of pure braid groups [15, 16] on
triangulations with edges marked by Laurent polynomials.

Similarly, the beautiful formula from section due to I.G. Korepanov [14]
originates from areas of quadrilaterals 111This solution was communicated to
the first named author by I.G. Korepanov, but the method of obtaining it was
recognized by the first author by himself. , however for $\zeta_{i}$ one can
take variables $\zeta$ from some field. Similar problems were studied in [10,
11, 12, 13].

In two words: one takes a fact known from geometry (maybe school geometry like
the Ptolemy for an inscribed quadrilateral, obtains some corollaries from it
which can be proved elementarily from the geometrical point of view, and then
since the equations hold, they should have an algebraic proof!

6\. Summary of caveats and ways to overcome them. One of the main difficulties
the authors of [15] met in knot theory was an attempt to pass from braid
invariants (there are lot of them coming from $\Gamma_{n}^{4}$) to knot
invariants. The problem was division by zero: the solution to the equation
$xy=ac+bd$ in $y$ is not unique once $ac+bd=0$.

However, one can apply either: a brute force methods (as I.M. Nikonov did in
[20], see also [18, 19]) when one deals with some rings, stabilisations,
equivalence types or by using some methods like blowing up in algebraic
geometry (there is more than one line passing through two coinciding points,
but we can make points distinct by some blow-up). This is not realised yet.

7\. Further directions. In Fig. 1 one can see the “Cluster algebra flower”.
Lots of leaves of it are already related to the groups $\Gamma$. In [20] one
gets braids and manifolds.

Among further directions which may appear in further papers, we mention just
two: In topology: passing from braids to knots, passing from 3-manifolds to
4-manifolds etc. In group theory: deeper understanding the structure of the
group $\Gamma$ themselves.

In this paper we work with the pentagon relation, hence, we do something with
3-manifolds. The reader can choose a subject according to his/her preferences
and try to find the photography method there!

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