numerically with the Plücker vector simply given from the Bloch eigenvectors
by $V(\boldsymbol{k})=u_{1}(\boldsymbol{k})\wedge\cdots\wedge
u_{p}(\boldsymbol{k})$. One advantage of using the Plücker vector
$V(\boldsymbol{k})$ is that there is no ambiguity of choosing consistent
(continuous) gauge phases for the each Bloch eigenvectors.

The metric is then

$\displaystyle g_{\mathbb{T}^{2}}$
$\displaystyle=\sum\limits_{m}(\partial_{k^{i}}V^{m})(\partial_{k^{j}}V^{m})\,dk^{i}\otimes
dk^{j}$ (115) $\displaystyle=v^{m}_{i}v^{m}_{j}\,dk^{i}\otimes dk^{j}\,,$

where we sum over $i,j\in\\{1,2\\}$. The volume form takes the same above as
above. Also, since $[\partial_{k^{1}},\partial_{k^{2}}]=0$, the form of the
sectional curvature Eq. (95) is preserved, and we only need to make
substitution
$(\partial_{\mu},\partial_{\nu})\rightarrow(\partial_{k^{1}},\partial_{k^{2}})$.

Euler form = sec dVol.

$(\partial_{k^{1}}v_{1})^{\top}(\partial_{k^{2}}v_{2})-(\partial_{k^{2}}v_{1})^{\top}(\partial_{k^{1}}v_{2})$
(116)

##  VII Riemannian structures of few-band models

A virtue of the outlined Plücker framework is that we can address all
Riemannan structures in a tractable manner, furnishing a tool that can be
widely applied. We showcase this in the remaining parts of the paper. The
analytical few-band models derived in the previous section prove a
particularly precious framework for which we derive explicitly and
analytically the successive geometric structures that are relevant in
realistic many-band systems. We will in particular focus on the real
Grassmannians due the rich multi-gap topological features, but we emphasize
that the results directly translate to the complex cases.

As in the previous section, it will be convenient to decompose the Bloch
Hamiltonian mapping from the Brillouin zone to the Grassmannian into two
steps, i.e. $H^{W}=f^{W}_{SG}\circ f_{TS}$, due to the motivation that the
topology of the system (i.e. the homotopy class of the Bloch Hamiltonian) may
be chosen to only depend on the winding of the second map $f^{W}_{SG}$. Using
the perspective and line of thought of the previous Section we directly
compute the Riemannian structures for concrete cases.

###  VII.1 Three-band Euler phases

We first turn to the three-band Euler phases. We recall that the Grassmannian
here is the embedding sphere in $\mathbb{R}^{3}$, such that the global angle
coordinates can be readily taken as

$\tilde{f}^{W}_{SG}:(\theta,\phi)\rightarrow(\theta^{1},\theta^{2})=(\theta,W\phi)\,,$
(117)

corresponding to the Plücker vector
$V(\theta,W\phi)=\check{\boldsymbol{e}}^{\top}\cdot\boldsymbol{n}(\theta,W\phi)$,
where $W\in\mathbb{Z}$ is the degree of the map. We then define the basis of
the tangent bundle by

$\displaystyle v_{\theta}$
$\displaystyle=\partial_{\theta}V(\theta,W\phi)=(\cos W\phi\cos\theta,\sin
W\phi\cos\theta,-\sin\theta)\,,$ (118) $\displaystyle v_{\phi}$
$\displaystyle=\partial_{\phi}V(\theta,W\phi)=W\sin\theta(-\sin W\phi,\cos
W\phi,0)\,.$

We need to be cautious however that this basis is not normalized. Defining
$e_{\theta}=v_{\theta}$ and $e_{\phi}=v_{\phi}/(W\sin\theta)$, the unit
vectors $(e_{\theta},e_{\phi})$ span the tangent space at any point of the
sphere. By including $e_{r}=V$, they simply correspond to the spherical
coordinates frame.

Since $v_{\theta}^{\top}\cdot v_{\theta}=1$, $v_{\phi}^{\top}\cdot
v_{\phi}=\sin\theta^{2}$, and $v_{\theta}^{\top}\cdot v_{\phi}=0$, the induced
metric

$g_{\mathbb{S}^{2}}=d\theta\otimes d\theta+W^{2}\sin^{2}\theta\,d\phi\otimes
d\phi\,,$ (119)

which for $W=1$ is nothing but the usual metric of the unit sphere. Then the
only nonzero Christoffel symbols are
$\Gamma_{\theta,\phi}^{\phi}=\Gamma_{\phi,\theta}^{\phi}=\cot\theta$, and
$\Gamma_{\phi,\phi}^{\theta}=-W^{2}\cos\theta\sin\theta$, leading to the
directional Riemannian operator

$R(\partial_{\theta},\partial_{\phi})\partial_{\phi}=W^{2}\sin^{2}\theta\;\partial_{\theta}\,,$
(120)

and the sectional curvature is

$sec(\partial_{\theta},\partial_{\phi})=1\,.$ (121)

For the volume form, we find

$d\text{Vol}_{g_{\mathbb{S}^{2}}}=|W||\sin\theta|\,d\theta\wedge d\phi\,.$
(122)

Evaluating now the integral of the Gauss-Bonnet theorem associated with the
metric $g_{\mathbb{S}^{2}}$, we find

$\displaystyle\chi_{g_{\mathbb{S}^{2}}}$
$\displaystyle=\dfrac{1}{2\pi}\int_{\mathbb{S}^{2}}sec(\partial_{\theta},\partial_{\phi})\,d\text{Vol}_{g_{\mathbb{S}^{2}}}$
(123)
$\displaystyle=\dfrac{1}{2\pi}\int_{\mathbb{S}^{2}}1\cdot|W||\sin\theta|d\theta\wedge\;d\phi\,,$
$\displaystyle=2|W|\in 2\mathbb{Z}\,.$

Since the map $f^{\boldsymbol{W}}_{SG}$ is not an immersion (when $W\neq 1$,
the differential of the map fails to be globally injective), the above result
can more rigorously be interpreted as the Rieman-Hurwitz relation. Indeed,
interpreting $f^{\boldsymbol{W}}_{SG}$ as a holomorphic map between two
Riemann surfaces, it defines a branched covering with two ramification points,
located at $\theta=0$ and $\pi$, each with the ramification order of $W-1$
Jost (2006). The Rieman-Hurwitz theorem says, given that the genus of the
sphere is zero, that the total ramification order of a map of degree $W$ is
$v_{f}=2(W-1)$ Jost (2006).

Since $\widetilde{\mathsf{G}}^{\mathbb{C}}_{1,2}\cong\mathbb{S}^{2}$, the
topology of the Chern phases of the complex $1+1$-phase can be obtained
similarly.

###  VII.2 Four-band Euler phases

####  VII.2.1 Whole Grassmannian

We now address the Riemannian structures for the whole of
$\widetilde{\mathsf{Gr}}_{2,4}^{\mathbb{R}}$.

From Eq. (48), we readily derive the tangent vectors

$\displaystyle\partial_{\theta_{+}}$
$\displaystyle\equiv\iota_{P*}(\partial_{\theta_{+}}^{\text{int}})=\partial_{\theta_{+}}V_{I}^{\prime}=\dfrac{1}{\sqrt{2}}\left(e_{\theta_{+}}\oplus\boldsymbol{0}\right)\,,$
(124a) $\displaystyle\partial_{\phi_{+}}$
$\displaystyle\equiv\iota_{P*}(\partial_{\phi_{+}}^{\text{int}})=\partial_{\phi_{+}}V_{I}^{\prime}=\dfrac{\sin\theta_{+}}{\sqrt{2}}\left(e_{\phi_{+}}\oplus\boldsymbol{0}\right)\,,$
$\displaystyle\partial_{\theta_{-}}$
$\displaystyle\equiv\iota_{P*}(\partial_{\theta_{-}}^{\text{int}})=\partial_{\theta_{-}}V_{I}^{\prime}=\dfrac{1}{\sqrt{2}}\left(\boldsymbol{0}\oplus
e_{\theta_{-}}\right)\,,$ $\displaystyle\partial_{\phi_{-}}$
$\displaystyle\equiv\iota_{P*}(\partial_{\phi_{-}}^{\text{int}})=\partial_{\theta_{-}}V_{I}^{\prime}=\dfrac{\sin\theta_{-}}{\sqrt{2}}\left(\boldsymbol{0}\oplus
e_{\phi_{-}}\right)\,,$ with $\left\\{\begin{aligned}
e_{\theta_{\pm}}&=(\cos\phi_{\pm}\cos\theta_{\pm},\sin\phi_{\pm}\cos\theta_{\pm},-\sin\theta_{\pm}),\\\
e_{\phi_{\pm}}&=(-\sin\phi_{\pm},\cos\phi_{\pm},0),\end{aligned}\right.$
(124b)

are the unit vectors of the spherical frame of reference
$(e_{\theta}(\theta,\phi),e_{\phi}(\theta,\phi),e_{r}(\theta,\phi)=\boldsymbol{n}(\theta,\phi))$
on a two-sphere. We then readily find that the frame
$(\partial_{\theta_{+}}\;\partial_{\phi_{+}}\;\partial_{\theta_{-}}\;\partial_{\phi_{-}})$
has vanishing Lie brackets, i.e. $[\partial_{\mu},\partial_{\nu}]=0$ for all
$\mu,\nu\in\\{\theta_{+},\phi_{+},\theta_{-},\phi_{-}\\}$.

From the ansatz Eq. (39b) and the change of variables Eq. (47), we find the
metric

$g_{\mathsf{G}_{2,4}}=\dfrac{1}{2}\left(d\theta_{+}\otimes
d\theta_{+}+\sin^{2}\theta_{+}\,d\phi_{+}\otimes d\phi_{+}\right.\\\
\left.+d\theta_{-}\otimes d\theta_{-}+\sin^{2}\theta_{-}\,d\phi_{-}\otimes
d\phi_{-}\right)\,,$ (125)

as was expected from Eq. (48c). The nonvanishing Christoffel symbols are then

$\displaystyle\Gamma_{\theta_{+}\phi_{+}}^{\phi_{+}}=\Gamma_{\phi_{+}\theta_{+}}^{\phi_{+}}$
$\displaystyle=\cot\theta_{+}\,,$ (126)
$\displaystyle\Gamma_{\phi_{+}\phi_{+}}^{\theta_{+}}$
$\displaystyle=-\cos\theta_{+}\sin\theta_{+}\,,$
$\displaystyle\Gamma_{\theta_{-}\phi_{-}}^{\phi_{-}}=\Gamma_{\phi_{-}\theta_{-}}^{\phi_{-}}$
$\displaystyle=\cot\theta_{-}\,,$
$\displaystyle\Gamma_{\phi_{-}\phi_{-}}^{\theta_{-}}$
$\displaystyle=-\cos\theta_{-}\sin\theta_{-}\,,$

and the non-vanishing elements of the directional curvature tensor are

$\displaystyle
R(\partial_{\theta_{+}},\partial_{\phi_{+}})\partial_{\phi_{+}}$
$\displaystyle=\sin^{2}\theta_{+}\,\partial_{\theta_{+}}\,,$ (127)
$\displaystyle
R(\partial_{\theta_{-}},\partial_{\phi_{-}})\partial_{\phi_{-}}$
$\displaystyle=\sin^{2}\theta_{-}\,\partial_{\theta_{-}}\,.$

We finally find a non-vanishing sectional curvature only for the pairs
$\\{\theta_{+},\phi_{+}\\}$ and $\\{\theta_{-},\phi_{-}\\}$,

$sec_{g_{\mathsf{G}_{2,4}}}(\partial_{\theta_{+}},\partial_{\phi_{+}})=sec_{g_{\mathsf{G}_{2,4}}}(\partial_{\theta_{-}},\partial_{\phi_{-}})=2\,,$
(128)

(this agrees with Kozlov (2000)) and the volume form

$d\text{Vol}_{g_{\mathsf{G}_{2,4}}}=\dfrac{1}{2}|\sin\theta_{+}||\sin\theta_{-}|\,d\theta_{+}\wedge
d\phi_{+}\wedge d\theta_{-}\wedge d\phi_{-}\,.$ (129)

####  VII.2.2 Restriction to
$\mathcal{M}=f^{(W_{+},W_{-})}_{SG}(\mathbb{S}^{2}_{0})$

We have defined the map $\tilde{f}^{(W_{+},W_{-})}_{SG}$ in Eq. (130) of
Section V.4,

$\left\\{\begin{aligned}
\theta_{+}&=C_{+}\theta\,,\quad&&\phi_{+}&&=W_{+}\phi\,,\\\
\theta_{-}&=C_{-}\theta\,,&&\phi_{-}&&=W_{-}\phi\,,\end{aligned}\right.$
(130a) that can rewritten in terms of the Euler numbers $(\chi_{I},\chi_{II})$
according to Eq. (51), i.e. using $\left\\{\begin{aligned}
W_{+}&=\frac{-\chi_{I}-\chi_{II}}{2}\,,\quad&&C_{+}&&=[1-\delta_{\chi_{II},-\chi_{I}}]\,,\\\
W_{-}&=\frac{\chi_{I}-\chi_{II}}{2}\,,&&C_{-}&&=[1-\delta_{\chi_{II},\chi_{I}}]\,.\end{aligned}\right.$
(130b)

From Eq. (125), and through Eq. (107), the induced metric is found

$g_{\mathcal{M}_{2,4}}=\dfrac{1}{2}\left(C_{+}+C_{-}\right)\;d\theta\otimes
d\theta\\\ +\dfrac{1}{8}\big{[}(\chi_{I}+\chi_{II})^{2}\sin^{2}C_{+}\theta\\\
+(\chi_{I}-\chi_{II})^{2}\sin^{2}C_{-}\theta\big{]}\;d\phi\otimes d\phi\,.$
(131)

Let first assume $|\chi_{II}|=|\chi_{I}|=\chi$ and $\chi_{I(II)}\neq 0$. We
call this the balanced condition since the absolute Euler number is the same
below and above the energy gap. We find the metric

$g^{(\text{bal})}_{\mathcal{M}_{2,4}}=\dfrac{1}{2}\;d\theta\otimes
d\theta+\dfrac{1}{2}\chi^{2}\sin^{2}\theta\;d\phi\otimes d\phi\,,$ (132)

the sectional curvature

$sec(\partial_{\theta},\partial_{\phi})=2\,,$ (133)

and the volume form

$d\text{Vol}_{\mathcal{M}_{2,4}}=\dfrac{1}{2}\chi|\sin\theta|\;d\theta\wedge
d\phi\,.$ (134)

Integrating the sectional curvature, we get an effective Euler characteristic

$\chi_{g_{\mathcal{M}_{2,4}}}=2\chi=2|\chi_{I}|=2|\chi_{II}|\in
2\mathbb{Z}\,.$ (135)

The interpretation for this result is that the balanced phases are realized
whenever one of the winding numbers $(W_{+},W_{-})$ is zero, which means that
the map $f^{\boldsymbol{W}}_{SG}$ only covers one sphere of
$\mathbb{S}^{2}_{+}(\frac{1}{\sqrt{2}})\times\mathbb{S}^{2}_{-}(\frac{1}{\sqrt{2}})$.
Therefore, we recover the Euler characteristic of the sphere, as derived above
for the three-band model (when $|\chi_{I}|=|\chi_{II}|=1$), and the Riemann-
Huwitz relation between two Riemann surfaces of genus zero (when
$|\chi_{I}|=|\chi_{II}|>1$), see the discussion below Eq. (123).

When $|\chi_{II}|\neq|\chi_{I}|$, which we call the imbalance case, we have
$C_{+}=C_{-}=1$ and the induced metric is

$g^{(\text{imb})}_{\mathcal{M}_{2,4}}=d\theta\otimes
d\theta+\dfrac{1}{4}\left(\chi_{I}^{2}+\chi_{II}^{2}\right)\sin^{2}\theta\;d\phi\otimes
d\phi\,.$ (136)

We then find the sectional curvature

$sec(\partial_{\theta},\partial_{\phi})=1\,,$ (137)

and the volume form

$d\text{Vol}_{g_{\mathcal{M}_{2,4}}}=\dfrac{1}{2}\sqrt{\chi_{I}^{2}+\chi_{II}^{2}}\;|\sin\theta|\;d\theta\wedge
d\phi\,.$ (138)

Finally, the integration of the sectional curvature gives an effective Euler
characteristic

$\chi_{g_{\mathcal{M}_{2,4}}}=\sqrt{\chi_{I}^{2}+\chi_{II}^{2}}\,.$ (139)

When the two winding numbers have equal norm one Euler number vanishes, i.e.
$\chi_{II}=0$ when $W_{-}=W_{+}$ such that

$\chi^{(W_{-}=W_{+})}_{g_{\mathcal{M}_{2,4}}}=|\chi_{I}|=|W_{+}|=|W_{-}|\in\mathbb{Z}\,,$
(140)

and $\chi_{I}=0$ when $W_{-}=-W_{+}$ such that

$\chi^{(W_{-}=-W_{+})}_{g_{\mathcal{M}_{2,4}}}=|\chi_{II}|=|W_{+}|=|W_{-}|\in\mathbb{Z}\,.$
(141)

Interestingly, when $|W_{+}|=|W_{-}|=1$, the Euler characteristic of
$\mathcal{M}$ in such imbalanced phases is $1$, i.e. the same as for a two-
dimensional disc. When $W_{+},W_{-}\neq 0$ and $|W_{-}|\neq|W_{+}|$, which
implies that both Euler numbers are nonzero, the Euler characteristic of
$\mathcal{M}$ is not even an integer. While this can be used as a geometric
indication of this special imbalanced topology, the topological interpretation
of the non-integer value is not clear, but poses an intriguing direction of
investigation.

####  VII.2.3 Euler form and Euler number of $T\mathcal{M}$

Alternatively, we can define the Euler form from the tangent subbundle
$T\mathcal{M}$, since it is of rank 2, and compute an Euler number similarly
to Section V. It turns out that this quantity is identical to the effective
Euler characteristic computed above from the sectional curvature, and this
other derivation can be seen as short cut to the effective Euler
characteristic. In order not to confuse it with the topological Euler numbers
derived in Section V, we will continue to call it an effective Euler
characteristic.

From Eq. (106) and Eq. (124), we obtain the globally defined basis of the
tangent subbundle

$\left\\{\begin{aligned}
\partial_{\theta}&=\dfrac{1}{\sqrt{2}}\big{[}C_{+}\,e_{\theta}(C_{+}\theta,W_{+}\phi)\oplus
C_{-}\,e_{\theta}(C_{-}\theta,W_{-}\phi)\big{]}\,,\\\
\partial_{\phi}&=\dfrac{1}{\sqrt{2}}\big{[}W_{+}\sin
C_{+}\theta\,e_{\phi}(C_{+}\theta,W_{+}\phi)\\\ &\quad\quad\quad\quad\oplus
W_{-}\sin
C_{-}\theta\,e_{\phi}(C_{-}\theta,W_{-}\phi)\big{]}\,.\end{aligned}\right.$
(142)

Defining the unit tangent vectors
$\widetilde{e}_{\theta}=\partial_{\theta}/|\partial_{\theta}|$ and
$\widetilde{e}_{\phi}=\partial_{\phi}/|\partial_{\phi}|$, we can now compute
the Euler form [Eq. (35b)]

$\displaystyle\mathtt{Eu}$
$\displaystyle=\left[\left(\partial_{\theta}\widetilde{e}_{\theta}\right)^{\top}\left(\partial_{\phi}\widetilde{e}_{\phi}\right)-\left(\partial_{\phi}\widetilde{e}_{\theta}\right)^{\top}\left(\partial_{\theta}\widetilde{e}_{\phi}\right)\right]d\theta\wedge
d\phi\,,$ (143)

from which we can compute the Euler number, $1/(2\pi)\int\mathtt{Eu}$, using
Eq. (35c). In the case of balanced phases, setting
$\chi=|\chi_{I}|=|\chi_{II}|$, and $(C_{+},C_{-})=(1,0)$ or
$(C_{+},C_{-})=(0,1)$, we find $\mathtt{Eu}=\chi\sin\theta\;d\theta\wedge
d\phi$, such that the effective Euler characteristic is $2\chi$. When the
phase is imbalanced, setting $(C_{+},C_{-})=(1,1)$, we get
$1/2\sqrt{(\chi_{I}+\chi_{II})^{2}}\sin\theta$, such that the effective Euler
characteristic is $\sqrt{\chi_{I}^{2}+\chi_{II}^{2}}$. These results are
identical to the above derivation via the sectional curvature, which shows the
consistency of the Riemanniann structures. Again, the topological
interpretation of the non-integer value of the effective Euler characteristic
of $T\mathcal{M}$ poses an interesting future pursuit.

###  VII.3 Five-band Euler and second Stiefel-Whitney phases

We finally address the generalization to five bands. We start with the
Riemannian structures of the whole of
$\widetilde{\mathsf{G}}^{\mathbb{R}}_{3,5}$. Moreover, we then derive the
geometric structures induced by the restriction from the whole Grassmannian to
$\mathcal{M}$, the image of the two-sphere, obtained via our pullback
construction. There, we essentially retrieve the above results for the
restricted band subspaces. As such these tractable examples set the stage for
full generalizations to arbitrary $N$-band systems and isolated band sub-
spaces thereof, showing the universal power of the Plücker framework.

####  VII.3.1 Whole Grassmannian

We here evaluate the Riemannian structures for the whole of
$\widetilde{\mathsf{G}}^{\mathbb{R}}_{3,5}$. From the ansatz Eq. (55a),
together with the change of variables Eq. (47) and the parametrization Eq.
(130), we find the metric

$g_{\mathsf{G}_{3,5}}=d\boldsymbol{\theta}^{\top}\cdot
M_{g_{\mathsf{G}_{3,5}}}^{(\otimes)}\cdot d\boldsymbol{\theta}\,,$ (144a) with
$\displaystyle d\boldsymbol{\theta}$
$\displaystyle=\left(d\theta^{1},d\theta^{2},d\theta^{3},d\theta^{4},d\theta^{5},d\theta^{6}\right)\,,$
(144b)
$\displaystyle=\left(d\theta_{+},d\phi_{+},d\theta_{-},d\phi_{-},d\theta^{5},d\theta^{6}\right)\,,$

where we combine the matrix and tensor products as

$d\theta^{\mu}\left[M_{g_{\mathsf{G}_{3,5}}}^{(\otimes)}\right]_{\mu\nu}d\theta^{\nu}=g_{\mathsf{G}_{3,5},\mu\nu}\;d\theta^{\mu}\otimes
d\theta^{\nu}\,,$ (145)

and evaluate only the non-vanishing elements given by

$\displaystyle\begin{aligned}
g_{\mathsf{G}_{3,5},11}&=\dfrac{1}{2}\,,\quad&&g_{\mathsf{G}_{3,5},22}&&=\dfrac{1}{2}\sin^{2}\theta_{+}\,,\\\
g_{\mathsf{G}_{3,5},33}&=\dfrac{1}{2}\,,\quad&&g_{\mathsf{G}_{3,5},44}&&=\dfrac{1}{2}\sin^{2}\theta_{-}\,,\end{aligned}$
(146) $\displaystyle
g_{\mathsf{G}_{3,5},55}=\dfrac{1+n_{+}^{1}n_{-}^{1}-n^{3}_{+}n^{3}_{-}+n_{+}^{2}n_{-}^{2}}{2},$
$\displaystyle\begin{aligned}
g_{\mathsf{G}_{3,5},16}&=g_{\mathsf{G}_{3,5},61}&&g_{\mathsf{G}_{3,5},26}&&=g_{\mathsf{G}_{3,5},62}\,,\\\
&=-\dfrac{\cos\phi_{+}\sin\theta^{5}}{2}\,,&&&&=\dfrac{n_{+}^{2}n_{+}^{3}\sin\theta^{5}}{2}\,,\\\
g_{\mathsf{G}_{3,5},36}&=g_{\mathsf{G}_{3,5},63}&&g_{\mathsf{G}_{3,5},46}&&=g_{\mathsf{G}_{3,5},64}\,,\\\
&=-\dfrac{\cos\phi_{-}\sin\theta^{5}}{2}\,,&&&&=\dfrac{n_{-}^{2}n_{-}^{3}\sin\theta^{5}}{2}\,,\end{aligned}$
$\displaystyle
g_{\mathsf{G}_{3,5},56}=g_{\mathsf{G}_{3,5},65}=-\cos\theta_{5}\dfrac{n_{+}^{3}n_{-}^{1}+n_{-}^{3}n_{+}^{1}}{2}\,,$

where we substituted the elements of the unit vectors defined in Eq. (48b).
Putting this all together we verify that

$\displaystyle
g_{\mathsf{G}_{3,5},66}=\frac{1}{64}\left[-32\cos^{2}\theta^{5}\sin\theta_{-}\sin\theta_{+}\cos(\phi_{-}+\phi_{+})+4\cos(2\theta^{5}-\theta_{-}-\theta_{+})+4\cos(2\theta^{5}+\theta_{-}-\theta_{+})\right.$
(147)
$\displaystyle+4\cos(2\theta^{5}-\theta_{-}+\theta_{+})+4\cos(2\theta_{a}+\theta_{-}+\theta_{+})+\cos
2(\theta^{5}-\theta_{-}-\phi_{-})+\cos 2(\theta^{5}+\theta_{-}-\phi_{-})+\cos
2(\theta^{5}-\theta_{-}+\phi_{-})$ $\displaystyle+\cos
2(\theta^{5}+\theta_{-}+\phi_{-})-2\cos 2(\theta^{5}-\theta_{-})-2\cos
2(\theta^{5}+\theta_{-})+16\sin^{2}\theta^{5}\sin^{2}\theta_{+}\cos
2\phi_{+}-2\cos 2(\theta^{5}-\theta_{+})$ $\displaystyle-2\cos
2(\theta^{5}+\theta_{+})-2\cos 2(\theta^{5}-\phi_{-})-2\cos
2(\theta^{5}+\phi_{-})-8\cos
2\theta^{5}+8\cos(\theta_{-}-\theta_{+})+8\cos(\theta_{-}+\theta_{+})$
$\displaystyle\left.-2\cos 2(\theta_{-}-\phi_{-})-2\cos
2(\theta_{-}+\phi_{-})+4\cos 2\theta_{-}+4\cos 2\theta_{+}+4\cos
2\phi_{-}+40\right]\,.$

While the Christoffel symbols can now be readily derived, their expressions
are cumbersome and we do not write them here. We just note that we find the
volume form

$d\text{Vol}_{g_{\mathsf{G}_{3,5}}}=\dfrac{1}{8}|\sin\theta_{+}\sin\theta_{-}\cos\theta^{5}\left(n^{2}_{+}+n^{2}_{-}\right)|\\\
d\theta_{+}\wedge d\phi_{+}\wedge d\theta_{-}\wedge d\phi_{-}\wedge
d\theta^{5}\wedge d\theta^{6}\,.$ (148)

We obtain the constant nonzero sectional curvature

$sec_{g_{\mathsf{G}_{3,5}}}(\partial_{\theta_{+}},\partial_{\phi_{+}})=sec_{g_{\mathsf{G}_{3,5}}}(\partial_{\theta_{-}},\partial_{\phi_{-}})=2\,,$
(149)

and

$sec_{g_{\mathsf{G}_{3,5}}}(\partial_{\theta^{\mu}},\partial_{\theta^{\nu}})=\dfrac{1}{2}\,,$
(150)

for the pairs of Plücker tangent vectors,
$(\theta^{\mu},\theta^{\nu})\in\\{(\theta_{+},\theta_{5}),(\phi_{+},\theta_{5}),(\theta_{-},\theta_{5}),(\phi_{-},\theta_{5})\\}$.
As in the four-band case, the sectional curvature is zero for
$(\theta^{\mu},\theta^{\nu})\in\\{(\theta_{+},\theta_{-}),(\theta_{+},\phi_{-}),(\phi_{+},\theta_{-}),(\phi_{+},\phi_{-})\\}$.

Finally, we obtain a variable and bounded sectional curvature

$0\leq
sec_{g_{\mathsf{G}_{3,5}}}(\partial_{\theta^{\mu}},\partial_{\theta^{\nu}})\leq
2\,,$ (151)

for the remaining pairs
$(\theta^{\mu},\theta^{\nu})\in\\{(\theta_{+},\theta^{6}),(\phi_{+},\theta^{6}),(\theta_{-},\theta^{6}),(\phi_{-},\theta^{6}),(\theta^{5},\theta^{6})\\}$.

We note that these results are in agreement with the classical result Wong
(1968) derived from the projector matrix representation of the Grassmannians
Bendokat  _et al._ (2020). We are not aware of any other source where the
Plücker representation of Grassmannians is directly used to obtain global
analytical expressions of the Riemannian structures beyond the case
$\mathsf{G}^{\mathbb{R}}_{2,4}$. We thus here demonstrate with the example
$\mathsf{G}^{\mathbb{R}}_{3,5}$ that our approach is tractable and can be
generalized. Moreover, our approach provides a very efficient framework for
the design of new material phases with non-trivial geometric signatures.

####  VII.3.2 Restriction to
$\mathcal{M}=f^{(W_{+},W_{-})}_{SG}(\mathbb{S}^{2}_{0})$

We now take the pullback by $f^{(W_{+},W_{-})}_{SG}$, still using Eq. (130),
i.e. we assume that the angles $\\{\theta^{5},\theta^{6}\\}$ are independent
of $\\{\theta,\phi\\}$. The latter assumption is justified since we have shown
that these directions do not affect the two-dimensional topologies. As a
consequence, the Riemannian metric and other structures on $\mathcal{M}$ are
identically the same as in Section VII.2.2 for the four-band systems.

Very interestingly, as for the four-band case, the tangent subbundle
$T\mathcal{M}$ is of rank 2, which allows us to compute the associated Euler
form and integrate it as an Euler number. Again making the same assumptions on
$\\{\theta^{5},\theta^{6}\\}$, we find the same expressions of the effective
Euler characteristics as in Section VII.2.3.

We emphasize that this result has a non-trivial consequence. Namely, while the
(topological) Euler number introduced in Section V is only defined for rank-2
Bloch vector bundles, i.e. strictly for a two-band subspace (see the
discussion on the Euler-to-Stiefel-Whitney reduction in Section V.5.1), we now
can characterize the topology of the three-band subspace (i.e. a rank-3 vector
bundle) in term of an effective Euler characteristic of the two-dimensional
submanifold $\mathcal{M}$, that is computed either via the sectional
curvature, or via the Euler form of the rank-2 tangent bundle [Section
VII.2.3].

###  VII.4 Towards a fully general description

We stress that our discussion can be readily extended to systems with
arbitrary many bands. Indeed, the restriction to the sub-manifold
$\mathcal{M}$, image of the sphere $\mathbb{S}^{2}_{0}$ within an arbitrary
large Grassmannian $\mathsf{G}^{\mathbb{R}}_{p,N}$ (for $p\geq 2$ and $N-p\geq
2$), always induces a (maximally) rank-2 tangent subbundle, $T\mathcal{M}$,
for which there is a well defined Euler form and Euler number. The obtained
Euler number corresponds to the effective Euler characteristic obtained from
the integration of sectional curvature over $\mathcal{M}$. Contrary to the
topological Euler number introduced in Section V, the effective Euler
characteristic exists for band-subspaces with an arbitrary number of bands.

##  VIII Physical applications

We already noted the quantum geometric tensor has recently been appearing in
multiple physical contexts that range from bounding superfluid densities
Peotta and Törmä (2015); Törmä  _et al._ (2021); Rhim  _et al._ (2020);
Herzog-Arbeitman  _et al._ (2022); Parker  _et al._ (2021); Julku  _et al._
(2016); Peri  _et al._ (2021) to probing topological band invariants as
responses to perturbations such as light, quench dynamics and in other
metrological setups Ozawa and Goldman (2018); Tran  _et al._ (2017); de Juan
_et al._ (2017); Mera and Ozawa (2021); Yu  _et al._ (2022); Li  _et al._
(2022). While the application of the introduced Plücker technology within
these contexts opens up many routes for new research initiatives that deserve
separate treatments, we here outline some preliminary points of view that
already firmly underpin the promised potential.

###  VIII.1 Operators and Response functions

We first consider how operators are defined within the Plücker setting. As a
result, we can then directly relate to perturbations and response theories,
setting the stage for a universal framework that will be applicable for a wide
range of physical settings as alluded to above.

####  VIII.1.1 Operator in the Plücker setting

As a first step we outline how to map operators $O$ from the original Hilbert
space to operators $\check{O}$ in the Plücker embedding. Evidently, such a map
needs to preserve expectation values over the occupied manifold

$\left\langle
V\right|\check{O}\left|V\right\rangle=\mathrm{Tr}{U^{\dagger}OU}.$ (152)

Analyzing
$\mathinner{\langle{V}|{\partial_{i}V}\rangle}=\mathrm{Tr}{U^{\dagger}\partial_{i}U}$
subsequently gives an insight on how to accomplish this. Replacing
$\partial_{i}$ with $O$, it can be deduced that we obtain the desired relation
if $\check{O}$ obeys a Leibniz rule on the wedge product of $u_{i}$.

$\check{O}\left|V\right\rangle=Ou_{1}\wedge u_{2}\wedge\ldots\wedge
u_{k}+\ldots+u_{1}\wedge\ldots\wedge Ou_{k}.$

We can now write $\check{O}$ explicitly in the $V_{i}$ basis

$\check{O}_{nm}=\sum_{I_{n}/i_{j}=I_{m}/i_{\ell}}(-1)^{j-\ell}O_{i_{j}i_{\ell}},$
(153)

where $n,m$ index the energy eigenbasis of the Plücker embedding,
$I_{n},I_{m}$ are their corresponding ordered index sets, and the sum runs
over all $i_{j},i_{\ell}$ labelling the $j$th ($\ell$th) element of $I_{n}$
($I_{m}$) for which the relation $I_{n}/i_{j}=I_{m}/i_{\ell}$ is true. The
number of terms in the sum will either be zero or one for $n\neq m$, and $k$
for $n=m$. Specifically, for the diagonal elements of $\check{O}$, we simply
obtain

$\check{O}_{nn}=\sum_{j=1}^{k}O_{jj}.$ (154)

Therefore, we can easily see that the groundstate expectation value amount to
the desired value

$\left\langle V\right|\check{O}\left|V\right\rangle=\sum_{j=1}^{k}\left\langle
u_{j}\right|O\left|u_{j}\right\rangle$ (155)

We note here that one can interpret the off-diagonal elements $n\neq m$ as
single band excitations out of the occupied manifold. Indeed, suppose $V_{m}$
can be obtained from $V_{n}$ by removing $u_{j}$ and appending $u_{\ell}$.
Then $\check{O}_{nm}$ is precisely the amplitude (up to a sign) for $O$ to
excite an electron from $u_{j}$ to $u_{\ell}$. That is,

$\check{O}_{nm}=(-1)^{k-j}O_{j\ell}$ (156)

If $V_{n}$ differs from $V_{m}$ by more than one $u_{j}$, then
$\check{O}_{nm}$ is simply zero as expected.

####  VIII.1.2 Perturbation theory and response functions

Since the expression to map operators into the Plücker embedding is linear, we
can directly analyze perturbations $\lambda H_{1}$ to the Hamiltonian $H$ that
thus carry over as

$H+\lambda H^{\prime}\mapsto\check{H}+\lambda\check{H}^{\prime}.$ (157)

To make this more concrete let us consider the perturbations to the original
states

$\left|u_{j}\right\rangle\rightarrow\left|u_{j}\right\rangle+\lambda\sum_{\ell\neq
j}\frac{\left|u_{\ell}\right\rangle\left\langle
u_{\ell}\right|H^{\prime}\left|u_{j}\right\rangle}{\epsilon_{j}-\epsilon_{\ell}}.$
(158)

When we now take the wedge product of the perturbed $u_{1},\ldots,u_{k}$, the
term at first order in $\lambda$ amounts to

$\lambda\sum_{j=1}^{k}\sum_{\ell=k+1}^{N}u_{1}\wedge\ldots\wedge\hat{u}_{j}\wedge\ldots\wedge
u_{k}\wedge u_{\ell}(-1)^{k-\ell}\frac{\left\langle
u_{\ell}\right|H^{\prime}\left|u_{j}\right\rangle}{\epsilon_{j}-\epsilon_{\ell}},$
(159)

where $\hat{u}_{j}$ denotes omission of $u_{j}$ and the minus sign arises from
reordering $u_{\ell}$ from where it is inserted at the $j$th position to the
end. We see that $(-1)^{k-j}\left\langle
u_{\ell}\right|H^{\prime}\left|u_{j}\right\rangle$ is the matrix element of
$\check{H}^{\prime}_{nm}$ for energy eigenstates $V_{n}$ and $V_{m}$ in the
Plücker embedding that differ by exactly one band index. In such a case, the
energies take the form

$\check{\epsilon}_{n}-\check{\epsilon}_{m}=\sum_{i_{n}\in
I_{n}}\epsilon_{i_{n}}-\sum_{i_{m}\in
I_{m}}\epsilon_{i_{m}}=\epsilon_{j}-\epsilon_{\ell}.$ (160)

When $I_{n}$ and $I_{m}$ differ by more than one index, we have
$\check{H}^{\prime}_{nm}=0$. Therefore we can rewrite the perturbation as a
sum over all $n\neq m$

$\left|V_{n}\right\rangle\rightarrow\left|V_{n}\right\rangle+\lambda\sum_{n\neq
m}\frac{\left|V_{m}\right\rangle\left\langle
V_{m}\right|\check{H}^{\prime}\left|V_{n}\right\rangle}{\check{\epsilon}_{n}-\check{\epsilon}_{m}}$
(161)

This exactly corresponds to the expected shifts of the wavefunctions if we
were to calculate the perturbation entirely in the Plücker embedding. It
follows that the linear order expression for a response function $R$ measured
by $O$ to a perturbation $\lambda H^{\prime}$ can be calculated in the Plücker
embedding as

$R=\lambda\sum_{n\neq m}\frac{\left\langle
V_{n}\right|\check{O}\left|V_{m}\right\rangle\left\langle
V_{m}\right|\check{H}^{\prime}\left|V_{n}\right\rangle}{\check{\epsilon}_{n}-\check{\epsilon}_{m}}+c.c.$
(162)

Usually, however, we are interested in operators $O$ of the form
$\frac{1}{i\hbar}[A,H]$, imposing a slight subtlety, since products $AH$ of
operators do not in general map to products $\check{A}\check{H}$ under the
Plücker embedding. That is, we have

$\check{O}_{nm}=\sum_{I_{n}/i_{j}=I_{m}/i_{\ell}}(-1)^{j-\ell}(A_{i_{j}i_{p}}H_{i_{p}i_{\ell}}-H_{i_{j}i_{p}}A_{i_{p}i_{\ell}}).$
(163)

The only relevant matrix elements in the response function must have $I_{n}$
and $I_{m}$ differing by one band index. Furthermore, we note that $H$ is
diagonal. As a result, we thus arrive at

$\check{O}_{nm}=\delta_{I_{n}/i_{j},I_{m}/i_{\ell}}(-1)^{j-\ell}(A_{i_{j}i_{\ell}}H_{i_{\ell}i_{\ell}}-H_{i_{j}i_{j}}A_{i_{j}i_{\ell}}).$
(164)

We therefore directly infer the impact of the mentioned subtlety, being that
$[\check{A},\check{H}]$ would include a sum of $H_{ii}$ over the occupied
sector, while there is only a single $H_{ii}$. However, since $I_{n}$ and
$I_{m}$ differ by only one band index, we can add and subtract these
additional terms. Since $(-1)^{j-\ell}A_{i_{j}i_{\ell}}=\check{A}_{nm}$ for
$n\neq m$, we obtain

$\check{O}_{nm}=\check{A}_{nm}\check{H}_{mm}-\check{H}_{nn}\check{A}_{nm}=[\check{A},\check{H}]_{nm},\;\;\;n\neq
m.$ (165)

As a last step, we may follow the usual process for turning the response
function into a curvature. Concretely, if $A=i\partial_{i}$ and
$H^{\prime}=i\partial_{j}$, we get

$\displaystyle R$ $\displaystyle=$ $\displaystyle\sum_{n\neq
m}\frac{\left\langle
V_{n}\right|[\partial_{j},\check{H}]\left|V_{m}\right\rangle\left\langle
V_{m}\right|i\partial_{j}\left|V_{n}\right\rangle}{\check{\epsilon}_{n}-\check{\epsilon}_{m}}+c.c.$
(166) $\displaystyle=$ $\displaystyle
i(\mathinner{\langle{\partial_{i}V_{n}}|{\partial_{j}V_{n}}\rangle}-c.c.)$

Since the Berry curvature within the Plücker embedding gives the sum of
single-band Berry curvatures over the occupied manifold, we know that the
curvature can be interpreted as the sum of band-diagonal responses. With the
above result we therefore confirm the anticipation that this response can
furthermore be expressed in the form of a Kubo formula.

####  VIII.1.3 Example of probing Chern numbers with dichroism

As a specific example of the above response theory we may connect to recent
predictions that suggest how to probe Chern numbers with circularly polarized
light Ozawa and Goldman (2018); Tran  _et al._ (2017); de Juan  _et al._
(2017); Mera and Ozawa (2021). To couple with circularly polarized light, we
consider a standard minimal coupling and take $H(\vec{k})\rightarrow
H(\vec{k}+\vec{A})$, where $\vec{A}=\frac{E}{\omega}(\cos(\omega
t),\pm\sin(\omega t))$ in the plane of the material. Viewing the original
Hamiltonian $H(k)$ within a rotating frame, $R^{\dagger}_{\pm}H(k)R_{\pm}$ in
terms of Tran  _et al._ (2017)

$\quad R_{\pm}=\exp\left(-i\tfrac{E}{\hbar\omega}(\cos(\omega
t)\hat{x}\pm\sin(\omega t)\hat{y})\right),$ (167)

we obtain a time-dependent perturbation that, to first order $E$, reads

$\displaystyle R^{\dagger}_{\pm}H(k)R_{\pm}\approx$ $\displaystyle
H_{0}+\tfrac{E}{\hbar\omega}\left(\cos(\omega t)\tfrac{\partial H}{\partial
k_{x}}\pm\sin(\omega t)\tfrac{\partial H}{\partial k_{y}}\right).$ (168)

The probability of transition from the ground state manifold to higher bands
over long times is given by the sum of transition rates between every pair of
occupied and unoccupied bands, which evaluates to

$\Gamma_{\pm}(\omega)=\sum_{n\in\text{gs}}\sum_{m\in\text{ex}}\frac{E}{2\hbar\omega}|\mathinner{\langle{m}|}(\partial_{k_{x}}\mp
i\partial_{k_{y}})\mathinner{|{n}\rangle}|^{2}\delta(\epsilon_{n}-\epsilon_{m}-\omega).$
(169)

This expression is now directly in terms of off-diagonal elements of
$\partial_{k}$ in the Plücker embedding. A Kubo formula, i.e. the curvature
form in the Plücker embedding, can be obtained by finding the integrated rate
$\tilde{\Gamma}=\int\Gamma\mathrm{d}\omega$ over a suitable range of $\omega$
and the taking difference of the two chiralities
$\Delta\Gamma=\Gamma_{+}-\Gamma_{-}$ to produce

$\Delta\tilde{\Gamma}=\sum_{n\in\text{gs}}\frac{iE}{2\hbar\omega}\left(\mathinner{\langle{\partial_{k_{x}}n}|{\partial_{k_{y}}n}\rangle}-\mathinner{\langle{\partial_{k_{y}}n}|{\partial_{k_{x}}n}\rangle}\right).$
(170)

This physical interpretation provides a useful intuition for the important
role that transitions between occupied and unoccupied bands take in the
Plücker embedding. We moreover emphasize that the embedding precisely renders
physical degrees of freedom as all gauge degrees of freedom have by
construction been eliminated, meaning that for this complex $n$-band system we
directly evaluate the describing Chern number and carry over no extra redudant
information.

###  VIII.2 Quantum volumes and bounds

The metric directly affects various other physical quantities. These relations
have notably resulted in revived perspectives on quantum metrology Yu  _et
al._ (2022); Li  _et al._ (2022), for example in the context of quantum
simulators, and Cramér-Rao information bounds. In addition, given the surge of
interest in flatband physics, especially in the context of twisted multi-
layered Van der Waals materials and superfluid (ultracold) systems, also
relations that bound the superfluid density by general correspondences to the
Fubini-Study metric at zero temperature have been receiving increasing
interest Peotta and Törmä (2015); Törmä  _et al._ (2021); Begholtz and Liu
(2013); Rhim  _et al._ (2020); Herzog-Arbeitman  _et al._ (2022); Parker  _et
al._ (2021); Julku  _et al._ (2016); Peri  _et al._ (2021). In particular,
using the quantum geometric tensor various trace and determinant identities
that relate the imaginary (entailing generalized Berry curvatures) and real
part (being the quantum distance part) can be derived. Indeed, using the
relation between the geometric and arithmetic mean, one readily obtains
$\text{Tr}g(\mathbf{k})\geq|\Omega(\mathbf{k})|$, paving a saturation
condition that facilitates the formation of fractional Chern insulating states
Parker  _et al._ (2021); Begholtz and Liu (2013); Repellin and Senthil (2020).
In addition, a similar application of such identities was shown to be
applicable in superfluid systems. That is, it was found the integral over the
Brillouin-zone of the quantum metric renders the superfluid weight in a flat
band and hence is bounded by the presence of invariants (the integral of
$\Omega$) such as Chern numbers or Euler invariants Peotta and Törmä (2015);
Törmä  _et al._ (2021); Peri  _et al._ (2021). As such our technology of
expressing quantum metric formulations and finding quantum volumes of
arbitrary multi-band systems also promises a versatile approach in these
contexts. We therefore close this section of applications of our perspective
by evaluating such quantum volumes and bounds for two representative model
settings. Namely, we derive analytically that the integrated quantum metric is
bounded from below by the topological Euler number in the 3-band and the
4-band cases when pulled-back on the two-sphere $S^{2}_{0}$. We then close the
discussion by addressing the effect of pulling-back the phase on the Brillouin
zone in terms of the tight-binding models, whose numerical form is obtained as
discussed above and in Refs. Bouhon  _et al._ (2020b); Bouhon and Slager
(2022). We reemphasize that these two cases serve as an illustration that can
be generlized to a varity of model settings.

As before, the Plücker approach is powerful in the intricately-related dual
aspects that concern the modelling aspect as well as retrieving the full
Riemannian structure. Given the generality of the framework we again focus on
the real topological phases due to their rich multi-gap nature, while the
complex counterparts can readily be derived analogously. In the subsequent it
will be of use to unify the discussion. To this end we write
$R_{p}=(u_{1}\cdots u_{p})\in\mathbb{R}^{N}\times\mathbb{R}^{p}$ to denote the
rectangular matrix with the occupied column-Bloch eigenvectors. The quantum
geometric tensor may then be defined as

$\boldsymbol{\sigma}_{ij}=\partial_{i}R_{p}^{\top}\left(\mathbb{1}_{N}-R_{p}R_{p}^{\top}\right)\partial_{j}R_{p}\,,$
(171)

that is an element of $\mathbb{R}^{p}\times\mathbb{R}^{p}$. The symmetric and
the anti-symmetric parts accordingly read

$\boldsymbol{g}_{ij}=\dfrac{1}{2}(\boldsymbol{\sigma}_{ij}+\boldsymbol{\sigma}_{ij}^{\top})\,,~{}\boldsymbol{\omega}_{ij}=\dfrac{1}{2}(\boldsymbol{\sigma}_{ij}-\boldsymbol{\sigma}_{ij}^{\top})\,.$
(172)

An important feature of $\sigma^{mn}_{ij}$ is its positive definiteness, see
e.g. Peotta and Törmä (2015); Xie  _et al._ (2020b), i.e. it satisfies

$\sum_{ij}v_{i}^{\top}\boldsymbol{\sigma}_{ij}v_{j}\geq 0\,,$ (173)

for any pair of real vectors $v_{i},v_{j}\in\mathbb{R}^{p}$.

####  VIII.2.1 Three-band $2+1$-Euler phases

We depart from the ansatz of the $2+1$-Euler phases of Section V.3.
Considering the two-band occupied subspace, $\boldsymbol{\sigma}_{ij}$ is a
2-by-2 matrix in the band space and the labels $\\{i,j\\}$ run through the
coordinates $\\{\theta,\phi\\}$ of $\mathcal{M}=\mathsf{G}=\mathbb{S}^{2}$.
Choosing the vectors $v_{\theta}=(1,1)$ and $v_{\phi}=(1,-1)$, the inequality
Eq. (173) gives

$\text{tr}\,\boldsymbol{g}-4\boldsymbol{\omega}^{12}_{\theta\phi}\geq 0\,,$
(174)

where the trace is taken over all the degrees of freedom, i.e.
$\text{tr}\,\boldsymbol{g}=\boldsymbol{g}^{11}_{\theta\theta}+\boldsymbol{g}^{22}_{\theta\theta}+\boldsymbol{g}^{11}_{\phi\phi}+\boldsymbol{g}^{22}_{\phi\phi}$.
If we take $v_{\theta}=(1,-1)$ and $v_{\phi}=(1,1)$ instead, we get
$\text{tr}\,\boldsymbol{g}+4\boldsymbol{\omega}^{12}_{\theta\phi}\geq 0$, such
that in general there a non-negative lower bound on the quantum metric

$\text{tr}\,\boldsymbol{g}\geq 4|\boldsymbol{\omega}^{12}_{\theta\phi}|\,.$
(175)

From the analytic ansatz of Section V.3, where $W$ fixes the winding of
$f^{W}_{SG}$ and the Euler number $\chi=2|W|$ of the phase, we find

$\left\\{\begin{aligned}
\text{tr}\,\boldsymbol{g}&=1+W^{2}\sin^{2}\theta=1+\dfrac{\chi^{2}}{4}\sin^{2}\theta\,,\\\
|\boldsymbol{\omega}^{12}_{\theta\phi}|&=|2W\sin\theta|=\chi|\sin\theta|\,,\end{aligned}\right.$
(176)

which after integration gives

$\left\\{\begin{aligned}
I_{1}&=\dfrac{1}{4\pi}\int\text{tr}\,\boldsymbol{g}\,d\theta\wedge
d\phi=\dfrac{\pi}{16}(8+\chi^{2})\,,\\\
I_{2}&=\dfrac{1}{\pi}\int|\boldsymbol{\omega}^{12}_{\theta\phi}|\,d\theta\wedge
d\phi=\chi\,,\end{aligned}\right.$ (177)

with the allowed values $\chi\in 2\mathbb{N}$. We plot in Fig. 1(a) the two
integrals as a function of the Euler number of the phase $\chi$, where we see
that the ansatz actually gives a strict inequality $I_{1}>I_{2}$.

| (a) | (b)
---|---
|
(c)
Figure 1:  Integrated quantum metric (blue) bounded by the topological Euler
number $\chi\geq 0$ (red). (a) Three-band $2+1$-Euler phases. (b) Four-band
$2+2$-Euler phases with balanced Euler numbers $|\chi_{I}|=|\chi_{II}|=\chi$.
(c) Four-band $2+2$-Euler phases with the imbalanced Euler numbers
$|\chi_{I}|$ (green), $|\chi_{II}|$ (orange).

####  VIII.2.2 Four-band $2+2$-Euler phases

Considering again two-band subspaces, we find the same general bound on the
metric

$\text{tr}\,\boldsymbol{g}\geq 4|\boldsymbol{\omega}^{12}_{\theta\phi}|\,.$
(178)

As a next step, using the four-band ansatz of Section V.4 for the balanced
phases ($|\chi_{I}|=|\chi_{II}|=\chi$), we find

$\left\\{\begin{aligned}
\text{tr}\,\boldsymbol{g}^{I}=\text{tr}\,\boldsymbol{g}^{II}&=\dfrac{1+\chi^{2}\sin^{2}\theta}{2}\,,\\\
|\boldsymbol{\omega}^{I,12}_{\theta\phi}|=|\boldsymbol{\omega}^{II,12}_{\theta\phi}|&=\chi|\sin\theta|\,,\end{aligned}\right.$
(179)

where $\\{\boldsymbol{g}^{I}\,\boldsymbol{\omega}^{I,12}_{\theta\phi}\\}$
($\\{\boldsymbol{g}^{II}\,\boldsymbol{\omega}^{II,12}_{\theta\phi}\\}$) are
the quantities obtained for the occupied (respectively, unoccupied) two-band
subspace, which after integration gives

$\left\\{\begin{aligned}
I^{I}_{1}&=I^{II}_{1}=\dfrac{\pi}{8}(2+\chi^{2})\,,\\\
I^{I}_{2}&=I^{II}_{2}=\chi\,,\end{aligned}\right.$ (180)

where here $\chi\in\mathbb{N}$. We again have a strict bound $I_{1}>I_{2}$.

If we however instead consider the imbalanced Euler phases
($|\chi_{I}|\neq|\chi_{II}|$), we find

$\left\\{\begin{aligned}
\text{tr}\,\boldsymbol{g}^{I}&=\text{tr}\,\boldsymbol{g}^{II}=1+\dfrac{1}{4}(\chi_{I}^{2}+\chi_{II}^{2})\sin^{2}\theta\,,\\\
|\boldsymbol{\omega}^{I,12}_{\theta\phi}|&=|\chi_{I}||\sin\theta|\,,~{}|\boldsymbol{\omega}^{II,12}_{\theta\phi}|=|\chi_{II}||\sin\theta|\,,\end{aligned}\right.$
(181)

which, after integration, gives

$\left\\{\begin{aligned}
I^{I}_{1}&=I^{II}_{1}=\dfrac{\pi}{16}(8+\chi_{I}^{2}+\chi_{II}^{2})\,,\\\
I^{I}_{2}&=|\chi_{I}|\,,~{}I^{II}_{2}=|\chi_{II}|\,.\end{aligned}\right.$
(182)

####  VIII.2.3 General numerical settings

The above analytical study can now be brought to the context of tight-binding
settings, where the geometric structures must be computed numerically. In that
respect, the fact that the sectional curvature of the submanifold
$\mathcal{M}$, which is defined numerically from the evaluations of the
Plücker vector, can be computed most efficiently from the Euler form of the
numerically obtained tangent bundle $T\mathcal{M}$, represents a great
advantage. In upcoming future work, we will show that the numerically
integrated metric is its self bounded from below by the analytical expressions
obtained above. This can be interpreted as the consequence that the metric of
tight-binding models is degenerate since the mapping of the Brillouin zone to
$\mathcal{M}$ fails to be an immersion, see the discussion in Section VI.3.
This will be reported in detail elsewhere, but its general applicability shows
the promise of the introduced perspective.

##  IX Conclusions and Discussion

We have shown that the Plücker embedding entails a versatile route to analyze
geometric tensors and Riemannian structures for arbitrary $n$-band systems.
Departing from simple two-band Chern models, we demonstrate how this embedding
universally generalizes the quantum metric tensor in terms of a simple
geometric viewpoint that formulates all relevant information. This concrete,
non-redundant, approach however becomes even more far-reaching in evaluating
general topological many-state systems hosting recently discovered multi-gap
phases that arise by rather generic reality conditions. As such, our universal
approach that manifests all relevant geometrical identities by appealing to a
simple vector description, presents a powerful universal benchmark to evaluate
physical topological systems.

The generic framework also provides a direct route towards a systematic
modelling, highlighting the interplay of topology, geometry and direct
descriptions in various situations. We outline how this approach accordingly
can by utilized in numerous physical contexts. Indeed, as preliminary
applications, we highlighted its manifestation in general perturbation theory
that in turn can be used to analyze optical responses of many-band systems and
outlined its use in defining quantum volumes that are of active interest to
derive bounds on superfluidity or formulate ideal conditions to host
fractional Chern states.

The above directions promise eminent potential to use the presented framework
and explore several novel directions. These not only include analyzing novel
interplays between topology and optical responses in many-band systems or
utilizing the generalized metric and its bounds in different settings, but
even reach to quantum computation and information. We already mentioned the
relation to Cramér-Rao information bounds and reemphasize that the active
scene of quantum metrology may directly profit from our framework. Moreover,
we anticipate that the direct handle of many-band systems, also in terms of
modelling, could induce new holonomic approaches to quantum compuation Pachos
_et al._ (1999). In such approaches one essentially encodes information in a
degenerate eigenspace of a parametric family of Hamiltonians, which basically
thrives on defining generalized non-Abelian Berry connections that can be
readily computed within our approach. We therefore believe that our results
can set a benchmark for a wide range of novel fundamental insights as well as
concrete physically relevant pursuits.

###### Acknowledgements.

A. B. was funded by a Marie-Curie fellowship, grant no. 101025315. R. J. S
acknowledges funding from a New Investigator Award, EPSRC grant EP/W00187X/1,
as well as Trinity college, Cambridge. We thank F. Nur Ünal for valuable
discussions.

##  Appendix A Derivation of Equation (III)

We here detail the derivation of Eq. (III). To define a notion of distance $D$
between two states $V_{1}$ and $V_{2}$ in the Plücker embedding is we consider
the canonical expression

$D^{2}_{\text{Pl}}=1-\mathinner{\langle{V_{2}}|{V_{1}}\rangle}\mathinner{\langle{V_{1}}|{V_{2}}\rangle}$
(183)

Assuming that $V_{2}$ differs from $V_{1}$ by an infinitesimal change
$d\vec{x}$ in the parameters of the Hamiltonian, we then obtain

$D^{2}_{\text{Pl}}=1-\mathinner{\langle{V(\vec{x}+\mathrm{d}\vec{x})}|{V(\vec{x})}\rangle}\mathinner{\langle{V(\vec{x})}|{V(\vec{x}+\mathrm{d}\vec{x})}\rangle}$
(184)

Expressing this result in term of the eigenstates, the above implies

$\displaystyle
D^{2}_{\text{Pl}}=1-\mathinner{\langle{u_{1}(\vec{x}+\mathrm{d}\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x}+\mathrm{d}\vec{x})}|{u_{1}(\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x})}\rangle}$
$\displaystyle\mathinner{\langle{u_{1}(\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x})}|{u_{1}(\vec{x}+\mathrm{d}\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x}+\mathrm{d}\vec{x})}\rangle}$

Expanding to second order in $\mathrm{d}\vec{x}$ then results in

$\displaystyle
D^{2}_{\text{Pl}}=1-\left\langle\left(\textstyle\sum_{i,j}u_{1}+\partial_{i}u_{1}\mathrm{d}x_{i}+\tfrac{1}{2}\partial_{i}\partial_{j}u_{1}\mathrm{d}x_{i}\mathrm{d}x_{j}\right)\wedge\ldots\wedge\left(\textstyle\sum_{i,j}u_{k}+\partial_{i}u_{k}\mathrm{d}x_{i}+\tfrac{1}{2}\partial_{i}\partial_{j}u_{k}\mathrm{d}x_{i}\mathrm{d}x_{j}\right)\right|$
$\displaystyle\left|u_{1}(\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x})\right\rangle\left\langle u_{1}(\vec{x})\wedge\ldots\wedge
u_{k}(\vec{x})\right|$
$\displaystyle\left|\left(\textstyle\sum_{i,j}u_{1}+\partial_{i}u_{1}\mathrm{d}x_{i}+\tfrac{1}{2}\partial_{i}\partial_{j}u_{1}\mathrm{d}x_{i}\mathrm{d}x_{j}\right)\wedge\ldots\wedge\left(\textstyle\sum_{i,j}u_{k}+\partial_{i}u_{k}\mathrm{d}x_{i}+\tfrac{1}{2}\partial_{i}\partial_{j}u_{k}\mathrm{d}x_{i}\mathrm{d}x_{j}\right)\right\rangle,$

where $i$ and $j$ indices denote components of the parameters $\vec{x}$.
Multiplying out each inner product and keeping only terms to second order, we
will have a few different categories of terms:

  * •

The zeroth order term $\mathinner{\langle{u_{1}\wedge\ldots\wedge
u_{k}}|{u_{1}\wedge\ldots\wedge u_{k}}\rangle}=1$

  * •

First order terms containing a single $\partial_{i}u_{n}\mathrm{d}x_{i}$. A
simple determinant calculation shows that these evaluate to
$(\partial_{i}u_{n})^{\dagger}u_{n}\mathrm{d}x_{i}$

  * •

Second order terms containing two instances of
$\partial_{i}u_{n}\mathrm{d}x_{i}$. For these, we have not only
$(\partial_{i}u_{n})^{\dagger}u_{n}(\partial_{j}u_{m})^{\dagger}u_{m}\mathrm{d}x_{i}\mathrm{d}x_{j}$
from the diagonal contribution to the determinant, but also an off-diagonal
contribution
$-(\partial_{i}u_{n})^{\dagger}u_{m}(\partial_{j}u_{m})^{\dagger}u_{n}\mathrm{d}x_{i}\mathrm{d}x_{j}$.
To get the correct count, we require $n>m$ for these.

  * •

Second order terms containing one instance of
$\frac{1}{2}\partial_{i}\partial_{j}u_{n}\mathrm{d}x_{i}\mathrm{d}x_{j}$. This
again has only a diagonal contribution to the determinant and thus evaluates
to
$\frac{1}{2}(\partial_{i}\partial_{j}u_{n})^{\dagger}u_{n}\mathrm{d}x_{i}\mathrm{d}x_{j}$

Applying the above, we arrive at

$\displaystyle
D^{2}_{\text{Pl}}=1-\left|1+\textstyle\sum_{i,j,n>m}(\partial_{i}u_{n})^{\dagger}u_{n}\mathrm{d}x_{i}+(\partial_{i}u_{n})^{\dagger}u_{n}(\partial_{j}u_{m})^{\dagger}u_{m}\mathrm{d}x_{i}\mathrm{d}x_{j}-(\partial_{i}u_{n})^{\dagger}u_{m}(\partial_{j}u_{m})^{\dagger}u_{n}\mathrm{d}x_{i}\mathrm{d}x_{j}\right.$
$\displaystyle\left.+\tfrac{1}{2}(\partial_{i}\partial_{j}u_{n})^{\dagger}u_{n}\mathrm{d}x_{i}\mathrm{d}x_{j}\right|^{2}.$

Multiplying out the square and keeping only terms to second order, we then get

$\displaystyle
D^{2}_{\text{Pl}}=1-\left(1+\textstyle\sum_{i,j,n,m}2\mathrm{Re}\,\left((\partial_{i}u_{n})^{\dagger}u_{n}\right)\mathrm{d}x_{i}+(\partial_{i}u_{n})^{\dagger}u_{n}u_{m}^{\dagger}(\partial_{j}u_{m})\mathrm{d}x_{i}\mathrm{d}x_{j}\right.$
$\displaystyle+2\mathrm{Re}\,\left((\partial_{i}u_{n})^{\dagger}u_{n}(\partial_{j}u_{m})^{\dagger}u_{m}\right)_{n>m}\mathrm{d}x_{i}\mathrm{d}x_{j}-2\mathrm{Re}\,\left((\partial_{i}u_{n})^{\dagger}u_{m}(\partial_{j}u_{m})^{\dagger}u_{n}\right)_{n>m}\mathrm{d}x_{i}\mathrm{d}x_{j}$
$\displaystyle+\left.\tfrac{1}{2}\cdot
2\mathrm{Re}\,\left((\partial_{i}\partial_{j}u_{n})^{\dagger}u_{n}\right)\mathrm{d}x_{i}\mathrm{d}x_{j}\right)$

The first order terms vanish because $(\partial_{i}u_{n})^{\dagger}u_{n}$ is
pure imaginary. We can drop the $\mathrm{Re}\,$ on most of the second order
terms because they are products of two such imaginary terms. For the $n>m$
terms, we extend to a sum over all $n,m$ by absorbing a factor of $2$ and
noticing that the $n=m$ terms internally cancel.

$\displaystyle
D^{2}_{\text{Pl}}=-\textstyle\sum_{i,j,n,m}\left((\partial_{i}u_{n})^{\dagger}u_{n}u_{m}^{\dagger}(\partial_{j}u_{m})+(\partial_{i}u_{n})^{\dagger}u_{n}(\partial_{j}u_{m})^{\dagger}u_{m}-(\partial_{i}u_{n})^{\dagger}u_{m}(\partial_{j}u_{m})^{\dagger}u_{n}\right.$
$\displaystyle+\left.\mathrm{Re}\,\left((\partial_{i}\partial_{j}u_{n})^{\dagger}u_{n}\right)\right)\mathrm{d}x_{i}\mathrm{d}x_{j}$

We also have that
$(\partial_{i}u_{n})^{\dagger}u_{n}u_{m}^{\dagger}(\partial_{j}u_{m})=-(\partial_{i}u_{n})^{\dagger}u_{n}(\partial_{j}u_{m})^{\dagger}u_{m}$,
so the first two terms cancel. We are left with

$D^{2}_{\text{Pl}}=-\textstyle\sum_{i,j,n,m}\left(-(\partial_{i}u_{n})^{\dagger}u_{m}(\partial_{j}u_{m})^{\dagger}u_{n}+\mathrm{Re}\,\left((\partial_{i}\partial_{j}u_{n})^{\dagger}u_{n}\right)\right)\mathrm{d}x_{i}\mathrm{d}x_{j}$

Integrating by parts, this can be written

$D^{2}_{\text{Pl}}=-\textstyle\sum_{i,j,n,m}\left((\partial_{i}u_{n})^{\dagger}u_{m}u_{m}^{\dagger}\partial_{j}u_{n}+\mathrm{Re}\,\left(\partial_{j}((\partial_{i}u_{n})^{\dagger}u_{n})-(\partial_{i}u_{n})^{\dagger}\partial_{j}u_{n}\right)\right)\mathrm{d}x_{i}\mathrm{d}x_{j}$

The first term is pure real and the first term inside the $\mathrm{Re}\,$ is
pure imaginary, so this reduces to

$D^{2}_{\text{Pl}}=\textstyle\sum_{i,j,n,m}\mathrm{Re}\,\left((\partial_{i}u_{n})^{\dagger}\partial_{j}u_{n}-(\partial_{i}u_{n})^{\dagger}u_{m}u_{m}^{\dagger}\partial_{j}u_{n}\right)\mathrm{d}x_{i}\mathrm{d}x_{j}$

Switching to Dirac notation, we discern the usual quantum metric

$g_{ij}=\sum_{n,m\in\text{occ}}\mathrm{Re}\,\mathinner{\langle{\partial_{i}u_{n}}|}\left(\mathbb{I}-\mathinner{|{u_{m}}\rangle}\mathinner{\langle{u_{m}}|}\right)\mathinner{|{\partial_{j}u_{n}}\rangle},\quad
D^{2}_{\text{Pl}}=\sum_{i,j}g_{ij}\mathrm{d}x_{i}\mathrm{d}x_{j}$

We thus observe that the usual notion of infinitesimal distance between two
manifolds of states $D^{2}$ corresponds exactly to the distance we defined in
the Plücker embedding,

$\displaystyle D^{2}$ $\displaystyle=$ $\displaystyle D^{2}_{\text{Pl}}$
$\displaystyle=$ $\displaystyle
1-\mathinner{\langle{V(x+\mathrm{d}x)}|{V(x)}\rangle}\mathinner{\langle{V(x)}|{V(x+\mathrm{d}x)}\rangle}.$

To write the quantum metric in the Plücker embedding, we would then go through
the exercise of expanding $V(x+dx)$ and rearranging $D^{2}_{\text{Pl}}$ into
the form $g_{ij}\mathrm{d}x_{i}\mathrm{d}x_{j}$. This however amounts to the
same derivation one would do for the single band case in the standard
formalism. Therefore we irectly infer that the usual quantum metric expressed
in the Plücker embedding is

$g_{ij}=\mathrm{Re}\,\left(\mathinner{\langle{\partial_{i}V}|{\partial_{j}V}\rangle}-\mathinner{\langle{\partial_{i}V}|{V}\rangle}\mathinner{\langle{V}|{\partial_{j}V}\rangle}\right).$
(185)

##  Appendix B Alternative parametrization of
$\widetilde{\mathsf{Gr}}_{2,4}^{\mathbb{R}}$

We start with the parametrization of a generic element of $\mathsf{SO}(4)$ as
de Guise  _et al._ (2018)

$R^{4B}(\theta_{1}\cdots\theta_{6})=e^{\theta_{1}L_{12}}e^{\theta_{2}L_{13}}e^{\theta_{3}L_{14}}e^{\theta_{4}L_{23}}e^{\theta_{5}L_{24}}e^{\theta_{6}L_{34}}\,,$
(186)

with the angular momentum matrices $[L_{ij}]_{\alpha\beta}=-\delta_{\alpha
i}\delta_{\beta j}+\delta_{\alpha j}\delta_{\beta i}$ for all pairs $(i,j)\in
I_{2}=\\{(a,b)|1\leq a<b\leq 4\\}=\\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\\}$,
that form a basis of $\mathsf{so}(4)$ (dim $\mathsf{so}(4)$=6). A direct
inspection shows that $\theta_{6}$ is a pure gauge sign and we thus set
$\theta_{6}=0$. In the end, we need a reduction to a maximum of $4$ principal
angles that parametrize the four-dimensional Grassmannian
$\widetilde{\mathsf{Gr}}_{2,4}^{\mathbb{R}}$. For this we take the wedge
products

$V_{I}=u_{1}\wedge u_{2}\,,\;V_{II}=u_{3}\wedge u_{4}\,,$ (187)

that both define six-dimensional vectors in
$\bigwedge^{2}\left(\mathbb{R}^{4}\right)$ and written in the basis
$\\{\check{\boldsymbol{e}}_{ij}=e_{i}\wedge e_{j}\\}_{(i,j)\in I_{2}}$ (again
with $\\{e_{i}\\}_{i=1}^{4}$ the Cartersian basis of $\mathbb{R}^{4}$). We now
take the linear combinations

$V_{\pm}=V_{I}\pm V_{II}\,,$ (188)

and write these as

$V_{\pm}=\check{\boldsymbol{e}}_{ij}\,[V_{\pm}]_{ij}=\check{\boldsymbol{e}}_{ij}^{\prime}\,[V_{\pm}^{\prime}]_{ij}\,,$
(189)

in the new basis,

$\left[\begin{array}[]{c}\check{\boldsymbol{e}}_{12}^{\prime}\\\
\check{\boldsymbol{e}}_{13}^{\prime}\\\
\check{\boldsymbol{e}}_{14}^{\prime}\\\
\check{\boldsymbol{e}}_{23}^{\prime}\\\
\check{\boldsymbol{e}}_{24}^{\prime}\\\
\check{\boldsymbol{e}}_{34}^{\prime}\end{array}\right]^{\top}=\left[\begin{array}[]{c}\check{\boldsymbol{e}}_{12}\\\
\check{\boldsymbol{e}}_{13}\\\ \check{\boldsymbol{e}}_{14}\\\
\check{\boldsymbol{e}}_{23}\\\ \check{\boldsymbol{e}}_{24}\\\
\check{\boldsymbol{e}}_{34}\end{array}\right]^{\top}\cdot\dfrac{1}{2}\left[\begin{array}[]{rrrrrr}0&1&0&0&-1&0\\\
0&0&1&1&0&0\\\ 1&0&0&0&0&1\\\ 0&1&0&0&1&0\\\ 0&0&1&-1&0&0\\\
1&0&0&0&0&-1\end{array}\right]\,.$ (190)

Then, by setting

$\displaystyle(\theta_{1},\theta_{2})$ $\displaystyle=(0,0)\,,$ (191)
$\displaystyle(\theta_{3},\theta_{4})$
$\displaystyle=\dfrac{1}{2}(\phi_{+}+\phi_{-},\phi_{+}-\phi_{-})\,,$
$\displaystyle\theta_{5}$ $\displaystyle=-\theta+\pi/2\,,$

we finally obtain

$\displaystyle V_{+}^{\prime}$
$\displaystyle=(\sin\phi_{+}\sin\theta,\cos\theta,\cos\phi_{+}\sin\theta,0,0,0)\,,$
(192) $\displaystyle V_{-}^{\prime}$
$\displaystyle=(0,0,0,-\sin\phi_{-}\sin\theta,\cos\theta,\cos\phi_{-}\sin\theta)\,,$

such that each Plücker vector ($V_{\pm}^{\prime}$) defines a two-sphere
($\mathbb{S}^{2}_{\pm}$) within one three-dimensional half of the six-
dimensional vector space $\bigwedge^{2}(\mathbb{R}^{4})=V_{+}\oplus
V_{-}\cong\mathbb{R}^{3}\oplus\mathbb{R}^{3}$, i.e.
$V_{\pm}^{\prime}\in\mathbb{S}^{2}_{\pm}\in V_{\pm}$. We remark that by only
keeping three angles, $V_{\pm}^{\prime}$ do not cover the whole of
$\mathsf{Gr}_{2,4}^{\mathbb{R}}$. Nevertheless, we do capture the two sub-
dimensional spheres contained in the four-dimensional Grassmannian and this is
sufficient to fully characterize the 2D Euler topology, as we now show.

Defining the unit vectors

$\displaystyle\boldsymbol{n}_{+}$
$\displaystyle=(\cos\phi_{+}\sin\theta,\sin\phi_{+}\sin\theta,\cos\theta)\in\mathbb{S}^{2}_{+}\,,$
(193a) $\displaystyle\boldsymbol{n}_{-}$
$\displaystyle=(\cos\phi_{+}\sin\theta,\sin\phi_{+}\sin\theta,\cos\theta)\in\mathbb{S}^{2}_{-}\,,$
and substituting the parameters Eq. (191) in the $2+2$-Hamiltonian form Eq.
(39), we get, after setting $E_{1}=E_{2}=-E_{3}=-E_{4}=-1$, $\displaystyle
H^{\mathbb{R},2+2}[\boldsymbol{n}_{+},\boldsymbol{n}_{-}]=$
$\displaystyle\,n^{3}_{-}\left(-n^{3}_{+}\Gamma_{33}+n^{2}_{+}\Gamma_{31}-n^{1}_{+}\Gamma_{10}\right)$
(193b) $\displaystyle+$
$\displaystyle\,n^{1}_{-}\left(+n^{3}_{+}\Gamma_{13}-n^{2}_{+}\Gamma_{11}-n^{1}_{+}\Gamma_{30}\right)$
$\displaystyle+$
$\displaystyle\,n^{2}_{-}\left(+n^{3}_{+}\Gamma_{01}+n^{2}_{+}\Gamma_{03}+n^{1}_{+}\Gamma_{22}\right)\,,$
$\displaystyle=$
$\displaystyle\,\boldsymbol{n}_{+}^{\top}\cdot\underline{\Gamma}\cdot\boldsymbol{n}_{-}\,,$
with the tensor
$\underline{\Gamma}=\left(\begin{array}[]{rrr}-\Gamma_{30}&\Gamma_{22}&-\Gamma_{10}\\\
-\Gamma_{11}&\Gamma_{03}&\Gamma_{31}\\\
\Gamma_{13}&\Gamma_{01}&-\Gamma_{33}\end{array}\right)\,.$ (193c)

We note that the above derivation differs from the previous ones exposed in
Bouhon  _et al._ (2020b); Bouhon and Slager (2022) by the initial choice of
the parametrization of the $\mathsf{SO}(4)$ matrix representing the frame of
eigenvectors.

If we set

$\displaystyle(\phi_{+},\phi_{-})$
$\displaystyle=(q_{+},q_{-})\,\phi_{0}\,,\;\theta=\theta_{0}\,,$ (194)
$\displaystyle q_{+},q_{-}$ $\displaystyle\in\mathbb{Z}\,,$

such that $\boldsymbol{n}_{+}$ ($\boldsymbol{n}_{-}$) wraps the sphere
$\mathbb{S}^{2}_{+}$ ($\mathbb{S}^{2}_{-}$) a number of times $q_{+}$
($q_{-}$) whenever $(\phi_{0},\theta_{0})$ covers one time the base sphere
$\mathbb{S}^{2}_{0}$. The direct computation of the Euler two-form for the
occupied and unoccupied two-band subspaces then gives

$\displaystyle\mathtt{F}_{\mathbb{S}^{2}_{0}}[\\{u_{1},u_{2}\\}]$
$\displaystyle=-\dfrac{q_{+}+q_{-}}{2}\sin\theta_{0}\,,$ (195)
$\displaystyle\mathtt{F}_{\mathbb{S}^{2}_{0}}[\\{u_{3},u_{4}\\}]$
$\displaystyle=-\dfrac{q_{+}-q_{-}}{2}\sin\theta_{0}\,,$

and accordingly Euler classes

$\displaystyle\chi_{I,\mathbb{S}^{2}_{0}}$
$\displaystyle=-(q_{+}+q_{-})\in\mathbb{Z}\,,$ (196)
$\displaystyle\chi_{II,\mathbb{S}^{2}_{0}}$
$\displaystyle=-(q_{+}-q_{-})\in\mathbb{Z}\,.$

##  Appendix C Orientability of Euler homotopy classes

The fact that we have used the oriented Grassmannian for the Plücker embedding
must now be corrected since the Bloch Hamiltonian are only orientable, and the
phases are classified by the free homotopy set (i.e. no base point), together
allowing the reversal of the orientation through an adiabatic transformation
(automorphism of $\pi_{2}$ by the action of $\pi_{1}$) Bouhon  _et al._
(2020b). The strict homotopy classification gives the following equivalence

$(\chi_{I},\chi_{II})\simeq(-\chi_{I},-\chi_{II})\,.$ (197)

Whenever the partial gap between two bands of the same subspace (that is
closed by nodal points) closes completely (i.e. at every momentum), or in
other words, when a band inversion takes place between these two bands such
that they become fully degenerate, there is yet a further homotopy equivalence
of signed Euler classes, namely

$\chi_{I}\simeq-\chi_{I}\,,\;\chi_{II}\simeq-\chi_{II}\,,$ (198)

see Bouhon and Slager (2022) for a detailed discussion.

There only remains to pullback the base sphere $\mathbb{S}^{2}_{0}$ to the
torus Brillouin zone via the map Eq. (27) such that the resulting $2+2$-Bloch
Hamiltonian has an Euler topology exhaustively captures by

$\displaystyle(\chi_{I,\mathbb{T}^{2}},\chi_{II,\mathbb{T}^{2}})=(\chi_{I,\mathbb{S}^{2}_{0}},\chi_{II,\mathbb{S}^{2}_{0}})\in\mathbb{Z}^{2}/\sim\,,$
(199)

with $\sim$ for the above homotopy equivalences.

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  * Note (3) Notwithstanding a subtle refined homotopy inequivalence $(\chi_{I},\chi_{II})\sim(-\chi_{I},-\chi_{II})\not\sim(\chi_{I},-\chi_{II})\sim(-\chi_{I},\chi_{II})$ discussed in detail in Bouhon and Slager (2022).
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