#  Riemannian geometry of resonant optical responses

Junyeong Ahn<EMAIL_ADDRESS>Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Guang-Yu Guo<EMAIL_ADDRESS>Department of Physics and Center for Theoretical Physics, National Taiwan
University, Taipei 10617, Taiwan  Physics Division, National Center for
Theoretical Sciences, Taipei 10617, Taiwan  Naoto Nagaosa<EMAIL_ADDRESS>RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo
113-8656, Japan  Ashvin Vishwanath<EMAIL_ADDRESS>Department of
Physics, Harvard University, Cambridge, MA 02138, USA

###### Abstract

The geometry of quantum states is well-established as a basis for
understanding the response of electronic systems to static electromagnetic
fields, as exemplified by the theory of the quantum and anomalous Hall
effects. However, it has been challenging to relate quantum geometry to
resonant optical responses. The main obstacle is that optical transitions
involve a pair of states, while existing geometrical properties are defined
for a single state. As a result, a concrete geometric understanding of optical
responses has so far been limited to two-level systems, where the Hilbert
space is completely determined by a single state and its orthogonal
complement. Here, we construct a general theory of Riemannian geometry for
resonant optical processes by identifying transition dipole moment matrix
elements as tangent vectors. This theory applies to arbitrarily high-order
responses, suggesting that optical responses can generally be thought of as
manifestations of the Riemannian geometry of quantum states. We use our theory
to show that third-order photovoltaic Hall effects are related to the Riemann
curvature tensor and demonstrate an experimentally accessible regime where
they dominate the response.

A common feature of geometric responses [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15] is that they originate from inter-band hybridizations, such
that they appear as quantum corrections to semiclassical descriptions. Since
resonant optical responses are inherently inter-band quantum-mechanical
processes, they are ideal candidates for a geometrical interpretation. Indeed,
nonlinear optical responses, in particular, have attracted much attention
recently for this reason [16, 17, 18, 19, 20, 21, 22, 23, 24] in addition to
the possibility of diverse physical applications [25, 26, 27]. Quantized
injection photocurrent responses were found in chiral semimetals [20, 22, 21],
and the shift photovoltaic responses of semimetals have been found to be
described by the Levi-Civita connection [19], a geometric quantity
distinguished from the typical Berry curvature and quantum metric.

However, the study of the relationship between quantum geometry and optical
responses has been limited to the vicinity of crossing points in semimetals,
where two-band descriptions are good approximations [20, 21, 19] or special
selection rules exist due to linear dispersion relations [22]. The fundamental
reason for this is the absence of a proper theory explaining the geometry of
optical transitions; the geometric quantities like the quantum metric and the
Berry curvature are usually defined for a single quantum state [28], while
optical transitions involve a pair of states. While the wave function of one
state completely determines the other in two-band systems, in general the two
quantum states involved in an optical transition are independent, meaning that
the geometry of a pair of states is different from the geometry of a single
state. Additional constraints like the strictly linear dispersion relations
was required for a geometric interpretation of optical responses in more-than-
two-band systems such as chiral multifold fermions [22].

Figure 1:  Geometry of the cell-periodic Bloch state and optical transitions.
a, Cell-periodic Bloch wave function as an $N$ by $N$ unitary matrix. The
limit $N\rightarrow\infty$ is assumed. b, Transition dipole moment matrix
elements as tangent basis vectors. $\hat{e}^{pq}$s and $\hat{e}^{mn}$ are
tangent vectors on the manifold ${\rm U}(N)/{\rm U}(1)^{N}$ of cell-periodic
Bloch states, and $\hat{u}$ presents normal vectors. Here, ${\rm U}(1)^{N}$
corresponds to the gauge degrees of freedom. Optical transitions between $m$
and $n$ states reveal geometrical information on the curve generated by
$\hat{e}^{mn}$. c, Physical meaning of the Hermitian metric, connection, and
curvature in optical processes for non-degenerate states. $R_{a}^{n}$ and
$F^{n}_{ad}$ are the Wannier center and the Berry curvature of the state
$|u_{n}\rangle$.

In this Article, we introduce a general theoretical description of the
Riemannian geometry for resonant optical transitions. We consider independent
electronic quasiparticles at zero-temperature in clean crystals and neglect
photon momentum. Let us note that optical responses are described by the
electric dipole Hamiltonian [29, 30]

$\displaystyle\hat{H}_{\rm int}=e{\bf E}(t)\cdot\hat{\bf r},$ (1)

where $-e$ is the electric charge (Methods 1 summarizes our conventions and
definitions), and $\hat{\bf r}$ is the position operator of electrons, and
${\bf E}(t)$ is the external electric field. Since all information on the
unperturbed quantum states is encoded in the overlap matrices
$\braket{\psi_{m\bf k}}{H_{\rm int}}{\psi_{n\bf k}}$ in perturbative response
theory, the transition dipole moment matrix element (per unit charge)

$\displaystyle{\bf r}_{mn}({\bf k})=\braket{\psi_{m\bf k}}{\hat{\bf
r}}{\psi_{n\bf k}}$ (2)

is the fundamental building block of the $k$th-order nonlinear optical
conductivity tensors [27] defined by (see Methods 2)

$\displaystyle j^{a}(\omega)$
$\displaystyle=\sum_{k}\sum_{a_{i},\omega_{i}}\sigma^{a;a_{1},\ldots,a_{k}}(\omega;\omega_{1},\ldots,\omega_{k})E^{a_{1}}(\omega_{1})\ldots
E^{a_{k}}(\omega_{k}).$ (3)

Our key observation is that the transition dipole moment matrix elements
define tangent vectors on the manifold of quantum states. In differential
geometry, the tangent vector associated with a coordinate $k^{a}$ is written
in an abstract way as $\frac{\partial}{\partial k_{a}}$ for $a=1,\ldots,d$,
where $d$ is the spatial dimension. Therefore, one can expect that the
position operator $\hat{r}^{a}\sim i\partial_{a}\equiv
i\frac{\partial}{\partial k_{a}}$ defines the tangent vector for the momentum
coordinate $k^{a}$ defined on a manifold. Then, defining a metric tensor
defines the Riemannian structure. We make this idea concrete for the
representation of $\hat{r}^{a}$ on the cell-periodic Bloch states. The same
procedure may also be applied to the energy eigenstates in a ${\bf k\cdot p}$
model or a tight-binding model.

This construction allows us to show that resonant optical responses are
geometric in nature. We show that linear optical responses and second-order
bulk photovoltaic effects are manifestations of the metric and connection. In
particular, the non-quantized injection photocurrents in chiral semimetals due
to higher-band corrections to the two-band description [20, 21] or a quadratic
dispersion in chiral multifold fermion models [22] can still be regarded as
geometric responses, although not as topological responses. Furthermore, our
geometric perspective can serve as a useful organizing principle for more
complicated higher-order optical conductivity tensors. As an example, we show
that Riemann curvature characterizes the third-order photovoltaic Hall
conductivity near the band edge of gapped topological materials.

To simplify the presentation of our theory, we mainly consider non-degenerate
states without spacetime inversion $(PT)$ symmetry in the main text. See
Methods 5 and 6 for details on degenerate states and symmetry constraints.

Transition dipole moment as a tangent vector. We begin by revisiting the
geometric meaning of the position operator. The position operator is
represented in the Bloch state basis as [31, 32]

$\displaystyle r^{a}_{mn}({\bf k})=\delta_{mn}i\partial_{a}+\braket{u_{m\bf
k}}{i\partial_{a}}{u_{n\bf k}}.$ (4)

The second term is a geometric quantity widely known as the non-abelian Berry
connection. Here, instead of viewing it as a connection, we regard it as the
Maurer-Cartan form $\xi_{a}({\bf k})=U^{\dagger}({\bf k})i\partial_{a}U({\bf
k})$ (see Methods 4) of the $N\times N$ unitary matrix $U_{\alpha m}({\bf
k})=\braket{{\bf x}_{\alpha}}{u_{m\bf k}}$, where $|{\bf x}_{\alpha}\rangle$
is a position eigenstate within the unit cell with eigenvalue ${\bf
x}_{\alpha}$. Here, we take both the number of bands and the number of
position eigenstates in the unit cell to be a finite number $N$ for
definiteness [Fig. 1(a)]. The limit $N\rightarrow\infty$ is taken in the end.
The Maurer-Cartan form has one-to-one correspondence with the tangent vector
$i\partial_{a}U({\bf k})$ at $U({\bf k})$, so it can be simply regarded as a
tangent vector. After modding out the ${\rm U}(1)$ gauge degrees of freedom of
each state from ${\rm U}(N)$, we obtain the manifold ${\cal M}={\rm U}(N)/{\rm
U}(1)^{N}$ of the cell-periodic Bloch states. The tangent vectors in ${\cal
M}$ are given $\xi^{\cal M}_{a}({\bf k})=U^{\dagger}({\bf
k})i\partial_{a}U({\bf k})-U^{\dagger}({\bf k})i\partial_{a}U({\bf k})|_{{\rm
U}(1)^{N}}$, where $[U^{\dagger}({\bf k})i\partial_{a}U({\bf k})|_{{\rm
U}(1)^{N}}]_{mn}=\delta_{mn}\braket{u_{m\bf k}}{i\partial_{a}}{u_{n\bf k}}$ is
the projection of $\xi_{a}$ along the ${\rm U}(1)^{N}$ direction. In the
operator form, the tangent vector is $\hat{\xi}^{\cal M}_{a}({\bf
k})=\sum_{m\neq n}r^{a}_{mn}({\bf k})|u_{m\bf k}\rangle\langle u_{n\bf k}|$.

Now we arrive at our main result: the transition dipole moment $r^{a}_{mn}$
between $m$ and $n$ states is one particular component of the
$N(N-1)/2$-dimensional complex-valued tangent vector $\hat{\xi}_{a}^{\cal M}$
(i.e., $r^{a}_{mn}$ is a complex-valued vielbein). Namely, an optical
transition between $m$ and $n$ states probes the one-dimensional complex
vector space spanned by

$\displaystyle\hat{e}^{mn}_{a}({\bf k})\equiv r^{a}_{mn}({\bf k})|u_{m\bf
k}\rangle\langle u_{n\bf k}|,\quad a=1,\ldots,d,$ (5)

with given $m$ and $n$ with $m\neq n$ [Fig. 1(b)]. In the following, we
describe the Riemannian structure of the subspace spanned by
$\hat{e}^{mn}_{a}$s. Here, at most two $\hat{e}^{mn}_{a=1,\ldots,d}$s are
linearly independent for a given pair $(m,n)$ because one complex dimension is
fully covered by two independent real coordinates.

Complex Riemannian structure. The complex Riemannian structure is induced by
the natural inner product in the space of $N\times N$ matrices, called the
Hilbert-Schmidt inner product:

$\displaystyle(A,B)=\sum_{\alpha,\beta}A^{*}_{\alpha\beta}B_{\alpha\beta}={\rm
Tr}\left[A^{\dagger}B\right].$ (6)

It is the complex Euclidean inner product with $|{\bf
x}_{\alpha}\rangle\langle{\bf x}_{\beta}|$s as basis vectors (equivalently,
$|u_{p\bf k}\rangle\langle u_{q\bf k}|$s at any fixed ${\bf k}$ as basis
vectors). This inner product is Hermitian as it satisfies $(B,A)^{*}=(A,B)$.
The Hermitian metric tensor (a.k.a. quantum geometric tensor) in the tangent
subspace spanned by $\hat{e}^{mn}_{a}$s is defined by Hilbert-Schmidt inner
product of the tangent basis vectors:

$\displaystyle
Q^{mn}_{ba}\equiv(\hat{e}^{mn}_{b},\hat{e}^{mn}_{a})=r^{b}_{nm}r^{a}_{mn}.$
(7)

This metric is identical to the Fubini-Study metric of the state
$|u_{n}\rangle$ in two-band systems, but the two are generally different; the
Fubini-Study metric is given by summing $Q^{mn}_{ba}$ over all possible
intermediate states $|u_{m}\rangle$.

The covariant derivative of $\hat{e}^{mn}_{a}$s define other geometric
quantities such as the connection and curvature. By taking a derivative on
$\hat{e}^{mn}_{a}$, one obtains parallel and perpendicular components:
$\partial_{c}\hat{e}^{mn}_{a}=\sum_{b}(C^{mn})^{b}_{ca}\hat{e}^{mn}_{b}+\ldots$
where the ellipsis indicates the components perpendicular to
$\hat{e}^{mn}_{b}$s. [Fig. 1]. Then, the covariant derivative $\nabla$ is
defined as the parallel-transported part by
$\nabla_{c}\hat{e}^{mn}_{a}=\sum_{b}(C^{mn})^{b}_{ca}e_{b}$. The Hermitian
connection is defined by

$\displaystyle
C^{mn}_{bca}\equiv\sum_{e}Q^{mn}_{be}(C^{mn})^{e}_{ca}=(\hat{e}^{mn}_{b},\nabla_{c}\hat{e}^{mn}_{a})=r^{b}_{nm}r^{a}_{mn,c},$
(8)

where $O_{mn,c}\equiv\partial_{c}O_{mn}-i[{\cal A}_{c},O]_{mn}$ is called the
generalized derivative [29], where $({\cal
A}_{c})_{mn}=\delta_{mn}\braket{u_{m}}{i\partial_{c}}{u_{n}}$ is the ${\rm
U}(1)^{N}$ Berry connection. Our construction gives a definite geometric
meaning to the generalized derivative. This Hermitian connection is in general
different from the Levi-Civita connection, the unique torsionless connection
determined by the metric, because it has a nontrivial torsion
$T^{mn}_{bca}=C^{mn}_{bca}-C^{mn}_{bac}=ir^{b}_{nm}\sum_{p\neq
m,n}\left(r^{c}_{mp}r^{a}_{pn}-r^{a}_{mp}r^{c}_{pn}\right)$ originating from
virtual transitions among three states. The Hermitian curvature tensor is
defined by antisymmetrizing the second-order covariant derivatives.

$\displaystyle
K^{mn}_{badc}\equiv(\hat{e}^{mn}_{b},(\nabla_{d}\nabla_{c}-\nabla_{c}\nabla_{d})\hat{e}^{mn}_{a})=-ir^{b}_{nm}[{\cal
F}_{dc},r^{a}]_{mn},$ (9)

where $({\cal F}_{dc})_{pq}=\partial_{d}({\cal A}_{c})_{pq}-\partial_{c}({\cal
A}_{d})_{pq}$ is the ${\rm U}(1)^{N}$ Berry curvature.

Meaning of metric, connection, and curvature in optical processes. $Q$, $C$,
and $K$ defined here are properties of a one-dimensional complex vector space.
Their real part define the Riemannian metric tensor (a.k.a quantum metric
tensor), metric connection, and the Riemann curvature tensor of the
corresponding two-dimensional real vector space spanned by ${\rm
Re}[\hat{e}_{a}]$s and ${\rm Im}[\hat{e}_{a}]$s. On the other hand, the
(minus) imaginary parts define the symplectic form, almost symplectic
connection, and the symplectic curvature tensor [33]. All are gauge invariant.

Figure 2:  Third-order photovoltaic Hall conductivity of a Dirac fermion. a,
Band structure of a massive Dirac fermion. The blue shaded region shows the
occupied states. We take the asymptotic velocity $v=8\times 10^{6}\;{\rm m/s}$
and the band gap $2|m|=17\;{\rm meV}$ for both 2D and 3D Dirac fermions. b and
c, Two dimensional Dirac fermion. Incident light is circularly polarized in b
and linearly polarized in c. d, Three dimensional Dirac fermion. The
conductivity of a single Weyl fermion is the half of that in d. We take the
relaxation rate of $\hbar\Gamma=1\;{\rm meV}$ in all calculations. All non-
vanishing tensor components not shown in the figures are related to the shown
components by rotational symmetries.

As the geometric quantities defined above are given by the transition dipole
moment and its gauge-invariant derivatives, they are basic building blocks of
the optical conductivity tensors. The Hermitian metric appears in the linear
optical conductivity tensor as

$\displaystyle\sigma^{b;a}=\frac{\pi\omega e^{2}}{h}\sum_{m,n}\int_{\bf
k}\delta\left(\omega-\omega_{mn}\right)f_{nm}Q^{mn}_{ba},$ (10)

where $\int_{\bf k}=\int d^{d}k/(2\pi)^{d}$, $\hbar\omega_{mn}$ is the energy
difference between $m$ and $n$ bands, and $f_{nm}=f_{n}-f_{m}$ is the
difference between the Fermi-Dirac distribution of the $n$ and $m$ states.

Also, the Hermitian metric is directly related to the optical transition rate
by the Fermi’s Golden rule since $|\braket{\psi_{m\bf k}}{e\hat{\bf
r}\cdot{\bf E}(\omega)}{\psi_{n\bf k}}|^{2}\propto
Q^{mn}_{ba}E_{b}(\omega)E_{a}^{*}(\omega)$. This relation is particularly
useful for the interpretation of geometric quantities [Fig. 1(c)]. For
example, let us consider the injection photovoltaic effect, the photocurrent
generation by the optical transition of the electron velocity. The
corresponding second-order conductivity tensor can be expressed in terms of
the Hermitian metric as

$\displaystyle\sigma^{c;ab}_{\rm inj}=-\frac{\pi
e^{3}}{\hbar^{2}\Gamma}\sum_{m,n}\int_{\bf
k}\delta\left(\omega-\omega_{mn}\right)f_{nm}Q^{mn}_{ba}(v^{c}_{mm}-v^{c}_{nn}),$
(11)

where $v^{c}_{mm}$ and $v^{c}_{nn}$ are the group velocities of bands $m$ and
$n$, respectively, which involve the band dispersion.

To explain the Hermitian connection, we write it as
$C_{bca}=r^{b}_{nm}r^{a}_{mn,c}=-ir^{b}_{nm}r^{a}_{mn}R^{c,a}_{mn}$, where
$R^{c,a}_{mn}={\cal A}^{c}_{mm}-{\cal A}^{c}_{nn}+i\partial_{c}\log
r^{a}_{mn}$ is the shift vector describing the shift of the electron position
during the optical excitation (note that $R^{c,a}_{mn}\sim
r^{c}_{mm}-r^{c}_{nn}$) [25, 34]. As $r^{b}_{nm}r^{a}_{mn}$ is proportional to
the transition rate, $C_{bca}$ is responsible for the shift photovoltaic
effect, a generation of the direct current $j^{c}_{\rm shift}\propto\partial
R^{c}/dt$ by illumination of light:

$\displaystyle\sigma^{c;ab}_{\rm shift}=-\frac{\pi
e^{3}}{2\hbar^{2}}\sum_{m,n}\int_{\bf
k}\delta\left(\omega-\omega_{mn}\right)f_{nm}i(C^{mn}_{bca}-(C^{mn}_{acb})^{*}).$
(12)

While the above relations generalize the ones found in two-band systems [20,
21, 19], the optical manifestation of the Hermitian curvature has not been
known even in two-band systems. However, we can also interpret the Hermitian
curvature $K_{badc}=-ir^{b}_{nm}[{\cal
F}_{dc},r^{a}]_{mn}=-ir^{b}_{nm}r^{a}_{mn}({\cal F}^{dc}_{mm}-{\cal
F}^{dc}_{nn})$ in the same vein as the optical transition of the Berry
curvature [Fig. 1(c)]. Since the Berry curvature is the source of the Hall
effect, the Hermitian curvature is expected to be responsible for the light-
induced dc Hall effect, also called the photovoltaic Hall effect [35], which
grows linearly in time with constant light intensity. At finite relaxation
rate $\Gamma$ of electronic quasiparticles, the saturated photovoltaic Hall
conductivity is proportional to $\Gamma^{-1}$; we call this as the injection
photovoltaic Hall conductivity following Ref. [36]. We elaborate more on this
response below.

Hermitian curvature in the photovoltaic Hall effect. The expression of the
third-order injection conductivity tensor in time-reversal-symmetric systems
was derived in Ref. [36]. When generalized to include time-reversal-breaking
systems, the third-order injection conductivity tensor contains the Hermitian
curvature.

$\displaystyle\sigma^{d;abc}_{\rm inj}$ $\displaystyle=\frac{\pi
e^{4}}{6\Gamma\hbar^{3}}\sum_{m,n}\int_{\bf
k}\delta\left(\omega-\omega_{mn}\right)f_{nm}iK^{mn}_{cbad}+\ldots,$ (13)

where the index $a$ is for the static electric field, indices $b$ and $c$ are
for the oscillating electric field of light, the ellipsis includes the second-
order connection, connection, metric, and virtual transitions among three
states (see Methods 3). The real (imaginary) part of the tensor is responsible
for the response independent (dependent) on the light helicity, which we call
linear (circular) photoconductivity. The photovoltaic Hall response is
characterized by the anti-symmetric part $\sigma^{[d;a]bc}_{\rm
inj}=(\sigma^{d;abc}_{\rm inj}-\sigma^{a;dbc}_{\rm inj})/2$. In the clean
limit where $\hbar\Gamma$ is much smaller than the photon energy and band
gaps, injection response is the largest contribution to the photovoltaic Hall
response.

Response | Jerk | Injection | Shift
---|---|---|---
Linear | Linear | Circular | Linear | Circular
Photovoltaic Hall effect | No | Yes
$T$ or $PT$ symmetry | Yes | No | Yes | Yes | No
Table 1:  Properties of the third-order photoconductivity tensors. Following
Ref. [36], the third-order photoconductivity is classified into jerk,
injection, and shift according to their dependence on the relaxation rate;
they are proportional to $\Gamma^{-2}$, $\Gamma^{-1}$, and $\Gamma^{0}$,
respectively. Linear (circular) means the light-helicity-independent
(-dependent) response. $T$ and $P$ indicate time reversal and spatial
inversion. Since third-order optical conductivity tensors are invariant under
spatial inversion, time-reversal-symmetric responses are spacetime-inversion-
symmetric also.

In time-reversal-symmetric systems, the injection response depends on the
helicity of the circularly polarized light [Table. 1]. Similarly, spacetime-
inversion symmetry also allows only circular photoconductivity because the
third-order optical conductivity tensor is invariant under spatial inversion.
On the other hand, when time reversal symmetry and spacetime inversion
symmetry are both broken, linearly polarized light can also induce injection
photovoltaic Hall conductivity.

Figure 3:  First-principles calculations on massive Dirac materials. Band
structures of (a) monolayer germanene and ferromagnetic (LaOsO3)2 bilayer and
(b) bulk Bi2Se3 along the high-symmetry lines. The top of valence bands is set
to $0\;{\rm eV}$. c, Third-order circular photovoltaic Hall conductivity
tensors. Black arrows point to the sign change of the third-order conductivity
tensor. d, Third-order linear photovoltaic Hall conductivity tensor. Only
(LaOsO3)2 bilayer shows nonzero response. The Gaussian broadening of $5\;{\rm
meV}$ is introduced to numerically calculate the the delta function. We take
the relaxation energy scale to be $\hbar\Gamma=1{\rm\;meV}$ in all cases.

Equation (13) in general has a complicated form including various geometric
quantities. However, as we show now, the Hermitian curvature dominates the
photovoltaic Hall response near the band edge of topological materials such as
topological insulators and massive Dirac semimetals, characterized by massive
Dirac Hamiltonians.

Let us consider the model of a two-dimensional Dirac fermion [Fig. 2(a)].

$\displaystyle H({\bf k})=\hbar
v(k_{x}\sigma_{x}+k_{y}\sigma_{y})+m\sigma_{z}.$ (14)

As the nonzero mass term break time reversal $T=i\sigma_{y}K$ and spacetime
inversion $PT=\sigma_{x}K$ symmetries, both linear and circular photovoltaic
Hall effects can occur.

We first consider circularly polarized light. Figure 2(b) shows the third-
order photovoltaic Hall conductivity tensors calculated with
$\hbar\Gamma=1\;{\rm meV}$, $v=8\times 10^{5}\;{\rm m/s}$, and $2|m|=17\;{\rm
meV}$, which are relevant to graphene on a hexagonal boron nitride substrate
[37]. The Hermitian curvature dominates the response near the band edge, while
the other contributions having the opposite sign grow as the photon energy
goes higher. This leads to the sign change of the third-order conductivity at
$2\sqrt{2}|m|=24\;{\rm meV}$. Since the circular photovoltaic Hall
conductivity is $T$\- and $PT$-symmetric, it is independent of the sign of the
mass $m$. Note that the response at high photon energies follows Oka and
Aoki’s result [35] for massless graphene. Therefore, the opposite sign of the
response at the band edge is an experimentally observable signature that the
response is due to the Hermitian curvature and is not by the Oka-Aoki
mechanism.

On the other hand, the linear photovoltaic Hall conductivity tensor changes
sign as the mass sign changes, reflecting the fact that the linear
photovoltaic Hall effect is a time-reversal-breaking effect [Fig. 2(c)].
Another feature of the linear photovoltaic Hall conductivity in Fig. 2(c) is
that no sign change occurs as the photon energy increases above the band gap.
This is because the response is purely from the Hermitian curvature in our
model.

Three-dimensional massive Dirac fermions show responses similar to two-
dimensional Dirac fermions [Fig. 2(d)]. However, their linear photovoltaic
Hall conductivity is zero because of spacetime inversion symmetry. See Methods
8 for analytic expressions of conductivity tensors.

We test our model-based predictions by performing first-principles
calculations for monolayer germanene, bulk Bi2Se3, and (LaOsO3)2 bilayer [Fig.
3]. These materials are ${\mathbb{Z}}_{2}$ topological insulators in two and
three dimensions [38, 39], and a Chern insulator [40], respectively, described
by a massive Dirac Hamiltonian close to the band edge. As expected from model
calculations, the Hermitian curvature dominates the response near the band
edge, and the third-order circular photovoltaic Hall conductivity changes sign
as the photon energy increases [Fig. 3(c)]. Ferromagnetic (LaOsO3)2 bilayer
can additionally show the linear photovoltaic Hall effect because time
reversal symmetry is broken [Fig. 3(d)]. Our calculations demonstrate that the
Hermitian curvature is a useful measure of the photovoltaic Hall response in
topological materials.

Topological optical response. Given an understanding of geometry, one
interesting question is about its topology. Topological invariants can be
defined using standard methods in Riemannian geometry. Recall the target
geometry probed by the transition between a specific pair of states is complex
one dimension. The Euler number is then a natural topological invariant to
consider because it classifies all one-dimensional closed complex manifolds
(but note that we probe a one-dimensional complex subtangent space, which may
not be the tangent space of a one-dimensional complex manifold. See
Supplementary Note 1 for more discussions.). According to the Gauss-Bonnet
theorem, the Euler number for the transition between $n$ and $m$ states is
given by

$\displaystyle\chi^{mn}=\frac{1}{2\pi}\oint dk_{1}\wedge
dk_{2}\frac{R^{mn}_{1212}}{\sqrt{g}},$ (15)

where $k_{1}$ and $k_{2}$ are two independent coordinates parametrizing a two-
dimensional submanifold in momentum space, $dk_{1}\wedge
dk_{2}=dk_{1}dk_{2}{\rm sgn}(-{\rm Im}Q^{mn}_{12})$ is the oriented area where
the sign determines the orientation of the mapping from momentum space to the
manifold of quantum states, $g\equiv\det
g^{mn}=g^{mn}_{11}g^{mn}_{22}-(g^{mn}_{12})^{2}$, $g^{mn}_{ij}={\rm
Re}Q^{mn}_{ij}$ is the Riemannian metric, and $R^{mn}_{1212}={\rm
Re}K^{mn}_{1212}$ is the Riemann curvature tensor. This invariant is relevant
only in three dimensions because optical transitions in $d$ spatial dimensions
occurs over $(d-1)$-dimensional submanifolds in momentum space.

We note that the integrand in Eq. (15) cannot be written as a polynomial
function of geometric quantities defined for the transition between $n$ and
$m$ states. This property implies that the topological optical responses are
not perturbative responses in general, unless constraints on the number of
bands or the spectrum are imposed as for the low-energy models of chiral
fermions.

To see which non-perturbative optical response is characterized by
$\chi^{mn}$, we relate it to the first Chern number $c_{1}$ of $n$ and $m$
bands by

$\displaystyle\chi^{mn}=c^{n}_{1}-c^{m}_{1}.$ (16)

Since $c_{1}$ is proportional to the Hall conductivity, equation (16) shows
that the photovoltaic Hall effect by the complete population inversion between
$n$ and $m$ bands is an example of quantized optical responses due to
$\chi^{mn}$. Therefore, the complete population inversion by extreme optical
pumping is a universal mechanism for topological optical responses.

The above features of topological optical responses appear in $PT$-symmetric
systems also, where the reality or symplecticity condition is imposed on the
geometry. The relevant topological invariants are non-polynomial functions of
geometric quantities and have relations similar to Eq. (16). See Methods 9 and
Supplementary Notes 1-3 for details.

Conclusion. Our interpretation of transition dipole moment matrix elements as
tangent vectors provides a concrete conceptual ground for understanding
quantum geometry of electromagnetic responses in a unified way. This is
because geometric responses to static electromagnetic fields originate from
virtual transitions, which are also described by transition dipole moments
(see Methods 10 for details). For example, as we already note above, the
Fubini-Study metric of a state $|u_{n}\rangle$ is given by summing the
Hermitian metric $Q^{mn}$ over all possible intermediate state
$|u_{m}\rangle$. This unified geometric perspective helps us relate seemingly
different physical phenomena. Since the geometric superfluid weight and the
linear optical conductivity are described by the quantum metric, one may
expect that there exists a physical relation between the two. Indeed, they are
intimately related by the optical frequency sum rule [41, 42, 43]. Similar
relations are expected for other geometric responses also. Optical
measurements are thus promising universal probes of quantum geometry, in
practice with a reasonable cutoff on the number of intermediate states for
static responses.

Moreover, our theoretical approach can be applied beyond electromagnetic
responses to understand other geometric responses such as the photovoltaic
thermal Hall and Seebeck effects. It is also possible to study the real-space
quantum geometry by identifying momentum operators as tangent vectors. We
leave these directions for future studies.

Acknowledgements
We appreciate Eslam Khalaf and Daniel Parker for helpful discussions and thank
Maine Christos for useful comments on the manuscript. J.A. was supported by
the Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education (Grant No.
2020R1A6A3A03037129). J.A. and A.V. were supported by the Center for
Advancement of Topological Semimetals, an Energy Frontier Research Center
funded by the US Department of Energy Office of Science, Office of Basic
Energy Sciences, through the Ames Laboratory under contract No. DE-
AC02-07CH11358. G.-Y. G. acknowledges the support from the Ministry of Science
and Technology and National Center for Theoretical Sciences in Taiwan and
thanks the National Center for High-performance Computing in Taiwan for the
computing time. N.N. was supported by JST CREST Grant Number JPMJCR1874 and
JPMJCR16F1, Japan, and JSPS KAKENHI Grant Number 18H03676.

Author contributions
J.A. conceived the original idea and performed theoretical analysis. G.-Y.G.
performed first-principles calculations. N.N. and A.V. supervised the project.
All authors discussed results and contributed to the formulation of theory and
writing of the manuscript.

Competing interests
The authors declare no competing financial interests.

Additional information
Correspondence and requests for materials may be addressed to any of the
authors.

## References

  * [1] Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M.  Quantized Hall conductance in a two-dimensional periodic potential.  _Phys. Rev. Lett._ 49, 405 (1982).
  * [2] Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P.  Anomalous Hall effect.  _Rev. Mod. Phys._ 82, 1539 (2010).
  * [3] Xiao, D., Chang, M.-C. & Niu, Q.  Berry phase effects on electronic properties.  _Rev. Mod. Phys._ 82, 1959 (2010).
  * [4] Sodemann, I. & Fu, L.  Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials.  _Phys. Rev. Lett._ 115, 216806 (2015).
  * [5] Hasan, M. Z. & Kane, C. L.  Colloquium: Topological insulators.  _Rev. Mod. Phys._ 82, 3045 (2010).
  * [6] Armitage, N. P., Mele, E. J. & Vishwanath, A.  Weyl and Dirac semimetals in three-dimensional solids.  _Rev. Mod. Phys._ 90, 015001 (2018).
  * [7] Neupert, T., Chamon, C. & Mudry, C.  Measuring the quantum geometry of Bloch bands with current noise.  _Phys. Rev. B_ 87, 245103 (2013).
  * [8] Peotta, S. & Törmä, P.  Superfluidity in topologically nontrivial flat bands.  _Nat. Commun._ 6, 8944 (2015).
  * [9] Xie, F., Song, Z., Lian, B. & Bernevig, B. A.  Topology-bounded superfluid weight in twisted bilayer graphene.  _Phys. Rev. Lett._ 124, 167002 (2020).
  * [10] Lapa, M. F. & Hughes, T. L.  Semiclassical wave packet dynamics in nonuniform electric fields.  _Phys. Rev. B_ 99, 121111 (2019).
  * [11] Gao, Y. & Xiao, D.  Nonreciprocal directional dichroism induced by the quantum metric dipole.  _Phys. Rev. Lett._ 122, 227402 (2019).
  * [12] Zhao, Y., Gao, Y. & Xiao, D.  Electric polarization in inhomogeneous crystals.  _Preprint at https://arxiv.org/abs/2009.09306_ (2020).
  * [13] Kozii, V., Avdoshkin, A., Zhong, S. & Moore, J. E.  Intrinsic anomalous Hall conductivity in non-uniform electric field.  _Preprint at https://arxiv.org/abs/2010.07322_ (2020).
  * [14] Gao, Y., Yang, S. A. & Niu, Q.  Geometrical effects in orbital magnetic susceptibility.  _Phys. Rev. B_ 91, 214405 (2015).
  * [15] Rhim, J.-W., Kim, K. & Yang, B.-J.  Quantum distance and anomalous Landau levels of flat bands.  _Nature_ 584, 59–63 (2020).
  * [16] Hosur, P.  Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response.  _Phys. Rev. B_ 83, 035309 (2011).
  * [17] Morimoto, T. & Nagaosa, N.  Topological nature of nonlinear optical effects in solids.  _Sci. Adv._ 2, e1501524 (2016).
  * [18] Nagaosa, N. & Morimoto, T.  Concept of quantum geometry in optoelectronic processes in solids: Application to solar cells.  _Adv. Mater._ 29, 1603345 (2017).
  * [19] Ahn, J., Guo, G.-Y. & Nagaosa, N.  Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals.  _Phys. Rev. X_ 10, 041041 (2020).
  * [20] de Juan, F., Grushin, A. G., Morimoto, T. & Moore, J. E.  Quantized circular photogalvanic effect in Weyl semimetals.  _Nat. Commun._ 8, 15995 (2017).
  * [21] de Juan, F. _et al._ Difference frequency generation in topological semimetals.  _Phys. Rev. Research_ 2, 012017 (2020).
  * [22] Flicker, F. _et al._ Chiral optical response of multifold fermions.  _Phys. Rev. B_ 98, 155145 (2018).
  * [23] Holder, T., Kaplan, D. & Yan, B.  Consequences of time-reversal-symmetry breaking in the light-matter interaction: Berry curvature, quantum metric, and diabatic motion.  _Phys. Rev. Research_ 2, 033100 (2020).
  * [24] Watanabe, H. & Yanase, Y.  Chiral photocurrent in parity-violating magnet and enhanced response in topological antiferromagnet.  _Phys. Rev. X_ 11, 011001 (2021).
  * [25] Sturman, B. I. & Fridkin, V. M.  _The Photovoltaic and Photo-refractive Effects in Noncentrosymmetric Materials_ , vol. 8 (CRC Press, 1992).
  * [26] Tokura, Y. & Nagaosa, N.  Nonreciprocal responses from non-centrosymmetric quantum materials.  _Nat. Commun._ 9, 3740 (2018).
  * [27] Boyd, R. W.  _Nonlinear optics_ (Academic press, 2020).
  * [28] Provost, J. P. & Vallee, G.  Riemannian structure on manifolds of quantum states.  _Commun. Math. Phys_ 76, 289–301 (1980).
  * [29] Aversa, C. & Sipe, J. E.  Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis.  _Phys. Rev. B_ 52, 14636 (1995).
  * [30] Ventura, G., Passos, D., dos Santos, J. L., Lopes, J. V. P. & Peres, N.  Gauge covariances and nonlinear optical responses.  _Phys. Rev. B_ 96, 035431 (2017).
  * [31] Karplus, R. & Luttinger, J.  Hall effect in ferromagnetics.  _Phys. Rev._ 95, 1154 (1954).
  * [32] Blount, E.  Formalisms of band theory.  In _Solid state physics_ , vol. 13, 305–373 (Elsevier, 1962).
  * [33] Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J. & Schwachhöfer, L.  Symplectic connections.  _Int. J. Geom. Methods Mod. Phys._ 3, 375–420 (2006).
  * [34] Sipe, J. E. & Shkrebtii, A. I.  Second-order optical response in semiconductors.  _Phys. Rev. B_ 61, 5337–5352 (2000).
  * [35] Oka, T. & Aoki, H.  Photovoltaic Hall effect in graphene.  _Phys. Rev. B_ 79, 081406 (2009).
  * [36] Fregoso, B. M.  Bulk photovoltaic effects in the presence of a static electric field.  _Phys. Rev. B_ 100, 064301 (2019).
  * [37] Kim, H. _et al._ Accurate gap determination in monolayer and bilayer graphene/h-BN Moiré superlattices.  _Nano Lett._ 18, 7732–7741 (2018).
  * [38] Acun, A. _et al._ Germanene: the germanium analogue of graphene.  _J. Phys. Condens. Matter_ 27, 443002 (2015).
  * [39] Xia, Y. _et al._ Observation of a large-gap topological-insulator class with a single dirac cone on the surface.  _Nat. Phys._ 5, 398–402 (2009).
  * [40] Chandra, H. K. & Guo, G.-Y.  Quantum anomalous hall phase and half-metallic phase in ferromagnetic (111) bilayers of 4 d and 5 d transition metal perovskites.  _Phys. Rev. B_ 95, 134448 (2017).
  * [41] Hazra, T., Verma, N. & Randeria, M.  Bounds on the superconducting transition temperature: Applications to twisted bilayer graphene and cold atoms.  _Physical Review X_ 9, 031049 (2019).
  * [42] Verma, N., Hazra, T. & Randeria, M.  Optical spectral weight, phase stiffness and tc bounds for trivial and topological flat band superconductors.  _Preprint at https://arxiv.org/abs/2103.08540_ (2021).
  * [43] Ahn, J. & Nagaosa, N.  Superconductivity-induced spectral weight transfer due to quantum geometry.  _Phys. Rev. B_ 104, L100501 (2021).

Methods
1\. Conventions and Definitions.

  * •

$-e$ is the electron charge. $e>0$.

  * •

Typical quantities in optical conductivity tensors.

$\displaystyle\partial_{a}$ $\displaystyle=\frac{\partial}{\partial_{k_{a}}},$
$\displaystyle\int_{\bf k}$ $\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}},$
$\displaystyle\hat{H}_{0}$ $\displaystyle=\text{single-electron Hamiltonian
without external field}$ $\displaystyle E_{n}$
$\displaystyle=\braket{\psi_{n\bf k}}{\hat{H}_{0}}{\psi_{n\bf
k}}=\braket{u_{n\bf k}}{\hat{H}_{0}({\bf k})}{u_{n\bf k}},$ $\displaystyle
v^{a}_{mn}$ $\displaystyle=\frac{i}{\hbar}\braket{\psi_{m\bf
k}}{[\hat{r}^{a},\hat{H}_{0}]}{\psi_{n\bf k}}=\hbar^{-1}\braket{u_{m\bf
k}}{\partial_{a}\hat{H}_{0}({\bf k})}{u_{n\bf k}},$ $\displaystyle f_{m}$
$\displaystyle=\text{Fermi-Dirac distribution of the $m$th band},$
$\displaystyle\omega_{mn}$ $\displaystyle=\hbar^{-1}\left(E_{m}-E_{n}\right),$
$\displaystyle f_{mn}$ $\displaystyle=f_{m}-f_{n},$ $\displaystyle r^{a}_{mn}$
$\displaystyle=i\delta_{mn}\partial_{a}+\braket{u_{m\bf
k}}{i\partial_{a}}{u_{n\bf k}},$ $\displaystyle({\cal A}_{a})_{mn}$
$\displaystyle=\braket{u_{m\bf k}}{i\partial_{a}}{u_{n\bf
k}}\delta_{E_{m},E_{n}}$ $\displaystyle{\cal D}_{a}$
$\displaystyle=\partial_{a}-i{\cal A}_{a},$ $\displaystyle r^{a}_{mn;c}$
$\displaystyle=[{\cal D}_{c},r^{a}]_{mn},$ $\displaystyle{\cal F}_{ba}$
$\displaystyle=i[{\cal D}_{b},{\cal D}_{a}]=\partial_{b}{\cal
A}_{a}-\partial_{a}{\cal A}_{b}-i[{\cal A}_{b},{\cal A}_{a}].$ (17)

  * •

Definition of geometric quantities in momentum space defined for a pair of
energy levels $E_{n}$ and $E_{m}$. We consider degenerate states
$|u_{n_{i}}\rangle$s with $E_{n_{i}}=E_{n}$ and $|u_{m_{j}}\rangle$s with
$E_{m_{j}}=E_{m}$.

$\displaystyle Q_{ba}$
$\displaystyle=\sum_{i,j}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i}}\equiv
g_{ba}-iF_{ba}/2,$ $\displaystyle C_{abc}$
$\displaystyle=\sum_{i,j}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i},c}\equiv\Gamma_{bca}-i\tilde{\Gamma}_{bca},$
$\displaystyle K_{badc}$
$\displaystyle=\sum_{i,j}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i},dc}-r^{b}_{nm}r^{a}_{mn,cd}\equiv
R_{badc}-i\tilde{R}_{badc},$ $\displaystyle D_{badc}$
$\displaystyle=\sum_{i,j}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i},dc}.$ (18)

  * •

Name of geometric quantities

$\displaystyle Q_{ba}=g_{ba}-iF_{ba}/2$ Hermitian metric $\displaystyle
g_{ba}$ Riemannian metric $\displaystyle F_{ba}/2$ symplectic form
$\displaystyle C_{bca}=\Gamma_{bca}-i\tilde{\Gamma}_{bca}$ Hermitian
connection $\displaystyle\Gamma_{bca}$ metric connection
$\displaystyle\tilde{\Gamma}_{bca}$ symplectic connection $\displaystyle
K_{badc}=R_{badc}-i\tilde{R}_{badc}$ Hermitian curvature $\displaystyle
R_{badc}$ Riemann curvature tensor $\displaystyle\tilde{R}_{badc}$ symplectic
curvature tensor $\displaystyle D_{badc}$ second-order connection (19)

2\. Nonlinear conductivity tensors. By perturbatively expanding the current
expectation value in terms of external electric fields, one obtains

$\displaystyle j^{a}(t)$
$\displaystyle=-e\sum_{k}\sum_{a_{1},\ldots,a_{k}}\left[\prod_{i=1}^{k}\int^{t_{i+1}}_{-\infty}\frac{ie}{\hbar}E^{a_{i}}(t_{i})\right]$
$\displaystyle\qquad\times\braket{[\hat{r}^{a_{k}},\ldots[\hat{r}^{a_{1}},\hat{v}^{a}]]},$
(20)

where $t_{k+1}=t$ at each order $k$. The Fourier components of the $k$th-order
optical conductivity tensors $\tilde{\sigma}^{a;a_{1},\ldots,a_{k}}$ are
defined by

$\displaystyle j^{a}(\Omega)$
$\displaystyle=\sum_{k}\sum_{\begin{subarray}{c}a_{1},\ldots,a_{k}\\\
\omega_{1},\ldots,\omega_{k}\end{subarray}}E^{a_{1}}(\omega_{1})\ldots
E^{a_{k}}(\omega_{k})$
$\displaystyle\qquad\times\tilde{\sigma}^{a;a_{1},\ldots,a_{k}}(\Omega;\omega_{1},\ldots,\omega_{k})\delta_{\Omega,\sum_{i=1}^{k}\omega_{i}},$
(21)

where we Fourier transform the current and electric fields using
$f(t)=\sum_{\omega}f(\omega)e^{-i\omega t}$, and the Fourier component
satisfies $f(-\omega)=[f(\omega)]^{*}$ because $f(t)$ we consider is real
valued. The conductivity tensor (unsymmetrized) has the form [30]

$\displaystyle\tilde{\sigma}^{a;a_{1},\ldots,a_{k}}\left(\Omega,\omega_{1},\ldots,\omega_{k}\right)=-e\left(\frac{ie}{\hbar}\right)^{k}\times$
$\displaystyle\int_{\bf k}\sum_{n}f_{n}\langle\psi_{n\bf
k}|[\hat{r}^{a_{k}},G_{k}\circ\ldots[\hat{r}^{a_{1}},G_{1}\circ
v^{a_{k+1}}]]|\psi_{n\bf k}\rangle,$ (22)

where $f_{n}$ is the Fermi-Dirac distribution of the $n$th band,
$\braket{\psi_{m\bf k^{\prime}}}{\hat{\bf r}^{a}}{\psi_{n\bf
k}}=\delta_{mn}i\partial_{a}\delta_{{\bf k}^{\prime},{\bf k}}+\braket{u_{m\bf
k^{\prime}}}{i\partial_{a}}{u_{n\bf k}}$ in momentum space, $A\circ
B=\sum_{\alpha,\beta}A_{\alpha\beta}B_{\alpha\beta}$ is the Hadamard product
in the Hilbert space, and
$G_{i}=[-i(\hat{\omega}+\sum_{i=1}^{k}\omega_{i})]^{-1}$, where
$\braket{\psi_{m\bf k^{\prime}}}{\hat{\omega}}{\psi_{n\bf
k}}=\omega_{mn}\delta_{\bf k^{\prime},k}$, and
$\omega_{mn}=\hbar^{-1}(E_{m}-E_{n})$. When the first operator
$\hat{r}^{a_{k}}$ acts as a derivative, it produces total derivative terms,
which are Fermi surface contributions. The other terms are inter-band
contributions.

In this work, we use the symmetrized conductivity tensor defined by [27]

$\displaystyle\sigma^{a;a_{1},\ldots,a_{k}}\left(\Omega,\omega_{1},\ldots,\omega_{k}\right)$
$\displaystyle=\frac{1}{N_{P}}\sum_{\rm
Perm}\tilde{\sigma}^{a;a_{1},\ldots,a_{k}}\left(\Omega,\omega_{1},\ldots,\omega_{k}\right),$
(23)

where $\sum_{\rm Perm}$ is the summation over all permutations of the $k$
pairs $(a_{i},\omega_{i})$ for $i=1,\ldots,k$, and $N_{P}=\sum_{\rm Perm}1$ is
the number of all possible permutations.

3\. Photoconductivity tensors. We are interested in the third-order
photoconductivity described by

$\displaystyle j^{d}(0)$ $\displaystyle=\left[\sigma^{d;a}_{\rm
dark}+\sigma^{d;a}_{\rm photo}(\omega)\right]E^{a}(0),$ (24)

where $\sigma^{d;a}_{\rm dark}$ is the linear conductivity without incident
light,

$\displaystyle\sigma^{d;a}_{\rm
photo}(\omega)=6\sum_{b,c}\sigma^{d;abc}(0,0,\omega,-\omega)E^{b}(\omega)E^{c}(-\omega)$
(25)

is the photoconductivity, and

$\displaystyle\sigma^{d;abc}(0;0,\omega,-\omega)$
$\displaystyle=\frac{1}{6}\bigg{[}\tilde{\sigma}^{d;abc}(0;0,\omega,-\omega)+\tilde{\sigma}^{d;acb}(0,0,-\omega,\omega)$
$\displaystyle+\tilde{\sigma}^{d;cab}(0;-\omega,0,\omega)+\tilde{\sigma}^{d;bac}(0;\omega,0,-\omega)$
$\displaystyle+\tilde{\sigma}^{d;bca}(0;\omega,-\omega,0)+\tilde{\sigma}^{d;cba}(0;-\omega,\omega,0)\bigg{]}$
(26)

is the symmetrized third-order photoconductivity tensor. It consists of jerk,
injection, and shift responses that are proportional to $\Gamma^{-2}$, and
$\Gamma^{-1}$, and $\Gamma^{0}$ [36]. In the clean limit where the relaxation
energy scale $\hbar\Gamma$ is much smaller than the photon energy
$\hbar\omega$ and the band gaps $\hbar\omega_{mn}$s, the jerk response is the
most dominant response. However, it does not generate Hall conductivity
because $\sigma^{d;abc}_{\rm jerk}\left(0;0,\omega,-\omega\right)=\frac{\pi
e^{4}}{\hbar^{3}}\frac{1}{6\Gamma^{2}}\int_{\bf
k}\sum_{n,m}f_{nm}\delta(-\omega+\omega_{mn})2r^{c}_{nm}r^{b}_{mn}\partial_{a}\partial_{d}\omega_{mn}$
is symmetric with respect to indices $a$ and $d$. Accordingly, the most
dominant photovoltaic Hall conductivity comes from the injection response.

$\displaystyle\sigma^{d;abc}_{\rm
inj}\left(0;0,\omega,-\omega\right)=\frac{\pi
e^{4}}{\hbar^{3}}\frac{1}{6\Gamma}\int_{\bf
k}\sum_{n,m}f_{nm}\delta(-\omega+\omega_{mn})\times$ $\displaystyle
i\Bigg{\\{}K^{mn}_{cbad}+D^{mn}_{cb(ad)}-(D^{mn}_{bc(ad)})^{*}$
$\displaystyle+\sum_{e}\left((C^{mn}_{edc})^{*}(C^{mn})^{e}_{ab}-(C^{mn}_{eac})^{*}(C^{mn})^{e}_{db}\right)$
$\displaystyle+2\frac{\partial_{d}\omega_{mn}}{\omega_{mn}}\left[C^{mn}_{cba}-(C^{mn}_{bca})^{*}\right]$
$\displaystyle-2\frac{\partial_{d}\omega_{mn}}{\omega_{mn}}\left(Q^{mn}_{ca}\frac{\partial_{b}\omega_{mn}}{\omega_{mn}}-(Q^{mn}_{ba})^{*}\frac{\partial_{c}\omega_{mn}}{\omega_{mn}}\right)$
$\displaystyle+2\partial_{d}\omega_{mn}\sum_{p:E_{p}\neq
E_{m},E_{n}}\bigg{[}r^{c}_{nm}\left(r^{b}_{mp}\frac{r^{a}_{pn}}{\omega_{pn}}-\frac{r^{a}_{mp}}{\omega_{mp}}r^{b}_{pn}\right)$
$\displaystyle+\left(r^{c}_{np}\frac{r^{a}_{pm}}{\omega_{pm}}-\frac{r^{a}_{np}}{\omega_{np}}r^{c}_{pm}\right)r^{b}_{mn}\bigg{]}\Bigg{\\}},$
(27)

where $K^{mn}_{cbad}=r^{c}_{nm}(r^{b}_{mn,da}-r^{b}_{mn,ad})$,
$D^{mn}_{cb(ad)}=\frac{1}{2}r^{c}_{nm}(r^{b}_{mn,da}+r^{b}_{mn,ad})$,
$Q^{mn}_{ba}=r^{b}_{nm}r^{a}_{mn}$,
$\sum_{e}Q^{mn}_{ce}(C^{mn})^{e}_{ba}=C^{mn}_{cba}=r^{c}_{nm}r^{a}_{mn,b}$,
$\sum_{e}(C^{mn}_{edc})^{*}(C^{mn})^{e}_{ab}=r^{c}_{nm,d}r^{b}_{mn,a}$,
$O_{mn,a}=\partial_{a}O_{mn}-i[{\cal A},O]_{mn}$, and ${\cal
A}_{mn}=\delta_{E_{m},E_{n}}\braket{u_{m}}{\partial_{a}}{u_{n}}$ is the Berry
connection. $\delta_{E_{m},E_{n}}=\delta_{mn}$ when bands are all non-
degenerate, as we assume in the most of part of the main text.

4\. Maurer-Cartan form. Here we explain the basic concept of the Maurer-Cartan
form (See Ref. [2] for further information). Let us consider the space of
complex $N\times N$ matrices ${\rm M}(N,{\mathbb{C}})$, defining an
$N^{2}$-dimensional complex Euclidean space ${\mathbb{C}}^{N^{2}}$. The
geometrical structure of subspaces such as ${\rm U}(N)$ or ${\rm U}(N)/{\rm
U}(1)^{N}$ is induced from the structure of ${\rm M}(N,{\mathbb{C}})$. Given a
matrix representation, we can define Euclidean coordinates $x^{ij}$ with
$i,j=1,\ldots,N$ by

$\displaystyle x^{ij}(g)=g^{ij},\quad g\in{\rm M}(N,{\mathbb{C}}),$ (28)

where $g^{ij}$ is the matrix element of $g$. The tangent space at $g$ is
spanned by $\frac{\partial}{\partial x^{ij}}|_{g}$. Here the symbol $|_{g}$
indicate that it is defined at the point $g\in{\rm M}(N,{\mathbb{C}})$. Two
tangent basis vectors $\frac{\partial}{\partial x^{ij}}|_{g}$ and
$\frac{\partial}{\partial x^{ij}}|_{g^{\prime}}$ are different objects when
$g\neq g^{\prime}$ because they live at distinct points.

Let us restrict our attention to the space of invertible matrices ${\rm
GL}(N,{\mathbb{C}})$. The left multiplication $L_{f}:g\rightarrow fg$ for
invertible matrices $f$ and $g$ induces a push-forward mapping $L_{f*}$
between their tangent vectors

$\displaystyle L_{f*}\partial_{x^{ij}}|_{g}=\frac{\partial
x^{kl}(fg)}{\partial x^{ij}(g)}\partial_{x^{kl}}|_{fg}$ (29)

Using the property of the coordinates we choose in equation (28), we have

$\displaystyle L_{f*}X|_{g}$
$\displaystyle=\sum_{k,j}\left[fX(g)\right]^{kj}\partial_{x^{kj}(fg)}$ (30)

for any vector field $X$, where
$X|_{g}=\sum_{i,j}X^{ij}(g)\partial_{x^{ij}(g)}$ is a tangent vector at $g$
defined by $X$, and $(fX)^{kl}=\sum_{m}f^{km}X^{ml}$ is the matrix product. If
we take $f=g^{-1}$ and consider the vector field describing the tangent basis
vectors obtained by momentum space coordinates $k^{a}$s, i.e.,
$X(g)=\partial_{a}|_{g}=\sum_{ij}\left[\partial_{a}x^{ij}(g)\right]\partial_{x^{ij}}(g)$,

$\displaystyle
L_{g^{-1}*}\left[\sum_{i,j}(\partial_{a}g^{ij})\partial_{x^{ij}(g)}\right]$
$\displaystyle=\sum_{k,j}\left(g^{-1}\partial_{a}g\right)^{kj}\partial_{x^{kj}(1)},$
(31)

where $1$ is the identity matrix. This shows that the Maurer-Cartan form
$\xi_{a}=g^{-1}\partial_{a}g$ is the tangent vector at $g$ that is sent back
to the Lie algebra ${\rm gl}(N,{\mathbb{C}})$, which is the tangent space
defined at the identity matrix. More formally, as a differential form, the
Maurer-Cartan form is defined as a linear map from tangent vectors to the Lie
algebra by $\xi(X|_{g})\equiv L_{g^{-1}*}X|_{g}$, from which
$\xi(\partial_{a}|_{g})=g^{-1}\partial_{a}g$. We simply call the Lie algebras
$g^{-1}\partial_{a}g$s Maurer-Cartan forms as is typical.

The Lie bracket of the Maurer-Cartan forms is evaluated using the matrix
commutator. $[V,W]^{ij}_{\rm LB}=-(V^{ik}W^{kj}-W^{ik}V^{kj})=-[V,W]^{ij}$,
where $V=g^{-1}\partial_{b}g$ and $W=g^{-1}\partial_{a}g$.

5\. Riemannian geometry with degenerate states. Let us consider two distinct
energy levels $E_{n}$ and $E_{m}$ with $N_{d}$\- and $M_{d}$-fold degenerate
states $|u_{n_{i}}\rangle$ and $|u_{m_{j}}\rangle$, respectively, where
$i=1,\ldots,N_{d}$ and $j=1,\ldots,M_{d}$. Then, the tangent vector associated
with the momentum coordinate $k^{a}$ is defined by

$\displaystyle\hat{e}^{mn}_{a}\equiv\sum_{i,j}r^{a}_{m_{j}n_{i}}|u_{m_{j}}\rangle\langle
u_{n_{i}}|.$ (32)

The Hermitian metric is evaluated from the Hilbert-Schmidt inner product of
two such basis vectors.

$\displaystyle Q^{mn}_{ba}$
$\displaystyle\equiv(\hat{e}^{mn}_{b},\hat{e}^{mn}_{a})$
$\displaystyle=\sum_{ij}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i}}.$ (33)

The covariant derivative is obtained by projecting to the basis
$|u_{m_{j}}\rangle\langle u_{n_{i}}|$s after taking an ordinary derivative.

$\displaystyle\nabla_{c}\hat{e}^{mn}_{a}\equiv
P_{m}(\partial_{c}\hat{e}_{a})P_{n}=\sum_{i,j}r^{a}_{m_{j}n_{i},c}|u_{m_{j}}\rangle\langle
n_{i}|,$ (34)

where $P_{m}=\sum_{j}|u_{m_{j}}\rangle\langle u_{m_{j}}|$,
$P_{n}=\sum_{i}|u_{n_{i}}\rangle\langle u_{n_{i}}|$, and

$\displaystyle r^{a}_{m_{j}n_{i},c}=[{\cal D}_{c},r^{a}]_{m_{j}n_{i}},$ (35)

where $[A,B]=AB=BA$ is the commutator, ${\cal D}_{c}=\partial_{c}-i{\cal
A}^{c}$, and ${\cal
A}_{pq}=\delta_{E_{p},E_{q}}\braket{u_{p}}{\partial_{c}}{u_{q}}$ is the ${\rm
U}(N_{d})\times{\rm U}(M_{d})$ Berry connection [${\rm U}(N_{d})$ for $n$ and
${\rm U}(M_{d})$ for $m$, and forget about the gauge group of the other states
here]. The Hermitian connection is then defined by

$\displaystyle
C^{mn}_{bca}\equiv(\hat{e}^{mn}_{b},\nabla_{c}\hat{e}^{mn}_{a})=\sum_{ij}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i},c},$
(36)

This connection satisfies the metric compatibility
$\nabla_{c}(\hat{e}^{mn}_{b},\hat{e}^{mn}_{a})=(\hat{e}^{mn}_{b},\nabla_{c}\hat{e}^{mn}_{a})+(\nabla_{c}\hat{e}^{mn}_{b},\hat{e}^{mn}_{a})$
because
$(\hat{e}^{mn}_{b},\nabla_{c}\hat{e}^{mn}_{a})=(\hat{e}^{mn}_{b},\partial_{c}\hat{e}^{mn}_{a})$,
and $\nabla_{c}$ acts as an ordinary derivative on scalars. Here, the
generalized derivative preserves the Hermiticity $(O_{mn,c})^{*}=O_{nm,c}$ for
a Hermitian operator $O$.

The Hermitian connection we define has a non-zero torsion in general when the
number of bands exceeds two. The torsion tensor is given by virtual
transitions among three bands.

$\displaystyle T^{mn}_{bca}$
$\displaystyle\equiv(\hat{e}^{mn}_{b},\nabla_{c}\hat{e}^{mn}_{a}-\nabla_{a}\hat{e}^{mn}_{c}-i[\hat{e}^{mn}_{c},\hat{e}^{mn}_{a}])$
$\displaystyle=C^{mn}_{bca}-C^{mn}_{bac}$
$\displaystyle=i\sum_{i,j}r^{b}_{n_{i}m_{j}}\sum_{p:E_{p}\neq
E_{m},E_{n}}\left(r^{c}_{m_{j}p}r^{a}_{pn_{i}}-r^{a}_{m_{j}p}r^{c}_{pn_{i}}\right),$
(37)

where we use that $[\hat{e}^{mn}_{c},\hat{e}^{mn}_{a}]$=0.

The Hermitian curvature tensor is defined by

$\displaystyle
K^{mn}_{badc}\equiv(\hat{e}^{mn}_{b},(\nabla_{d}\nabla_{c}-\nabla_{c}\nabla_{d}-\nabla_{[\hat{e}^{mn}_{d},\hat{e}^{mn}_{c}]})\hat{e}^{mn}_{a}).$
(38)

It is identical to $D^{mn}_{badc}-D^{mn}_{badc}$ because
$[\hat{e}^{mn}_{d},\hat{e}^{mn}_{c}]=0$, where

$\displaystyle D^{mn}_{badc}$
$\displaystyle\equiv(\hat{e}^{mn}_{b},\nabla_{d}\nabla_{c}\hat{e}^{mn}_{a})=\sum_{ij}^{N_{d}}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{i},cd}$
(39)

is the second-order connection, and

$\displaystyle r^{a}_{m_{j}n_{i},cd}$ $\displaystyle=[{\cal D}_{d},[{\cal
D}_{c},r^{a}]]_{m_{j}n_{i}}$ (40)

is the double covariant derivative. The index $c$ of the double covariant
derivative does not transform tensorially under general coordinate
transformations, in contrast to case of the second covariant derivative
$S^{mn}_{badc}=(\hat{e}^{mn}_{b},(\nabla_{d}\nabla_{c}-\nabla_{\nabla_{d}\hat{e}^{mn}_{c}})\hat{e}^{mn}_{a})$.
However, the double covariant derivative is more convenient for our purpose.
The Hermitian curvature tensor can be written as

$\displaystyle K^{mn}_{badc}$
$\displaystyle=-i\sum_{i,j}r^{b}_{n_{i}m_{j}}[{\cal
F}_{dc},r^{a}]_{m_{j}n_{i}},$ (41)

where ${\cal F}_{dc}$ is the Berry curvature $({\cal F}_{dc})_{pq}=i[{\cal
D}_{d},{\cal D}_{c}]_{pq}=\left(\partial_{d}{\cal A}_{c}-\partial_{c}{\cal
A}_{d}-i[{\cal A}_{d},{\cal A}_{c}]\right)_{pq}.$ It follows from

$\displaystyle r^{a}_{m_{j}n_{i},cd}-r^{a}_{m_{j}n_{i},dc}$
$\displaystyle=-i[{\cal F}_{dc},r^{a}]_{m_{j}n_{i}},$ (42)

where we use $[{\cal D}_{d},[{\cal D}_{c},r^{a}]]_{m_{i}n_{j}}-[{\cal
D}_{c},[{\cal D}_{d},r^{a}]]_{m_{j}n_{i}}=[[{\cal D}_{d},{\cal
D}_{c}],r^{a}]_{m_{j}n_{i}}$. A useful property of the Hermitian curvature is

$\displaystyle(K^{mn}_{badc})^{*}=K^{mn}_{abdc},$ (43)

which can be shown from equation (41) and the Hermitian property
$(K^{mn}_{badc})^{*}=K^{nm}_{badc}$.

6\. Symmetry constraints on the geometric quantities. To study symmetry
properties, it is convenient to write $\hat{e}^{mn}_{a}$ using projection
operators as $\hat{e}^{mn}_{a}({\bf k})=P_{m}({\bf
k})i\partial_{k_{a}}P_{n}({\bf k})$, where $P_{n}({\bf
k})=\sum_{i=1}^{N_{d}}|u_{n_{i}\bf k}\rangle\langle u_{n_{i}\bf k}|$ is the
projection to $N_{d}$-fold degenerate states with $E_{n_{i}}=E_{n}$.

Let us first consider time reversal symmetry. It imposes $TP_{m}({\bf
k})T^{-1}=P_{m}(-{\bf k})$, where $T$ is the time reversal operator acting on
the cell-periodic Bloch states. Accordingly, $e_{a}$ follows the corresponding
symmetry constraint. $T\hat{e}^{mn}_{a}({\bf k})T^{-1}=P_{m}({-\bf
k})i\partial_{-k_{a}}[P_{n}(-{\bf k})]=\hat{e}^{mn}_{a}(-{\bf k})$. Using
this, one can show that time reversal symmetry imposes

$\displaystyle G({\bf k})=G^{*}(-{\bf k})$ (44)

for $G=Q^{mn}$, $C^{mn}$, $D^{mn}$, or $K^{mn}$

Spatial symmetry under $g:{\bf r}\rightarrow R_{g}{\bf r}+{\bf a}_{g}$ imposes
$U_{g}P_{m}({\bf k})U_{g}^{-1}=P_{m}(R_{g}{\bf k})$, where $U_{g}$ is the
unitary representation of $g$ for cell-periodic Bloch states. The $g$ symmetry
constraint on the tangent vector is $U_{g}\hat{e}^{mn}_{a}({\bf
k})U_{g}^{-1}=P_{m}({R_{g}\bf k})i\partial_{k_{a}}[P_{n}(R_{g}{\bf
k})]=\sum_{b}(R_{g})_{ba}\hat{e}^{mn}_{b}(-{\bf k})$. Geometric quantities
then satisfy

$\displaystyle G_{a_{1}\ldots a_{k}}({\bf
k})=(R_{g})_{b_{1}a_{1}}\ldots(R_{g})_{b_{k}a_{k}}G_{b_{1}\ldots
b_{k}}(R_{g}{\bf k})$ (45)

for $G=Q^{mn}$, $C^{mn}$, $D^{mn}$, or $K^{mn}$.

Spacetime inversion $PT$ symmetry plays a distinguished role because it gives
constraints at each ${\bf k}$ point. The constraint $PT\hat{e}^{mn}_{a}({\bf
k})(PT)^{-1}=\hat{e}^{mn}_{a}({\bf k})$ gives a real structure when
$(PT)^{2}=1$. This is most clearly seen by taking a gauge where $PT$ is
represented by a complex conjugation $K$, such that
$(\hat{e}^{mn}_{a})^{*}=\hat{e}^{mn}_{a}$ is real-valued. It forms a one-
dimensional real subspace of the tangent space of the manifold ${\rm
O}(N)/{\rm O}(1)^{N}$ of non-degenerate $PT$-symmetric cell-periodic Bloch
states. On the other hand, $PT$ symmetry gives a quaternion structure when
$(PT)^{2}=-1$. To see this, let us write down the matrix element of
$\hat{e}^{mn}_{a}$. Each band becomes twofold degenerate by the $PT$ symmetry
satisfying $(PT)^{2}=-1$ (Kramers theorem). We can fix the gauge such that
$PT$ is represented by $i\sigma_{y}K$ within each set of twofold degenerate
states. Then $PT$ symmetry
$(r^{a})_{m_{i}n_{j}}^{*}=-(i\sigma_{y}^{-1}r^{a}i\sigma_{y})_{m_{i}n_{j}}$
requires that
$\left(r^{a}_{m_{i}n_{j}}\right)=-i(f^{a}_{1}\sigma_{0}+f^{a}_{2}i\sigma_{x}+f^{a}_{3}i\sigma_{y}+f^{a}_{4}i\sigma_{z})$,
where $\sigma_{0}$ is the $2\times 2$ identity matrix, and $\sigma_{i=x,y,z}$
are Pauli matrices, and $f^{a}_{i=1,2,3,4}$ are real-valued functions. The
matrices $\\{\sigma_{0},i\sigma_{x},i\sigma_{y},i\sigma_{z}\\}$ satisfy the
quaternion algebra, defining a quaternion structure. They form a one-
dimensional quaternionic subspace of the tangent space of the manifold ${\rm
Sp}(N)/{\rm Sp}(1)^{N}$ of Kramers degenerate $PT$-symmetric cell-periodic
Bloch states. Independent of the sign of $(PT)^{2}$, the geometric quantities
satisfy $G({\bf k})=G^{*}({\bf k})$ for $G=Q^{mn}$, $C^{mn}$, $D^{mn}$, or
$K^{mn}$.

7\. Riemann curvature tensor of two-level systems. Let us consider a model
with two energy levels $E_{n}$ and $E_{m}$ which are $N_{d}$\- and
$M_{d}$-fold degenerate, respectively. Using that

$\displaystyle({\cal
Q}^{n}_{ba})_{n_{i}n_{k}}=\sum_{j=1}^{M_{d}}r^{b}_{n_{i}m_{j}}r^{a}_{m_{j}n_{k}}$
(46)

defines the non-abelian Hermitian metric of the level $n$ (similarly for the
level $m$), where $E_{n_{i}}=E_{n_{k}}=E_{n}$ and $E_{m_{j}}=E_{m}$, we obatin

$\displaystyle K^{mn}_{badc}$ $\displaystyle=i{\rm Tr}_{n}{\cal
Q}^{n}_{ba}{\cal F}^{n}_{dc}-i{\rm Tr}_{m}{\cal F}^{m}_{dc}{\cal Q}^{m}_{ab},$
(47)

where ${\cal Q}^{n}_{ba}={\cal G}^{n}_{ba}-(i/2){\cal F}^{n}_{ba}$ is the non-
abelian Hermitian metric of the level $E_{n}$, and ${\rm Tr}_{n}$ is the trace
within the energy level $E_{n}$. The non-abelian Riemannian metric ${\cal
G}^{n}_{ba}$ and the Berry curvature ${\cal F}^{n}_{ba}$ are respectively
symmetric and anti-symmetric with respect to the exchange of $a$ and $b$.

Now we suppose that the system has chiral $S$ symmetry $SH({\bf
k})S^{-1}=-H({\bf k})$ or particle-hole-times-inversion $CP$ symmetry
$CPH({\bf k})(CP)^{-1}=-H^{*}({\bf k})$. Those symmetries impose

$\displaystyle{\cal Q}^{m}_{ba}({\bf k})$ $\displaystyle=U_{S}^{-1}{\cal
G}^{n}_{ba}({\bf k})U_{S},$ $\displaystyle{\cal Q}^{m}_{ba}({\bf k})$
$\displaystyle=U_{CP}^{-1}[{\cal Q}^{n}_{ba}({\bf k})]^{*}U_{CP}.$ (48)

The expression of the Hermitian curvature tensor then reduces to,

$\displaystyle K^{mn}_{badc}$ $\displaystyle=2i{\rm Tr}_{n}{\cal
Q}^{n}_{ba}{\cal F}^{n}_{dc}.$ (49)

This formula can be applied to the linearized model of Dirac or Weyl point
whether it is massive or massless, because such a model has $CP$ or chiral
symmetry. One interesting special case is ${\rm
Re}K^{mn}_{baba}=R^{mn}_{baba}={\rm Tr}_{n}{\cal F}^{n}_{ba}{\cal
F}^{n}_{ba}$, which measures the norm of the non-abelian Berry curvature
matrix ${\cal F}^{n}_{ba}$.

8\. Third-order photovoltaic Hall conductivity of Dirac and Weyl fermions. Let
us consider a two-dimensional Dirac fermion described by equation (14). The
Riemann curvature term is

$\displaystyle{\rm Im}\sigma^{[x;y]xy}_{K}$
$\displaystyle=\frac{e^{4}}{12\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right)^{2}\left(\frac{2m}{\hbar\omega}\right)^{2}\Theta(|\omega|-2|m|),$
(50)

while the remaining term has the opposite sign

$\displaystyle{\rm Im}\sigma^{[x;y]xy}-{\rm Im}\sigma^{[x;y]xy}_{K}$
$\displaystyle=-\frac{e^{4}}{12\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right)^{2}\left[1-\left(\frac{2m}{\hbar\omega}\right)^{2}\right]\
\Theta(|\omega|-2|m|).$ (51)

The Riemann curvature leads to the dominant photovoltaic Hall response near
the band edge $\hbar|\omega|=2|m|$. Other contributions become dominant when
$\hbar|\omega|>2\sqrt{2}|m|$, where the third-order conductivity changes sign.
When we take parameters used in the main text, $v=8\times 10^{6}\;{\rm m/s}$
and $2|m|=17\;{\rm meV}$, the peak value $1.2\times 10^{-15}{\rm
Am^{2}V^{-3}}$ at the band edge. It corresponds to the photoconductivity
$\sigma^{x;y}_{\rm photo}=(e^{2}/h)\times 3.5\times 10^{-4}I_{\rm light}/({\rm
1\;Wcm^{-2}})$, where $e^{2}/h=3.874\times 10^{-5}\Omega^{-1}$ is the
conductance quantum per spin. This magnitude is the same as the
photoconductivity for a massless Dirac fermion at $\hbar\omega=17\;{\rm meV}$,
but the sign is opposite.

Nonzero components of the linear photovoltaic Hall conductivity tensor in two
dimensions are

$\displaystyle{\rm Re}\sigma^{[x;y]yy}={\rm Re}\sigma^{[x;y]xx}$
$\displaystyle=\frac{e^{4}}{24\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right)^{2}\left(\frac{2m}{\hbar\omega}\right)\left[1+\left(\frac{2m}{\hbar\omega}\right)^{2}\right]\Theta(|\omega|-2|m|),$
(52)

while $\sigma^{[x;y]xy}=\sigma^{[x;y]yx}=0$ by rotational symmetry. Here, the
whole response is from the symplectic curvature tensor, and no other
contributions exist.

For a Weyl fermion in three dimensions described by

$\displaystyle H_{\rm Weyl}=\hbar
v(k_{x}\sigma_{x}+k_{y}\sigma_{y}+k_{z}\sigma_{z}),$ (53)

there is only one non-vanishing linearly independent component of the
photovoltaic Hall conductivity tensor, which is

$\displaystyle{\rm Im}\sigma^{[x;y]xy}$
$\displaystyle=-\frac{e^{4}}{72\pi\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right),$
(54)

and ${\rm Im}\sigma^{[x;y]xy}_{K}=-{\rm Im}\sigma^{[x;y]xy}$. Other non-zero
components are related by $SO(3)$ rotational symmetries

$\displaystyle{\rm Im}\sigma^{[x;y]xy}={\rm Im}\sigma^{[yz];yz}={\rm
Im}\sigma^{[zx];zx}$ (55)

and the inherent relations ${\rm Im}\sigma^{[a;d]cb}=-{\rm
Im}\sigma^{[d;a]cb}=-{\rm Im}\sigma^{[a;d]bc}$ by definition. The linear
photovoltaic Hall conductivity is zero is due to time reversal
$T=i\sigma_{y}K$ symmetry of a Weyl fermion around the band-crossing point.

Massless Dirac fermions in three dimensions have the photovoltaic Hall
conductivity tensors twice that of a Weyl fermion, as it consists of two
copies of Weyl fermions. When the mass gap $2m$ is introduced by

$\displaystyle H_{\rm Dirac}=\hbar
v(k_{x}\tau_{z}\sigma_{x}+k_{y}\tau_{z}\sigma_{y}+k_{z}\tau_{z}\sigma_{z})+m\tau_{x},$
(56)

the non-vanishing linearly independent photovoltaic Hall conductivity
component is

$\displaystyle{\rm Im}\sigma^{[x;y]xy}$
$\displaystyle=\frac{e^{4}}{36\pi\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right)\left[4\left(\frac{2m}{\hbar\omega}\right)^{2}-1\right]\Theta(|\omega|-2|m|),$
(57)

and

$\displaystyle{\rm Im}\sigma^{[x;y]xy}_{K}$
$\displaystyle=\frac{e^{4}}{36\pi\hbar^{3}}\frac{1}{\Gamma}\frac{1}{\omega}\left(\frac{v}{\omega}\right)\left[1+2\left(\frac{2m}{\hbar\omega}\right)^{2}\right]\Theta(|\omega|-2|m|).$
(58)

Here, we use $R^{cv}_{yxyx}={\rm Tr}\left[({\cal
F}^{n}_{xy})^{2}\right]=8(v/\omega_{cv})^{4}[(2\hbar
vk_{z})^{2}+(2m)^{2}]/(\hbar\omega_{cv})^{2}$ ($c$ and $v$ indicating the
upper- and lower-energy states). See Methods 8. Since the mass preserves
$PT=i\tau_{x}\sigma_{y}K$, the linear photovoltaic Hall conductivity is zero.
It does not depend on the choice of the matrix representation. Any Dirac mass
term preserves $PT$ symmetry.

9\. Topology of the transition matrix dipole moment. When the spatial
dimension $d$ is even, the Euler number is given by the generalized Gauss-
Bonnet theorem [2]

$\displaystyle\chi^{mn}$
$\displaystyle=\oint{\wedge^{d}}dk\sqrt{g}\sum_{a_{i},b_{j}}\frac{\epsilon^{a_{1}a_{2}\ldots
a_{d-1}a_{d}}\epsilon^{b_{1}b_{2}\ldots
b_{d-1}b_{d}}}{(2\pi)^{d/2}2^{d}(d/2)!g}$ $\displaystyle\qquad
R^{mn}_{a_{1}a_{2}b_{1}b_{2}}\ldots R^{mn}_{a_{d-1}a_{d}b_{d-1}b_{d}},$ (59)

where $\epsilon$ is the Levi-Civita symbol, $g\equiv\det g^{mn}$ is the
determinant of the metric $g^{mn}$ with respect to its momentum indices that
are implicit in this notation, ${\wedge^{d}}dk$ is the oriented integral
measure, and $R^{mn}_{a_{1}a_{2}b_{1}b_{2}}={\rm
Re}K^{mn}_{a_{1}a_{2}b_{1}b_{2}}$ is the Riemann curvature tensor.

In two-dimensions, we have

$\displaystyle\chi^{mn}$ $\displaystyle=\frac{1}{2\pi}\oint dk_{1}dk_{2}{\rm
sgn}(F^{mn}_{12})\frac{R^{mn}_{1212}}{\sqrt{g}}$
$\displaystyle=\frac{1}{2\pi}\oint dk_{1}dk_{2}\frac{|F^{mn}_{12}|({\cal
F}_{12}^{n}-{\cal F}_{12}^{m})}{2\sqrt{g}}$ $\displaystyle=\frac{1}{2\pi}\oint
dk_{1}dk_{2}({\cal F}_{12}^{n}-{\cal F}_{12}^{m})$
$\displaystyle=c_{1}^{n}-c_{1}^{m},$ (60)

where we use $|F^{mn}_{12}|/2=g$ following from $\det
Q^{mn}=Q^{mn}_{11}Q^{mn}_{22}-Q^{mn}_{12}Q^{mn}_{21}=0$. Here,
$F^{mn}_{12}\equiv-2{\rm Im}Q^{mn}_{12}$ should be distinguished from the
Berry curvature ${\cal F}$ (except in two-band systems). $\det Q^{mn}=0$
because we introduce two momentum coordinates while $Q^{mn}$ is defined on
one-dimensional tangent space (cf. $\det g$ is nonzero because the target
space is two-dimensional as a real space). Here, the sign ${\rm
sgn}(F^{mn}_{12})$ due to the oriented nature of the integral measure should
not be discarded. Without this factor, one obtains a non-quantized value as in
Refs. [4, 5, 6, 7]

In spin-orbit coupled $PT$-symmetric systems, where $(PT)^{2}=-1$, the
transition dipole moments $\hat{e}^{m_{i}n_{j}}_{a}$s between two pairs of
Kramers-degenerate states $|u_{m_{i=1,2}}\rangle$ and $|u_{n_{j=1,2}}\rangle$
with $E_{m_{1}}=E_{m_{2}}\neq E_{n_{1}}=E_{n_{2}}$ define four-dimensional
tangent subspace. Therefore, the Gauss-Bonnet theorem can be applied in a
four-dimensional momentum space. The result is

$\displaystyle\chi^{mn}$ $\displaystyle=\oint
dk_{1}dk_{2}dk_{3}dk_{4}\sum_{a_{i},b_{j}}\frac{{\rm sgn}(\wedge
d^{4}k)}{(2\pi)^{2}2^{4}2\sqrt{g}}\epsilon^{a_{1}a_{2}a_{3}a_{4}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}$
$\displaystyle\qquad
R^{nm}_{a_{1}a_{2}b_{1}b_{2}}R^{nm}_{a_{3}a_{4}b_{1}b_{4}}$
$\displaystyle=\oint\frac{d^{4}k}{32\pi^{2}}\sum_{b_{j}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}$
$\displaystyle\qquad\sum_{i=1}^{2}\left[({\cal F}_{b_{1}b_{2}}{\cal
F}_{b_{3}b_{4}})_{n_{i}n_{i}}-({\cal F}_{b_{1}b_{2}}{\cal
F}_{b_{3}b_{4}})_{m_{i}m_{i}}\right]$ $\displaystyle=p_{1}^{n}-p_{1}^{m},$
(61)

where $p_{1}$ is the first Pontryagin number. See Supplementary Note 2 for a
derivation.

In spinless $PT$-symmetric systems, where $(PT)^{2}=1$, $\hat{e}^{mn}_{a}$
define a one-dimensional real tangent vector. Since it is odd-dimensional, the
generalized Gauss-Bonnet theorem does not apply. Instead, we can define the
first Stiefel-Whitney number associated with it by

$\displaystyle w_{1}^{mn}$ $\displaystyle=\frac{1}{\pi}\oint d{\bf k}\cdot
i\partial_{\bf k}\log r^{a}_{mn}\mod 2.$ (62)

It measures the orientability of the tangent vector $\hat{e}^{mn}_{a}$ over
the one-dimensional momentum space. $w_{1}^{mn}=0$ ($w_{1}^{mn}=1$) indicates
that $\hat{e}^{mn}_{a}$ is orientable (non-orientable). As we show in
Supplementary Note 3,

$\displaystyle w_{1}^{mn}=w^{n}_{1}-w^{m}_{1},$ (63)

where $w^{n}_{1}$ and $w^{m}_{1}$ are the first Stiefel-Whitney numbers of
bands $n$ and $m$, or equivalently, the Berry phases of bands $n$ and $m$
divided by $\pi$ [3, 4].

10\. Geometry of a single state and others. A single state $|u_{n}\rangle$
lives on the complex projective manifold

$\displaystyle{\rm{\mathbb{C}}P}^{N-1}=\frac{{\rm U}(N)}{{\rm U}(1)\times{\rm
U}(N-1)},$ (64)

where $N$ is the number of all bands, ${\rm U}(N)$ describes the degrees of
freedom of the cell-periodic Bloch state $U_{\alpha n}=\braket{{\bf
r}_{\alpha}}{u_{n}}$, ${\rm U}(1)$ is the phase rotation of the state
$|u_{n}\rangle$, and ${\rm U}(N-1)$ is the unitary transformation of the other
states. The tangent vectors on this manifold is described by the Maurer-Cartan
form

$\displaystyle\hat{\xi}^{n}_{a}$
$\displaystyle=U^{-1}i\partial_{a}U-U^{-1}i\partial_{a}U|_{{\rm
U}(1)}-U^{-1}i\partial_{a}U|_{{\rm U}(N-1)}$
$\displaystyle\equiv\hat{e}^{n}_{a}+(\hat{e}^{n})_{a}^{\dagger},$ (65)

where $U^{-1}i\partial_{a}U=\sum_{m,n}|u_{m}\rangle r^{a}_{mn}\langle u_{n}|$
is the Maurer-Cartan form of ${\rm U}(N)$, $U^{-1}i\partial_{a}U|_{{\rm
U}(1)}=|u_{n}\rangle\braket{u_{n}}{i\partial_{a}}{u_{n}}\langle u_{n}|$ is its
projection to the ${\rm U}(1)$ direction, $U^{-1}i\partial_{a}U|_{{\rm
U}(N-1)}=\sum_{m\neq n,p\neq
n}|u_{m}\rangle\braket{u_{m}}{i\partial_{a}}{u_{p}}\langle u_{p}|$ is the
projection to ${\rm U}(N-1)$ direction, and

$\displaystyle\hat{e}^{n}_{a}$ $\displaystyle=\sum_{m:m\neq n}|u_{m}\rangle
r^{a}_{mn}\langle u_{n}|.$ (66)

The Hermitian metric tensor of the $n$th band is

$\displaystyle Q^{n}_{ba}=(\hat{e}^{n}_{b},\hat{e}^{n}_{a})=\sum_{m:m\neq
n}r^{b}_{nm}r^{a}_{mn}.$ (67)

This is the Fubini-Study metric. Equivalently, it can be defined by
$Q^{n}_{ba}=\frac{1}{2}\left[(\hat{\xi}_{b},\hat{\xi}_{a})-i(\hat{\xi}_{b},i\hat{\xi}_{a})\right]$.
In two-band systems, the Fubini-Study metric is identical to the metric in
equation (7) because there is only one pair of bands $(m,n)$; this is the
reason why the geometric understanding of optical responses in two-band
systems was possible through the Fubini-Study metric in the previous works
[20, 21, 19]. The imaginary part of the Fubini-Study metric, the symplectic
form, is identical to the Berry curvature of the band $n$. The Hermitian
connection, torsion, and curvature can be defined from the covariant
derivative

$\displaystyle O_{mn,c}$ $\displaystyle=\left[\partial_{c}-i{\cal A}_{c}^{{\rm
U}(1)}-i{\cal A}_{c}^{{\rm U}(N-1)},O\right]_{mn},$ (68)

where $({\cal A}_{c}^{{\rm
U}(1)})_{mp}=\delta_{mn}\delta_{pn}\braket{u_{n}}{i\partial_{c}}{u_{n}}$ is
the ${\rm U}(1)$ Berry connection of band $n$, $({\cal A}_{c}^{{\rm
U}(N-1)})_{mp}=(1-\delta_{mn})(1-\delta_{pn})\braket{u_{m}}{i\partial_{c}}{u_{p}}$
is the ${\rm U}(N-1)$ Berry connection of the other bands.

Note that the torsion tensor is zero because
$r^{a}_{mn,c}-r^{c}_{mn,a}=\partial_{c}r^{a}_{mn}-\partial_{a}r^{c}_{mn}-i[r^{c},r^{a}]_{mn}=0$.
It is due to the fact that we consider the tangent space of a Kähler manifold,
which has zero torsion [2]. Any of its complex submanifolds also has zero
torsion because it is a Kähler manifold [2]. Let us compare this with that
$\hat{e}^{mn}_{a}$ defines a subspace of the tangent space of the projective
space. The non-zero torsion (see Methods 5) of our connection for
$\hat{e}^{mn}_{a}$ is the manifestation that the subspace of a tangent space
may not define a tangent space of a submanifold by a topological obstruction
(see Supplementary Note 1), i.e., not every vector bundle of dimension $D$
defines a tangent bundle of dimension $D$. The torsionless connection is
called the Levi-Civita connection.

The manifold of the whole occupied states is described by the complex
Grassmannian manifold

$\displaystyle{\rm Gr}_{\mathbb{C}}(N_{\rm occ},N)=\frac{U(N)}{{\rm U}(N_{\rm
occ})\times U(N-N_{\rm occ})}.$ (69)

In this case, $\hat{\xi}_{a}=\hat{e}_{a}+\hat{e}_{a}^{\dagger}$, where

$\displaystyle\hat{e}_{a}$ $\displaystyle=\sum_{m\in{\rm unocc}}\sum_{n\in{\rm
occ}}|u_{m}\rangle r^{a}_{mn}\langle u_{n}|.$ (70)

This geometric structure appears in dc responses because, in the limit
$\omega\ll\omega_{mn}$ for all $m$ and $n$, all excitation channels contribute
to the response non-trivailly through the Lorentzian
$i/(\omega-\omega_{mn}+i\Gamma)\sim i/\omega_{mn}\neq 0$, which on the other
hand behaves as the delta function $\delta(\omega-\omega_{mn})$ that chooses a
specific pair of bands $(m,n)$ for resonant frequencies. The Hermitian metric
and other quantities are defined similarly. The torsion is zero.

Let us remark that the formula equation (47), used for two-level systems,
applies also to the Hermitian curvature of a single state or occupied states.
Instead of considering two energy levels $E_{n}$ and $E_{m}$, we just need to
consider two bipartite states (a single state vs the rest, or the occupied
states vs the unoccupied states) and the corresponding change in the gauge
group. For example, the Hermitian curvature tensor of the $n$th state is
$K^{n}_{badc}=i{\cal Q}^{n}_{ba}{\cal F}^{n}_{dc}-i{\rm Tr}_{n^{\prime}}{\cal
Q}^{n^{\prime}}_{dc}{\cal F}^{n^{\prime}}_{ab}$, where $n^{\prime}$ indicates
the set of all states excluding the $n$th state.

We can also consider general complex flag manifolds

$\displaystyle{\rm Fl}_{\mathbb{C}}(N_{1},\ldots,N_{k})=\frac{U(N)}{{\rm
U}(1)\times\ldots\times{\rm U}(N_{k})}$ (71)

with $\sum_{i=1}^{k}N_{i}=N$. These are all Kähler manifolds, so they have
zero torsion tensor. It is straightforward to calculate the geometric
quantities of the flag manifolds and their real and quaternion counterparts.
One just needs to consider different gauge groups when the Maurer-Cartan form
and the covariant derivative are defined. Then, one can show that
$T_{bca}=C_{bca}-C_{bac}-i(\hat{e}_{b},[\hat{e}_{c},\hat{e}_{a}])=0$ (see
Supplementary Note 4).

11\. First-principles calculations. We perform first-principles calculations
based on the density functional theory with the generalized gradient
approximation (GGA) [10]. We consider monolayer germanene modelled by a slab-
supercell, ferromagnetic (LaOsO3)2 bilayer in the (111) (LaOsO3)2/(LaAlO3)10
superlattice (see [40] for the superlattice structure), and bulk Bi2Se3. For
germanene, the separation between two neighboring slabs is at least 15 Å in
order to minimize the artificial inter-slab interaction. The electronic
structure calculations are carried out using the accurate projector-augmented
wave (PAW) method, as implemented in the Vienna ab-initio simulation package
(VASP) [11, 12]. The fully relativistic PAW potentials are adopted in order to
include the spin-orbit coupling (SOC) effect. Large plane-wave cutoff energies
of $450\;\rm eV$, $400\;\rm eV$, and 350 $\rm eV$ are used for germanene,
(LaOsO3)2 bilayer, and Bi2Se3, respectively. For the Brillouin zone (BZ)
integration, a $k$-point mesh of $20\times 20\times 1$, $12\times 12\times 2$
and $12\times 12\times 12$ is used respectively. All the self-consistent
electronic structure calculations are performed with an energy convergence
within $10^{-6}{\;\rm eV}$ between the successive iterations.

The third-order injection photovoltaic Hall conductivity is calculated using
Eq. (Riemannian geometry of resonant optical responses) in Methods 3. Since a
large number of $k$-points are needed to get accurate NLO responses [19], we
use the efficient Wannier function interpolation method based on maximally
localized Wannier functions (MLWFs) [13, 14, 15]. MLWFs of $p$ orbitals for
the Ge, Bi and Se atoms and $d$ orbitals for the Os atom are constructed by
fitting to the GGA+SOC band structures [Fig. 3(a,b)]. The band structures
generated by the Wannier function interpolation are indistinguishable from the
corresponding GGA+SOC band structures. The third-order injection current
conductivity spectra [Fig. 3(c,d)] are evaluated by using dense $k$-meshes of
$10000\times 10000\times 1$, $300\times 300\times 50$, and $500\times
500\times 500$ for germanene, (LaOsO3)2 bilayer, and Bi2Se3, respectively. We
find that the third-order conductivity obtained using such dense $k$-point
meshes converge within a few percent. Here we consider the ”cold” materials,
i.e., the Fermi-Dirac function in Eq. (Riemannian geometry of resonant optical
responses) is taken to be a step function. Furthermore, the Dirac $\delta$
function is replaced by a Gaussian function with broadening width of
$\varepsilon=5\;{\rm meV}$. We use the relaxation rate $\Gamma$ given by
$\hbar\Gamma=1{\rm\;meV}$.

Data Availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

Code Availability
The code that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * [1]
  * [2] Nakahara, M.  _Geometry, Topology and Physics_ (CRC Press, 2003).
  * [3] Allendoerfer, C. B. & Weil, A.  The Gauss-Bonnet theorem for Riemannian polyhedra.  _Trans. Am. Math. Soc._ 53, 101–129 (1943).
  * [4] Ma, Y.-Q., Gu, S.-J., Chen, S., Fan, H. & Liu, W.-M.  The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems.  _EPL_ 103, 10008 (2013).
  * [5] Tan, X. _et al._ Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit.  _Phys. Rev. Lett._ 122, 210401 (2019).
  * [6] Zhu, Y.-Q. _et al._ Note on “Experimental measurement of quantum metric tensor and related topological phase transition with a superconducting qubit”.  _Preprint at https://arxiv.org/abs/1908.06462_ (2019).
  * [7] Ma, Y.-Q.  Euler characteristic number of the energy band and the reason for its non-integer values.  _Preprint at https://arxiv.org/abs/2001.05946_ (2020).
  * [8] Ahn, J., Kim, D., Kim, Y. & Yang, B.-J.  Band topology and linking structure of nodal line semimetals with $Z_{2}$ monopole charges.  _Phys. Rev. Lett._ 121, 106403 (2018).
  * [9] Ahn, J., Park, S., Kim, D., Kim, Y. & Yang, B.-J.  Stiefel-Whitney classes and topological phases in band theory.  _Chin. Phys. B_ 28, 117101 (2019).
  * [10] Perdew, J. P., Burke, K. & Ernzerhof, M.  Generalized gradient approximation made simple.  _Phys. Rev. Lett._ 77, 3865 (1996).
  * [11] Kresse, G. & Hafner, J.  Ab initio molecular dynamics for liquid metals.  _Phys. Rev. B_ 47, 558 (1993).
  * [12] Kresse, G. & Furthmüller, J.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.  _Phys. Rev. B_ 54, 11169 (1996).
  * [13] Wang, X., Yates, J. R., Souza, I. & Vanderbilt, D.  Ab initio calculation of the anomalous hall conductivity by Wannier interpolation.  _Phys. Rev. B_ 74, 195118 (2006).
  * [14] Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D.  Maximally localized Wannier functions: Theory and applications.  _Rev. Mod. Phys._ 84, 1419 (2012).
  * [15] Ibañez-Azpiroz, J., Tsirkin, S. S. & Souza, I.  Ab initio calculation of the shift photocurrent by Wannier interpolation.  _Phys. Rev. B_ 97, 245143 (2018).

##  Supplementary Note 1 Obstruction to finding an associated submanifold

Here, we show that there can be a topological obstruction for
$\hat{e}^{mn}_{a}$ to define a submanifold of ${\cal M}={\rm U}(N)/{\rm
U}(1)^{N}$. Let us first note that the complex projective line
${\rm{\mathbb{C}}P}^{1}\simeq S^{2}$ is the only compact and closed one-
dimensional complex manifold. Since ${\rm{\mathbb{C}}P}^{1}$ has the Euler
characteristic of two, the Euler characteristic calculated in the Brillouin
zone should be $\chi^{mn}=N_{w}\chi({\rm{\mathbb{C}}P}^{1})=2N_{w}$, where
$N_{w}$ is the winding number of the map from the Brillouin zone to
${\rm{\mathbb{C}}P}^{1}$. However, our formula for the Riemann curvature
tensor and the Gauss-Bonnet theorem gives

$\displaystyle\chi^{mn}$ $\displaystyle=c_{1}^{n}-c_{1}^{m}$
$\displaystyle=2c_{1}^{n}-(c_{1}^{n}+c_{1}^{m}),$ (S1)

which can be an odd integer when the total Chern number of the $n$ and $m$
bands is an odd number. This shows that there is a topological obstruction to
defining a submanifold generated by $\hat{e}^{mn}_{a}$.

We can derive a similar result for real manifolds ($PT$-symmetric cases with
$(PT)^{2}=1$). The real projective line ${\rm{\mathbb{R}}P}^{1}\simeq S^{1}$
is the only compact and connected one-dimensional manifold [1]. While
$\chi({\rm{\mathbb{R}}P}^{1})=0$, meaning that $\chi^{mn}=0$, we have in
general [See Appendix Supplementary Note 3]

$\displaystyle\chi^{mn}\mod 2$ $\displaystyle=w_{1}^{n}-w_{1}^{m}$
$\displaystyle=w_{1}^{n}+w_{1}^{m},$ (S2)

where $w_{1}=0$ or $1$ is the first Stiefel-Whitney number (Berry phase
divided by $\pi$).

We can extend these to the quaternionic manifolds [$(PT)^{2}=-1$]. In this
case, the quaternionic projective line ${\rm{\mathbb{H}}P}^{1}\simeq S^{4}$
has the Euler characteristic of two. So,
$\chi^{mn}=N_{w}\chi({\rm{\mathbb{H}}P}^{1})=2N_{w}$ is expected. However, an
obstruction can arise due to the relation

$\displaystyle\chi^{mn}$ $\displaystyle=p_{1}^{n}-p_{1}^{m}$
$\displaystyle=2p_{1}^{n}-(p_{1}^{n}+p_{1}^{m}),$ (S3)

where $p_{1}$ is the Pontryagin number, when $p_{1}^{n}+p_{1}^{m}$ is odd.

##  Supplementary Note 2 Generalized Gauss-Bonnet theorem in four dimensions

The generalized Gauss-Bonnet theorem [2] in four dimensions are relevant to
the case with twofold degeneracy by $(PT)^{2}=-1$, in which case we have

$\displaystyle\chi^{mn}$ $\displaystyle=\int
dk_{1}dk_{2}dk_{3}dk_{4}\sum_{a_{i},b_{j}}\frac{{\rm sgn}(\wedge
d^{4}k)}{(2\pi)^{2}2^{4}2\sqrt{g}}\epsilon^{a_{1}a_{2}a_{3}a_{4}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}$
$\displaystyle\times
R^{mn}_{a_{1}a_{2}b_{1}b_{2}}R^{mn}_{a_{3}a_{4}b_{1}b_{4}}.$ (S4)

We show that the Euler number can be related to the difference of the
Pontryagin number of bands $n$ and $m$.

First, we investigate the symmetry constraints. $PT$ symmetry
$PT|u_{q}\rangle=G_{pq}|u_{p}\rangle$ imposes
$(r^{a})_{m_{i}n_{j}}^{*}=-(G^{-1}r^{a}G)_{m_{i}n_{j}}$. If we choose a basis
where $G_{n_{i}n_{j}}=(i\sigma_{y})_{n_{i}n_{j}}$ and
$G_{m_{i}m_{j}}=(i\sigma_{y})_{m_{i}m_{j}}$,

$\displaystyle\left(r^{a}_{m_{i}n_{j}}\right)$
$\displaystyle=\begin{pmatrix}r^{a}_{m_{1}n_{1}}&r^{a}_{m_{1}n_{2}}\\\
r^{a}_{m_{2}n_{1}}&r^{a}_{m_{2}n_{2}}.\end{pmatrix}$
$\displaystyle=-if^{a}_{1}\sigma_{0}+f^{a}_{2}\sigma_{x}+f^{a}_{3}\sigma_{y}+f^{a}_{4}\sigma_{z},$
(S5)

where

$\displaystyle f^{a}_{1}$
$\displaystyle=\frac{i}{2}\left(r^{a}_{m_{1}n_{1}}+r^{a}_{m_{2}n_{2}}\right),$
$\displaystyle f^{a}_{2}$
$\displaystyle=\frac{1}{2}\left(r^{a}_{m_{1}n_{1}}+r^{a}_{m_{2}n_{2}}\right),$
$\displaystyle f^{a}_{3}$
$\displaystyle=\frac{i}{2}\left(r^{a}_{m_{1}n_{2}}-r^{a}_{m_{2}n_{1}}\right),$
$\displaystyle f^{a}_{4}$
$\displaystyle=\frac{1}{2}\left(r^{a}_{m_{1}n_{2}}-r^{a}_{m_{2}n_{1}}\right).$
(S6)

For our purpose, it is convenient to introduce the notation

$\displaystyle\left(r^{a}\right)$
$\displaystyle=\begin{pmatrix}0&0&r^{a}_{m_{1}n_{1}}&r^{a}_{m_{1}n_{2}}\\\
0&0&r^{a}_{m_{2}n_{1}}&r^{a}_{m_{2}n_{2}}\\\
r^{a}_{n_{1}m_{1}}&r^{a}_{n_{1}m_{2}}&0&0\\\
r^{a}_{n_{2}m_{1}}&r^{a}_{n_{2}m_{2}}&0&0\end{pmatrix}$
$\displaystyle=f_{1}^{a}\tau_{y}\sigma_{0}+f_{2}^{a}\tau_{x}\sigma_{x}+f_{3}^{a}\tau_{x}\sigma_{y}+f_{4}^{a}\tau_{x}\sigma_{z}$
$\displaystyle\equiv
f_{1}^{a}\Gamma_{1}+f_{2}^{a}\Gamma_{2}+f_{3}^{a}\Gamma_{3}+f_{4}^{a}\Gamma_{4}$
(S7)

where
$\Gamma_{1}=\tau_{y}\sigma_{0},\Gamma_{2}=\tau_{x}\sigma_{x},\Gamma_{3}=\tau_{x}\sigma_{y},\Gamma_{4}=\tau_{x}\sigma_{z},\Gamma_{5}=\tau_{z}\sigma_{0}$.
The Gamma matrices satisfy
$\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}\Gamma_{5}=1$ and thus ${\rm
Tr}\left[\Gamma_{i}\Gamma_{j}\Gamma_{k}\Gamma_{l}\Gamma_{5}\right]=4\epsilon^{ijkl}$.

The determinant of the metric is evaluated to be.

$\displaystyle g\equiv\det
g^{mn}=\left(4\sum_{a_{1},a_{2},a_{3},a_{4}}\epsilon^{a_{1}a_{2}a_{3}a_{4}}f^{a_{1}}_{1}f^{a_{2}}_{2}f^{a_{3}}_{3}f^{a_{4}}_{4}\right)^{2}.$
(S8)

The Riemann curvature tensor can be written simply as

$\displaystyle R^{mn}_{a_{1}a_{2}b_{1}b_{2}}$
$\displaystyle=\frac{1}{2}\sum_{i,j}{\rm Tr}\left[F_{a_{1}a_{2}}{\cal
F}_{b_{1}b_{2}}\right],$ (S9)

where we define

$\displaystyle(F_{a_{1}a_{2}})$
$\displaystyle=\begin{pmatrix}(F_{a_{1}a_{2}}^{nm})_{m_{i}m_{j}}&0\\\
0&(F_{a_{1}a_{2}}^{mn})_{n_{i}n_{j}}\end{pmatrix},$ $\displaystyle({\cal
F}_{a_{1}a_{2}})$ $\displaystyle=\begin{pmatrix}({\cal
F}_{a_{1}a_{2}})_{m_{i}m_{j}}&0\\\ 0&({\cal
F}_{a_{1}a_{2}})_{n_{i}n_{j}}\end{pmatrix},$ (S10)

and

$\displaystyle(F_{a_{1}a_{2}}^{mn})_{n_{i}n_{k}}$
$\displaystyle=i\sum_{j}(r^{a_{1}}_{n_{i}m_{j}}r^{a_{2}}_{m_{j}n_{k}}-r^{a_{2}}_{n_{i}m_{j}}r^{a_{1}}_{m_{j}n_{k}}),$
$\displaystyle(F_{a_{1}a_{2}}^{nm})_{m_{i}m_{k}}$
$\displaystyle=i\sum_{j}(r^{a_{1}}_{m_{i}n_{j}}r^{a_{2}}_{n_{j}m_{k}}-r^{a_{2}}_{m_{i}n_{j}}r^{a_{1}}_{n_{j}m_{k}}).$
(S11)

Therefore,

$\displaystyle\epsilon^{a_{1}a_{2}a_{3}a_{4}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}R^{mn}_{a_{1}a_{2}b_{1}b_{2}}R^{mn}_{a_{3}a_{4}b_{1}b_{4}}$
$\displaystyle=\frac{1}{4}\sum_{\mu,\nu,\rho,\sigma=1}^{4}\left[\epsilon^{a_{1}a_{2}a_{3}a_{4}}(F_{a_{1}a_{2}})_{\mu\nu}(F_{a_{3}a_{4}})_{\rho\sigma}\right]$
$\displaystyle\qquad\times\left[\epsilon^{b_{1}b_{2}b_{3}b_{4}}({\cal
F}_{b_{1}b_{2}})_{\mu\nu}({\cal F}_{b_{3}b_{4}})_{\rho\sigma}\right]$ (S12)

The $FF$ term is

$\displaystyle\sum_{a_{1},\ldots,a_{4}}\epsilon^{a_{1}a_{2}a_{3}a_{4}}(F_{a_{1}a_{2}})_{\mu\nu}(F_{a_{3}a_{4}})_{\rho\sigma}$
$\displaystyle=-\left(4\sum_{a_{1},a_{2},a_{3},a_{4}}\epsilon^{a_{1}a_{2}a_{3}a_{4}}f^{a_{1}}_{1}f^{a_{2}}_{2}f^{a_{3}}_{3}f^{a_{4}}_{4}\right)$
$\displaystyle\qquad\times\sum_{i,j,k,l}\epsilon^{ijkl}(\Gamma_{i}\Gamma_{j})_{\mu\nu}(\Gamma_{k}\Gamma_{l})_{\rho\sigma}.$
(S13)

Now the Gamma matrix part is

$\displaystyle\sum_{i,j,k,l}\epsilon^{ijkl}(\Gamma_{i}\Gamma_{j})_{\mu\nu}(\Gamma_{k}\Gamma_{l})_{\rho\sigma}$
$\displaystyle=\frac{3}{2}\left[\delta_{\mu\sigma}(\Gamma_{5})_{\rho\nu}+(\Gamma_{5})_{\mu\sigma}\delta_{\rho\nu}-\frac{1}{2}\delta_{\mu\nu}(\Gamma_{5})_{\rho\sigma}-\frac{1}{2}(\Gamma_{5})_{\mu\nu}\delta_{\rho\sigma}\right]$
$\displaystyle\quad+\frac{1}{4}\sum_{i,j,k,l}\epsilon^{ijkl}(\Gamma_{i}\Gamma_{j})_{\mu\nu}(\Gamma_{k}\Gamma_{l})_{\rho\sigma},$
(S14)

where we apply the Fierz identity twice (see A). It can be equivalently
written as

$\displaystyle\sum_{i,j,k,l}\epsilon^{ijkl}(\Gamma_{i}\Gamma_{j})_{\mu\nu}(\Gamma_{k}\Gamma_{l})_{\rho\sigma}$
$\displaystyle=2\left[\delta_{\mu\sigma}(\Gamma_{5})_{\rho\nu}+(\Gamma_{5})_{\mu\sigma}\delta_{\rho\nu}-\frac{1}{2}\delta_{\mu\nu}(\Gamma_{5})_{\rho\sigma}-\frac{1}{2}(\Gamma_{5})_{\mu\nu}\delta_{\rho\sigma}\right].$
(S15)

Using this result, we have

$\displaystyle\epsilon^{a_{1}a_{2}a_{3}a_{4}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}R^{mn}_{a_{1}a_{2}b_{1}b_{2}}R^{mn}_{a_{3}a_{4}b_{1}b_{4}}$
$\displaystyle=-2\det(f^{a}_{i})\sum_{\mu,\nu,\rho,\sigma=1}^{4}\bigg{[}\delta_{\mu\sigma}(\Gamma_{5})_{\rho\nu}+(\Gamma_{5})_{\mu\sigma}\delta_{\rho\nu}$
$\displaystyle-\frac{1}{2}\delta_{\mu\nu}(\Gamma_{5})_{\rho\sigma}-\frac{1}{2}(\Gamma_{5})_{\mu\nu}\delta_{\rho\sigma}\bigg{]}\left[\epsilon^{b_{1}b_{2}b_{3}b_{4}}({\cal
F}_{b_{1}b_{2}})_{\mu\nu}({\cal F}_{b_{3}b_{4}})_{\rho\sigma}\right]$
$\displaystyle=4\det(f^{a}_{i})\epsilon^{b_{1}b_{2}b_{3}b_{4}}{\rm
Tr}\left(-\Gamma_{5}{\cal F}_{b_{1}b_{2}}{\cal F}_{b_{3}b_{4}}\right).$ (S16)

By combining this expression and and the expression of the determinant of the
metric and using ${\rm sgn}[\det(f^{a}_{i})]={\rm sgn}(\wedge d^{4}k)$, we
finally obtain

$\displaystyle\chi^{mn}$ $\displaystyle=\int
dk_{1}dk_{2}dk_{3}dk_{4}\sum_{b_{j}}\frac{2^{2}}{(2\pi)^{2}2^{4}2}\epsilon^{b_{1}b_{2}b_{3}b_{4}}$
$\displaystyle\quad\times{\rm Tr}\left(-\Gamma_{5}{\cal F}_{b_{1}b_{2}}{\cal
F}_{b_{3}b_{4}}\right)$
$\displaystyle=\int\frac{d^{4}k}{32\pi^{2}}\sum_{b_{j}}\epsilon^{b_{1}b_{2}b_{3}b_{4}}$
$\displaystyle\quad\times\sum_{i}\left[({\cal F}_{b_{1}b_{2}}{\cal
F}_{b_{3}b_{4}})_{n_{i}n_{i}}-({\cal F}_{b_{1}b_{2}}{\cal
F}_{b_{3}b_{4}})_{m_{i}m_{i}}\right]$ $\displaystyle=p_{1}^{n}-p_{1}^{m}.$
(S17)

##  Supplementary Note 3 First Stiefel-Whitney number in one dimension

Here we consider $PT$-symmetric systems with $(PT)^{2}=1$. Let us choose a
direction $a$ such that $r^{a}_{mn}$ is nonvanishing along a one-dimensional
curve in the Brillouin zone, then the first Stiefel-Whitney number on the
curve is given by

$\displaystyle w_{1}^{mn}$ $\displaystyle=\frac{1}{\pi}\oint d{\bf k}\cdot
i\partial_{\bf k}\log r^{a}_{mn}.$ (S18)

This formula is calculated in a complex smooth and periodic gauge.

The first Stiefel-Whitney number $w_{1}^{mn}$ measures whether $r^{a}_{mn}$ in
a real gauge (where eigenstates $|n\rangle$ and $|m\rangle$ are real such that
$r^{a}_{mn}$ is also real) reverses its direction as it goes around the one-
dimensional closed curve. To see this, since $r^{a}_{mn}$ is real-valued in a
real gauge, it has zero phase winding. It means that, if it had a nonzero
phase winding $w_{1}^{mn}\neq 0$ in the smooth gauge, the gauge transformation
$\\{|n\rangle,|m\rangle\\}\rightarrow\\{e^{i\phi_{n}}|n\rangle,e^{i\phi_{m}}|m\rangle\\}$
from the complex gauge to real gauge (i.e., $r^{a}_{mn}\rightarrow
e^{-i(\phi_{n}-\phi_{m})}r^{a}_{mn}$) should satisfy $\frac{1}{\pi}\oint d{\bf
k}\cdot\partial_{\bf k}(\phi_{n}-\phi_{m})=w_{1}^{mn}$ to cancel the
nontrivial phase winding. However, this kind of gauge transformation induces a
nontrivial boundary condition on the real-valued tangent vector,
$r^{a}_{mn}(2\pi)=e^{i[\phi_{nm}(2\pi)-\phi_{nm}(0)]}r^{a}_{mn}(0)=e^{i\pi
w_{1}^{mn}}r^{a}_{mn}(0)$ in real gauge, since
$r^{a}_{mn}(2\pi)=r^{a}_{mn}(0)$ in the initial complex smooth and periodic
gauge. This means that $r^{a}_{mn}$ in the $w_{1}^{mn}=1$ case do not form a
tangent vector of a real manifold, because it is not a smooth and single-
valued vector on a circle.

Let us now show that $w_{1}^{mn}$ can be related to topological invariants of
bands $n$ and $m$ as in the case of the Euler characteristic of complex
manifolds. That is, we show that

$\displaystyle w_{1}^{mn}=w_{1}^{n}-w_{1}^{m},$ (S19)

where $w_{1}^{n}$ and $w_{1}^{m}$ are the first Stiefel-Whitney numbers of
bands $n$ and $m$. Since $w_{1}^{n}$ is identical to the Berry phase of band
$n$ divided by $\pi$ [3, 4], i.e.,

$\displaystyle w_{1}^{n}=\frac{1}{\pi}\oint d{\bf k}\cdot{\bf\cal A},$ (S20)

Eq. (S19) follows from that the real part of the following shift vector
$R^{c,a}=i(r^{a}_{mn})^{-1}\partial_{c}r^{a}_{mn}+({\cal A}^{c}_{mm}-{\cal
A}^{c}_{nn})$ vanishes in $PT$-symmetric systems.

We can show ${\rm Re}R^{c,a}=0$ as follows. The $PT$ symmetry constraints
$PT|n\rangle=e^{i\theta_{n}}|n\rangle$ and
$PT|m\rangle=e^{i\theta_{m}}|m\rangle$ lead to
$r_{mn}=i|r_{mn}|e^{i(\theta_{m}-\theta_{n})/2}$, ${\cal
A}^{a}_{mm}=\frac{1}{2}\partial_{a}\theta_{m}$, ${\cal
A}^{a}_{nn}=\frac{1}{2}\partial_{a}\theta_{n}$ for $m\neq n$. Therefore, we
have
$i(r^{a}_{mn})^{-1}\partial_{c}r^{a}_{mn}=-\frac{1}{2}(\partial_{c}\theta_{m}-\partial_{c}\theta_{n})+i|r^{a}_{mn}|^{-1}\partial_{c}|r^{a}_{mn}|$.

##  Supplementary Note 4 Vanishing of the torsion tensor on generalized flag
manifolds

Let us define

$\displaystyle\hat{e}_{a}$
$\displaystyle=\sum_{m>n}|u_{m}\rangle(r^{a}_{mn}-{\cal A}^{a}_{mn})\langle
u_{n}|,$ (S21)

which is the complexification of the Maurer-Cartan form
$\hat{\xi}_{a}=\hat{e}_{a}+\hat{e}^{\dagger}_{a}$ of a generalized complex
flag manifold

$\displaystyle{\rm Fl}_{\mathbb{C}}(N_{1},\ldots,N_{k})=\frac{U(N)}{{\rm
U}(1)\times\ldots\times{\rm U}(N_{k})},$ (S22)

and ${\cal A}^{a}$ is the ${\rm U}(1)\times\ldots\times{\rm U}(N_{k})$ Berry
connection. The Hermitian metric is defined by

$\displaystyle Q_{ba}=(\hat{e}_{b},\hat{e}_{a})={\rm
Tr}[\hat{e}^{\dagger}_{b},\hat{e}_{a}],$ (S23)

and the Hermitian connection is defined by

$\displaystyle
C_{bca}=(\hat{e}_{b},\nabla_{c}\hat{e}_{a})=(\hat{e}_{b},\partial_{c}\hat{e}_{a}).$
(S24)

The torsion tensor is then defined by

$\displaystyle T_{bca}$
$\displaystyle=C_{bca}-C_{bac}-i(\hat{e}_{b},[\hat{e}_{c},\hat{e}_{a}]).$
(S25)

To show that torsion is zero. let us calculate the commutator.

$\displaystyle(\partial_{c}\hat{e}_{a}-\partial_{a}\hat{e}_{c})_{mn}$
$\displaystyle=\partial_{c}(r^{a}_{mn}-{\cal A}^{a}_{mn})$
$\displaystyle\qquad-i\sum_{p}\left[{\cal A}^{c}_{mp}(r^{a}_{pn}-{\cal
A}^{a}_{pn})-(r^{a}_{mp}-{\cal A}^{a}_{mp}){\cal
A}^{c}_{pn}\right]-(c\leftrightarrow a)$
$\displaystyle=i\sum_{p}(r^{c}_{mp}r^{a}_{pn}-r^{a}_{mp}r^{c}_{pn})-{\cal
F}^{ca}_{mn}$ $\displaystyle\qquad-i\sum_{p}\bigg{[}{\cal
A}^{c}_{mp}r^{a}_{pn}+r^{c}_{mp}{\cal A}^{a}_{pn}-r^{a}_{mp}{\cal
A}^{c}_{pn}-{\cal A}^{a}_{mp}r^{c}_{pn}$ $\displaystyle\qquad+({\cal
A}^{c}_{mp}{\cal A}^{a}_{pn}-{\cal A}^{a}_{mp}{\cal A}^{c}_{pn})\bigg{]}$
$\displaystyle=-{\cal F}^{ca}_{mn}+i\sum_{p}\bigg{[}(r^{c}_{mp}-{\cal
A}^{c}_{mp})(r^{a}_{pn}-{\cal A}^{a}_{pn})$
$\displaystyle\qquad-(r^{a}_{mp}-{\cal A}^{a}_{mp})(r^{c}_{pn}-{\cal
A}^{c}_{pn})\bigg{]}$ $\displaystyle=-{\cal
F}^{ca}_{mn}+i[\hat{e}_{c},\hat{e}_{a}]_{mn}.$ (S26)

Because $(\hat{e}_{b},{\cal F}^{ca})=0$, we have
$C_{bca}-C_{bac}=(\hat{e}_{b},\partial_{c}\hat{e}_{a}-\partial_{c}\hat{e}_{a})=i(\hat{e}_{b},[\hat{e}_{c},\hat{e}_{a}])$,
and thus

$\displaystyle T_{bca}$ $\displaystyle=0.$ (S27)

One can also show that $T_{bca}=0$ on generalized real and quaternionic flag
manifolds.

##  Appendix A Fierz identities

Here we follow Ref. [5]. Let us introduce the following notation
$\Gamma^{\mu}_{A}$ for the 16 generators of the $4\times 4$ matrices. Here,
$A$ indicates the representation under the action of $O(4)$, and $\mu$
indicates the components in it.

$\displaystyle\Gamma^{\mu=1}_{S}$ $\displaystyle=1,$
$\displaystyle\Gamma^{\mu=1,\ldots,4}_{V}$ $\displaystyle=\Gamma_{\mu},$
$\displaystyle\Gamma^{\mu=1,\ldots,6}_{T}$
$\displaystyle=-i\Gamma_{i}\Gamma_{j},$
$\displaystyle\Gamma^{\mu=1,\ldots,4}_{A}$
$\displaystyle=i\Gamma_{5}\Gamma_{\mu},$ $\displaystyle\Gamma^{\mu=1}_{P}$
$\displaystyle=\Gamma_{5},$ (S28)

where $\\{\Gamma_{i},\Gamma_{j}\\}=\delta_{ij}$ for $i,j=1,\ldots,5$.

The Fierz identity follows from the completeness of these basis matrices. That
is, for any $4\times 4$ matrix $M$, we can express it as

$\displaystyle M=\sum_{\mu,A}m^{\mu}_{A}\Gamma^{\mu}_{A}$ (S29)

for some $m^{\mu}_{A}$. It’s value given by

$\displaystyle m^{\mu}_{A}=\frac{1}{4}{\rm Tr}(\Gamma^{\mu}_{A}M),$ (S30)

where we use (no Einstein summation convention below)

$\displaystyle{\rm Tr}(\Gamma^{\mu}_{A}\Gamma^{\nu}_{B})$
$\displaystyle=4\delta_{\mu\nu}\delta_{AB}.$ (S31)

Rewriting the above completeness relation,

$\displaystyle M_{ab}$ $\displaystyle=\sum_{\mu,A}\frac{1}{4}{\rm
Tr}(\Gamma^{\mu}_{A}M)(\Gamma^{\mu}_{A})_{ab}$
$\displaystyle=\frac{1}{4}\sum_{\mu,A}\sum_{c,d}(\Gamma^{\mu}_{A})_{cd}M_{dc}(\Gamma^{\mu}_{A})_{ab}$
$\displaystyle=\sum_{c,d}\left[\frac{1}{4}\sum_{\mu,A}\Gamma^{\mu}_{A}\Gamma^{\mu}_{A}\right]_{ab;cd}M_{dc},$
(S32)

one can find

$\displaystyle\frac{1}{4}\sum_{\mu,A}(\Gamma^{\mu}_{A})_{ab}(\Gamma^{\mu}_{A})_{cd}=\delta_{ad}\delta_{bc}.$
(S33)

Let us multiply
$(\Gamma^{\nu}_{B})_{b^{\prime}b}(\Gamma^{\nu}_{B})_{d^{\prime}d}$ and sum
over $\nu,a,c$.

$\displaystyle\sum_{\nu}(\Gamma^{\nu}_{B})_{ad^{\prime}}(\Gamma^{\nu}_{B})_{cb^{\prime}}$
$\displaystyle=\frac{1}{4}\sum_{\mu,\nu,A}(\Gamma^{\mu}_{A}\Gamma^{\nu}_{B})_{ab^{\prime}}(\Gamma^{\mu}_{A}\Gamma^{\nu}_{B})_{cd^{\prime}}$
$\displaystyle=\sum_{\rho,\sigma,C}(\Gamma^{\rho}_{C})_{ab^{\prime}}(\Gamma^{\sigma}_{C})_{cd^{\prime}}\left(\frac{1}{4}\sum_{\mu,\nu,A}f_{CAB}^{\rho\mu\nu}f_{CAB}^{\sigma\mu\nu}\right)$
$\displaystyle=\sum_{\rho,C}(\Gamma^{\rho}_{C})_{ab^{\prime}}(\Gamma^{\rho}_{C})_{cd^{\prime}}K_{BC},$
(S34)

where we use that
$\sum_{\mu,\nu,A}f_{CAB}^{\rho\mu\nu}f_{CAB}^{\sigma\mu\nu}\propto\delta_{\rho\sigma}$
because it should be $O(4)$-invariant. The value of $K$ can be calculated by
contracting $a$ with $b^{\prime}$ and $c$ with $d^{\prime}$:

$\displaystyle{\rm
Tr}(\Gamma^{\nu}_{B}\Gamma^{\sigma}_{D}\Gamma^{\nu}_{B}\Gamma^{\sigma}_{D})=\sum_{\rho,C}\left[{\rm
Tr}(\Gamma^{\rho}_{C}\Gamma^{\sigma}_{D})\right]^{2}K_{BC}=16K_{BD}$ (S35)

So,

$\displaystyle K_{AB}=\frac{1}{16}\sum_{\mu,\nu}{\rm
Tr}(\Gamma^{\mu}_{A}\Gamma^{\nu}_{B}\Gamma^{\mu}_{A}\Gamma^{\nu}_{B}).$ (S36)

$\displaystyle\left(K_{AB}\right)=\frac{1}{4}\begin{pmatrix}1&1&1&1&1\\\
4&-2&0&2&-4\\\ 6&0&-2&0&6\\\ 4&2&0&-2&-4\\\ 1&-1&1&-1&1\end{pmatrix}.$ (S37)

## References

  * Milnor and Weaver [1997] J. Milnor and D. W. Weaver,  _Topology from the differentiable viewpoint_ (Princeton university press, 1997).
  * Allendoerfer and Weil [1943] C. B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Am. Math. Soc. 53, 101 (1943).
  * Ahn  _et al._ [2018] J. Ahn, D. Kim, Y. Kim, and B.-J. Yang, Band topology and linking structure of nodal line semimetals with $Z_{2}$ monopole charges, Phys. Rev. Lett. 121, 106403 (2018).
  * Ahn  _et al._ [2019] J. Ahn, S. Park, D. Kim, Y. Kim, and B.-J. Yang, Stiefel-Whitney classes and topological phases in band theory, Chin. Phys. B 28, 117101 (2019).
  * Nieves and Pal [2004] J. F. Nieves and P. B. Pal, Generalized fierz identities, American journal of physics 72, 1100 (2004).