# Quantized shift response in multi-gap topological phases

Wojciech J. Jankowski<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory,
Department of Physics, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
Robert-Jan Slager<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory,
Department of Physics, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

We show that certain 3D multi-gap topological insulators can host quantized
shift photoconductivities due to bulk invariants that are defined under
reality conditions imposed by additional symmetries. We recast the
quantization in terms of the integrated torsion tensor and the non-Abelian
Berry connection constituting Chern-Simons forms. Physically, we recognize
that the topological quantization emerges purely from virtual transitions
contributing to the optical response. Our findings provide another quantized
electromagnetic DC response due to the non-trivial band topology, beyond the
quantum anomalous Hall effect of Chern insulators and quantized circular
photogalvanic effect found in Weyl semimetals.

††preprint: APS/123-QED

Introduction.— Topological insulators and semimetals constitute an active
research area in condensed matter physics [1, 2, 3, 4]. Amid a fast track of
progress, especially the past years have seen the emergence of a rather
uniform view on a large fraction of topological materials upon including
crystalline symmetries [5, 6, 7, 8, 9, 10]. In particular, considering how
irreducible representations at high symmetry points need to consistently glue
together over the whole Brillouin zone, one may obtain a set of exhaustive
consistency equations [7] that upon comparing to real space Wannierization
conditions leverages universally implementable schemes to discern topological
materials [8, 10].

More recently, however, further advances have revealed a new class of
topological phases that depend on multi-gap conditions [11, 12, 13, 14, 15]
and cannot be a priori captured by symmetry-indicated methods. In such phases,
band spaces can acquire novel multi-gap invariants. In particular, in presence
of a reality condition of the Hamiltonian, such as the presence of
$\mathcal{C}_{2}\mathcal{T}$ (two-fold rotations and time reversal) or
$\mathcal{P}\mathcal{T}$ (parity and time reversal) symmetries, band
degeneracies can carry non-Abelian charges [16], akin to disclination charges
in a bi-axial nematic [17, 18, 19, 20]. As a result of braiding processes,
band subspaces can subsequently come to host a set of similarly-valued charges
that can not be removed, posing an obstruction to annihilate as long as the
gaps with the other parts of spectrum are maintained [12, 21]. This
obstruction to annihilate degeneracies within the band subspace is moreover
quantified by a topological invariant, the Euler class [12, 22, 11].

While the multi-gap point of view opens up novel theoretical pursuits,
including deeper relations with homotopy theory [11], recent developments have
shown distinctive physical signatures of these kinds of phases [23]. In the
out-of-equilibrium context, novel Floquet phases were postulated [24] and
monopole–anti-monopole creation in the quench dynamics of Euler phases
captured by a Hopf map was predicted [25]. The latter was in fact recently
observed [26]. Similarly, stress or strain has been predicted to lead to
braiding and Euler invariants in the context of phononic [27, 28] and
electronic spectra [12, 29] of real materials, where in the electronic case it
was also shown that multi-gap physics can arise by virtue of structural phase
transitions [30]. Finally, multi-gap topologies are increasingly finding their
way to metamaterial experiments [31, 32, 33, 34, 35].

Figure 1:  (a) Quantized bulk photovoltaic shift currents induced by incident
light. A second-order DC current density response $j_{z}(0)$ of a multi-gap
non-Abelian topological insulator with photoconductivity quantized in terms of
$2\frac{e^{3}}{\hbar^{2}}$, on coupling circularly-polarized light with photon
energy $\hbar\omega$ and electric field
$\mathcal{E}(\frac{\hat{x}+i\hat{y}}{\sqrt{2}})$. The shift response is due to
virtual transitions, as captured by the torsion tensor $T^{nm}_{abc}$ between
the occupied and unoccupied bands. (b) Diagrammatic representation of the
corresponding virtual transitions between topological bands, inducing a shift
response encoded by the torsion tensor. Transition dipole amplitudes
$r^{x}_{24},r^{y}_{43}$ from band $\ket{u_{2,{\bf k}}}$ to $\ket{u_{3,{\bf
k}}}$, virtually through $\ket{u_{4,{\bf k}}}$, are indicated. These are
proportional to the elements of non-Abelian Berry connection $A^{x}_{24}$,
$A^{y}_{43}$, contributing to the torsion tensor element $T^{23}_{xyz}$.

A hallmark of topological materials is that they can render quantized
responses. Chern bands for example were shown to exhibit quantized anomalous
Hall (QAH) and quantized dichroism responses [36, 37, 38, 39]. Similarly, Weyl
semimetals can be probed by second-order quantized circular photogalvanic
effects (QCPGE) [40, 41, 42, 43]. We note that these phenomena are presently
more and more understood from the upcoming perspective of quantum geometry.
Following formulations of quantum geometric tensors (QGT) quite some time ago
[44, 45], these concepts have recently been reinvigorated in several contexts
that range from superconductivity and superfluidity to transport properties
[46, 47, 48, 49]. Response theories fit particularly well in this context [50]
as they relate directly to elements of QGTs (see also Appendix A). Recent
progress includes rephrasing matrix elements of optical processes in terms of
Riemannian geometry [51], and moreover a universal strategy in terms of
Plücker embeddings [52], to effectively calculate geometric entities and QGTs,
and thus accordingly optical responses, in general multi-band systems.

The mentioned mathematical Plücker framework works in two directions. Apart
from quantifying geometry, it allows for an efficient method to model
topological phases for arbitrarily partitioned band spaces. As such, this
approach has stood at the basis of generalizing multi-gap topologies to fully-
generalized settings, as it allows for a direct evaluation of the classifying
space set by the Hamiltonian [11, 14]. Particular examples include three
dimensional (3D) phases of real-valued 4-band Hamiltonians, displaying a
double Hopf invariant when the bands are partitioned in two 2-band subspaces
[53], a Pontryagin index if the bands partition into a 3-band and a 1-band
subspace [54], or two winding numbers relating to isoclinic rotations when all
bands are non-degenerate, referred to as so-called generalized multi-gap flag
phases [11, 14, 54] (for more details, see Appendix B).

It is in this field of interplay of geometrical insights, non-Abelian multi-
gap notions in three-dimensional systems, and non-linear optical responses,
that we find our results. Namely, in this work, we discover quantized
photovoltaic shift effect in three-dimensional non-Abelian topological
insulators enjoying a reality condition. In particular, we show that (i) the
shift response in these systems is purely governed by the virtual transitions
in the topological multi-band subspaces, which (ii) we identify with the non-
triviality of the torsion tensor introduced in the Riemannian-geometric
reformulation of the optical responses [51]. Correspondingly, we retrieve its
quantization from the real Chern-Simons forms (iii), directly related to the
aforementioned Hopf indices. We numerically validate these findings in the
minimal models capturing the relevant non-Abelian topologies (iv). In this
context, we identify (v) the quantized changes of the shift response induced
by the virtual transitions over topological phase transitions (TPTs). Finally,
we discuss the novel $quantized$ shift effect, and its changes across TPTs, in
the context of candidate materials.

Figure 2:  (a) Topological band structure of a Real Hopf Insulator
($k_{z}=0$). (b) Band structure of RHI model, Eq. (9), for $M=3/2$, exhibiting
a semimetallic node emerging at the BZ edge $(k_{x},k_{y},k_{z})=(\pi,\pi,0)$,
across the topological phase transition. (c) The variation of the quantized
shift effect represented by $F_{\text{sym}}$, indicating TPTs at $M=3/2$ and
$M=3$, across which both topological bulk invariant and the associated
quantized integrated shift photoconductivities $F_{\text{sym}}$ change
sharply. Namely, in the central, low $|M|$ region,
$(\chi_{z},\chi_{w})=(0,2w)$, and for $3/2<|M|<3$,
$(\chi_{z},\chi_{w})=(0,-w)$.

Non-linear optical shift responses.— The non-linear second-order optical shift
responses central to this work are manifested by bulk DC photovoltaic
currents,

$j_{c}(0)=\sum_{a,b=x,y,z}\sigma^{cab}_{\text{shift}}(\omega)\mathcal{E}_{a}(\omega)\mathcal{E}_{b}(-\omega),$
(1)

induced on coupling a 3D material to AC electric field with components of
magnitude $\mathcal{E}_{a}(\omega)$, where $\omega$ is the frequency of
incident electromagnetic radiation. The second-order shift photoconductivity
$\sigma^{cab}_{\text{shift}}(\omega)$, which can be decomposed in terms of
linear $\sigma^{cab}_{\text{shift,L}}(\omega)$ and circular
$\sigma^{cab}_{\text{shift,C}}(\omega)$ contributions picked by the
corresponding light polarizations
(${\sigma^{cab}_{\text{shift}}(\omega)=\sigma^{cab}_{\text{shift,L}}(\omega)+\mathrm{i}\sigma^{cab}_{\text{shift,C}}(\omega)}$)
[22] was formulated in terms of the momentum-space derivatives of the QGT,
encoded by a Hermitian connection $C^{mn}_{abc}$ [51] (see also Appendix A).
More explicitly, $C^{mn}_{bca}=-\mathrm{i}r^{b}_{nm}r^{a}_{mn}R^{c,a}_{mn}$
[55, 51], in terms of the transition dipole moments
$r^{a}_{nm}=\mathrm{i}(1-\delta_{mn})A^{a}_{nm}$, given by the non-Abelian
Berry connection $A^{a}_{nm}=\bra{u_{n,{\bf k}}}\ket{\partial_{k_{a}}u_{m,{\bf
k}}}$, with $\ket{\psi_{n,{\bf k}}}=e^{i{\bf k}\cdot{\bf r}}\ket{u_{n,{\bf
k}}}$, the Bloch states. Here, $R^{c,a}_{mn}\equiv
A^{c}_{mm}-A^{c}_{nn}+\mathrm{i}\partial_{k_{c}}\text{log}~{}r^{a}_{mn}$ is
the ‘shift vector’, associated with the name of the effect. Intuitively,
$R^{c,a}_{mn}$ reflects a positional shift of an electron upon optical
transition [56, 57]. In terms of $C^{mn}_{abc}$, the photoconductivity reads
[51],

$\sigma^{cab}_{\text{shift}}(\omega)=\frac{\pi
e^{3}}{2\hbar^{2}}\sum_{m,n}\int_{\text{BZ}}\frac{\differential^{3}k}{(2\pi)^{3}}\delta(\omega-\omega_{mn})f_{nm}\mathrm{i}\left(C^{nm}_{bca}-C^{mn}_{acb}\right).$
(2)

In the above, an integration of the Hermitian connection over the entire
Brillouin zone (BZ) is assumed, $f_{nm}$ are the differences in the Fermi
occupation factors of correspondingly occupied and unoccupied bands $n,m$,
which are split by energies $\hbar\omega_{mn}$. Importantly, the key
ingredient of the response, $C^{mn}_{abc}$, defines the torsion tensor
$T^{mn}_{abc}$, a quantity central to this work,

$T^{mn}_{abc}\equiv C^{mn}_{abc}-C^{mn}_{acb}.$ (3)

Following Ref. [51], we notice that physically the torsion tensor is
associated with virtual transitions between three bands $n,p,m$ (for
reference, see Fig. 1). The question is, if such transitions can result in a
topological quantized response in certain scenarios. In the following, we
provide a positive answer.

Results.— The central result of this work is to recognize that the integrated
$circular$ shift photoconductivity relevant for the net bulk photovoltaic
responses [58], can be expressed in terms of the torsion tensor (see Appendix
C for a full derivation),

$\begin{split}\int\differential\omega~{}(\sigma^{abc}_{\text{shift,C}}(\omega)+\sigma^{bca}_{\text{shift,C}}(\omega)+\sigma^{cab}_{\text{shift,C}}(\omega))\\\
=-\frac{e^{3}}{8\pi^{2}\hbar^{2}}\int_{\text{BZ}}\differential^{3}k\sum_{m,n}f_{mn}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab}).\end{split}$
(4)

Centrally, for non-Abelian multi-gap topological insulators as considered in
this work, we recognize that it is naturally possible to induce torsion in any
four real Bloch bands close to the Fermi level $\\{\ket{u_{1,{\bf
k}}},\ket{u_{2,{\bf k}}},\ket{u_{3,{\bf k}}},\ket{u_{4,{\bf k}}}\\}$. Here,
two bands are occupied and two are unoccupied. Correspondingly, we provide
schematics for the virtual transitions associated with the torsion in Fig. 1.
Furthermore, on additionally considering the zero-temperature limit,
$f_{nm}=1$, we retrieve (see Appendix D):

$\begin{split}F_{\text{sym}}\equiv-\mathrm{i}\int\differential\omega~{}(\sigma^{xyz}_{\text{shift}}(\omega)+\sigma^{yzx}_{\text{shift}}(\omega)+\sigma^{zxy}_{\text{shift}}(\omega))\\\
=\frac{e^{3}}{4\pi^{2}\hbar^{2}}\int_{\text{BZ}}\differential^{3}k~{}({\bf
a}^{v}\cdot\text{{\bf Eu}}^{v}+{\bf a}^{c}\cdot\text{{\bf
Eu}}^{c}),\end{split}$ (5)

as ${\sigma^{cab}_{\text{shift,L}}(\omega)=0}$ vanishes by the reality
condition. Following the notion of the ‘figure of merit’ of Ref. [58], here we
spectrally integrate the shift photoconductivities to obtain a symmetrized
analogue, $F_{\text{sym}}$. Furthermore, we define Euler connections and
curvatures [12] in terms of the pairs of conduction and valence bands ${{\bf
a}^{v/c}=\bra{u_{2/4,{\bf k}}}\nabla_{\bf k}\ket{u_{1/3,{\bf k}}}}$,
${\text{{\bf Eu}}^{v/c}=\nabla_{\bf k}\crossproduct{\bf a}^{v/c}}$. As shown
in Appendix D, the integrand can be written as the sum of Chern-Simons
3-forms, given by non-Abelian Berry connection matrices $A^{a}_{nm}$ as
$\text{Tr}[{\epsilon_{abc}(A^{a}_{nm}\partial_{k_{b}}A^{c}_{mn}+\frac{2}{3}A^{a}_{nm}A^{b}_{mp}A^{c}_{pn})]}$,
constrained under the reality condition, which provides a quantization upon
integrating over the compact BZ 3-torus. Thus, we obtain a quantized second-
order photovoltaic shift effect on coupling to circularly-polarized light,
purely due to the virtual transitions in the real non-Abelian topological
bands. We thus identify a second-order shift current cousin of the QCPGE
manifested by the second-order injection currents in the Weyl semimetals [40,
59].

We recognize that the quantized response is present in the aforementioned real
Hopf insulators [53] (RHIs) classified by homotopy,
$\pi_{3}[S^{2}\crossproduct
S^{2}]\cong~{}\mathbb{Z}\oplus\mathbb{Z}~{}\equiv~{}\mathbb{Z}^{2}$, of a
double Hopf fibration (see also Appendix B), as the integrated 3-form
introduced in Eq. (5) can be recast in terms of two real Hopf invariants. The
Hopf invariants $\chi_{z},\chi_{w}$ require two occupied, and unoccupied
bands, which can be taken fully degenerate (see Fig. 2). In particular, these
invariants can be expressed by adding/subtracting products of the Euler
curvature and connection elements mixing both valence and conduction bands
[53]: ${\bf a}^{c}\cdot\text{{\bf Eu}}^{v}$ and ${\bf a}^{v}\cdot\text{{\bf
Eu}}^{c}$, in the integrand of Eq. (5) (see Appendix B for further details).
In that case, we ultimately obtain:

$\begin{split}F_{\text{sym}}=\frac{2e^{3}}{\hbar^{2}}(\chi_{z}+\chi_{w}).\end{split}$
(6)

Furthermore, we recognize that beyond the RHIs, the quantization condition is
expected to hold in a broader class of non-Abelian topological insulators
satisfying the reality condition imposed by $\mathcal{P}\mathcal{T}$
symmetries. In particular, this includes the recently introduced multi-gap
topological phases classified by real flag manifolds
($\pi_{3}[\mathsf{Fl}_{1,1,1,1}(\mathbb{R})]\cong\mathbb{Z}^{2}$) being
isoclinic rotation winding numbers $w_{L},w_{R}$. Here, all four bands are
isolated by topological gaps, all admitting topological edge modes induced by
the bulk invariant and $\pi$-Zak phases [54]. Analogously to the RHIs, the
invariants $w_{L},w_{R}$ can be expressed in terms of the non-Abelian Berry
connections defining Chern-Simons 3-forms of Eq. (5), see also Appendix B for
more details on the classification of flag manifolds and multiply gapped non-
Abelian insulators. In this case, we similarly obtain:

$\begin{split}F_{\text{sym}}=\frac{2e^{3}}{\hbar^{2}}(w_{L}+w_{R}).\end{split}$
(7)

While $w_{L}$ and $w_{R}$ can be obtained analogously to the real Hopf
invariants from the non-Abelian Berry connections, these are not associated
with the Hopf fibrations, which characterize the mappings of two-band
subbundles to 2-spheres $S^{2}\times S^{2}$. In other words, each winding
number corresponds to a non-trivial element of $\pi_{3}[S^{3}]$, rather than
$\pi_{3}[S^{2}]$. In particular, the multiply-gapped flag insulators support
edge states in all three topological gaps between four topological bands [54].
Finally, we retrieve a quantized response due to a single topological index,
rather than two, under the $3\oplus 1$ band partitioning of
$\mathcal{PT}$-symmetric phases with Pontryagin index $Q\in\mathbb{Z}$ [54];
where $Q$ corresponds to the single homotopy, $\pi_{3}[S^{3}]\cong\mathbb{Z}$
(see Appendix B). Here, to access all transitions necessary for the
quantization of the response, one band below the principal gap needs to become
depopulated, up to the band degeneracies between the second and third band,
yielding a semimetallic state. Correspondingly, such non-Abelian semimetals
enjoy a simple quantization as,

$\begin{split}F_{\text{sym}}=\frac{2e^{3}}{\hbar^{2}}Q,\end{split}$ (8)

completing the glossary of quantized shift responses due to different types of
non-Abelian four-band topologies, as identified in this work.

Model realizations.— We now concretize the above ideas in the context of
simple models. For details on general classes of models capturing the
considered band topologies, we furthermore refer to Appendix B.

In Fig. 2, we demonstrate the results for RHIs. Here, we present the evolution
of quantized shift responses in the RHI models with
$(\chi_{z},\chi_{w})=(0,w)$, taking the form $H({\bf k})=R_{w}({\bf
k})~{}\text{diag}(-1,-1,1,1)~{}R^{\text{T}}_{w}({\bf k})$ generated with a
matrix of Bloch eigenvectors,

$\begin{split}R_{w}({\bf k})=\sin wk_{x}\Gamma_{11}+\mathrm{i}\sin
k_{y}\Gamma_{23}-\sin k_{z}\Gamma_{31}+\\\ +(M-\cos wk_{x}-\cos k_{y}-\cos
k_{z})\Gamma_{03},\end{split}$ (9)

where $\Gamma_{ij}=\sigma_{i}\otimes\sigma_{j}$ are $4\crossproduct 4$
Kronecker products of identity and Pauli matrices
$(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3})\equiv(\mathbf{1}_{2},\sigma_{x},\sigma_{y},\sigma_{z})$.
Here, $M$ is the topological mass, changes of which can cause phase
transitions trivializing the RHI. In Fig. 3, we show analogous results for the
multi-gap flag phases. Here, we also employ models with tunable topological
mass $M$. In particular, we choose $H({\bf
k})=V_{R}V^{\text{T}}_{L}~{}\text{diag}[-2,-1,1,2]V_{L}V^{\text{T}}_{R}$ with,

$\begin{split}V_{L/R}=\sin w_{L/R}k_{x}\Gamma_{11}+\mathrm{i}\sin
k_{y}\Gamma_{23}-\sin k_{z}\Gamma_{31}+\\\ +(M-\cos w_{L/R}k_{x}-\cos
k_{y}-\cos k_{z})\Gamma_{03},\end{split}$ (10)

in terms of the isoclinic winding numbers $w_{L},w_{R}$. In both cases, we
observe sharp quantized jumps in the integrated shift responses
$F_{\text{sym}}$, coinciding with the values of topological masses
corresponding to TPTs. Naturally, we expect $F_{\text{sym}}$ to vanish, as the
topological numbers dictating the optical shift response in one-to-one
correspondence are trivialized. At the TPT points, the systems become metallic
on gap closings, which results in crossover $F_{\text{sym}}$ values across the
jumps. Notably, the shift photoconductivities in topological semimetals,
including $\mathcal{PT}$-symmetric models, were extensively studied in Refs.
[22, 60, 61, 62].

Figure 3:  (a) Topological band structure of the non-Abelian multi-gap flag
phase for $k_{z}=\pi/2$. (b) Band structure at the critical topological mass
$M=3$ of the flag phase with a semimetallic node emerging at the BZ centre
$(k_{x},k_{y},k_{z})=(0,0,0)$ across the TPT. (c) The quantized shift effect
represented by $F_{\text{sym}}$, changing sharply with the topological mass,
across the values $|M|=3/2$, ${|M|=3}$. Correspondingly, the bulk topological
invariant changes, and trivializes further,
$(w_{L},w_{R})=(-2w,0)\rightarrow(w,0)\rightarrow(0,0)$.

Discussion.— Having introduced the topological quantized phenomenology of the
shift responses in non-Abelian phases, we address the possibilities of
observing such physical effects and how our results galvanize new areas of
investigation. Firstly, similarly to the behaviour of QCPGE associated with
the second-order injection (rather than shift) currents, we expect that on
observing such a characteristic sharp response, the quantization of the
proposed effect might be mildly violated on introducing the corrections due to
the electron-electron interactions [42]. Additionally, one could also wonder
about how significant a contribution to the shift currents due to the 2D
surface modes correspondent to the non-trivial non-Abelian 3D topological bulk
could be. Here it is important to notice that the surface bands are unlikely
to contribute to $F_{\text{sym}}$, because by construction it necessarily
involves combinations of all three spatial dimensions in each constituent
photoconductivity term. Finally, it remains an open question, beyond the
momentum-conserving formulation, of how robust the quantized effect is in real
disorder materials, lacking translational symmetries. While the aspects of
disorder and interactions motivate a separate extensive study, which we leave
for future, at the same time we naturally identify $\mathcal{PT}$-symmetric 3D
topological phases, amongst the candidate materials to realize the effect.
Such phases include CuMnAs, a system modelled with a four-band low-energy
theory, already known to support chiral gyration photocurrents, which include
$j_{x}$ induced by Hermitian connection elements $C_{xxy},C_{yxy}$, under
circularly-polarized light [63, 64], or phases obtainable from anti-
ferromagnetic MnBi2nTe3n+1 heterostructures [65, 66, 67], which were recently
related to non-trivial quantum geometry [68]. In particular, a measurable
quantized jump in $F_{\text{sym}}$ is expected across TPT, if the considered
multi-gap topologies were present. Importantly, to obtain the non-trivial
second-order shift effect, the material needs to be non-centrosymmetric [56],
i.e. break $\mathcal{P}$ symmetry. Hence to preserve $\mathcal{PT}$, it also
necessarily needs to break $\mathcal{T}$ symmetry, i.e. be magnetic. Indeed,
for RHI, $\mathcal{P}$ symmetry is admitted only if $\chi_{w}+\chi_{z}=0$
[53], which yields no shift response, consistently with Eq. (6).

We finally note that recently it was suggested that virtual transitions can
constitute a significant contribution to shift currents in the multi-band
systems, in particular, layered moiré materials [69], and there might exist
interesting connections to the band topology. Here, we precisely retrieve such
a connection, manifestly captured by a quantization encoded in the virtual
multi-band transitions. Additionally, we show that indeed, the changes in the
band topology across TPTs directly result in the quantized jumps of the shift
responses present in non-Abelian multi-band topological insulators. Relating
the above results and multi-gap phases to moiré van der Waals platforms
therefore generally entails for a promising future pursuit.

Summary and Outlook.— We identify a new quantized photovoltaic response in
terms of (circular) shift photoconductivities manifested by associated second-
order currents, which can be induced by non-trivial non-Abelian multi-gap band
topology. We analytically and numerically retrieve this response in
topological phases with real Hopf, flag manifold, or Pontryagin-indexed,
multi-gap topology, captured by minimal four Bloch band models under reality
conditions. In particular, we show that the integrated total shift effect,
purely contributed by the virtual interband transitions, can indicate
topological phase transitions in non-Abelian topological phases. Hence, we
find another unique topological quantized DC response, beyond QAH effect in
Chern insulators and QCPGE in Weyl semimetals, providing a smoking-gun
criterion for the search of exotic non-Abelian multi-gap topological phases.

###### Acknowledgements.

W.J.J. acknowledges funding from the Rod Smallwood Studentship at Trinity
College, Cambridge. R.-J.S. acknowledges funding from a New Investigator
Award, EPSRC grant EP/W00187X/1, a EPSRC ERC underwrite grant EP/X025829/1,
and a Royal Society exchange grant IES/R1/221060, as well as Trinity College,
Cambridge. We cordially thank Zory Davoyan, Adrien Bouhon, and F. Nur Ünal for
numerous discussions concerning non-Abelian and multi-gap topological phases.
R.-J. S thanks Joel E. Moore for discussions.
Note added: While completing this manuscript, we became aware of the work [69]
on enhancing shift currents via multiband transitions, although in the 2D
context of moiré materials. Here, beyond retrieving an unprecedented quantized
shift response due to the virtual transitions in 3D systems, we explictly
demonstrate shift responses for probing TPTs in 3D multiband topological
materials.

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##  Appendix A Basic quantum-geometric definitions

In this section, for completeness, we define the relevant quantum-geometric
objects within the single-particle picture of Bloch states, as originally
introduced in Ref. [44] and extended towards optical responses in Ref. [51].
We start with the quantum-geometric tensor (QGT) for bands $m,n$ mentioned in
the main text,

$Q^{mn}_{ab}\equiv r^{a}_{nm}r^{b}_{mn}.$ (11)

Here, the transition dipole moment vector, ${\bf r}_{mn}({\bf
k})=\bra{\psi_{m,{\bf k}}}\hat{{\bf r}}\ket{\psi_{n,{\bf k}}}$, provides a
probability amplitude for an electric dipole (E1) transition between Bloch
states $\ket{\psi_{n,{\bf k}}}$ and $\ket{\psi_{m,{\bf k}}}$. More abstractly,
${\bf r}_{mn}$ defines a tangent vector on the manifold of quantum states
[51], with components

${\bf r}^{a}_{mn}({\bf k})=\mathrm{i}(1-\delta_{mn})A^{a}_{nm}({\bf k})\;,$
(12)

where $A^{a}_{nm}({\bf k})\equiv\bra{u_{n,{\bf k}}}\ket{\partial_{a}u_{m,{\bf
k}}}$ is the non-Abelian Berry connection, and we employ a simplified notation
$\partial_{a}\equiv\partial_{k_{a}}$.

The QGT, known otherwise as Hermitian metric, decomposes as

$Q^{mn}_{ab}=g^{mn}_{ab}-\frac{\mathrm{i}}{2}F^{mn}_{ab},$ (13)

and equips the vector bundle of Bloch states [52] over a Brillouin zone (BZ),
$\text{BZ}\cong T^{d}$, where $T^{d}$ is a $d$-dimensional torus, with a
Riemannian structure through its real part
$g^{mn}_{ab}=\mathfrak{Re}~{}Q^{mn}_{ab}$, the quantum metric. In particular
$g^{mn}_{ab}$ defines a Fubini-Study metric through
$g^{ab}_{n}=\sum^{\text{unocc}}_{m}g^{mn}_{ab}$,

$ds^{2}_{n}\equiv 1-|\bra{u_{n,{\bf k}}}\ket{u_{n,{\bf k}+\delta{\bf
k}}}|^{2}=g^{ab}_{n}dk_{a}dk_{b},$ (14)

for any band $n$. The symplectic part $F^{mn}_{ab}$ yields the Abelian Berry
curvature $F^{n}_{ab}=\sum^{\text{unocc}}_{m}F^{mn}_{ab}$ on performing a
summation over the unoccupied bands.

Furthermore, for higher order in $k$-space derivatives, the Hermitian
connection can be defined. The advantage of the described representation is
that it allows to easily introduce a Hermitian connection as

$C^{mn}_{bca}\equiv r^{b}_{nm}r^{c;a}_{mn},$ (15)

where we include the covariant derivative $r^{b;c}_{nm}$ of the transition
dipole, defined in terms of the Berry connection as

$\begin{split}r^{c;a}_{nm}({\bf k})=\nabla_{a}r^{c}_{nm}({\bf
k})=\partial_{a}r^{c}_{nm}({\bf k})-\mathrm{i}(A^{a}_{nn}({\bf
k})-A^{a}_{mm}({\bf k}))r^{c}_{nm}({\bf k}).\end{split}$ (16)

The formal definition in terms of covariant derivative can be directly recast
into the shift vector definition introduced in the main text.

Furthermore, the Hermitian connection can be decomposed in terms of the metric
connection $\mathfrak{Re}~{}C^{mn}_{abc}$ and symplectic connection
$\mathfrak{Im}~{}C^{mn}_{abc}$ as

$C^{mn}_{abc}=\mathfrak{Re}~{}C^{mn}_{abc}+i\mathfrak{Im}~{}C^{mn}_{abc}.$
(17)

The torsion-free part of the metric connection $\mathfrak{Re}~{}C^{mn}_{abc}$
reads [22],

$\Gamma^{mn}_{abc}=\frac{1}{2}(\partial_{b}g^{mn}_{ac}+\partial_{c}g^{mn}_{ab}-\partial_{a}g^{mn}_{bc}),$
(18)

but this part crucially omits the virtual transitions, essential for the
optical responses of the non-Abelian topological insulators. The torsion
tensor $T^{mn}_{abc}\equiv C^{mn}_{abc}-C^{mn}_{acb}$, as defined in the main
text, explicitly captures the virtual transitions as [51],

$T^{mn}_{abc}=i\sum_{i,j}r^{a}_{n_{i}m_{j}}\sum_{E_{p}\neq
E_{m},E_{n}}(r^{b}_{m_{j}p}r^{c}_{pn_{i}}-r^{c}_{m_{j}p}r^{b}_{pn_{i}}),$ (19)

on direct substition in terms of $r^{a}_{mn}$ transition dipole matrix
elements. Here, we account for the degeneracies, by labelling energies of the
(degenerate) electronic eigenstates at given momentum ${\bf k}$, as
$E_{n}=E_{n_{i}}$. Here, explicitly, the torsion tensor elements combine the
E1 transition amplitudes given by the transition dipole matrix elements
$r^{a}_{nm}$ for (i) starting from a occupied band $n$, (ii) virtually
exciting an electron through band $p$, and (iii) finally ending up
photoexcited in initially unoccupied band $m$.

We note that in particular, if the Bloch states can be chosen in the real
gauge, i.e. $\ket{u_{n,{\bf k}}}$ become real vectors, under the so-called
reality condition (see Appendix B), $C_{abc}=\mathfrak{Re}~{}C_{abc}$, as
follows from a direct substitution of Bloch states to the transition dipole
moment matrix elements.

##  Appendix B Non-Abelian three-dimensional topologies under the reality
condition

In this section, we first clarify the constraints due to the reality condition
imposed by the $\mathcal{P}\mathcal{T}$ symmetry imposed on three-dimensional
non-Abelian topological phases. Regarding the latter, we provide simple models
and elaborate on the topologies of (i) multi-gap flag phases, (ii) the real
Hopf insulator, (iii) Pontryagin-indexed insulator, all of which can be
generated and characterized within the Plücker formalism framework [11, 52].

###  B.1 Reality condition

The action of the $\mathcal{P}\mathcal{T}$ symmetry (here, we consider
spinless $(\mathcal{P}\mathcal{T})^{2}=+1$ symmetry) ensures that the Bloch
Hamiltonian of a three-dimensional solid treated within an independent-
particle approximation is real, $H({\bf k})=H^{*}({\bf k})$, and there exists
a gauge in which the associated eigenstates labelled with quasimomenta ${\bf
k}=(k_{x},k_{y},k_{z})^{\text{T}}$ are also real vectors. In particular, this
follows from the symmetry transformations of the Hamiltonian [11]:

$(\mathcal{PT})H(k_{x},k_{y},k_{z})(\mathcal{PT})^{-1}=H^{*}(k_{x},k_{y},k_{z})=H(k_{x},k_{y},k_{z}),$
(20)

which is related to the fact that the individual symmetries of the composed
symmetry act as:

$\mathcal{T}H(k_{x},k_{y},k_{z})\mathcal{T}^{-1}=H^{*}(-k_{x},-k_{y},-k_{z}),$
(21)
$\mathcal{P}H(k_{x},k_{y},k_{z})\mathcal{P}^{-1}=H(-k_{x},-k_{y},-k_{z}).$
(22)

Hence, the Bloch Hamiltonian, $H(k_{x},k_{y},k_{z})=H^{*}(k_{x},k_{y},k_{z})$,
can be represented as a real matrix [11]. Combined with the Hermiticity of the
Hamiltonian, the Hamiltonian indeed reduces to a real $symmetric$ matrix,
therefore it enjoys a well-known property that its eigenvectors can be taken
real, $\ket{u_{n,{\bf k}}}\in\mathbb{R}^{N}$.

Therefore, we note that under such reality condition imposed by the
$\mathcal{P}\mathcal{T}$ symmetry,

${\bf A}_{nn}({\bf k})=i\bra{u_{n,{\bf k}}}\ket{\nabla_{\bf k}u_{n,{\bf
k}}}=i\nabla_{\bf k}\bra{u_{n,{\bf k}}}\ket{u_{n,{\bf
k}}}-\mathrm{i}\bra{\nabla_{\bf k}u_{n,{\bf k}}}\ket{u_{n,{\bf
k}}}=-i\bra{u_{n,{\bf k}}}\ket{\nabla_{\bf k}u_{n,{\bf k}}}=-{\bf A}_{nn}({\bf
k}),$ (23)

using (i) a product rule, (ii) normalization condition imposed on the
eigenvectors, and (iii) complex conjugation of a real inner product. Hence,
${\bf A}_{nn}({\bf k})=0$, i.e. the diagonal elements of the non-Abelian Berry
connection, as well as its trace, vanish identically; as a result of imposing
the reality condition on the Hamiltonian.

###  B.2 Multi-gap flag phases

Here, we provide details on non-Abelian real topology of multi-gap flag
phases, including the construction of relevant models encoded in corresponding
Bloch Hamiltonians. As introduced in Ref. [54], the multi-gap flag phases with
four isolated bands are classified by the homotopy of flag varieties,
$\widetilde{\mathsf{Fl}}_{p_{1},\ldots,p_{k}}(\mathbb{R})\equiv
SO(\sum^{k}_{i=1}p_{i})/[SO(p_{1})\times\ldots\times SO(p_{k})]$, where
$p_{1},\ldots,p_{k}\in\mathbb{N}$ are numbers of bands in the gapped
subbundles. Here, $p_{1}=p_{2}=p_{3}=p_{4}=1$, with:

$\widetilde{\mathsf{Fl}}_{1,1,1,1}(\mathbb{R})=SO(4)\cong\frac{S^{3}\times
S^{3}}{\mathbb{Z}_{2}}.$ (24)

Namely, we have
$\pi_{3}(\mathsf{Fl}_{1,1,1,1}(\mathbb{R}))\cong\pi_{3}(SO(4))\cong\mathbb{Z}\oplus\mathbb{Z}=\mathbb{Z}^{2}$.

To interpret the meaning of the $\mathbb{Z}^{2}$ invariant, we recall the
well-known isomorphism,

$SO(4)\cong\frac{S^{3}_{L}\times S^{3}_{R}}{\mathbb{Z}_{2}},$ (25)

which on recognizing that the $\mathbb{Z}_{2}$ quotient does not affect
$\pi_{n}~{}\text{for}~{}n\geq 2$, allows to interpret the invariant as a pair
of windings on two copies of 3-spheres,

$\pi_{3}(S^{3}_{L}\times
S^{3}_{R})\cong\pi_{3}(S^{3}_{L})\oplus\pi_{3}(S^{3}_{R})\cong\mathbb{Z}\oplus\mathbb{Z}.$
(26)

We recognize that the non-trivial $S^{3}\rightarrow S^{3}$ maps corresponding
to the non-trivial elements of $\pi_{3}(S^{3}_{L/R})\cong\mathbb{Z}$ can be
interpreted as winding numbers $w_{L},w_{R}$.

Correspondingly, we construct the flag Hamiltonian by taking:

$H(\textbf{k})=V(\textbf{k})~{}\text{diag}[E_{1},E_{2},E_{3},E_{4}]~{}V(\textbf{k})^{\text{T}},$
(27)

with $E_{1}<E_{2}<E_{3}<E_{4}$ introducing three gaps, and
$V=V_{R}V^{\text{T}}_{L}$ which can be factored in terms of two matrices
introducing the isoclinic windings. These can be constructed as

$V_{L}=\left(\begin{array}[]{rrrr}l_{0}&-l_{3}&l_{2}&-l_{1}\\\
l_{3}&l_{0}&-l_{1}&-l_{2}\\\ -l_{2}&l_{1}&l_{0}&-l_{3}\\\
l_{1}&l_{2}&l_{3}&l_{0}\end{array}\right),$ (28)
$V_{R}=\left(\begin{array}[]{rrrr}r_{0}&-r_{3}&r_{2}&r_{1}\\\
r_{3}&r_{0}&-r_{1}&r_{2}\\\ -r_{2}&r_{1}&r_{0}&r_{3}\\\
-r_{1}&-r_{2}&-r_{3}&r_{0}\end{array}\right),$ (29)

with $l_{0}^{2}+l_{1}^{2}+l_{2}^{2}+l_{3}^{2}=1$ and
$r_{0}^{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1$. On such factorization, the
invariants of the flag phases can be directly deduced from the following
formulae of the winding invariants:

$\displaystyle w_{L}=\frac{1}{2\pi^{2}}\int_{\text{BZ}}\differential^{3}{\bf
k}~{}\epsilon_{ijpq}l^{i}\partial_{k_{x}}l^{j}\partial_{k_{y}}l^{p}\partial_{k_{z}}l^{q},$
(30) $\displaystyle
w_{R}=\frac{1}{2\pi^{2}}\int_{\text{BZ}}\differential^{3}{\bf
k}~{}\epsilon_{ijpq}r^{i}\partial_{k_{x}}r^{j}\partial_{k_{y}}r^{p}\partial_{k_{z}}r^{q}.$

In particular, we can model these by a choice

$(l_{0},l_{1},l_{2},l_{3})=(\sin{w_{L}k_{x}},\sin{k_{y}},\sin{k_{z}},M-\cos{w_{L}k_{x}}-\cos{k_{y}}-\cos{k_{z}})^{\text{T}},$
(31)
$(r_{0},r_{1},r_{2},r_{3})=(\sin{w_{R}k_{x}},\sin{k_{y}},\sin{k_{z}},M-\cos{w_{R}k_{x}}-\cos{k_{y}}-\cos{k_{z}})^{\text{T}}.$
(32)

where we set the topological mass to $M=2$. Additionally, changing the
parameter $M$ allows to induce topological phase transitions (TPTs), on gap
closings, as shown explicitly in the main text.

###  B.3 Real Hopf insulator

In the case of a real Hopf insulator we consider the $2\oplus 2$ band
partitioning. In terms of homotopy classification of the topological phases
encoded with real topologies [11], the band partitioning yields a
classification on an oriented real Grassmannian manifold,

$\widetilde{\mathsf{Gr}}_{2,4}(\mathbb{R})=SO(4)/(SO(2)\crossproduct
SO(2))\cong S^{2}\crossproduct S^{2}.$ (33)

Therefore, the invariant can be obtained as
$\pi_{3}(\widetilde{\mathsf{Gr}}_{2,4})\cong\pi_{3}(S^{2}\crossproduct
S^{2})\cong\pi_{3}(S^{2})~{}\oplus~{}\pi_{3}(S^{2})\cong\mathbb{Z}~{}\oplus~{}\mathbb{Z}$,
where each $\pi_{3}(S^{2})\cong\mathbb{Z}$ defines a Hopf fibration
($S^{3}\rightarrow S^{2}$) and an associated Hopf index.

Following Ref. [53], these Hopf invariants can be written as:

$\displaystyle\chi_{z}=-\frac{1}{16\pi^{2}}\int_{BZ}\mathrm{a}^{c}\wedge\mathrm{Eu}^{c}+\mathrm{a}^{v}\wedge\mathrm{Eu}^{v}+\mathrm{a}^{c}\wedge\mathrm{Eu}^{v}+\mathrm{a}^{v}\wedge\mathrm{Eu}^{c}\in\mathbb{Z},$
(34)
$\displaystyle\chi_{w}=-\frac{1}{16\pi^{2}}\int_{BZ}\mathrm{a}^{c}\wedge\mathrm{Eu}^{c}+\mathrm{a}^{v}\wedge\mathrm{Eu}^{v}-\mathrm{a}^{c}\wedge\mathrm{Eu}^{v}-\mathrm{a}^{v}\wedge\mathrm{Eu}^{c}\in\mathbb{Z}.$

In particular, we notice that on summation, the invariants yield

$\chi_{z}+\chi_{w}=-\frac{1}{8\pi^{2}}\int_{\text{BZ}}\mathrm{a}^{c}\wedge\mathrm{Eu}^{c}+\mathrm{a}^{v}\wedge\mathrm{Eu}^{v}\in\mathbb{Z}$
(35)

which is the quantized number entering the integrated shift response
introduced in the main text.

The models for real Hopf insulators can be obtained from a flattened
Hamiltonian ensuring a correct band partitioning

$H^{\chi_{z},\chi_{w}}({\bf k})=R_{z,w}({\bf
k})~{}\text{diag}[-1,-1,1,1]~{}R^{\text{T}}_{z,w}({\bf k}),$ (36)

where $R_{z,w}({\bf k})$ encodes the double Hopf fibration inducing indices
($\chi_{w},\chi_{z}$), as introduced and discussed in detail in Ref. [53].

###  B.4 Pontryagin-indexed insulator

To define the insulator with Pontryagin index, we demand a $3\oplus 1$
partitioning of four topological bands. From the homotopy perspective, the
effective classifying space of interest becomes a oriented real Grassmannian,

$\widetilde{\mathsf{Gr}}_{1,4}(\mathbb{R})=SO(4)/SO(3)\cong S^{3},$ (37)

as discussed in Ref. [54]. The homotopy classification in three spatial
dimensions then gives

$\pi_{3}(\widetilde{\mathsf{Gr}}_{1,4}(\mathbb{R}))\cong\pi_{3}(S^{3})\cong\mathbb{Z},$
(38)

where indeed $\pi_{3}(S^{3})\cong\mathbb{Z}$ defines the Pontryagin index $Q$,
which classifies homotopy-inequivalent maps: $S^{3}\rightarrow S^{3}$ [70].

The Pontryagin index can be computed from the non-Abelian Berry connection
elements as

$Q=\frac{1}{2\pi^{2}}\int_{T^{3}}\differential^{3}k~{}\epsilon^{ijk}A^{i}_{41}A^{j}_{42}A^{k}_{43}.$
(39)

Equivalently [54], the invariant can be computed as the winding of the fourth
band ${\bf u}_{4}\equiv\ket{u_{4,{\bf k}}}$ as,

$Q=\frac{1}{2\pi^{2}}\int_{S^{3}}\differential^{3}k~{}\epsilon_{ijpq}({\bf
u}_{4})^{i}\partial_{k_{x}}({\bf u}_{4})^{j}\partial_{k_{y}}({\bf
u}_{4})^{p}\partial_{k_{z}}({\bf u}_{4})^{q}\;,$ (40)

pushed forward to a 3-sphere $S^{3}$ from the BZ 3-torus ($T^{3}\rightarrow
S^{3}$). As shown in Ref. [54], to generate a concrete model, one could take a
flattened Hamiltonian and a Bloch band matrix $R({\bf k})$ as

$H^{Q}({\bf k})=R({\bf k})\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{pmatrix}R({\bf k})^{\text{T}}.$ (41)

Here, the matrix $R({\bf k})\in SO(4)$ can be obtained by a three-angle
parametrization $(\phi,\psi,\theta)$ obtained from Plücker formalism, namely
[11, 54]:

$\psi({\bf k})=\pi~{}\text{max}\\{|k_{x}|,|k_{y}|,|k_{z}|\\}\;,$ (42)
$\theta({\bf k})=\cos^{-1}(k_{z}/\sqrt{k^{2}_{x}+k^{2}_{y}+k^{2}_{z}})\;,$
(43)
$\phi({\bf k})=Q\tan^{-1}(k_{x}/k_{y})\;,$ (44)

deduced for any $k$-point in the BZ. Correspondingly, the Bloch matrix
inducing $Q\neq 0$ can be generated as

$R({\bf
k})=\text{e}^{\mathrm{i}\theta\Gamma_{20}}\text{e}^{\mathrm{i}\phi\Gamma_{21}}\text{e}^{\mathrm{i}\psi\Gamma_{12}}.$
(45)

Having revised the possible non-Abelian four-band topology in three spatial
dimensions we note however that the optical shift responses induced by the
Pontryagin index on occupying the entire three-band subspace do not yield a
3-form resulting in a quantized response, contrary to the topology of real
Hopf insulators and the multi-gap flag phases discussed above. On the other
hand, the Pontryagin-indexed insulators can be doped, such that only two bands
are occupied, as the Fermi level is set at band crossings, and in that case
the quantized shift response can be realized on photoexcitations. For the
derivation of the quantized responses, see Appendix D.

##  Appendix C Torsional representation of symmetrized circular shift
photoconductivities

As introduced in the main text, the circular shift photoconductivity
$\sigma^{cab}_{\text{shift,C}}(\omega)$ in terms of the metric connection
$C^{mn}_{bca}=(C^{mn}_{bca})^{*}=\mathfrak{Re}~{}C^{mn}_{bca}$, under the
reality condition (see also Appendix A), reads

$\sigma^{cab}_{\text{shift,C}}(\omega)=-\frac{e^{2}}{8\pi^{2}\hbar^{2}}\int\differential^{3}k\sum_{m,n}f_{mn}\mathfrak{Re}~{}(C^{mn}_{bca}-C^{mn}_{acb})\delta(\omega_{mn}-\omega).$
(46)

Now, to derive the torsional representation of circular shift
photoconductivities, we start by recognizing that the sum of circular shift
photoconductivities can be written as:

$\begin{split}\int\differential\omega~{}(\sigma^{abc}_{\text{shift,C}}(\omega)+\sigma^{bca}_{\text{shift,C}}(\omega)+\sigma^{cab}_{\text{shift,C}}(\omega))=\\\
=-\frac{e^{2}}{8\pi^{2}\hbar^{2}}\int\differential^{3}k\sum_{m,n}f_{mn}\Big{(}(\mathfrak{Re}~{}C^{mn}_{cab}-\mathfrak{Re}~{}C^{mn}_{bac})+(\mathfrak{Re}~{}C^{mn}_{abc}-\mathfrak{Re}~{}C^{mn}_{cba})+(\mathfrak{Re}~{}C^{mn}_{bca}-\mathfrak{Re}~{}C^{mn}_{acb})\Big{)}=\\\
=\frac{e^{2}}{8\pi^{2}\hbar^{2}}\int\differential^{3}k\sum_{m,n}f_{mn}\Big{(}(\mathfrak{Re}~{}C^{mn}_{acb}-\mathfrak{Re}~{}C^{mn}_{abc})+(\mathfrak{Re}~{}C^{mn}_{cba}-\mathfrak{Re}~{}C^{mn}_{cab})+(\mathfrak{Re}~{}C^{mn}_{bac}-\mathfrak{Re}~{}C^{mn}_{bca})\Big{)}=\\\
=\frac{e^{3}}{8\pi^{2}\hbar^{2}}\int\differential^{3}k\sum_{m,n}f_{mn}\mathfrak{Re}~{}(T^{mn}_{acb}+T^{mn}_{cba}+T^{mn}_{bac}),\end{split}$
(47)

where the first equality follows directly from the Eq. (46) of the main text
on performing the integration over the frequencies, and second equality
corresponds identically to reordering the terms. The third equality follows
from the definition of the torsion tensor and decomposition of Hermitian
connection into real (metric) and imaginary (symplectic) parts. Therefore, the
sum of circular shift photoconductivities can be captured purely in terms of
the torsion tensor.

##  Appendix D Torsion tensor in non-Abelian topological phases satisfying the
reality condition

First, we rewrite the torsion tensor in terms of the non-Abelian Berry
connection, assuming that a symmetry imposing the reality condition is
present,

$T^{mn}_{abc}=A^{a}_{nm}\sum_{m\neq p\neq
n}(A^{c}_{mp}A^{b}_{pn}-A^{b}_{mp}A^{c}_{pn}).$ (48)

Correspondingly, we have:

$\begin{split}T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab}=\sum_{m\neq p\neq
n}(A^{a}_{nm}A^{c}_{mp}A^{b}_{pn}-A^{a}_{nm}A^{b}_{mp}A^{c}_{pn}+A^{b}_{nm}A^{a}_{mp}A^{c}_{pn}+\\\
-A^{b}_{nm}A^{c}_{mp}A^{a}_{pn}+A^{c}_{nm}A^{b}_{mp}A^{a}_{pn}-A^{c}_{nm}A^{a}_{mp}A^{b}_{pn}),\end{split}$
(49)

which explicitly reflects the definitional nature of the torsion tensor
capturing possible virtual transitions, here in terms of the non-Abelian Berry
connection elements.

In the subsequent, we evaluate the symmetrized sum of the torsion tensor
elements in the four-band and three-band setups of the non-Abelian topological
insulators satisfying the reality condition, cf. Appendix B.

###  D.1 Torsion in four-band models

We start with the evaluation of the torsion tensor in the four-band models,
relevant for the non-Abelian topologies studied in Refs. [53, 54].

Here, we assume that there are 2 occupied bands ($n=1,2$) and 2 unoccupied
bands ($m=3,4$), obtaining:

$\begin{split}T^{31}_{abc}+T^{31}_{bca}+T^{31}_{cab}=(A^{a}_{13}A^{c}_{32}A^{b}_{21}-A^{a}_{13}A^{b}_{32}A^{c}_{21}+A^{b}_{13}A^{a}_{32}A^{c}_{21}-A^{b}_{13}A^{c}_{32}A^{a}_{21}+A^{c}_{13}A^{b}_{32}A^{a}_{21}-A^{c}_{13}A^{a}_{32}A^{b}_{21})+\\\
+(A^{a}_{13}A^{c}_{34}A^{b}_{41}-A^{a}_{13}A^{b}_{34}A^{c}_{41}+A^{b}_{13}A^{a}_{34}A^{c}_{41}-A^{b}_{13}A^{c}_{34}A^{a}_{41}+A^{c}_{13}A^{b}_{34}A^{a}_{41}-A^{c}_{13}A^{a}_{34}A^{b}_{41}),\end{split}$
(50)
$\begin{split}T^{41}_{abc}+T^{41}_{bca}+T^{41}_{cab}=(A^{a}_{14}A^{c}_{42}A^{b}_{21}-A^{a}_{14}A^{b}_{42}A^{c}_{21}+A^{b}_{14}A^{a}_{42}A^{c}_{21}-A^{b}_{14}A^{c}_{42}A^{a}_{21}+A^{c}_{14}A^{b}_{42}A^{a}_{21}-A^{c}_{14}A^{a}_{42}A^{b}_{21})+\\\
+(A^{a}_{14}A^{c}_{43}A^{b}_{31}-A^{a}_{14}A^{b}_{43}A^{c}_{31}+A^{b}_{14}A^{a}_{43}A^{c}_{31}-A^{b}_{14}A^{c}_{43}A^{a}_{31}+A^{c}_{14}A^{b}_{43}A^{a}_{31}-A^{c}_{14}A^{a}_{43}A^{b}_{31}),\end{split}$
(51)
$\begin{split}T^{32}_{abc}+T^{32}_{bca}+T^{32}_{cab}=(A^{a}_{23}A^{c}_{31}A^{b}_{12}-A^{a}_{23}A^{b}_{31}A^{c}_{12}+A^{b}_{23}A^{a}_{31}A^{c}_{12}-A^{b}_{23}A^{c}_{31}A^{a}_{12}+A^{c}_{23}A^{b}_{31}A^{a}_{12}-A^{c}_{23}A^{a}_{31}A^{b}_{12})+\\\
+(A^{a}_{23}A^{c}_{34}A^{b}_{42}-A^{a}_{23}A^{b}_{34}A^{c}_{42}+A^{b}_{23}A^{a}_{34}A^{c}_{42}-A^{b}_{23}A^{c}_{34}A^{a}_{42}+A^{c}_{23}A^{b}_{34}A^{a}_{42}-A^{c}_{23}A^{a}_{34}A^{b}_{42}),\end{split}$
(52)
$\begin{split}T^{42}_{abc}+T^{42}_{bca}+T^{42}_{cab}=(A^{a}_{24}A^{c}_{41}A^{b}_{12}-A^{a}_{24}A^{b}_{41}A^{c}_{12}+A^{b}_{24}A^{a}_{41}A^{c}_{12}-A^{b}_{24}A^{c}_{41}A^{a}_{12}+A^{c}_{24}A^{b}_{41}A^{a}_{12}-A^{c}_{24}A^{a}_{41}A^{b}_{12})+\\\
+(A^{a}_{24}A^{c}_{34}A^{b}_{32}-A^{a}_{24}A^{b}_{43}A^{c}_{32}+A^{b}_{24}A^{a}_{43}A^{c}_{32}-A^{b}_{24}A^{c}_{43}A^{a}_{32}+A^{c}_{24}A^{b}_{43}A^{a}_{32}-A^{c}_{24}A^{a}_{43}A^{b}_{32}).\end{split}$
(53)

On summing all the terms and simplifying using $A^{a}_{ij}=-A^{a}_{ji}$
consistently with the reality condition (see Appendix B):

$\begin{split}\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2A^{a}_{12}(A^{b}_{13}A^{c}_{32}-A^{c}_{13}A^{b}_{32}+A^{b}_{14}A^{c}_{42}-A^{c}_{14}A^{b}_{42})+\\\
-2A^{b}_{12}(A^{c}_{13}A^{a}_{32}-A^{a}_{13}A^{c}_{32}+A^{c}_{14}A^{a}_{42}-A^{a}_{14}A^{c}_{42})-2A^{c}_{12}(A^{a}_{13}A^{b}_{32}-A^{b}_{13}A^{a}_{32}+A^{a}_{14}A^{b}_{42}-A^{b}_{14}A^{a}_{42})+\\\
-2A^{a}_{34}(A^{b}_{31}A^{c}_{14}-A^{c}_{31}A^{b}_{14}+A^{b}_{32}A^{c}_{24}-A^{c}_{32}A^{b}_{24})+\\\
-2A^{b}_{34}(A^{c}_{31}A^{a}_{14}-A^{a}_{31}A^{c}_{14}+A^{c}_{32}A^{a}_{24}-A^{a}_{32}A^{c}_{24})-2A^{c}_{34}(A^{a}_{31}A^{b}_{14}-A^{b}_{31}A^{a}_{14}+A^{a}_{32}A^{b}_{24}-A^{b}_{32}A^{a}_{24}).\end{split}$
(54)

In terms of the two-band Euler connection:
$\text{Eu}^{ab}_{nm}=\bra{\partial_{a}u_{n}}\ket{\partial_{b}u_{m}}-\bra{\partial_{b}u_{n}}\ket{\partial_{a}u_{m}}=\sum_{n\neq
p\neq m}A^{a}_{np}A^{b}_{pm}-A^{b}_{np}A^{a}_{pm}$, where the second equality
follows from resolution of identity $1=\sum_{p}\outerproduct{u_{p}}{u_{p}}$,
and the reality condition ($A^{a}_{ii}=-A^{a}_{ii}=0$). Hence:

$\begin{split}\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2A^{a}_{12}\text{Eu}^{bc}_{12}-2A^{b}_{12}\text{Eu}^{ca}_{12}-2A^{c}_{12}\text{Eu}^{ab}_{12}-2A^{a}_{34}\text{Eu}^{bc}_{34}-2A^{b}_{34}\text{Eu}^{ca}_{34}-2A^{c}_{34}\text{Eu}^{ab}_{34}.\end{split}$
(55)

On identifying $\textbf{a}^{v}={\bf A}_{21}=-{\bf A}_{12}$,
$\textbf{a}^{c}={\bf A}_{43}=-{\bf A}_{34}$, and finally
$(\text{{Eu}}^{v/c})_{a}=\epsilon_{abc}\text{Eu}^{bc}_{1/3~{}2/4}$, we
retrieve:

$\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2({\bf
a}^{v}\cdot\text{{\bf Eu}}^{v}+{\bf a}^{c}\cdot\text{{\bf Eu}}^{c}).$ (56)

In terms of the language of differential forms:

$\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})~{}dk_{x}\wedge
dk_{y}\wedge dk_{z}=-2(a^{v}\wedge\text{Eu}^{v}+a^{c}\wedge\text{Eu}^{c}),$
(57)

which is the 3-form integrated over the $\text{BZ}\cong T^{3}$ to find the sum
of real Hopf invariants in the four-band case. Here, the Euler connections
(1-forms) and curvatures (2-forms) are defined in terms of the pairs of
conduction and valence bands
${a^{v/c}=\bra{u_{2/4}}\differential\ket{u_{1/3}}}$,
${\text{Eu}^{v/c}=\differential a^{v/c}}$. To demonstrate the non-triviality
of the 3-form components, we plot the relevant torsion tensor components
summed over bands $T_{xyz}\equiv\sum_{m,n}T^{mn}_{xyz}$,
$T_{yzx}\equiv\sum_{m,n}T^{mn}_{yzx}$, $T_{zxy}\equiv\sum_{m,n}T^{mn}_{zxy}$
in the form of a vector field $(T_{xyz},T_{yzx},T_{zxy})$ over BZ torus, see
Fig. 4. For the demonstration, we choose the real Hopf insulator model from
the main text. We notice that significant contributions appear from the edge
of BZ.

The BZ integral of the sum of the torsion tensor components yields quantized
numbers reflected by the circular shift photoconductivity, as introduced in
the main text.

Figure 4: Momentum space fields of the torsion tensor components, represented
as vectors ($T_{xyz},T_{yzx},T_{zxy}$) over the 3D $\text{BZ}\cong T^{3}$. We
choose the RHI model studied for the quantized shift effect in the main text,
with topological mass $M$ as a free parameter, and $w=1$. (a) Setting $M=0$
yields Hopf indices $(\chi_{z},\chi_{w})=(0,2w)$, which is reflected by the
high magnitudes of the torsion tensor components, (b) Crossing the TPT at
$M=3/2$ reduces the Hopf indices to $(\chi_{z},\chi_{w})=(0,-w)$, and the
torsion tensor components individually decrease, yielding a reduced quantized
circular shift effect on summation. (c) $M=5$ trivializes the indices to
$(\chi_{z},\chi_{w})=(0,0)$, almost completely removing the torsion tensor
components. The leftover values of $T_{xyz},T_{yzx},T_{zxy}$ cancel on
performing a full summation over the BZ, yielding a vanishing effect quantized
by the Hopf indices.

It should be stressed again that a four-band bundle, as modelled here in terms
of the virtual transitions between four-bands (with amplitudes given by the
elements of the non-Abelian Berry connection), can be selectively accessed by
tuning the range of frequencies involved in non-linear optical transitions
generating the circular shift current. Moreover, the four-band subspace needs
to be isolated from the rest of the valence and conduction bands. Otherwise,
the studied homotopy invariants can be trivialized [11], which emphasizes the
multi-gap nature of these topological invariants. Finally, we notice that if
the material is isotropic, then:
$\sigma^{xyz}_{\text{shift}}(\omega)=\sigma^{yzx}_{\text{shift}}(\omega)=\sigma^{zxy}_{\text{shift}}(\omega)$.
In that case, on combining with the final result of Appendix C, the shift
photoconductivity integrates to a quantized value for $each$ of the
$individual$ directions of light incidence, i.e. the necessity of combining
all symmetrized directions to achieve the quantization condition is removed.

###  D.2 Torsion in three-band models

For completeness, we also introduce the derivation of the symmetrized sums of
the torsion tensor elements for 3-band models under the reality condition
(Appendix B). Now, we assume that there are 2 occupied bands ($n=1,2$) and 1
unoccupied band ($m=3$), obtaining:

$\begin{split}T^{31}_{abc}+T^{31}_{bca}+T^{31}_{cab}=(A^{a}_{13}A^{c}_{32}A^{b}_{21}-A^{a}_{13}A^{b}_{32}A^{c}_{21}+A^{b}_{13}A^{a}_{32}A^{c}_{21}-A^{b}_{13}A^{c}_{32}A^{a}_{21}+A^{c}_{13}A^{b}_{32}A^{a}_{21}-A^{c}_{13}A^{a}_{32}A^{b}_{21}),\end{split}$
(58)
$\begin{split}T^{32}_{abc}+T^{32}_{bca}+T^{32}_{cab}=(A^{a}_{23}A^{c}_{31}A^{b}_{12}-A^{a}_{23}A^{b}_{31}A^{c}_{12}+A^{b}_{23}A^{a}_{31}A^{c}_{12}-A^{b}_{23}A^{c}_{31}A^{a}_{12}+A^{c}_{23}A^{b}_{31}A^{a}_{12}-A^{c}_{23}A^{a}_{31}A^{b}_{12}).\end{split}$
(59)

On summing all the terms and simplifying, using $A^{a}_{ij}=-A^{a}_{ji}$:

$\begin{split}\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2A^{a}_{12}(A^{b}_{13}A^{c}_{32}-A^{c}_{13}A^{b}_{32})+\\\
-2A^{b}_{12}(A^{c}_{13}A^{a}_{32}-A^{a}_{13}A^{c}_{32})-2A^{c}_{12}(A^{a}_{13}A^{b}_{32}-A^{b}_{13}A^{a}_{32}).\end{split}$
(60)

In terms of the two-band Euler connection:
$\text{Eu}^{ab}_{nm}=\bra{\partial_{a}u_{n}}\ket{\partial_{b}u_{m}}-\bra{\partial_{b}u_{n}}\ket{\partial_{a}u_{m}}=\sum_{n\neq
p\neq m}A^{a}_{np}A^{b}_{pm}-A^{b}_{np}A^{a}_{pm}$. Hence:

$\begin{split}\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2A^{a}_{12}\text{Eu}^{bc}_{12}-2A^{b}_{12}\text{Eu}^{ca}_{12}-2A^{c}_{12}\text{Eu}^{ab}_{12}.\end{split}$
(61)

On identifying $\textbf{a}^{v}={\bf A}_{21}=-{\bf A}_{12}$, and
$(\text{{Eu}}^{v})_{a}=\epsilon_{abc}\text{Eu}^{bc}_{12}$, we retrieve:

$\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})=-2{\bf
a}^{v}\cdot\text{{\bf Eu}}^{v}.$ (62)

In terms of the differential forms:

$\sum_{m,n}(T^{mn}_{abc}+T^{mn}_{bca}+T^{mn}_{cab})~{}dk_{x}\wedge
dk_{y}\wedge dk_{z}=-2a^{v}\wedge\text{Eu}^{v},$ (63)

which is the 3-form integrated to find the possible quantization in the three-
band case. However, contrary to the four-band case, the topology of three-band
models in three spatial dimensions has not been studied extensively, as for
the time being. In particular, in three dimensions, a partitioning of three
real bands into 2-band and 1-band subspaces can in principle yield a single
$\mathbb{Z}$ invariant, captured by the homotopy
$\pi_{3}[\widetilde{\mathsf{Gr}}_{1,3}(\mathbb{R})]\cong\pi_{3}[SO(3)/SO(2)]\cong\pi_{3}[S^{2}]\cong\mathbb{Z}$
associated with a single Hopf index [25]. We leave the exhaustive study of
such topological phases, supporting the non-trivial real 3-form from Eq. (63),
as well as the torsional circular shift photoconductivity response, for future
investigation.