††thanks: Refer for correspondence to<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
# Multi-gap topological conversion of Euler class via band-node braiding:
minimal models, $PT$-linked nodal rings, and chiral heirs

Adrien Bouhon1,2 Robert-Jan Slager1 1TCM Group, Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United
Kingdom  2Nordita, Stockholm University and KTH Royal Institute of Technology,
Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

###### Abstract

The past few years have seen rapid progress in characterizing topological band
structures using symmetry eigenvalue indicated methods. Recently, however,
there has been increasing theoretical and experimental interest in multi-gap
dependent topological phases that cannot be captured by this paradigm. These
topologies arise by braiding band degeneracies that reside between different
bands and carry non-Abelian charges due to the presence of either $C_{2}T$ or
$PT$ symmetry, culminating in different invariants such as $\mathbb{Z}$-valued
Euler class. Here, we present a universal formulation for Euler phases
motivated by their homotopy classification that is related to the Skyrmion-
profile of a single unit-vector in three-level systems, and that of two unit-
vectors in four-level systems. In addition, upon employing the strategy of
systematically building 3D models from a pair of sub-dimensional Euler phases,
we show that phase transitions between any two inequivalent Euler phases are
mediated by the presence of adjacent (in-gap) nodal rings linked with sub-gap
nodal lines, forming trajectories corresponding to the braiding or debraiding
of nodal points. The stability of the linked adjacent nodal rings is
furthermore demonstrated to be indicated by an Euler class monopole charge
matching with its $\mathbb{Z}$-valued linking numbers. We finally also
systematically address the conversion of Euler phases into descendant Chern
phases upon breaking the $C_{2}T$ or $PT$ symmetry. All the topological phases
discussed in this work are corroborated with explicit minimal lattice models.
These models can themselves directly serve as an extra impetus for
experimental searches or be employed for theoretical studies, thereby
underpinning the upcoming of this nascent pursuit.

##  I Introduction

Topological materials Qi and Zhang (2011); Hasan and Kane (2010); Armitage
_et al._ (2018) entail an active field in condensed matter, encompassing
studies that range from theoretical pursuits to material science impetuses.
These intensive efforts have resulted in a wide charted field of phases and a
plethora of topological characterizations Volovik (2003); Kruthoff  _et al._
(2017); Bouhon and Black-Schaffer (2017); Hughes  _et al._ (2011); Fu (2011);
Turner  _et al._ (2012); Slager  _et al._ (2013); Juričić  _et al._ (2012);
Shiozaki and Sato (2014); Slager (2019); Chiu  _et al._ (2016);
Alexandradinata  _et al._ (2014, 2016); Scheurer and Slager (2020); Shiozaki
_et al._ (2017); Po  _et al._ (2017); Ran  _et al._ (2009); Rhim  _et al._
(2018); Teo and Kane (2010); Bradlyn  _et al._ (2017); Slager  _et al._
(2015); Bouhon  _et al._ (2018); Slager  _et al._ (2017); Alexandradinata  _et
al._ (2020); Slager  _et al._ (2014); Fang  _et al._ (2012); Ünal  _et al._
(2019); Cornfeld and Carmeli (2021). Much of this progress has been rooted in
symmetry eigenvalue analyses. Using the information of representations at high
symmetry points in the Brillouin zone Kruthoff  _et al._ (2017); Bouhon and
Black-Schaffer (2017) a significant fraction of topological phases in momentum
space can be efficiently characterized and, upon comparing which of these
combinations have an atomic limit, versatile classification schemes have been
formulated Po  _et al._ (2017); Bradlyn  _et al._ (2017).

These endeavours have also transpired in the retrieval of different
topological phases exhibiting different features. First, within the symmetry
eigenvalue setting, it was found that certain topologies phases can be fragile
Po  _et al._ (2018). Such phases have led to new explorations into their
properties Bouhon  _et al._ (2019); Bradlyn  _et al._ (2019); Hwang  _et al._
(2019); Song  _et al._ (2020); Palumbo (2021); Lange  _et al._ (2021); Wieder
and Bernevig (2018) and also resulted in experimental signatures Peri  _et
al._ (2020). More recently, however, a new class of topological phases, that
depend on multi-gap conditions, have increasingly been gaining interest. A
prominent example in this regard occur in phases enjoying a real Hamiltonian
representation by virtue of $C_{2}T$ symmetry or $PT$ symmetry. Band
degeneracies between different bands (which we will refer to as ‘gaps’) can
then carry non-Abelian frame charges Wu  _et al._ (2019); Tiwari and Bzdušek
(2020); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a), akin to
$\pi$-disclination defects in bi-axial nematics Alexander  _et al._ (2012);
Liu  _et al._ (2016); Volovik and Mineev (2018); Beekman  _et al._ (2017), and
braiding them around in momentum space leads to similarly-valued band
touchings within a certain gap. The resulting obstruction to annihilate these
band touchings is directly related to a multi-gap topological invariant, known
as Euler class Ahn  _et al._ (2018); Bouhon  _et al._ (2020a); Ahn  _et al._
(2019); Bouhon  _et al._ (2020b); Ünal  _et al._ (2020). This invariant
corresponds to a characteristic form being the real counterpart of the complex
variant that underlies Chern numbers.

We have recently shown that Euler phases can generally be understood as
arising from refined partition schemes and classified by specific homotopy
characterizations, which in turn can also be used reversely as general
strategy to construct models having desired Euler class Bouhon  _et al._
(2020b). Such Euler class models are increasingly becoming of importance and
have for example been proposed to induce monopole-antimonople generation in
quench setups Ünal  _et al._ (2020), while the observation of this physical
observable has just been reported in trapped-ion experiments Zhao  _et al._
(2022). In addition, the braiding and emerging of such non-Abelian charges and
its relation to Euler class have also been inspiring pursuits in other
experimental contexts that range from from phononic systems Park  _et al._
(2021); Lange  _et al._ (2022); Peng  _et al._ (2022a, b) and electronic
systems Chen  _et al._ (2022); Bouhon  _et al._ (2021, 2020a); Könye  _et al._
(2021); Lange  _et al._ (2021); Yu  _et al._ (2021); Lian  _et al._ (2020);
Guan  _et al._ (2021) to acoustic, photonic and electric circuit metamaterials
Guo  _et al._ (2021); Park  _et al._ (2022); Jiang  _et al._ (2021); Qiu  _et
al._ (2022); Ezawa (2021).

Given this interest, we here wish to further underpin these developments by
introducing simple models that exhibit non-trivial Euler topologies and
controlled band node formation. In particular this allows us to further
examine braiding processes Bouhon  _et al._ (2020a) and their interplay with
symmetries as well as relation to homotopy perspectives Bouhon  _et al._
(2020b). More precisely, we retrieve a simple formulation of tight-binding
models with Euler topology in term of Skyrmion winding numbers of a single
Bzdušek and Sigrist (2017) or double unit vector for the three and four band
cases, respectively. Remarkably, this allows us to formulate a great variety
of Euler phases by exploiting the phase diagram of a single two-band Chern
lattice model parametrized to produce a Chern number ranging from $-2$ to $2$
Sticlet  _et al._ (2012). Extending the above intrinsic considerations, we
find that the transition between inequivalent Euler phases, while preserving
the reality condition, are generically mediated by the presence of “adjacent”
nodal rings linked with “sub-gap” nodal lines, the former of which appearing
within the gap of the Euler phases, and the later being formed by the band
crossings of the connected two-band subspaces of the Euler phases. Running
through the transition, the nodal points extend into nodal braids forming
trajectories that correspond to the braiding or debraiding of nodal points
Tiwari and Bzdušek (2020). The stability of the adjacent nodal rings is
moreover found to be indicated by specific monopole charges dictated by the
“difference” of Euler classes and corresponding to the linking numbers Ahn
_et al._ (2018) of the nodal rings. This point of view thus culminates in the
systematic building of 3D $PT$-symmetric models, obtained through the
embedding of pairs of 2D Euler phases within the 3D Brillouin zone, that host
linked adjacent nodal rings. As an other extension, we address the systematic
conversion of Euler phases into descendant Chern phases upon breaking the
$C_{2}T$ or $PT$ symmetry. Then, similarly to the $PT$-symmetric case, we
build 3D $C_{2}T$-symmetric chiral phases, obtained from pairs of Euler
phases, that trap a number of Weyl nodes that is again dictated by the same
“difference” of Euler classes. From a practical point of view, all these
models can directly be implemented by experimental and modeling pursuits with
the hope of further advancing this new field.

This paper is organized as follows. In Sec. II we begin our discussion by
reviewing the generic homotopy-induced strategy to model Euler phases Bouhon
_et al._ (2020b) for both three-level and four-level systems, where the latter
are specified by the relative balance between the Euler class of the two two-
band subspaces. In particular, we introduce the parametrization of three-band
and four-band models through the Skyrmion winding number of one unit vector,
and that of two independent unit vectors, respectively. Readers interested in
the concrete models may directly skip to Sec. III, where we formulate the
models of interest hosting the Euler topology of orientable phases. Given our
generic framework that can generate any kind of model, these examples are on
purpose taken as simple as possible. This however may generically induce
additional symmetries, which is the topic that we address systematically in
Sec. IV. In Sec. V we then expose the relation between inequivalent 2D Euler
phases and their relation to 3D $PT$ symmetric nodal lines structures when the
2D phases are seen as planar cuts of a 3D embedding and discuss the
quantification in terms of monopole charges and linking numbers. This general
point of view is then again made concrete with readily implementable models in
Sec. VI, for everyone of which we present the linked nodal ring structures
obtained numerically. Finally, we discuss how breaking symmetries can lead to
descendant topologies, such as 2D Chern phases, and the notion of 3D chiral
phases in Sec. VII, before concluding in Sec. VIII.

##  II Geometric and homotopic modeling of orientable Euler phases

In this section we review the geometry and homotopy frameworks that motivate
the derivation of explicit models with Euler class topology. In particular, we
obtain that flattened and two-by-two Euler Hamiltonians are fully parametrized
in terms of three-component unit vectors winding on a sphere. These homotopy
representative Hamiltonians are then used in the next Section to derive
explicit minimal tight-binding Hamiltonians for a variety of Euler phases. In
the whole work we assume that the system has a $C_{2}T$ symmetry (spinful or
spinless), with $C_{2}$ the $\pi$ rotation axis perpendicular to the system’s
basal plane and $T$ is time reversal, with $[C_{2}T]^{2}=+\mathbb{1}$.
(Equivalently, the system can host a $PT$ symmetry, with $P$ the inversion
symmetry, still with $[PT]^{2}=+\mathbb{1}$. In that case however, the system
must be spinless.) Because the anti-unitary symmetry squares to the identity,
it can be shown (through the Takagi factorization of the symmetric unitary
matrix that represents $C_{2}T$ in the Bloch orbital basis, see below) that
there exists a special basis for which the Bloch Hamiltonian matrix is real
and symmetric Bouhon  _et al._ (2020a); Chen  _et al._ (2022). In the
following we call it the reality condition of Euler phases. We again note that
this section is more technical in nature and can be skipped by readers
interested in the minimal models for direct implementation that are presented
in the subsequent Sections.

###  II.1 Homotopy classification of two-dimensional orientable Euler phases

We review the homotopy classification and modeling of Euler phases obtained in
Ref. Bouhon  _et al._ (2020b). Let us consider the Bloch Hamiltonian operator

$\mathcal{H}=\sum\limits_{\boldsymbol{k}\in\mathrm{BZ}}\sum\limits_{ab}|\phi_{a},\boldsymbol{k}\rangle
H_{ab}(\boldsymbol{k})\langle\phi_{b},\boldsymbol{k}|,$ (1)

where the wave-vector $\boldsymbol{k}$ of the two-dimensional system is a
point of the Brillouin zone
$\mathbb{T}^{2}=\\{(k_{1},k_{2})|k_{1}\in[-\pi,\pi),k_{2}\in[-\pi,\pi)\\}$,
and where $\\{|\phi_{a},\boldsymbol{k}\rangle\\}_{a}$ is assumed to be a Bloch
orbital basis obtained from the Fourier transform of a localized Wannier
basis. Our starting point is the spectral decomposition of the real and
symmetric Bloch Hamiltonian matrix, i.e.

$H(\boldsymbol{k})=R(\boldsymbol{k})\cdot D(\boldsymbol{k})\cdot
R(\boldsymbol{k})^{T},$ (2)

with the diagonal matrix of energy ordered eigenvalues
$D(\boldsymbol{k})=\mathrm{diag}[E_{1}(\boldsymbol{k}),\dots,E_{N}(\boldsymbol{k})]$,
such that $E_{n}(\boldsymbol{k})\leq E_{n+1}(\boldsymbol{k})$ for
$n=1,\dots,N-1$, the matrix of column eigenvectors
$R(\boldsymbol{k})=\left[u_{1}(\boldsymbol{k})~{}\cdots~{}u_{N}(\boldsymbol{k})\right]\in\mathsf{O}(N)$,
and where $N$ is the total number of bands.

Assuming that the first $p$ bands are separated from the higher $(N-p)$ bands
by an energy gap, the classifying space of the Hamiltonian takes the form of a
real Grassmannian,
$\mathsf{Gr}^{\mathbb{R}}_{p,N}=\mathsf{O}(N)/[\mathsf{O}(p)\times\mathsf{O}(N-p)]$.
In the following, we will use the flattened Hamiltonian

$Q(\boldsymbol{k})=R(\boldsymbol{k})\cdot(-\mathbb{1}_{p}\oplus\mathbb{1}_{N-p})\cdot
R(\boldsymbol{k})^{T},$ (3)

as the homotopy representative of the dispersive Hamiltonian
$H(\boldsymbol{k})$.

We start the homotopy characterization by noting that two-dimensional systems
can host nontrivial one-dimensional topologies as indicated by the nontrivial
first homotopy group,
$\pi^{(l)}_{1}[\mathsf{Gr}^{\mathbb{R}}_{p,N}]=\mathbb{Z}_{2}$, over one non-
contractible direction of the two-dimensional Brillouin zone torus, i.e.
$l\in\\{l^{(k_{2})}_{1},l^{(k_{1})}_{2}\\}$ where
$l^{(k_{2})}_{1}=\\{(k_{1},k_{2})|k_{1}\in[-\pi,\pi)\\}$, i.e. the path
crossing the Brillouin zone at a fixed $k_{2}$, and similarly for
$l^{(k_{1})}_{2}$. The one-dimensional topologies are indicated by the
quantized Berry phase $\gamma_{B}[l]\in\\{0,\pi\\}\;\mathrm{mod}\;2\pi$ or,
equivalently, by the first Stiefel-Whitney class Ahn  _et al._ (2018), which
characterize the orientability of the phase, i.e. whether the frame of
eigenvector $R(\boldsymbol{k})$ can be chosen to be continuous and periodic
across the Brillouin zone Bouhon  _et al._ (2020b); Ahn  _et al._ (2018). In
the following we write the homotopy classes of one-dimensional cuts as
$\alpha_{1(2)}=[l^{(k_{2(1)})}_{1(2)}]\in\pi_{1}[\mathsf{Gr}^{\mathbb{R}}_{2,3}]$.

In Ref. Bouhon  _et al._ (2020b), we have derived the general homotopy
classification of Euler phases with multiple energy gaps, in which case the
classifying space takes the form of a generalized real flag manifold. For this
work, it is sufficient to consider the topological classification of the two-
dimensional Euler phases with a single principal gap. We nevertheless relate
these phases to the braiding of multi-gap nodes and the conversion of their
non-Abelian homotopy charges. Indeed, we show in Section V that the mapping of
an Euler phase to another requires the braiding of nodes from an adjacent gap.
In the following, we label the occupied (unoccupied) band-subspace with the
roman letter $I$ ($II$), e.g. we write their respective Euler classes
$(\chi_{I},\chi_{II})$.

In this work, we only consider orientable phases, i.e. with
$R(\boldsymbol{k})$ periodic. The homotopy classification and modeling of
orientable phases (i.e. with trivial one-dimensional topology) is most
conveniently obtained from the classification of two-dimensional oriented
vector bundles, i.e. with the oriented Grassmannian as the classifying space,
$\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{p,N}=\mathsf{SO}(N)/[\mathsf{SO}(p)\times\mathsf{SO}(N-p)]$.
Indeed, we have for the orientable phases Bouhon  _et al._ (2020b)

$\displaystyle{}^{(\alpha_{1}=0,\alpha_{2}=0)}$
$\displaystyle=[\mathbb{S}^{2},\mathsf{Gr}^{\mathbb{R}}_{p,N}]$ (4)
$\displaystyle=\pi_{2}[\mathsf{Gr}^{\mathbb{R}}_{p,N}]/{\sim}$
$\displaystyle=\pi_{2}[\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{p,N}]/{\sim},$

where the equivalence relation $\sim$ corresponds to the Euler class reversal
map

$(\chi_{I},\chi_{II})\sim(-\chi_{I},-\chi_{II}).$ (5)

The important point here is that
$\pi_{2}[\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{p,N}]$ is known. (It can be
computed through the long exact sequence of homotopy groups associated to
fiber bundles Hatcher (2001).) The reduction of the homotopy classification
through the equivalence relation ${\sim}$ for the orientable phases, as
compared to the oriented phases classified by
$\pi_{2}[\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{p,N}]$, is due to the absence
of a fixed base point in the definition of the homotopy classes
$[\mathbb{T}^{2},\mathsf{Gr}^{\mathbb{R}}_{p,N}]$ that capture the topology of
Hamiltonians (contrary to homotopy groups that are defined assuming a fixed
base point). The absence of a fixed base point permits the nontrivial action
of a generator of the first homotopy group (i.e. the deformation of the
Hamiltonian along one non-contractible loop of the classifying space) on the
elements of the second homotopy group. More precisely, this action defines an
automorphism between distinct elements of the second homotopy group while
remaining within the same homotopy class Bouhon  _et al._ (2020b); Wojcik  _et
al._ (2020); Hatcher (2001).

We also use the oriented Grassmannian
$\widetilde{\mathsf{Gr}}_{p,N}^{\mathbb{R}}$ for the modelling of Euler
phases. Indeed, starting from the representative $R\in\mathsf{SO}(N)$ of a
point of the oriented Grassmannian (here defined as a coset)
$[R]=\\{R\cdot(O_{I}\oplus
O_{II})|O_{I}\in\mathsf{SO}(p),O_{II}\in\mathsf{SO}(N-p)\\}$, the flattened
Hamiltonian $Q=R\cdot(-\mathbb{1}\oplus\mathbb{1})\cdot R^{T}$ inherits the
equivalence relation $\sim$ defined above. This directly follows from the
higher gauge freedom of the Hamiltonian form as compared to the coset element
$[R]$. (Explicitly, the transformation $R\rightarrow R\cdot(O_{I}\oplus
O_{II})$ with $O_{I}\in\mathsf{O}(p)$ and $O_{II}\in\mathsf{O}(N-p)$, leaves
$Q$ invariant, while it maps to a frame that is not necessarily represented by
the coset $[R]$.)

In the following, we concentrate on the three-band and four-band Euler phases,
i.e. for $(p,N)=(2,3)$ and $(p,N)=(2,4)$.

###  II.2 Three-band Euler phases

The homotopy classification of (two-dimensional) orientable three-band Euler
phases that split into $2+1$-band-subspaces is given by

$\displaystyle\pi_{2}[\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{2,3}]/{\sim}$
$\displaystyle=\pi_{2}[\mathbb{S}^{2}]/{\sim}$ (6) $\displaystyle=\\{a\in
2\mathbb{Z}|a\sim-a\\}=2\mathbb{N}\ni\beta(H^{(2+1)}),$

where we have used the identities
$\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{2,3}=\mathsf{SO}(3)/\mathsf{SO}(2)=\mathbb{S}^{2}$,
and $\beta(H^{(2+1)})$ represents the homotopy class of the Bloch Hamiltonian
$H^{(2+1)}(\boldsymbol{k})$ with eigenvalues that split as
$E_{1}(\boldsymbol{k})\leq E_{2}(\boldsymbol{k})<E_{3}(\boldsymbol{k})$ for
all $\boldsymbol{k}\in\mathbb{T}^{2}$. The factor two in the classifying set
$2\mathbb{N}$ will become clear below. We define the Euler class of the two-
band subspace, i.e. Bouhon  _et al._ (2020a); Ahn  _et al._ (2018, 2019); Zhao
and Lu (2017)

$\displaystyle\chi_{I}[\\{u_{1},u_{2}\\}]$
$\displaystyle=\dfrac{1}{2\pi}\int_{\mathrm{BZ}}\mathsf{Eu}(\boldsymbol{k})\in\mathbb{Z},$
(7a) where the Euler form $\mathsf{Eu}$ is obtained from the connection
$\mathsf{a}=u_{1}^{T}\cdot du_{2}$ through $\mathsf{Eu}=d\mathsf{a}$, leading
to
$\mathsf{Eu}(\boldsymbol{k})=\left(\partial_{k_{1}}u_{1}^{T}\cdot\partial_{k_{2}}u_{2}-\partial_{k_{2}}u_{1}^{T}\cdot\partial_{k_{1}}u_{2}\right)dk_{1}\wedge
dk_{2}.$ (7b)

The homotopy invariant is then readily given as an equivalence class

$\displaystyle\beta\left(H^{(2+1)}\right)$
$\displaystyle=\left[\,\chi_{I}[\\{u_{1},u_{2}\\}]\,\right]\,,$ (8)
$\displaystyle=\left\\{|\,\chi_{I}[\\{u_{1},u_{2}\\}]\,|\,,-|\,\chi_{I}[\\{u_{1},u_{2}\\}]\,|\right\\}\,.$

We note that while we could simply take $|\chi_{I}|\in 2\mathbb{N}$ as a the
number representative of the equivalence class $[\chi_{I}]$, we will see in
Section V that the equivalence class must be used to predict the correct
expression of the monopole charge and linking number of $PT$-symmetry
protected linked nodal rings. Importantly, the above definition of the Euler
class holds for any orientable two-band subspace $\\{u_{n},u_{n+1}\\}$
isolated from all the other bands, i.e. $\chi_{\nu}[\\{u_{n},u_{n+1}\\}]$
($\nu=I,II,\dots$) is well defined whenever
$E_{n-1}(\boldsymbol{k})<E_{n}(\boldsymbol{k})\leq
E_{n+1}(\boldsymbol{k})<E_{n+2}(\boldsymbol{k})$ for all
$\boldsymbol{k}\in\mathbb{T}^{2}$.

The modeling of three-band Euler phases with two occupied bands can then be
readily obtained from a representative $R\in\mathsf{SO}(3)$ of the coset
$[R]\in\mathsf{SO}(3)/\mathsf{SO}(2)=\mathbb{S}^{2}$. The spherical frame
readily satisfies this condition, i.e.
$R(\phi,\theta)=(u_{1}~{}u_{2}~{}u_{3})=(e_{\theta}~{}e_{\phi}~{}e_{r})$, with
$e_{\theta}=(\cos\phi\cos\theta,\sin\phi,\cos\theta,-\sin\theta)$,
$e_{\phi}=(-\sin\phi,\cos\phi,0)$ and
$e_{r}=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$, from which we get
Bzdušek and Sigrist (2017); Wu  _et al._ (2019)

$Q^{(2+1)}[\boldsymbol{n}(\phi,\theta)]=2\boldsymbol{n}(\phi,\theta)\cdot\boldsymbol{n}(\phi,\theta)^{T}-\mathbb{1}_{3},$
(9)

with the unit vector $\boldsymbol{n}(\phi,\theta)=e_{r}$ (the superscript
‘$2+1$’ refers to the spectral decomposition into one two-band subspace and
one single band). Since for orientable phases we can simplify the Brillouin
zone to a sphere, see Eq. (4), let us represent a point of the Brillouin zone
by the angles
$(\phi_{\boldsymbol{k}},\theta_{\boldsymbol{k}})\in\mathbb{S}^{2}_{\mathrm{BZ}}$.
More concretely, this follows e.g. by choosing

$\displaystyle\phi_{\boldsymbol{k}}$
$\displaystyle=\mathrm{arg}\,(k_{1}+\mathrm{i}k_{2})\,,$ (10)
$\displaystyle\theta_{\boldsymbol{k}}$
$\displaystyle=\max\left(|k_{1}|,|k_{2}|\right)\,.$

We can then define the Euler phases through the ansatz

$\displaystyle\phi_{q}(\phi_{\boldsymbol{k}},\theta_{\boldsymbol{k}})$
$\displaystyle=q\phi_{\boldsymbol{k}}\,,$ (11)
$\displaystyle\theta_{q}(\phi_{\boldsymbol{k}},\theta_{\boldsymbol{k}})$
$\displaystyle=(1-\delta_{0,q})\theta_{\boldsymbol{k}}\,,$

where the integer $q\in\mathbb{Z}$ fixes the number of times the mapping
$(\phi_{q},\theta_{q})$ wraps the sphere as we cover the base sphere
$\mathbb{S}^{2}_{\mathrm{BZ}}$ one time, which is computed by the Skyrmion
winding number Bouhon  _et al._ (2020a); Ünal  _et al._ (2020)

$\displaystyle W[\boldsymbol{n}_{q}]$
$\displaystyle=\dfrac{1}{4\pi}\int_{\mathbb{S}^{2}_{\mathrm{BZ}}}\boldsymbol{n}_{q}\cdot\left(\partial_{\phi_{\boldsymbol{k}}}\boldsymbol{n}_{q}\times\partial_{\theta_{\boldsymbol{k}}}\boldsymbol{n}_{q}\right)d\phi_{\boldsymbol{k}}\wedge
d\theta_{\boldsymbol{k}},$ (12) $\displaystyle=q\in\mathbb{Z}\,,$

with $\boldsymbol{n}_{q}=\boldsymbol{n}(\phi_{q},\theta_{q})$. Substituting
the above ansatz in the expression for the connection
$\mathsf{a}(\phi_{q},\theta_{q})=e_{\theta}^{T}\cdot de_{\phi}$, we get the
Euler form

$\displaystyle\mathsf{Eu}(\phi_{q},\theta_{q})$
$\displaystyle=\left(\partial_{\phi_{\boldsymbol{k}}}u_{1}^{T}\cdot\partial_{\theta_{\boldsymbol{k}}}u_{2}-\partial_{\theta_{\boldsymbol{k}}}u_{1}^{T}\cdot\partial_{\phi_{\boldsymbol{k}}}u_{2}\right)d\phi_{\boldsymbol{k}}\wedge
d\theta_{\boldsymbol{k}},$ (13a)
$\displaystyle=-q(1-\delta_{0,q})\sin\left[(1-\delta_{0,q})\theta_{\boldsymbol{k}}\right]d\phi_{\boldsymbol{k}}\wedge
d\theta_{\boldsymbol{k}},$ and then the Euler class
$\displaystyle\chi_{I}[\\{e_{\theta},e_{\phi}\\}]$
$\displaystyle=\dfrac{1}{2\pi}\int_{\mathbb{S}_{\mathrm{BZ}}^{2}}\mathsf{Eu}(\phi_{q},\theta_{q}),$
(13b) $\displaystyle=q\left(-1+\cos[(1-\delta_{0,q})\pi]\right)=-2q,$

from which we see that the Euler class is doubled, i.e. only even values of
the Euler class are permitted. This is a direct consequence of the fact that
the Hamiltonian $Q^{(2+1)}$ is given by the “square” of the winding unit
vector $\boldsymbol{n}$. Taking the equivalence relation
$\chi_{I}\sim-\chi_{I}$ in account, the homotopy classes of three-band Euler
phases, splitting into $2+1$-band-subspaces, are thus classified by one even
number through

$\beta(H^{(2+1)})=[\chi_{I}]=[2q]\,,~{}\text{with}~{}|2q|\in 2\mathbb{N},$
(14)

such that the corresponding Euler phases are all represented up-to-homotopy by
the flattened Hamiltonian $Q^{(2+1)}(\boldsymbol{n}_{q})$, i.e. by Eq. (9)
with the ansazt Eq. (11) for $q\in\mathbb{N}$. We give minimal tight-binding
models for the phases $\chi_{I}\in\\{2,4\\}$ in Section III.1.

For completeness, we give an example of an Euler class reversal map Bouhon
_et al._ (2020b); Wojcik  _et al._ (2020). For this we first define a
representation of the nontrivial element of
$\pi_{1}[\mathbb{R}P^{2}]=\mathbb{Z}_{2}$ through the deformation of the
flattened Hamiltonian (noting
$\mathsf{Gr}^{\mathbb{R}}_{2,3}=\mathbb{R}P^{2}$) Bouhon  _et al._ (2020b);
Wojcik  _et al._ (2020)

$\displaystyle\ell_{\boldsymbol{n}}$
$\displaystyle:[0,1]\rightarrow\mathbb{R}P^{2}$ (15)
$\displaystyle:t\mapsto\ell_{\boldsymbol{n}}(t)=Q^{(2+1)}[S(t)\cdot\boldsymbol{n}(\phi,\theta)],$
$\displaystyle\text{with}~{}S(t)$
$\displaystyle=\left(\begin{array}[]{ccc}\cos\pi t&0&-\sin\pi t\\\ 0&1&0\\\
\sin\pi t&0&\cos\pi t\end{array}\right).$

The transformation acts non-trivially on all the points of the classifying
space, except at $\boldsymbol{n}(\pi/2,\pi/2)=(0,1,0)$ since
$S(t)\cdot(0,1,0)=(0,1,0)$. In particular, the deformation starting at
$\boldsymbol{n}_{0}=\boldsymbol{n}(\phi=0,\theta=0)=(0,0,1)$ defines a closed
loop in $\mathbb{R}P^{2}$ since
$\ell_{\boldsymbol{n}_{0}}(1)=Q^{(2+1)}[S(1)\cdot\boldsymbol{n}_{0}]=Q^{(2+1)}[-\boldsymbol{n}_{0}]=Q^{(2+1)}[\boldsymbol{n}_{0}]=\ell_{\boldsymbol{n}_{0}}(0)$.
Noting that $Q^{(2+1)}[S(t)\cdot\boldsymbol{n}]=S(t)\cdot
Q^{(2+1)}[\boldsymbol{n}]\cdot S(t)^{T}$, the transformation of the frame at
the reference base point $(\phi,\theta)=(0,0)$, is
$R_{t=1}(0,0)=(u_{1}(0,0)~{}u_{2}(0,0)~{}u_{3}(0,0))_{t=1}=(-u_{1}(0,0)~{}u_{2}(0,0)~{}-u_{3}(0,0))_{t=0}$,
such that the Berry phase factors over the loop $\ell_{\boldsymbol{n}_{0}}$
for the two band subspaces are
$\mathrm{e}^{-i\gamma_{B}[\ell_{\boldsymbol{n}_{0}};\\{u_{3}\\}]}=-1$ and
$\mathrm{e}^{-i\gamma_{B}[\ell_{\boldsymbol{n}_{0}};\\{u_{1},u_{2}\\}]}=-1$.
We then conclude that the homotopy class $[[0,1],\ell_{\boldsymbol{n}_{0}}]$,
with $\ell_{\boldsymbol{n}_{0}}(1)=\ell_{\boldsymbol{n}_{0}}(0)$, represents
the generator of $\pi_{1}[\mathbb{R}P^{2}]=\mathbb{Z}_{2}$, as indicated by
the $\pi$-Berry phase. We now want to compare the Euler class of the
Hamiltonians before, $Q^{(2+1)}[\boldsymbol{n}(\phi,\theta)]$, and after the
transformation, $Q^{(2+1)}[S(1)\cdot\boldsymbol{n}(\phi,\theta)]$. The Euler
classes can only be compared if we chose the same gauge with respect to the
same chosen reference point for both Hamiltonians. This reference point must
be taken as the base point $\boldsymbol{n}_{0}$ that generates the above
nontrivial loop. Keeping $\boldsymbol{n}_{0}$ fixed for the evaluation of the
Euler class, we must thus compare the winding number of $\boldsymbol{n}$ in
$Q^{(2+1)}[\boldsymbol{n}]$, with the winding number of
$-S(t)\cdot\boldsymbol{n}$ in
$Q^{(2+1)}[-S(t)\cdot\boldsymbol{n}](=Q^{(2+1)}[S(t)\cdot\boldsymbol{n}])$,
since $-S(1)\cdot\boldsymbol{n}_{0}=\boldsymbol{n}_{0}$. We conclude that the
transformation reverses the signed Euler class from
$\chi_{I}=2W[\boldsymbol{n}]=2q$ to
$\chi_{I}=2W[-S(1)\cdot\boldsymbol{n}]=-2q$.

We end this part by noting that $3$ is the minimal number of bands permitting
a nontrivial Euler phase. Indeed, in the case of a (orientable) two-band
system, the frame of eigenvectors $R\in\mathsf{SO}(2)$ can be written

$R(f)=\left(u_{1}(f)~{}u_{2}(f)\right)=\left(\begin{array}[]{cc}\cos f&-\sin
f\\\ \sin f&\cos f\end{array}\right),$ (16)

with $f\in\mathbb{S}^{1}$. Representing again the points of the Brillouin zone
by the points of a sphere
$\mathbb{S}^{2}_{\mathrm{BZ}}\ni(\phi_{\boldsymbol{k}},\theta_{\boldsymbol{k}})$
(assuming the orientability of the phase), we find that the Euler form Eq.
(13) is identically zero.

###  II.3 Four-band Euler phases

The homotopy classification of the two-dimensional orientable four-band Euler
phases that split into $2+2$-band subspaces is given by

$\displaystyle\pi_{2}[\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{2,4}]/{\sim}$
$\displaystyle=\pi_{2}[\mathbb{S}^{2}\times\mathbb{S}^{2}]/{\sim},$ (17)
$\displaystyle=\\{(a,b)\in\mathbb{Z}^{2}|(a,b){\sim}(-a,-b)\\}\ni\beta(H^{(2+2)}),$

where we have used the diffeomorphism
$\widetilde{\mathsf{Gr}}^{\mathbb{R}}_{2,4}\approx\mathbb{S}^{2}\times\mathbb{S}^{2}$,
and $\beta(H^{(2+2)})$ represents the homotopy class of the Bloch Hamiltonian
$H^{(2+2)}(\boldsymbol{k})$ with eigenvalues that split as
$E_{1}(\boldsymbol{k})\leq E_{2}(\boldsymbol{k})<E_{3}(\boldsymbol{k})\leq
E_{4}(\boldsymbol{k})$ for all $\boldsymbol{k}\in\mathbb{T}^{2}$. The homotopy
invariants are computed through the Euler classes of the two-band occupied and
unoccupied subspaces, $(\chi_{I},\chi_{II})$, modulo the homotopy equivalence
$(\chi_{I},\chi_{II})\sim(-\chi_{I},-\chi_{II})$, which we show is a
consequence of the existence of an adiabatic deformation of the Hamiltonian
reversing both Euler classes at the same time. (This is a consequence of the
facts that $(i)$ while the Euler class is a homotopy invariant of an oriented
vector bundle, the real Bloch Hamiltonians are only orientable, and $(ii)$ the
topology of Bloch Hamiltonians are captured by homotopy classes (i.e. with no
base point) rather than by homotopy groups (i.e. with a fixed base point)
Bouhon  _et al._ (2020b).) In the following, we write the homotopy invariant
in terms of an equivalence class of Euler classes, i.e.

$\displaystyle\beta\left(H^{(2+2)}\right)$
$\displaystyle=[\chi_{I},\chi_{II}]=[\boldsymbol{\chi}]$ (18)
$\displaystyle=\\{(\chi_{I},\chi_{II}),(-\chi_{I},-\chi_{II})\\}\,.$

The modeling of the four-band Euler phases is obtained from a representative
$R\in\mathsf{SO}(4)$ of the coset
$[R]\in\mathsf{SO}(4)/[\mathsf{SO}(2)\times\mathsf{SO}(2)]\cong\mathbb{S}^{2}_{+}\times\mathbb{S}^{2}_{-}$.
Using the Plücker embedding, we find (see Ref. Bouhon  _et al._ (2020b);
Bouhon (2020))

$R(\phi_{+},\theta_{+},\phi_{-},\theta_{-})=[\text{Eq.\;(\ref{eq_R_full})\;in\;Appendix}].$
(19)

Then, setting

$\displaystyle(\phi_{+},\theta_{+})$ $\displaystyle=(\phi,\theta),$ (20)
$\displaystyle(\phi_{-},\theta_{-})$
$\displaystyle=(\phi^{\prime}+\pi/2,\theta^{\prime}+\pi/2),$

we obtain the twofold degenerated Hamiltonian,

$\displaystyle
H[\boldsymbol{n},\boldsymbol{n}^{\prime};\epsilon_{1},\epsilon_{2}]$ (21a)
$\displaystyle=R(\phi,\theta,\phi^{\prime},\theta^{\prime})\cdot\left(\epsilon_{1}\mathbb{1}_{2}\oplus\epsilon_{2}\mathbb{1}_{2}\right)\cdot
R(\phi,\theta,\phi^{\prime},\theta^{\prime})^{T},$
$\displaystyle=\dfrac{1}{2}\left\\{(\epsilon_{1}+\epsilon_{2})\Gamma_{00}+(-\epsilon_{1}+\epsilon_{2})Q^{(2+2)}[\boldsymbol{n},\boldsymbol{n}^{\prime}]\right\\},$
such that the gap condition reads $\epsilon_{1}<\epsilon_{2}$, with the
flattened Hamiltonian (i.e. for $\epsilon_{2}=-\epsilon_{1}=+1$) given by
$\displaystyle Q^{(2+2)}[\boldsymbol{n},\boldsymbol{n}^{\prime}]$
$\displaystyle=H[\boldsymbol{n},\boldsymbol{n}^{\prime};-1,1]$ (21b)
$\displaystyle=n_{1}^{\prime}\left(-n_{1}\Gamma_{33}+n_{2}\Gamma_{31}+n_{3}\Gamma_{10}\right)$
$\displaystyle-
n_{2}^{\prime}\left(+n_{1}\Gamma_{13}-n_{2}\Gamma_{11}+n_{3}\Gamma_{30}\right)$
$\displaystyle+n_{3}^{\prime}\left(+n_{1}\Gamma_{01}+n_{2}\Gamma_{03}-n_{3}\Gamma_{22}\right),$
$\displaystyle=\boldsymbol{n}^{T}\cdot\underline{\Gamma}\cdot\boldsymbol{n}^{\prime},$
which is determined by two unit vectors
$\boldsymbol{n}(\phi,\theta)=(\cos\phi\sin\theta,\sin\phi,\sin\theta,\cos\theta)$
and $\boldsymbol{n}^{\prime}=\boldsymbol{n}(\phi^{\prime},\theta^{\prime})$,
with
$\underline{\Gamma}=\left(\begin{array}[]{rrr}-\Gamma_{33}&-\Gamma_{13}&\Gamma_{01}\\\
\Gamma_{31}&\Gamma_{11}&\Gamma_{03}\\\
\Gamma_{10}&-\Gamma_{30}&-\Gamma_{22}\end{array}\right)\;,$ (21c)

where the Dirac matrices $\Gamma_{ij}=\sigma_{i}\otimes\sigma_{j}$ are defined
from the Pauli matrices $\sigma_{1}=\begin{pmatrix}[r]0&1\\\
1&0\end{pmatrix}$, $\sigma_{2}=\begin{pmatrix}[r]0&-i\\\ i&0\end{pmatrix}$,
$\sigma_{3}=\begin{pmatrix}[r]1&0\\\ 0&-1\end{pmatrix}$, and
$\sigma_{0}=\mathbb{1}_{2}$. Inversely, inserting two generic vectors
$\boldsymbol{h}$ and $\boldsymbol{h}^{\prime}$ (i.e. non unit vectors), we get

$Q^{(2+2)}[\boldsymbol{h},\boldsymbol{h}^{\prime}]=H\left[\dfrac{\boldsymbol{h}}{|\boldsymbol{h}|},\dfrac{\boldsymbol{h}^{\prime}}{|\boldsymbol{h}^{\prime}|};|\boldsymbol{h}|,|\boldsymbol{h}^{\prime}|\right]\;.$
(22)

Simplifying the Brillouin zone to the sphere $\mathbb{S}_{\mathrm{BZ}}^{2}$
(without loss of generality for the orientable phases), we can define all
Euler phases in terms of two integers $(q,q^{\prime})\in\mathbb{Z}^{2}$
through the ansatz

$\displaystyle(\phi_{q},\theta_{q})$
$\displaystyle=(q\phi_{\boldsymbol{k}},[1-\delta_{0,q}]\theta_{\boldsymbol{k}}),$
(23) $\displaystyle(\phi^{\prime}_{q^{\prime}},\theta^{\prime}_{q^{\prime}})$
$\displaystyle=(q^{\prime}\phi_{\boldsymbol{k}},[1-\delta_{0,q^{\prime}}]\theta_{\boldsymbol{k}}).$

We note that this parametrization readily implies that the Euler phases are
characterized by the two winding numbers [Eq. (12)].

$q=W[\boldsymbol{n}_{q}],~{}q^{\prime}=W[\boldsymbol{n}^{\prime}_{q^{\prime}}].$
(24)

Substituting Eq. (23), we obtain the Euler forms

$\mathsf{Eu}_{I}=\dfrac{\sin\theta_{0}}{2}\left(q-q^{\prime}\right),~{}\mathsf{Eu}_{II}=\dfrac{\sin\theta_{0}}{2}\left(q+q^{\prime}\right),$

leading to the Euler classes

$\chi_{I}=q-q^{\prime},~{}\chi_{II}=q+q^{\prime}.$ (25)

We thus conclude that
$Q^{(2+2)}[\boldsymbol{n}_{q},\boldsymbol{n}^{\prime}_{q^{\prime}}]$ in Eq.
(21b) represents all the homotopy classes $[\chi_{I},\chi_{II}]$ of four-band
Euler phases, that is a pair of Euler classes modulo the equivalence relation
$(\chi_{I},\chi_{II})\sim(-\chi_{I},-\chi_{II})$, or, in terms of the winding
numbers, $(q,q^{\prime})\sim(-q,-q^{\prime})$. As discussed above, this
reduction comes from the existence of an adiabatic mapping that reverses both
Euler classes, see Section II.8 below. It is important to note that, as a
consequence, we can only keep track of the relative signs of the Euler classes
in Eq. (25).

Importantly, we find the sum rule

$\chi_{I}+\chi_{II}=0\mod 2,$ (26)

which guarantees the global cancellation of the second Stiefel Whitney class,
i.e. from the definition $w_{2,I(II)}=\chi_{I(II)}\mod 2$, we get
$w_{2,I}+w_{2,II}=0\mod 2$. This is actually a requirement for any total
oriented real vector bundle, i.e. here taking all $N$ bands of the Bloch
Hamiltonian. (Here, we implicitly assume that all the elements
$\\{H_{ab}(\boldsymbol{k})\\}_{a,b=1,\dots,N}$ of the Bloch Hamiltonian are
analytic functions of the momentum, or in words, that these are given by
finite Fourier series.)

###  II.4 Balanced vs imbalanced four-band phases

When one winding number is zero, we obtain $|\chi_{I}|=|\chi_{II}|$, i.e. the
absolute Euler classes are equal across the energy gap. We call these phases
the balanced Euler phases. Whenever both winding numbers are nonzero, i.e.
$|q|,|q^{\prime}|>0$, we get unequal absolute Euler classes across the energy
gap, i.e. $|\chi_{I}|\neq|\chi_{II}|$. We call these phases the imbalanced
Euler phases. The different responses under an external magnetic field between
the balanced and the imbalanced Euler phases via their Hofstadter spectrum
have been systematically studied in Ref. Guan  _et al._ (2021).

Fixing one constant unit vector, say
$\boldsymbol{n}^{\prime}_{q^{\prime}=0}=\boldsymbol{n}(\phi^{\prime}_{c},\theta^{\prime}_{c})$,
we importantly note the topological non-equivalence of the two balanced phases

$H_{a}=H[\boldsymbol{n}_{q},\boldsymbol{n}^{\prime}_{0};\epsilon_{1},\epsilon_{2}]\not\simeq
H[\boldsymbol{n}^{\prime}_{0},\boldsymbol{n}_{q};\epsilon_{1},\epsilon_{2}]=H_{b},$
(27a) as indicated by the two inequivalent homotopy invariants, i.e.
$\displaystyle\beta(H_{a})$
$\displaystyle=[\boldsymbol{\chi}_{a}]=\\{(q,q),(-q,-q)\\}$ (27b)
$\displaystyle\neq\beta(H_{b})$
$\displaystyle=[\boldsymbol{\chi}_{b}]=\\{(-q,q),(-q,q)\\}\,.$

We discuss in detail in Section V an indicator that distinguishes the phases
$[\chi_{I},\chi_{II}]$ and $[\chi_{I},-\chi_{II}]$.

###  II.5 Mirror Chern number of the balanced degenerate Euler phases

We show in Appendix B, see also Ref. Guan  _et al._ (2021), that all the
balanced and degenerate Euler phases possess an effective spinful basal mirror
symmetry (i.e.
$\sigma_{h}H_{\text{bal}}^{\text{deg}}\sigma_{h}=H_{\text{bal}}^{\text{deg}}$
with $\sigma_{h}^{2}=-1$) which permits the definition of a mirror Chern
number. Fixing $\boldsymbol{n}^{\prime}=(0,0,1)$, we obtain in Appendix B

$\displaystyle C^{(-\mathrm{i})}_{I}$
$\displaystyle=-C^{(\mathrm{i})}_{I}=C^{(\mathrm{i})}_{II}=-C^{(-\mathrm{i})}_{II},$
(28) $\displaystyle=W[\boldsymbol{n}]=q=\chi_{I}=\chi_{II}.$

The reverse is true, namely all Euler phases that are not twofold degenerate
have no basal mirror symmetry and the mirror Chern number is not defined,
leaving the Euler class as the unique fundamental topological invariant (i.e.
discarding crystalline topologies characterized by symmetry-indicators Bradlyn
_et al._ (2017); Po  _et al._ (2017); Khalaf  _et al._ (2018); Slager  _et
al._ (2013); Kruthoff  _et al._ (2017), see also Section IV).

###  II.6 Stable nodal points and nodal lines in 3D

Let us label every block of isolated bands ordered in energy from below with a
roman number $\nu=I,II,III,\dots$, i.e. $E_{\nu,m_{\nu}}<E_{\nu+I,1}$, where
$m_{\nu}$ is the number of bands in the $\nu$-th block of bands with the
eigenenergies $\\{E_{\nu,1}\leq\dots\leq E_{\nu,m_{\nu}}\\}$. Given that the
Euler class is only well defined for orientable two-band subspaces Bouhon  _et
al._ (2020a); Ahn  _et al._ (2018), the most striking observable of the
nontrivial Euler topology of a two-band subspace is the presence of stable
nodal points that cannot be annihilated as long as the two bands remain
separated from all the other bands (while preserving $C_{2}T$ symmetry). More
precisely, given a two-band subspace, say the $\nu$-th block, with the Bloch
eigenenergies $E_{\nu,1}\leq E_{\nu,2}$ and the Bloch eigenvectors
$\\{u_{\nu,1},u_{\nu,2}\\}$, it must host a number $2|\chi_{\nu}|\in
2\mathbb{N}$ of stable nodal points determined by its Euler class
$\chi_{\nu}[\\{u_{\nu,1},u_{\nu,2}\\}]$. This has for instance the consequence
that any nontrivial two-band subspace with flat energy levels must necessarily
be twofold degenerate. Non-trivial flat bands thus host mirror Chern numbers,
and pairs of anti-propagating chiral branches must appear on each edge of the
system. (We note that the situation is more subtle in the case of the
coexistence of flat bands with dispersive bands, e.g. as in twisted bilayer
Graphene.)

###  II.7 Nodal line continuations of nodal points in 3D

Upon the adiabatic deformation of any Euler phase, say by a term of the real
Hamiltonian scaling with the parameter $\lambda\in\mathbb{R}$, the nodal
points extend into nodal lines within the three-dimensional parameter space
$(\boldsymbol{k},\lambda)\in\mathbb{T}^{2}\times\mathbb{R}$. We show in
Section V, and in Section VI with concrete models, how this allows us to
systematically generate 3D tight-binding Hamiltonian with linked nodal rings
characterized by (1D) non-Abelian frame charges and (2D) linking numbers.

###  II.8 Euler class reversal map

For completeness, we here elaborate on the Euler class reversing map in the
four-band case Bouhon  _et al._ (2020b). Using the compact notation
$\boldsymbol{p}=(\boldsymbol{n},\boldsymbol{n}^{\prime})$, and taking
$\boldsymbol{p}_{0}=(\boldsymbol{n}_{0},\boldsymbol{n}^{\prime}_{0})$, with
$\boldsymbol{n}_{0}=\boldsymbol{n}^{\prime}_{0}=\boldsymbol{n}(\phi=0,\theta=0)=(0,0,1)$,
as a reference base point, the transformation

$\displaystyle\ell_{\boldsymbol{p}}$
$\displaystyle:[0,1]\rightarrow\mathsf{Gr}^{\mathbb{R}}_{2,4}$ (29)
$\displaystyle:t\mapsto\ell_{\boldsymbol{p}}(t)=Q^{(2+2)}[S(t)\cdot\boldsymbol{p}]\,.$

(the unit interval here, $[0,1]$, should not be confused with one equivalence
class), with $S(t)$ given in Eq. (15), induces a non-trivial closed loop in
the Hamiltonian space at $\boldsymbol{p}_{0}$, since
$\ell_{\boldsymbol{p}_{0}}(1)=Q^{(2+2)}[S(1)\cdot\boldsymbol{p}_{0}]=Q^{(2+2)}[-\boldsymbol{p}_{0}]=Q^{(2+2)}[\boldsymbol{p}_{0}]=\ell_{\boldsymbol{p}_{0}}(0)$.
The gapped spectrum of the Hamiltonian remains constant
($-\epsilon_{1}=\epsilon_{2}=1$) through the whole transformation, since
$|S(t)\cdot\boldsymbol{n}|=|S(t)\cdot\boldsymbol{n}^{\prime}|=1$ [by Eq. (21b)
and Eq. (21a)]. We thus conclude that there exists a similitude relation
$Q^{(2+2)}[S(t)\cdot\boldsymbol{p}_{0}]=O(t)\cdot
Q^{(2+2)}[\boldsymbol{p}_{0}]\cdot O(t)^{T}$ with $O(t)\in\mathsf{SO}(4)$,
from which we get the action on the eigen-frame, $O(t)\cdot
R[\boldsymbol{p}_{0}]=(u_{1}(\boldsymbol{p}_{0})~{}\cdots~{}u_{4}(\boldsymbol{p}_{0}))_{t}$.
This leads to the Berry phase factors
$\mathrm{e}^{-\gamma_{B}[\ell_{\boldsymbol{p}_{0}};\\{u_{1},u_{2}\\}]}=-1$ and
$\mathrm{e}^{-\gamma_{B}[\ell_{\boldsymbol{p}_{0}};\\{u_{3},u_{4}\\}]}=-1$,
indicating that the homotopy class $[[0,1],\ell_{\boldsymbol{p}_{0}}]$
represents the generator of
$\pi_{1}[\mathsf{Gr}^{\mathbb{R}}_{2,4}]=\mathbb{Z}_{2}$. As in the three-band
case, the Euler classes before and after the transformation must be evaluated
with respect to the same gauge at the fixed base point
$\boldsymbol{p}_{0}=(\boldsymbol{n}_{0},\boldsymbol{n}^{\prime}_{0})$. Since
$S(1)\cdot\boldsymbol{p}_{0}=-\boldsymbol{p}_{0}$, we compare the winding
numbers $(q,q^{\prime})$ for $Q^{(2+2)}[\boldsymbol{p}]$ with those for
$Q^{(2+2)}[-S(1)\cdot\boldsymbol{p}]$ (i.e. fixing the same reference point
with the same gauge). We conclude that the transformation reverses the winding
numbers $(q,q^{\prime})=(W[\boldsymbol{n}],W[\boldsymbol{n}^{\prime}])$ to
$(W[-S(1)\cdot\boldsymbol{n}],W[-S(1)\cdot\boldsymbol{n}^{\prime}])=(-q,-q^{\prime})$,
and thus reverses the Euler classes from
$(\chi_{I},\chi_{II})=(q-q^{\prime},q+q^{\prime})$ to
$(\chi_{I},\chi_{II})=(-q+q^{\prime},-q-q^{\prime})$.

##  III Minimal models with Euler class topology

In this section we formulate the models of interest hosting the Euler topology
of orientable phases. In particular we address concrete forms of several three
and four band models from lower to higher Euler classes.

As reviewed in the previous Section, while the three-band case is
characterized by a single even Euler class, the four-band case is classified
by two Euler classes, thus permitting a greater variety of inequivalent
topological phases, as we will detail in the following. Remarkably, we
formulate explicit tight-binding models for a great variety of Euler phases by
simply combining the generic forms derived in the previous Section together
with a minimal parametrization of a two-band Chern model with the Chern number
ranging from $-2$ to $2$. An alternative approach consists in truncating the
inverse Fourier transform (i.e. from the reciprocal space to the direct
lattice space) of the degenerate Bloch Hamiltonians given in the previous
Section, see Bouhon  _et al._ (2020b); Guan  _et al._ (2021) for more detail
and Ref. Bouhon (2020) for the implementation in a Mathematica notebook.

###  III.1 3-band case

The generic 3-band real Hermitian Hamiltonian can be written in terms of the
five real Gell-Mann matrices

$\begin{array}[]{l}\begin{array}[]{lll}\Lambda_{1}\,=\,\begin{pmatrix}[r]0&1&0\\\
1&0&0\\\ 0&0&0\end{pmatrix},&\Lambda_{3}\,=\,\begin{pmatrix}[r]1&0&0\\\
0&-1&0\\\ 0&0&0\end{pmatrix},&\Lambda_{4}\,=\,\begin{pmatrix}[r]0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},\end{array}\\\
\begin{array}[]{ll}\Lambda_{6}\,=\,\begin{pmatrix}[r]0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},&\Lambda_{8}\,=\,\frac{1}{\sqrt{3}}\begin{pmatrix}[r]1&0&0\\\
0&1&0\\\ 0&0&-2\end{pmatrix},\end{array}\end{array}$ (30)

to which we add the identity matrix $\Lambda_{0}=\mathbb{1}_{3}$, as

$\displaystyle H^{[\chi_{I}]}_{3\text{B}}(\boldsymbol{k})$
$\displaystyle=\boldsymbol{f}^{[\chi_{I}]}(\boldsymbol{k})\cdot\boldsymbol{\Lambda}$
(31)
$\displaystyle=\sum\limits_{i=0,1,3,4,6,8}f^{[\chi_{I}]}_{i}(\boldsymbol{k})\;\Lambda_{i},$

where $|\chi_{I}|$ is the maximum Euler class reachable for the given ansatz
(see below).

We give here minimal tight-binding models for the topological Euler phases
$\chi_{I}=2,4$. While the generalization to an arbitrary high Euler class is
straightforward, the distance in the hopping processes required in order to
achieve the nontrivial topology increases with the Euler class, making the
experimental realization of higher Euler classes more involving.

Taking advantage of the specific form of the flattened Hamiltonian Eq. (9), we
can readily use the ansatz of a two-band Chern model
$H_{\text{2B}}=\boldsymbol{h}^{\alpha}\cdot\boldsymbol{\sigma}$, for which the
Chern number is given by the winding of the vector
$\boldsymbol{h}^{\alpha}(\boldsymbol{k})=(h^{\alpha}_{1}(\boldsymbol{k}),h^{\alpha}_{2}(\boldsymbol{k}),h^{\alpha}_{3}(\boldsymbol{k}))$,
i.e. by $W[\boldsymbol{h}^{\alpha}/|\boldsymbol{h}^{\alpha}|]$ in Eq. (12).
Defining the functions

$\displaystyle a(\boldsymbol{k})$ $\displaystyle=\lambda\sin
k_{1},~{}b(\boldsymbol{k})=\lambda\sin k_{2}\,,$ (32) $\displaystyle
c(\boldsymbol{k})$ $\displaystyle=m-t_{1}(\cos k_{1}+\cos
k_{2})-t_{2}\cos(k_{1}+k_{2})\,,$

we set $\boldsymbol{h}^{\alpha}$ ($\alpha=A,B,C$) to be one of the cyclic
permutations of $(a,b,c)$, i.e.

$\boldsymbol{h}^{A}=(a,b,c)\,,~{}\boldsymbol{h}^{B}=(b,c,a)\,,~{}\boldsymbol{h}^{C}=(c,a,b)\,.$
(33)

Substituting this ansatz within the two-band Chern model
$\boldsymbol{h}^{\alpha}\cdot\boldsymbol{\sigma}$, the Chern number takes the
generic form (assuming $|\lambda|>0$) Sticlet  _et al._ (2012)

$c_{1}[\boldsymbol{h}^{\alpha}]=W\left[\dfrac{\boldsymbol{h}^{\alpha}(\boldsymbol{k})}{|\boldsymbol{h}^{\alpha}(\boldsymbol{k})|}\right]=\operatorname{sign}[-m-t_{2}]\\\
+\dfrac{1}{2}\left(\operatorname{sign}[m+2t_{1}-t_{2}]+\operatorname{sign}[m-2t_{1}-t_{2}]\right)\,,$
(34)

which is independent to the cyclic form $\alpha=A,B,C$. In the following we
set $\lambda=1$, without loos of generality. We show the generic phase diagram
for $c_{1}[\boldsymbol{h}]$ for a fixed parameter $m$ in Figure 1 (which is
slightly adapted from Ref. Sticlet  _et al._ (2012)).

---
Figure 1:  Phase diagram of the Chern number (winding number)
$c_{1}[\boldsymbol{h}]$ given by Eq. (34), adapted from Ref. Sticlet  _et al._
(2012).

We readily conclude that the Chern number is bounded as $-2\leq
c_{1}[\boldsymbol{h}]\leq 2$.

Making the substitution
$\boldsymbol{n}\rightarrow\boldsymbol{h}^{\alpha}(\boldsymbol{k})$ in Eq. (9),
we obtain the following degenerate Bloch Hamiltonian (non-flattened because
$\boldsymbol{h}^{\alpha}$ is not a unit vector)

$\displaystyle
H^{\alpha}_{\text{3B,deg}}[m,t_{1},t_{2},\lambda](\boldsymbol{k})$
$\displaystyle=\boldsymbol{h}^{\alpha}(\boldsymbol{k})\cdot\boldsymbol{h}^{\alpha}(\boldsymbol{k})^{T}\,$
(35a)
$\displaystyle=\boldsymbol{f}^{[\chi_{I}]}(\boldsymbol{k})\cdot\boldsymbol{\Lambda},$
where we have discarded the factor $2$ and the term $-\mathbb{1}_{3}$ used in
Eq. (9) that scale and shift the whole spectrum without changing the topology,
with $\begin{array}[]{rcl
rcl}f^{[\chi_{I}]}_{0}&=&\dfrac{|\boldsymbol{h}^{\alpha}|^{2}}{3}\,,&f^{[\chi_{I}]}_{1}&=&h_{1}^{\alpha}h_{2}^{\alpha}\,,\\\
f^{[\chi_{I}]}_{3}&=&\dfrac{1}{2}(h_{1}^{\alpha}-h_{2}^{\alpha})(h^{\alpha}_{1}+h^{\alpha}_{2})\,,&f^{[\chi_{I}]}_{4}&=&h^{\alpha}_{1}h^{\alpha}_{3}\,,\\\
f^{[\chi_{I}]}_{8}&=&\dfrac{|\boldsymbol{h}^{\alpha}|^{2}-3{h^{\alpha}_{3}}^{2}}{2\sqrt{3}}\,,&f^{[\chi_{I}]}_{6}&=&h^{\alpha}_{2}h^{\alpha}_{3}\,.\end{array}$
(35b)

We readily observe that any three-band model of a phase with nontrivial Euler
topology requires a winding vector $\boldsymbol{h}^{\alpha}(\boldsymbol{k})$,
such that the three components
$\\{h^{\alpha}_{i}(\boldsymbol{k})\\}_{i=1,2,3}$ cannot be identically zero,
which itself implies that all real Gell-Mann matrices must be present in Eq.
(35a).

The analytical eigenvalues are now

$\displaystyle E_{1}(\boldsymbol{k})$ $\displaystyle=E_{2}(\boldsymbol{k})=0,$
(36) $\displaystyle E_{3}(\boldsymbol{k})$
$\displaystyle=\boldsymbol{h}^{\alpha}\cdot\boldsymbol{h}^{\alpha}=a(\boldsymbol{k})^{2}+b(\boldsymbol{k})^{2}+c(\boldsymbol{k})^{2}\,,$

where the two-band subspace is still flat on top of being degenerate. Whenever
the phase is gapped, combining $|\chi_{I}|=|2q|$ [Eq. (13b)] with
$q=c_{1}[\boldsymbol{h}^{\alpha}]$ [Eq. (34)], we find that the topology of
the gapped Euler phase is given by

$[\chi_{I}]=[\,2c_{1}[\boldsymbol{h}^{\alpha}]\,]\,,$ (37)

if $E_{3}(\boldsymbol{k})>0$ for all $\boldsymbol{k}\in\text{BZ}$.

####  III.1.1 $[\chi_{I}]=[2]$

Setting $(t_{2},\lambda)=(0,1)$, we deduce from Eq. (34) and Eq. (37) that the
maximum Euler class is $\chi_{I}=2$. From Eq. (36), we readily find the energy
gap Ezawa (2021)

$\displaystyle\Delta_{(t_{2}=0,\lambda=1)}$
$\displaystyle=\min\limits_{\boldsymbol{k},m,t_{1}}\left\\{E_{3}(\boldsymbol{k})-E_{2}(\boldsymbol{k})\right\\},$
(38) $\displaystyle=\min\left\\{m^{2},(m\pm 2t_{1})^{2}\right\\}.$

We thus conclude, given Eq. (34) and (37), that the gapped phases and their
topologies are given by ($\lambda=1,t_{2}=0$)

$\displaystyle|\chi_{I}|$
$\displaystyle=2~{}\text{for}~{}|t_{1}|>\dfrac{|m|}{2}~{}\text{and}~{}|m|>0,$
(39) $\displaystyle\text{and}~{}|\chi_{I}|$
$\displaystyle=0~{}\text{for}~{}|t_{1}|<\dfrac{|m|}{2}~{}\text{and}~{}|m|>0\,.$

By choosing $\boldsymbol{h}^{A}$ and adding a small constant term that splits
the degeneracy of the lower two-band, we obtain the minimal model

$H^{[2]}_{\text{3B}}(\boldsymbol{k})=H^{A}_{\text{3B,deg}}[m,t_{1},0,1](\boldsymbol{k})+\delta\,\Lambda_{3}\,,$
(40)

where we choose $|\delta|>0$ such that the band gap remains open. For
instance, setting $m=t_{1}=1$, the gap remains open for $|\delta|<1/2$. We
show the band structure of $H^{[2]}_{\text{3B}}(\boldsymbol{k})$ for
$(m,t_{1},\delta)=(1,1,1/4)$ in Figure 2(a), and the winding of Wilson loop of
the lower two-band subspace in Figure 2(c) indicating an Euler class
$|\chi_{I}|=2$.

(a) | (b)
---|---
|
(c) | (d)
|
Figure 2:  (a,b) Band structures and (c,d) Wilson loop of the gapped
(orientable) Euler phases in three-level systems, obtained from the minimal
models (a,c) $H^{[2]}_{\text{3B}}$ for $(m,t_{1},\delta)=(1,1,1/4)$ [Eq. (40)]
for the Euler class $\chi_{I}=2$, and (b,d) $H^{[4]}_{\text{3B}}$ for
$(m,t_{1},t_{2},\delta)=(1/2,0,-3/2,1/4)$ [Eq. (43)] for the Euler class
$\chi_{I}=4$. We have taken $k_{2}$ in units of $\pi$.

####  III.1.2 $[\chi_{I}]=[4]$

Including the term in $t_{2}$ in Eq. (32), we find the maximum Euler class
$\chi_{I}=4$ [from Eq. (34) and Eq. (37)]. The values of the parameters for
which the phase is gapless are readily defined at the jumps of the step
function Eq. (34) (since these correspond to a transition between different
Chern phases, which requires the closing of the gap) Sticlet  _et al._ (2012),
i.e. ($\lambda=1$)

$\Delta=\min\left\\{E_{3}-E_{2}\right\\}=0\\\
\Leftrightarrow(t_{2}=-m)~{}\text{or}~{}(t_{2}=\pm 2t_{1}+m)\,,$ (41)

From the relation $|\chi_{I}|=|2c_{1}[\boldsymbol{h}]|$ and $\lambda=1$, we
find the following conditions for the gapped phase of maximum Euler class,
assuming $m\geq 0$, (see Figure 1)

$|\chi_{I}|=4\Leftrightarrow\\\
\begin{array}[]{cl}&\left(t_{2}>2t_{1}+m~{}\text{and}~{}t_{2}>-2t_{1}+m\right)\\\
\text{or}&\left(t_{2}<-m~{}\text{and}~{}t_{2}<2t_{1}+m~{}\text{and}~{}t_{2}<-2t_{2}+m\right)\,.\end{array}$
(42)

Choosing $\boldsymbol{h}^{A}$, we then define the minimal Bloch Hamiltonian as
($\lambda=1$)

$H^{[4]}_{\text{3B}}(\boldsymbol{k})=H^{A}_{\text{3B,deg}}[m,t_{1},t_{2},1](\boldsymbol{k})+\delta\,\Lambda_{3}\,,$
(43)

where $|\delta|>0$ lifts the degeneracy of the two-band subspace, leaving
eight stable nodal points connecting the bands 1 and 2. The parameter $\delta$
must be chosen as a function of $(m,t_{1},t_{2})$ under the condition that the
energy gap remains open. For instance, setting $(m,t_{1},t_{2})=(1/2,0,-3/2)$,
the gap remains open for $|\delta|<3/7$. We show the band structure of
$H^{[4]}_{\text{3B}}$ for $(m,t_{1},t_{2},\delta)=(1/2,0,-3/2,1/4)$ in Figure
2(a), and the winding of Wilson loop of the lower two-band subspace in Figure
2(c) indicating an Euler class $|\chi_{I}|=4$.

###  III.2 4-band case

We now turn to four-band models with a $2+2$-band splitting. We closely follow
the same strategy, as for the three-band models, of using the minimal
parametrization of a two-band Chern model Eq. (32) and with the Euler topology
inferred from the phase diagram in Figure 1 [Eq. (34) Sticlet  _et al._
(2012)]. Writing the Euler classes of the two subspaces as the vector
$\boldsymbol{\chi}=(\chi_{I},\chi_{II})$ and corresponding Homotopy class
$[\boldsymbol{\chi}]=[\chi_{I},\chi_{II}]$, the generic 4-band real Hermitian
Hamiltonian can be written in terms of nine real gamma matrices as

$\displaystyle H^{[\boldsymbol{\chi}]}_{4\text{B}}(\boldsymbol{k})$
$\displaystyle=\sum\limits_{\begin{subarray}{c}ij\in 01,03,10,30,\\\
11,13,22,31,33\end{subarray}}g^{[\boldsymbol{\chi}]}_{ij}(\boldsymbol{k})\Gamma_{ij}\,,$
(44)
$\displaystyle=\boldsymbol{g}^{[\boldsymbol{\chi}]}(\boldsymbol{k})\cdot\boldsymbol{\Gamma}\,,$

and where we have used the vector notation
$\boldsymbol{\Gamma}=(\Gamma_{01},\Gamma_{03},\Gamma_{10},\Gamma_{30},\Gamma_{11},\Gamma_{13},\Gamma_{22},\Gamma_{31},\Gamma_{33})$
[see below Eq. (21c)] and similarly for
$\boldsymbol{g}^{[\boldsymbol{\chi}]}$.

We make the substitution $\boldsymbol{n}\rightarrow\boldsymbol{h}^{\alpha}$
and $\boldsymbol{n}^{\prime}\rightarrow\boldsymbol{h}^{\prime\beta}$ in Eq.
(21b), defining the vectors $\boldsymbol{h}^{\alpha}$ and
$\boldsymbol{h}^{\prime\beta}$, for $\alpha,\beta=A,B,C$, as one of the cyclic
permutations

$\begin{array}[]{rcl rcl
rcl}\boldsymbol{h}^{A}&=&(a,b,c),&\boldsymbol{h}^{B}&=&(b,c,a),&\boldsymbol{h}^{C}&=&(c,a,b),\\\
\boldsymbol{h}^{\prime
A}&=&(a^{\prime},b^{\prime},c^{\prime}),&\boldsymbol{h}^{\prime
B}&=&(b^{\prime},c^{\prime},a^{\prime}),&\boldsymbol{h}^{\prime
C}&=&(c^{\prime},a^{\prime},b^{\prime}),\end{array}$ (45)

with $a(\boldsymbol{k})$, $b(\boldsymbol{k})$ and $c(\boldsymbol{k})$ defined
in Eq. (32), and similarly for $a^{\prime}(\boldsymbol{k})$,
$b^{\prime}(\boldsymbol{k})$ and $c^{\prime}(\boldsymbol{k})$ with the
substitution
$(m,t_{1},t_{2},\lambda)\rightarrow(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\lambda^{\prime})$.
This gives the degenerate (non-flattened) Bloch Hamiltonian

$\displaystyle
H^{\alpha\beta}_{\text{4B,deg}}\left[\begin{smallmatrix}m&t_{1}&t_{2}&\lambda\\\
m^{\prime}&t_{1}^{\prime}&t_{2}^{\prime}&\lambda^{\prime}\end{smallmatrix}\right](\boldsymbol{k})$
$\displaystyle=\boldsymbol{h}^{\alpha}(\boldsymbol{k})^{T}\cdot\underline{\Gamma}\cdot\boldsymbol{h}^{\prime\beta}(\boldsymbol{k})$
(46a)
$\displaystyle=\boldsymbol{g}^{[\boldsymbol{\chi}]}(\boldsymbol{k})\cdot\boldsymbol{\Gamma},$
with $\begin{array}[]{rcr rcr
rcr}g^{[\boldsymbol{\chi}]}_{01}&=&h_{1}^{\alpha}h_{3}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{03}&=&h_{2}^{\alpha}h_{3}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{10}&=&h_{3}^{\alpha}h_{1}^{\prime\beta}\,,\\\
g^{[\boldsymbol{\chi}]}_{30}&=&-h_{3}^{\alpha}h_{2}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{11}&=&h_{2}^{\alpha}h_{2}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{13}&=&-h_{1}^{\alpha}h_{2}^{\prime\beta}\,,\\\
g^{[\boldsymbol{\chi}]}_{22}&=&-h_{3}^{\alpha}h_{3}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{31}&=&h_{2}^{\alpha}h_{1}^{\prime\beta}\,,&g^{[\boldsymbol{\chi}]}_{33}&=&-h_{1}^{\alpha}h_{1}^{\prime\beta}\,.\end{array}$
(46b)

The eigenvalues are now

$\displaystyle
E_{1}(\boldsymbol{k})=E_{2}(\boldsymbol{k})=-\epsilon(\boldsymbol{k})\,,~{}E_{3}(\boldsymbol{k})=E_{4}(\boldsymbol{k})=\epsilon(\boldsymbol{k})\,,$
(47)
$\displaystyle\epsilon(\boldsymbol{k})=|\boldsymbol{h}(\boldsymbol{k})||\boldsymbol{h}^{\prime}(\boldsymbol{k})|\,,$

such that the spectrum is gapped whenever $\epsilon(\boldsymbol{k})>0$ for all
$\boldsymbol{k}\in\text{BZ}$.

From the identifications

$c_{1}[\boldsymbol{h}^{\alpha}]=q\,,~{}c_{1}[\boldsymbol{h}^{\prime\beta}]=q^{\prime}\,,$
(48)

and the phase diagram of Eq. (34), together with Eq. (25), we remarkably
obtain that the ansatz of the two-band Chern model Eq. (32) is sufficient to
generate all the following four-band ($2+2$)-Euler phases

$\displaystyle\in\left\\{\begin{array}[]{l}(q-q^{\prime},q+q^{\prime})\\\
\sim(-q+q^{\prime},-q-q^{\prime})\end{array}\left|\begin{array}[]{r}-2\leq
q\leq 2,\\\ -2\leq q^{\prime}\leq 2\end{array}\right.\right\\}\,,$ (49)
$\displaystyle\in\left\\{[0,0],[1,1],[1,-1],[2,2],[2,-2],\right.$
$\displaystyle\quad~{}~{}\,[2,0],[4,0],[3,1],[3,-1],$
$\displaystyle\quad~{}\;\left.[0,2],[0,4],[1,3],[1,-3]\right\\}\,.$

In the following, we only consider $\\{[2,0],[4,0],[3,1],[3,-1]\\}$ among the
imbalanced phases, since the imbalanced phases
$\\{[0,2],[0,4],[1,3],[1,-3]\\}$ can readily be obtained from the former
through the transformation $H\rightarrow-H$ of the Hamiltonian.

####  III.2.1 Balanced phase $[\boldsymbol{\chi}]=[1,1]$

We start with the balanced model for the homotopy class
$[\boldsymbol{\chi}]=[\chi_{I},\chi_{II}]=[1,1]=\\{(1,1),(-1,-1)\\}$ obtained
for $q=c_{1}[\boldsymbol{h}]=-1$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=0$ [by Eq. (25) and Eq. (48)].
Choosing $\boldsymbol{h}^{A}$ and $\boldsymbol{h}^{\prime A}$, and setting
$(t_{2},\lambda)=(0,1)$ and
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\lambda^{\prime})=(1,0,0,0)$
(leading to $\boldsymbol{h}^{\prime A}=(0,0,1)$ and
$c_{1}[\boldsymbol{h}^{\prime A}]=0$), we define from Eq. (46a) the minimal
model

$\displaystyle H^{[1,1]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}m&t_{1}&0&1\\\
1&0&0&0\end{smallmatrix}\right](\boldsymbol{k})+\delta\,\Gamma_{13}\,,$ (50)
$\displaystyle=\sin k_{1}\Gamma_{01}+\sin
k_{2}\Gamma_{03}+\delta\,\Gamma_{13}$ $\displaystyle-[m-t_{1}(\cos k_{1}+\cos
k_{2})]\Gamma_{22}\,,$

where the parameter $|\delta|>0$ is taken in order to lift the degeneracy of
the two two-band subspaces, while keeping the band gap open. For instance,
setting $(m,t_{1})=(1,1)$, the gap remains open for $|\delta|<1$. We show in
Figure 3(a) the band structure and (c) the Wilson loop for the model
$H^{[1,1]}_{\text{4B}}(\boldsymbol{k})$ for $(m,t_{1},\delta)=(3/2,1,1/2)$.
(Note that the choice $(m,t_{1})=(3/2,1)$ implies the gap condition
$|\delta|<2/\sqrt{5}$.) The winding of Wilson loop of the two-band subspaces
indicates the Euler classes $[\boldsymbol{\chi}]=[1,1]$ or
$[\boldsymbol{\chi}]=[1,-1]$. Indeed, we cannot read the relative signs of the
Euler classes $(\chi_{I},\chi_{II})$ from the Wilson loop only. We assign the
homotopy indicator $[1,1]$ from the fact that we know the winding numbers $q$
and $q^{\prime}$ by construction, i.e. here $(q,q^{\prime})=(-1,0)$. In the
next Section we present an indicator that differentiates the homotopy classes
$[1,1]$ and $[1,-1]$, in the form of a linked nodal ring with nontrivial
monopole Euler charge (equivalently a non-Abelian frame charge) at the
interface of the two phases.

(a) | (b)
---|---
|
(c) | (d)
|
Figure 3:  (a,b) Band structures and (c,d) Wilson loop of the gapped
(orientable) four-band balanced Euler phases belonging to the homotopy classes
$[\boldsymbol{\chi}]=[1,1]$ and $[2,2]$, obtained for the minimal models (a,c)
$H^{[1,1]}_{\text{4B}}$ with $(m,t_{1},\delta)=(1,3/2,1/2)$ in Eq. (40), and
(b,d) $H^{[2,2]}_{\text{4B}}$ with $(m,t_{1},t_{2},\delta)=(1/2,0,-3/2,1/2)$
in Eq. (43). We have taken $k_{2}$ in units of $\pi$. By the symmetry of Eq.
(47) under the permutations Eq. (45), (a,c) remain the same for
$H^{[1,-1]}_{\text{4B}}$ with
$(m^{\prime},t_{1}^{\prime},\delta)=(1,3/2,1/2)$, and (b,d) for
$H^{[2,-2]}_{\text{4B}}$ with
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\delta)=(1/2,0,-3/2,1/2)$.

####  III.2.2 Balanced phase $[\boldsymbol{\chi}]=[1,-1]$

The phase $[\boldsymbol{\chi}]=[1,-1]$ is obtained for
$q=c_{1}[\boldsymbol{h}]=0$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=-1$. Choosing $\boldsymbol{h}^{B}$
and $\boldsymbol{h}^{\prime A}$, and setting the parameters
$(t_{2}^{\prime},\lambda^{\prime})=(0,1)$ and
$(m,t_{1},t_{2},\lambda)=(1,0,0,0)$ (that gives
$\boldsymbol{h}^{B}=(b,c,a)=(0,0,1)$ and $c_{1}[\boldsymbol{h}^{B}]=0$), we
define the minimal model

$\displaystyle H^{[1,-1]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=H_{\text{4B,deg}}^{BA}\left[\begin{smallmatrix}1&0&0&0\\\
m^{\prime}&t_{1}^{\prime}&1&0\end{smallmatrix}\right](\boldsymbol{k})+\delta\,\Gamma_{13}\,,$
(51) $\displaystyle=\sin k_{1}\Gamma_{31}+\sin
k_{2}\Gamma_{11}+\delta\,\Gamma_{13}$
$\displaystyle+[m^{\prime}-t_{1}^{\prime}(\cos k_{1}+\cos
k_{2})]\Gamma_{03}\,.$

Noting the symmetry of Eq. (47) under the permutations of $\\{a,b,c\\}$ and
$\\{a^{\prime},b^{\prime},c^{\prime}\\}$ Eq. (45), the band structure for
$H^{[1,-1]}_{\text{4B}}$ is identical to the one of $H^{[1,1]}_{\text{4B}}$,
upon exchanging non-primed to primed parameters. The gap condition is thus the
same as in the previous case, e.g. $|\delta|<1$ if we set
$(m^{\prime},t_{1}^{\prime})=(1,1)$, or $|\delta|<2/\sqrt{5}$ if we set
$(m^{\prime},t_{1}^{\prime})=(1,3/2)$. For
$(m^{\prime},t_{1}^{\prime},\delta^{\prime})=(1,3/2,1/2)$, the band structure
is again given by Figure 3(a). Moreover, since the Wilson loop does not
capture the relative sign of the Euler classes it is also the same as in
Figure 3(c).

####  III.2.3 Balanced phase $[\boldsymbol{\chi}]=[2,2]$

For the Euler phase $[\boldsymbol{\chi}]=[2,2]$, we take
$q=c_{1}[\boldsymbol{h}]=2$ and $q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=0$,
and define the minimal model

$\displaystyle H^{[2,2]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}m&t_{1}&t_{2}&1\\\
1&0&0&0\end{smallmatrix}\right](\boldsymbol{k})+\delta\,\Gamma_{13}\,,$ (52)
$\displaystyle=\sin k_{1}\Gamma_{01}+\sin
k_{2}\Gamma_{03}+\delta\,\Gamma_{13}$ $\displaystyle-[m-t_{1}(\cos k_{1}+\cos
k_{2})-t_{2}\cos(k_{1}+k_{2})]\Gamma_{22}\,.$

Setting $(m,t_{1},t_{2},\delta)=(1/2,0,-3/2,1/2)$, we plot the band structure
and the Wilson loop in Figure 3(b,d).

####  III.2.4 Balanced phase $[\boldsymbol{\chi}]=[2,-2]$

For the Euler phase $[2,-2]$ we take $q=c_{1}[\boldsymbol{h}]=0$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=2$, which, similarly to the
previous case, leads to the minimal model

$\displaystyle H^{[2,-2]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=H^{BA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&0&0&0\\\
m^{\prime}&t_{1}^{\prime}&1&t_{2}^{\prime}\end{smallmatrix}\right](\boldsymbol{k})+\delta\,\Gamma_{13}\,,$
(53) $\displaystyle=\sin k_{1}\Gamma_{31}+\sin
k_{2}\Gamma_{11}+\delta\,\Gamma_{13}$
$\displaystyle+[m^{\prime}-t_{1}^{\prime}(\cos k_{1}+\cos
k_{2})-t_{2}^{\prime}\cos(k_{1}+k_{2})]\Gamma_{03}\,.$

Setting $(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\delta)=(1/2,0,-3/2,1/2)$,
again by the symmetry of Eq. (47) we obtain the same band structure obtained
for $H^{[2,-2]}_{\text{4B}}(\boldsymbol{k})$ in 3(b). The Wilson loop is also
the same as for the $[2,2]$-phase shown in Figure 3(d).

####  III.2.5 Imbalanced phase $[\boldsymbol{\chi}]=[2,0]$

For the Euler phase $[2,0]$, we take $q=c_{1}[\boldsymbol{h}]=-1$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=1$, that is compatible with the
minimal model

$\displaystyle H^{[2,0]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=\dfrac{1}{4}H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&1&1&2\\\
-1&-1&-1&2\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{11}\,,$
(54) $\displaystyle=\left(\begin{array}[]{c}\bar{a}\\\ \bar{b}\\\
\bar{c}\end{array}\right)^{T}\cdot\underline{\Gamma}\cdot\left(\begin{array}[]{c}\bar{a}\\\
\bar{b}\\\ -\bar{c}\end{array}\right)\,,$

with

$\displaystyle\bar{a}$ $\displaystyle=\sin k_{1}\,,~{}\bar{b}=\sin k_{2}\,,$
(55) $\displaystyle\bar{c}$ $\displaystyle=\frac{1}{2}(1-\cos k_{1}-\cos
k_{2}-\cos(k_{1}+k_{2}))\,,$

where we have set
$(m,t_{1},t_{2},\lambda)=(-m^{\prime},-t_{1}^{\prime},-t_{2}^{\prime},\lambda^{\prime})=(1/2,1/2,1/2,1)$,
and with $\underline{\Gamma}$ defined in Eq. (21b). We show the band structure
and the Wilson loop (blue for the lower two-band subspace, and dashed red for
the higher two-band subspace) in Figure 4(a,b).

(a) | (b)
---|---
|
(c) | (d)
|
(e) | (f)
|
Figure 4:  Band structure and Wilson loop of the imbalanced gapped
(orientable) Euler phases in four-level systems. The Wilson loop of the lower
(higher) two-band subspace is the blue full line (red dashed line). (a,b)
Imbalanced phase $[\boldsymbol{\chi}]=[2,0]$ defined by the model
$H^{[2,0]}_{\text{4B}}$ Eq. (54). (c,d) Imbalanced phase
$[\boldsymbol{\chi}]=[4,0]$ defined by the model $H^{[4,0]}_{\text{4B}}$ Eq.
(56). (e,f) Imbalanced phase $[\boldsymbol{\chi}]=[3,1]$ defined by the model
$H^{[3,1]}_{\text{4B}}$ Eq. (58).

####  III.2.6 Imbalanced phase $[\boldsymbol{\chi}]=[4,0]$

For the Euler phase $[4,0]$, we take $q=c_{1}[\boldsymbol{h}]=2$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=-2$, that is compatible with the
minimal model

$\displaystyle H^{[4,0]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=\dfrac{1}{4}H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&0&-3&2\\\
1&0&3&2\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,,$
(56) $\displaystyle=\left(\begin{array}[]{c}\bar{a}\\\ \bar{b}\\\
\bar{c}\end{array}\right)^{T}\cdot\underline{\Gamma}\cdot\left(\begin{array}[]{c}-\bar{a}\\\
-\bar{b}\\\ \bar{c}^{\prime}\end{array}\right)\,,$

with

$\displaystyle\bar{a}$ $\displaystyle=\sin
k_{1}\,,~{}\bar{c}=\frac{1}{2}(1+3\cos(k_{1}+k_{2}))\,,$ (57)
$\displaystyle\bar{b}$ $\displaystyle=\sin
k_{2}\,,~{}\bar{c}^{\prime}=\frac{1}{2}(1-3\cos(k_{1}+k_{2}))\,,$

where we have taken
$(m,t_{1},t_{2},\lambda)=(m^{\prime},t_{1}^{\prime},-t_{2}^{\prime},-\lambda^{\prime})=(1/2,0,-3/2,1)$.
We show the band structure and the Wilson loop in Figure 4(c,d).

####  III.2.7 Imbalanced phase $[\boldsymbol{\chi}]=[3,1]$

For the Euler phase $[3,1]$, we take $q=c_{1}[\boldsymbol{h}]=2$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=-1$, that is compatible with the
minimal model

$\displaystyle H^{[3,1]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=\dfrac{1}{4}H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&-1&-3&2\\\
3&2&1&2\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,,$
(58) $\displaystyle=\left(\begin{array}[]{c}\bar{a}\\\ \bar{b}\\\
\bar{c}\end{array}\right)^{T}\cdot\underline{\Gamma}\cdot\left(\begin{array}[]{c}\bar{a}\\\
\bar{b}\\\ \bar{c}^{\prime}\end{array}\right)\,,$

with

$\displaystyle\bar{a}$ $\displaystyle=\sin k_{1}\,,~{}\bar{b}=\sin k_{2}\,,$
(59) $\displaystyle\bar{c}$
$\displaystyle=\frac{1}{2}(1+1\cos(k_{1}+k_{2})+3\cos(k_{1}+k_{2}))\,,$
$\displaystyle\bar{c}^{\prime}$
$\displaystyle=\frac{1}{2}(3-2\cos(k_{1}+k_{2})-\cos(k_{1}+k_{2}))\,,$

where we have taken $(m,t_{1},t_{2},\lambda)=(1/2,-1/2,-3/2,1)$ and
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\lambda^{\prime})=(3/2,1,1/2,1)$.
We show the band structure and the Wilson loop in Figure 4(e,f).

####  III.2.8 Imbalanced phase $[\boldsymbol{\chi}]=[3,-1]$

For the Euler phase $[3,-1]$, we take $q=c_{1}[\boldsymbol{h}]=1$ and
$q^{\prime}=c_{1}[\boldsymbol{h}^{\prime}]=-2$, that is compatible with the
minimal model

$\displaystyle H^{[3,-1]}_{\text{4B}}(\boldsymbol{k})$
$\displaystyle=\dfrac{1}{4}H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}-3&-2&-1&2\\\
-1&1&3&2\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,,$
(60) $\displaystyle=\left(\begin{array}[]{c}\bar{a}\\\ \bar{b}\\\
\bar{c}\end{array}\right)^{T}\cdot\underline{\Gamma}\cdot\left(\begin{array}[]{c}\bar{a}\\\
\bar{b}\\\ \bar{c}^{\prime}\end{array}\right)\,,$

with

$\displaystyle\bar{a}$ $\displaystyle=\sin k_{1}\,,~{}\bar{b}=\sin k_{2}\,,$
(61) $\displaystyle\bar{c}$ $\displaystyle=\frac{1}{2}(-3+2(\cos k_{1}+\cos
k_{2})+\cos(k_{1}+k_{2}))\,,$ $\displaystyle\bar{c}^{\prime}$
$\displaystyle=-\frac{1}{2}(1+\cos k_{1}+\cos k_{2}+3\cos(k_{1}+k_{2}))\,,$

where we have taken $(m,t_{1},t_{2},\lambda)=(-3/2,-1,-1/2,1)$ and
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\lambda^{\prime})=(-1/2,1/2,3/2,1)$.
We obtain the same band structure and the Wilson loop as for the phase
$[3,1]$.

##  IV Symmetries and symmetry-breaking terms of the four-level systems

The purpose of defining minimal models is to simplify their realization in
experiments. While our strategy to systematically generate simple models is
solely conditioned by the targeted Euler topology, requiring the reality
condition (from a $C_{2}T$ or a $PT$ symmetry that squares to the identity),
the simplicity of the four-band models itself make them symmetric under
additional symmetries. While the presence of extra symmetries does not affect
the Euler topology, these bring their own phenomenology which should not be
confused with the manifestations of the Euler topology per se. We therefore
identify all the additional symmetries for each model and define the minimal
terms that break them.

The ansatz of the three-band models Eq. (35a) is general enough, by spanning
all the real Gell-Mann matrices [see the remark below Eq. (35b)], to break all
symmetries at the exception of the reality condition 111One caveat here comes
from the presence of some accidental symmetries in the three-band models and
in the imbalanced four-band models that merely come from the specific ansatz
in Eq. (35a). The effect of these accidental symmetries is further discussed
in Section VII..

$\begin{array}[]{ | l || c | c | c | c | c | c | c | c |}\hline\cr g(T)&[1,1]&[1,-1]&[2,2]&[2,-2]&[2,0]&[4,0]&[3,1]&[3,-1]\\\ \hline\cr C_{2z}T&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}&\mathbb{1}_{4}\mathcal{K}\\\ C_{2z}&\mathrm{i}\,\Gamma_{22}&\mathrm{i}\,\Gamma_{03}&\mathrm{i}\,\Gamma_{22}&\mathrm{i}\,\Gamma_{03}&\mathrm{i}\,\Gamma_{22}&\mathrm{i}\,\Gamma_{22}&\mathrm{i}\,\Gamma_{22}&\mathrm{i}\,\Gamma_{22}\\\ T&-\mathrm{i}\,\Gamma_{22}\mathcal{K}&-\mathrm{i}\,\Gamma_{03}\mathcal{K}&-\mathrm{i}\,\Gamma_{22}\mathcal{K}&-\mathrm{i}\,\Gamma_{03}\mathcal{K}&-\mathrm{i}\,\Gamma_{22}\mathcal{K}&-\mathrm{i}\,\Gamma_{22}\mathcal{K}&-\mathrm{i}\,\Gamma_{22}\mathcal{K}&-\mathrm{i}\,\Gamma_{22}\mathcal{K}\\\ m_{y}&\Gamma_{31}&\Gamma_{31}&-&-&-&-&-&-\\\ m_{x}&\Gamma_{13}&\Gamma_{10}&-&-&-&-&-&-\\\ m_{y}T&-\mathrm{i}\,\Gamma_{13}\mathcal{K}&-\mathrm{i}\,\Gamma_{10}\mathcal{K}&-&-&-&-&-&-\\\ m_{x}T&-\mathrm{i}\,\Gamma_{31}\mathcal{K}&-\mathrm{i}\,\Gamma_{13}\mathcal{K}&-&-&-&-&-&-\\\ \hline\cr Sg(T)&S=\Gamma_{12}&S=\Gamma_{02}&S=\Gamma_{12}&S=\Gamma_{02}&&&&\\\ \hline\cr SC_{2z}T&\Gamma_{12}\mathcal{K}&\Gamma_{02}\mathcal{K}&\Gamma_{12}\mathcal{K}&\Gamma_{02}\mathcal{K}&-&-&-&-\\\ SC_{2z}&-\Gamma_{30}&-\Gamma_{01}&-\Gamma_{30}&-\Gamma_{01}&-&-&-&-\\\ ST&\Gamma_{30}\mathcal{K}&\Gamma_{01}\mathcal{K}&\Gamma_{30}\mathcal{K}&\Gamma_{01}\mathcal{K}&-&-&-&-\\\ Sm_{y}&-\Gamma_{23}&-\mathrm{i}\,\Gamma_{33}&-&-&-&-&-&-\\\ Sm_{x}&\mathrm{i}\,\Gamma_{01}&\Gamma_{12}&-&-&-&-&-&-\\\ Sm_{y}T&\Gamma_{01}\mathcal{K}&-\mathrm{i}\,\Gamma_{12}\mathcal{K}&-&-&-&-&-&-\\\ Sm_{x}T&\mathrm{i}\,\Gamma_{23}\mathcal{K}&\Gamma_{11}\mathcal{K}&-&-&-&-&-&-\\\ \hline\cr\end{array}$ Table 1: Extra symmetries $(S)g(T)$ with their representation in the Bloch orbital basis, $U_{(S)g(T)}(\mathcal{K})$, for the minimal four-band models $H^{[\boldsymbol{\chi}]}_{\text{4B}}$ of Section III, listed for each Euler phase $[\boldsymbol{\chi}]=[\chi_{I},\chi_{II}]$. The symmetries are composed of the crystalline symmetries $g\in\\{E,C_{2z},m_{y},m_{x}\\}$ ($E$ is the identity), the spinless time reversal $T$ ($T^{2}=+1$), with $\mathcal{K}$ the complex conjugation, and the chiral symmetry $S$ ($S^{2}=+1$). We have imposed $U_{Sg(T)}=SU_{g(T)}$. The mirror symmetries hold when we take $(x_{1},x_{2})=(x,y)$ and $(k_{1},k_{2})=(k_{x},k_{y})$.

In our context, the symmetries of the system take the form a constraint to be
satisfied by the Bloch Hamiltonian. Symmetries can be of four types, depending
on whether it is unitary (i.e. no complex conjugation) or anti-unitary (i.e.
with complex conjugation), or if it is a “symmetry” (i.e. commuting with the
Hamiltonian) or an “anti-symmetry” (i.e. anti-commuting with the Hamiltonian)
Shiozaki and Sato (2014), see the examples for each case below. Since there
are only twofold (and symmorphic) crystalline symmetries in our models, all
the topological classes protected by these symmetries, when combined with the
time reversal, chiral, and particle-hole symmetries, have been classified in
Ref. Shiozaki and Sato (2014).

The symmetries of any (hermitian) four-band models, built from the Dirac
matrices $\Gamma_{ij}=\sigma_{i}\otimes\sigma_{j}$ ($i,j=0,1,2,3$), are easily
determined from the anti-commutation of the Pauli matrices. Indeed, we have
that $\Gamma_{ij}$ commutes with

$\Gamma_{ij},\Gamma_{i0},\Gamma_{0j},\text{\;and\;}\left(\Gamma_{kl}\right)_{k\neq
i,l\neq j}\,,$ (62)

and anti-commutes with

$\left(\Gamma_{k0}\right)_{k\neq i},\left(\Gamma_{0l}\right)_{l\neq
j},\left(\Gamma_{kj}\right)_{k\neq i},\left(\Gamma_{il}\right)_{l\neq j}\,,$
(63)

where take $\sigma_{0}=\mathbb{1}_{2}$. Taking the generic four-band model
$H_{\text{4B}}(\boldsymbol{k})=\boldsymbol{g}(\boldsymbol{k})\cdot\boldsymbol{\Gamma}$
in Eq. (44), the symmetries and anti-symmetries are then readily obtained from
whether $g_{ij}(\boldsymbol{k})$ is purely real or imaginary, and whether it
is even or odd under the independent flip of the momentum coordinates, i.e.
$k_{1}\rightarrow-k_{1}$ and $k_{2}\rightarrow-k_{2}$. In the following we
assume that the momentum coordinates match with the coordinates of an
orthorhombic Bravais lattice (i.e. a rectangular lattice) and take
$\boldsymbol{k}=(k_{1},k_{2})=(k_{x},k_{y})$.

We list in Table 1, for each minimal four-band model, all the symmetries and
anti-symmetries with their representation in the orbital basis of the Bloch
Hamiltonian, where $T$ is the spinless time-reversal ($T^{2}=\mathbb{1}_{4}$),
and $S$ is the operator of chiral symmetry ($S^{2}=\mathbb{1}_{4}$). An
example of unitary symmetry is $C_{2z}$ ($\pi$ rotation around $\hat{z}$)
acting on the Bloch orbital basis as

${}^{C_{2z}}|\boldsymbol{\phi},\boldsymbol{k}\rangle=|\boldsymbol{\phi},C_{2z}\boldsymbol{k}\rangle\cdot
U_{C_{2z}}\,,$ (64)

with $C_{2z}(k_{x},k_{y})=(-k_{x},-k_{y})$ and
$\boldsymbol{\phi}=(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$, leading to the
constraint of the Bloch Hamiltonian

$U_{C_{2z}}\cdot H^{[\boldsymbol{\chi}]}_{\text{4B}}(-\boldsymbol{k})\cdot
U^{\dagger}_{C_{2z}}=H^{[\boldsymbol{\chi}]}_{\text{4B}}(\boldsymbol{k})\,.$
(65)

where the unitary representation $U_{C_{2z}}$ is listed in Table 1 for each
Euler phase $[\boldsymbol{\chi}]$. An example of anti-unitary symmetry is time
reversal $T$, acting on the Bloch orbital basis as

${}^{T}|\boldsymbol{\phi},\boldsymbol{k}\rangle=|\boldsymbol{\phi},-\boldsymbol{k}\rangle\cdot
U_{T}\mathcal{K}\,,$ (66)

and leading to the constraint

$U_{T}\cdot H^{*}(-\boldsymbol{k})\cdot U^{\dagger}_{T}=H(\boldsymbol{k})\,,$
(67)

where $\mathcal{K}$ is complex conjugation. An example of unitary anti-
symmetry is $Sm_{y}$, with $S$ the chiral symmetry and the mirror symmetry
$m_{y}(k_{x},k_{y})=(k_{x},-k_{y})$, acting as

${}^{Sm_{y}}|\boldsymbol{\phi},\boldsymbol{k}\rangle=|\boldsymbol{\phi},m_{y}\boldsymbol{k}\rangle\cdot
U_{Sm_{y}}\,,$ (68)

and leading to the constraint

$U_{Sm_{y}}\cdot H(m_{y}\boldsymbol{k})\cdot
U^{\dagger}_{Sm_{y}}=-H(\boldsymbol{k})\,.$ (69)

Note that we have imposed for all unitary anti-symmetries $U_{Sg}=SU_{g}$.
Then, an example of anti-unitary anti-symmetry is $SC_{2z}T$, acting as

${}^{Sm_{y}T}|\boldsymbol{\phi},\boldsymbol{k}\rangle=|\boldsymbol{\phi},-m_{y}\boldsymbol{k}\rangle\cdot
U_{Sm_{y}T}\mathcal{K}\,,$ (70)

and leading to the constraint

$U_{Sm_{y}T}\cdot H^{*}(-m_{y}\boldsymbol{k})\cdot
U^{\dagger}_{Sm_{y}T}=-H(\boldsymbol{k})\,.$ (71)

Note that we have again imposed for all anti-unitary anti-symmetries that
$U_{SgT}=SU_{gT}$.

It is now straightforward to break any symmetry $g(T)$, or anti-symmetry
$Sg(T)$ with $g\in\\{E,C_{2z},m_{y},m_{x}\\}$, by adding a term that does not
commute, or anti-commute, with the representation of $g(T)$, or $Sg(T)$, in
the sense of Eq. (65) and Eq. (67), or of Eq. (69) and Eq. (71), respectively.

For instance, the model $H^{[1,1]}_{\text{4B}}(\boldsymbol{k})$ in Eq. (50)
has the mirror symmetry $m_{y}$ (taking $(k_{1},k_{2})=(k_{x},k_{y})$),
represented by $\Gamma_{31}$. The only Dirac matrix in Eq. (50) that does not
commute with $\Gamma_{31}$ is $\Gamma_{03}$, which comes with the factor $\sin
k_{y}$ that is odd under $m_{y}k_{y}\rightarrow-k_{y}$. The other terms of Eq.
(50) all have a Dirac matrix that commutes with $\Gamma_{31}$ (i.e. these are
$\\{\Gamma_{01},\Gamma_{13},\Gamma_{22}\\}$) and each with a factor that is
even under the reversal of $k_{2}=k_{y}$. Then, the breaking of $m_{y}$ is
achieved by adding any one of the following terms

$\widetilde{g}_{e}(\boldsymbol{k})\left\\{\Gamma_{02},\Gamma_{03},\Gamma_{10},\Gamma_{11},\Gamma_{20},\Gamma_{21},\Gamma_{32},\Gamma_{33}\right\\}\,,$
(72)

where $\widetilde{g}_{e}(\boldsymbol{k})$ is even under $k_{y}\rightarrow-
k_{y}$, or

$\widetilde{g}_{o}(\boldsymbol{k})\left\\{\Gamma_{00},\Gamma_{01},\Gamma_{12},\Gamma_{13},\Gamma_{22},\Gamma_{23},\Gamma_{30},\Gamma_{31}\right\\}\,,$
(73)

where $\widetilde{g}_{o}(\boldsymbol{k})$ is odd under $k_{y}\rightarrow-
k_{y}$. This analysis for the symmetry $m_{y}$ can be straightforwardly
extended to all other symmetries. We note that any term with a complex Dirac
matrix breaks $C_{2z}T$ symmetry, which we discuss in more detail in Section
VII.

While the additional symmetries do not play a special role for the intrinsic
manifestations of the Euler class topology, e.g. see the next Section where we
systematically generate linked nodal rings in 3D from 2D Euler phases, they
must be considered in Section VII where we address the conversion of the Euler
phases into Chern phases (e.g. a remaining mirror symmetry enforces zero Chern
numbers).

We conclude this section by noting that the spinfull basal mirror symmetry and
spinfull $PT$ symmetri ($[PT]^{2}=+1$), discussed in Section II.5 and in
Appendix B, are broken in all the models of Section III since the two-band
subspaces are not degenerate.

##  V From 2D Euler phases to $PT$-protected adjacent linked nodal rings in 3D

In this section we address the intricate interplay of non-Abelian multi-gap
topology and Euler class. In particular, we find that these notions directly
tie to linked nodal structures protected by $PT$-symmetry (i.e. inversion and
time reversal) in three-dimension, providing for a rather rich topological
underpinning. This section focuses on the conceptual aspects of the linked
nodal structures obtained from pairs of inequivalent 2D Euler phases, that is,
the explicit tight-binding models and their linked nodal structures obtained
numerically are discussed in the next section. In particular, we address the
relation between the non-Abelian frame charges Wu  _et al._ (2019); Tiwari and
Bzdušek (2020) and the refined patch Euler class characterization Bouhon  _et
al._ (2020a); Jiang  _et al._ (2021); Peng  _et al._ (2022a). We then
introduce the homotopy invariant of the linked nodal structure that is
invariant under all changes of gauge and all Euler class reversal maps. We
finally introduce linking numbers defined as the Euler class-valued monopole
charges of linked nodal rings, and show their relation with the homotopy
invariant. We note that this section has overlap with the earlier works Ref.
Ahn  _et al._ (2018), which introduced the Euler class for nodal rings while
focusing on its $\mathbb{Z}_{2}$ reduction (to the second Stiefel-Whitney
class) for occupied subspaces with more than two bands, and with Ref. Wu  _et
al._ (2019); Tiwari and Bzdušek (2020), which introduced the non-Abelian frame
charges for nodal rings. This section goes beyond these works by $(i)$ fully
exploiting the $\mathbb{Z}$ Euler class classification of two-band subspaces
separated by two energy gaps (from above and from below), which leads to many
topological configurations that have not been considered before, $(ii)$ by
providing a systematic method for the building of arbitrary linked nodal
structures obtained as a transition between two inequivalent Euler phases, and
$(iii)$ by clarifying the effect of the adiabatic Euler class reversal maps in
terms of the relation between the homotopy invariant of a linked nodal
structure and its linking numbers assuming a fixed gauge choice.

###  V.1 General 3D ansatz

We begin by noting that our construction of minimal two-dimensional Euler
models allows us to then systematically deform one topological Euler phase
($H^{[\chi^{0}]}$) into any other neighboring topological Euler phase
($H^{[\chi^{1}]}$). Assuming that such topological deformations are controlled
by a single parameter, say $\lambda_{PT}\in[0,1]$, that does not break the
reality condition, we can generically model the transition from one phase to
another through a linear combination of Hamiltonians, i.e.

$H^{\Delta[\chi]}(\boldsymbol{k},\lambda_{PT})=\cos\left(\dfrac{\lambda_{PT}\pi}{2}\right)H^{[\chi^{0}]}(\boldsymbol{k})+\\\
\sin\left(\dfrac{\lambda_{PT}\pi}{2}\right)H^{[\chi^{1}]}(\boldsymbol{k})\;.$
(74)

In the above we noted the “difference” of equivalence classes

$\Delta[\chi]=[\chi^{1}]-[\chi^{0}]\,,$ (75)

which we define more precisely and evaluate below, with the Euler classes
$\chi^{0,1}=\chi(\lambda_{PT}=1,0)$. The nodal points of the initial 2D phases
extend to nodal lines, or braids, in the enlarged parameter space
$(\boldsymbol{k},\lambda_{PT})\in\mathbb{T}^{2}\times[0,1]$. As an alternative
approach, which is the one we take below, one may also embed a pair of 2D
Euler Bloch Hamiltonians into one real 3D Bloch Hamiltonian through

$H^{\Delta[\chi]}_{PT}(\boldsymbol{k}_{\parallel},k_{z})=\dfrac{1+\cos
k_{z}}{2}H^{[\chi^{0}]}(\boldsymbol{k}_{\parallel})+\\\ \dfrac{1-\cos
k_{z}}{2}H^{[\chi^{\pi}]}(\boldsymbol{k}_{\parallel}),$ (76)

with $\Delta[\chi]=[\chi^{\pi}]-[\chi^{0}]$ and
$\chi^{0,\pi}=\chi(k_{z}=0,\pi)$, such that the section of the 3D model at
$k_{z}=0$ is given by $H^{[\chi^{0}]}(\boldsymbol{k}_{\parallel})$, and the
section at $k_{z}=\pi$ is $H^{[\chi^{\pi}]}(\boldsymbol{k}_{\parallel})$,
where $k_{z}$ is the momentum perpendicular to the
$\boldsymbol{k}_{\parallel}=(k_{1},k_{2})$-plane. Evidently, $k_{z}$ then acts
as the deformation parameter between the two Euler phases. By retaining the
reality condition, the resulting 3D Hamiltonian preserves an effective
spinless $PT$ symmetry, i.e. with $[PT]^{2}=+1$, which supports stable band
crossings in the form of nodal lines Volovik (2003); Burkov  _et al._ (2011);
Fang  _et al._ (2015); Bouhon and Black-Schaffer (2017). We find that all such
deformations produce nodal braids from adjacent energy gaps to be linked
together. While these nodal braids are characterized by complementary non-
Abelian charges Wu  _et al._ (2019); Tiwari and Bzdušek (2020); Ahn  _et al._
(2019); Bouhon  _et al._ (2020a), we show in the subsequent that the knowledge
of the initial ($k_{z}=0$) and final ($k_{z}=\pi$) Euler classes of the
deformation provides a greatly refined characterization of linked nodal rings.

The rationale for the presence of linked nodal lines directly relates to the
multi-gap nature of the Euler class. Indeed, a change in the Euler class
necessitates the creation or the removal of pairs of stable nodal points of a
two-band subspace, which can only happen via their braiding around nodes
present in one of the two adjacent gaps, i.e. the gaps above and below in
energy, while the reality condition (protected by $C_{2}T$ or $PT$) is
maintained Ahn  _et al._ (2019); Wu  _et al._ (2019); Tiwari and Bzdušek
(2020); Bouhon  _et al._ (2020a). Considering the total trajectory of the
nodes through a braiding, or through the path, we obtain nodal braids that
form linked nodal rings. Although nontrivial linked nodal rings and their non-
Abelian charges have been detailed conceptually Tiwari and Bzdušek (2020), no
explicit models have so far been formulated in the general context of Euler
topology. Furthermore, the present characterization of linked nodal rings in
terms of the $\mathbb{Z}$ Euler class for two-band subspaces Bzdušek and
Sigrist (2017); Bouhon  _et al._ (2019) (see below) constitutes a substantial
refinement compared to the finite non-Abelian group of loop charges of the
frame of eigenvectors Tiwari and Bzdušek (2020), i.e. computed over base loops
encircling the nodal rings (which provides an effective $\mathbb{Z}_{4}$
counting in each gap, see details below), and of the $\mathbb{Z}_{2}$ monopole
charge Fang  _et al._ (2015); Ahn  _et al._ (2018) when more than two bands
must be considered (corresponding to the reduction of the Euler class to the
$\mathbb{Z}_{2}$ second Stiefel-Whitney class). This work thus fills these
gaps by providing concrete minimal tight-binding models that can be readily
used as a guide for the design of acoustic metamaterials Jiang  _et al._
(2021); Guo  _et al._ (2021), photonic crystals Park  _et al._ (2022),
electronic circuits Ezawa (2021), and optical traps for cold atoms Ünal  _et
al._ (2020); Zhao  _et al._ (2022).

###  V.2 Non-Abelian frame charge of nodal braids

We here discuss in more detail the braiding of nodes taking place at the
transition between inequivalent Euler phases. For this we introduce the non-
Abelian frame charge of nodal braids that complements the Euler classes.

To this end, let us first consider the special case of changing Euler class in
the $I$-th two-band subspace from some finite value at $k_{z}=0$ to zero at
$k_{z}=\pi$, while the Euler class of the $II$-th block of bands remains
unchanged at zero, i.e. we have $|\chi^{0}_{I}|\neq|\chi^{\pi}_{I}|=0$ and
$\chi^{\pi}_{II}=\chi^{0}_{II}=0$. That is, a transition from the homotopy
class $[\chi^{0}_{I},0]=[\chi^{0}_{I}]$ to $[0,0]=[0]$. Then, by varying
$k_{z}$ from $0$ to $\pi$, the number of stable nodal points must change from
$2|\chi^{0}_{I}|$ to zero, implying that a number $|\chi^{0}_{I}|$ of nodes of
the $I$-th two-band subspace must be braided with some adjacent nodes located
in the energy gap between the $I$-th and the $II$-th blocks. We explicitly
illustrate this result with the example below.

In the following, we refer to nodes formed by the crossing of the energy
levels $E_{I,1}$ and $E_{I,2}$ as $I$-th nodes, and denote the adjacent nodes
in the gap between the $I$-th and $II$-th blocks (i.e. formed by the crossing
of the energy levels $E_{I,2}$ and $E_{II,1}$) as $(I,II)$-nodes. Since by
construction the $(I,II)$-energy gap is open (i.e. $E_{II,1}-E_{I,1}>0$) at
the initial ($k_{z}=0$) and at the final ($k_{z}=\pi$) phases, the complete
trajectory of the intermediary adjacent nodes must form closed rings, which we
refer to as
$\mathcal{L}_{(I,II),i}=\\{\boldsymbol{k}|E_{I,2}(\boldsymbol{k})=E_{II,1}(\boldsymbol{k})\\}_{i}$
where $i=1,2,\dots$ lists all the connected components (in the example below
we take a single adjacent nodal ring). We then denote the $I$-th braids by
$\mathcal{L}_{I,i}=\\{\boldsymbol{k}|E_{I,1}(\boldsymbol{k})=E_{I,2}(\boldsymbol{k})\\}_{i}$,
where $i=1,2,\dots$ again lists the distinct connected components.

We illustrate the intuitive picture for $[\chi^{0}_{I}]=[2]$ and
$[\chi^{\pi}_{I}]=[0]$ in Figure 5. Taking a plane at a fixed $k_{z}$-value in
the Figure, every nodal braid is crossed at a point, to which we attribute one
(loop) non-Abelian frame charge. We do this for every gap that hosts a nodal
line. For instance, in the three-band case, the non-Abelian frame charges are
given by the elements of the quaternion group, i.e. $\pm i$ for the $I$-th
nodes and $\pm j$ for the adjacent $(I,II)$-nodes Wu  _et al._ (2019); Tiwari
and Bzdušek (2020); Bouhon  _et al._ (2020a). We represent the sign of the
non-Abelian nodal charges with open and full symbols, i.e. open or full
circles for the $I$-nodes [green in Fig. 5], and open and full triangles for
the adjacent $(I,II)$-nodes [red in Fig. 5]. While these signs are gauge
dependent, we assume that a reference point has been chosen with a fixed
choice of gauge phases of the (real) eigenvectors. The presence of nodal
points constitutes an obstruction to assign a smooth choice of gauge over the
whole 2D cut. Indeed, each single node hosts a $\pi$-Berry phase
disinclination of the pair of eigenstates forming the node. This obstruction
to define a globally smooth gauge sign can be conveniently represented through
a Dirac string Ahn  _et al._ (2019) connecting every pair of nodes, i.e. the
two eigenvectors forming the nodes undergo sign-flip across the Dirac string.
See Jiang  _et al._ (2021); Peng  _et al._ (2022a) for a systematic method for
the consistent global attribution of non-Abelian frame charges in 2D systems.

---
Figure 5:  Linking nodal structure protected by $PT$ symmetry for a transition
of Euler phases from $[\chi^{0}_{I},\chi^{0}_{II}]=[2,0]$ at $k_{z}=0$ to
$[\chi^{\pi}_{I},\chi^{\pi}_{II}]=[0,0]$ at $k_{z}=\pi$. The green circles and
braids correspond to the trajectories of the $I$-th nodal points (see text),
with their signed non-Abelian frame charges (see text) represented by the open
and full circles at the $k_{z}=0,\pi$-planes, and by arrows along the braids.
The conversion of Euler class is mediated by the presence of a linked nodal
ring $\mathcal{L}_{(I,II)}$ (red dashed ring) formed by the trajectory of the
adjacent $(I,II)$-nodes, with their signed non-Abelian frame charges
represented by open and full triangles at $k_{z}=0,\pi$, and by arrows along
the braid. The non-Abelian frame charge of a braid is flipped whenever it runs
below one adjacent braid Tiwari and Bzdušek (2020); Jiang  _et al._ (2021);
Peng  _et al._ (2022a), in which case it must cross one adjacent Dirac sheet
(see text), here represented by the greenish, yellowish, and reddish surfaces
between pairs of braids. The linking numbers of the adjacent nodal ring (red
dashed) linked with the $I$-th nodal braids is defined by its $I$-th monopole
Euler class, i.e.
$\mathsf{Ln}_{I}[\mathcal{L}_{(I,II)}]=\chi_{I}[S]=\Delta\chi_{I}=+2$ (see
text), with $S$ the wrapping envelope (gray).

Since we can repeat the above analysis for any fixed $k_{z}$ value, we can
thus attribute a sign-dependent non-Abelian frame charge to each nodal braid.
Following Tiwari and Bzdušek (2020), we represent these charges by an oriented
arrow on each braid, see Figure 5, and use different symbols to represent
nodal braids from different gaps, i.e. green full lines for the $I$-th nodal
braids, $\\{\mathcal{L}_{I,i}\\}_{i=1,\dots,4}$, and red dashed lines for the
adjacent nodal braid, $\mathcal{L}_{(I,II)}$. Finally, we note that each Dirac
sting of a $k_{z}$-plane cut extends for varying $k_{z}$-values into a Dirac
sheet connecting pair of nodal braids, which we have represented in light
colors (greenish, yellowish and reddish) in Figure 5. Very importantly,
whenever one nodal braid, say $\mathcal{L}_{I,3}$, runs below an adjacent
nodal braid ($\mathcal{L}_{(I,II)}$) it must cross the adjacent Dirac sheet
(reddish) and its non-Abelian frame charge (green arrow) must be flipped
Tiwari and Bzdušek (2020); Ahn  _et al._ (2019).

We finish this part with a few comments on the strict relation between the
Euler class and the non-Abelian frame charges. Strictly speaking, the set of
non-Abelian frame charges available to characterize the nodes in each gap is
$\mathbb{Z}_{4}=\\{1,q_{(n,n+1)},-q_{(n,n+1)},-1\\}$, where $1$ is the trivial
frame charge, $\pm q_{(n,n+1)}$ are the charge for an odd number of nodes in
the gap between the bands $n$ and $n+1$, and $q_{(n,n+1)}^{2}=-1$ is the frame
charge for an even number modulo 4 of nodes with the same charge. The Euler
class of two-band subspace (a two-dimensional invariant) thus provides a
generalization of the counting of stable nodes from $\mathbb{Z}_{4}$ to an
arbitrary number in $\mathbb{Z}$. While we have assumed that there is no
adjacent nodes at $k_{z}=0$ and $k_{z}=\pi$ (where we have the gapped Euler
phases), we can readily generalize the $\mathbb{Z}$-counting even in the
presence of adjacent nodes. This is done using the patch Euler class Ahn  _et
al._ (2019); Bouhon  _et al._ (2020a); Jiang  _et al._ (2021); Peng  _et al._
(2022a)

$\chi[D]=\dfrac{1}{2\pi}\left(\displaystyle\int_{D}\mathsf{Eu}-\oint_{\partial
D}\mathsf{a}\right)\in\mathbb{Z}\,,$ (77)

where $D$ is a disk, in the $(k_{1},k_{2})$-plane at a fixed $k_{z}$, covering
one nodal point, and $\partial D$ is the oriented boundary of $D$. Assuming
that each patch Euler class is evaluated with respect to the same global
choice of gauge at a fixed reference base point, the patch Euler classes of
all the nodal points of a two-band subspace can be added together, leading to
the counting of arbitrary many stable nodal braids within each gap. We note
that, similarly, to the non-Abelian charge, the patch Euler class of one nodal
braid changes sign whenever the braid passes below one adjacent nodal braid,
namely when it crosses one adjacent Dirac sheet Ahn  _et al._ (2019); Jiang
_et al._ (2021); Peng  _et al._ (2022a).

###  V.3 Homotopy invariant of the linked nodal structures

The assignment of signed non-Abelian frame charges to all the nodal braids
relies on a choice of gauge at a reference base point. Yet, the Euler class
reversal maps of the Euler phases at $k_{z}=0$ and $k_{z}=\pi$, inducing the
equivalence
$(\chi^{k_{z}}_{I},\chi^{k_{z}}_{II})\sim(-\chi^{k_{z}}_{I},-\chi^{k_{z}}_{II})$,
would flip the signs of the charges at $k_{z}=0$ and $k_{z}=\pi$,
independently. These maps are adiabatic, in the sense that they don’t require
the closing of the $(I,II)$-gaps of the initial and final Euler phases. As a
consequence, the nodal structures must eventually be classified up-to-homotopy
by an equivalence class that does not depend on a specific choice of gauge.
Through the detail discussion of two simple examples in the next section, we
motivate that the homotopy invariant of Euler-generated linked nodal
structures can be defined by

$\displaystyle\Delta[\boldsymbol{\chi}]$
$\displaystyle=[\chi^{\pi}_{I},\chi^{\pi}_{II}]-[\chi^{0}_{I},\chi^{0}_{II}]$
(78)
$\displaystyle=\left\\{\,[\chi^{\pi}_{I}-\chi^{0}_{I},\chi^{\pi}_{II}-\chi^{0}_{II}]\sim[\chi^{\pi}_{I}+\chi^{0}_{I},\chi^{\pi}_{II}+\chi^{0}_{II}]\,\right\\}\,,$

i.e. the difference of equivalence classes is now explicitly valued as an
equivalence class of equivalence classes. While we used the “sign-forgetful”
function by considering the gauge invariant homotopy classes, i.e.

$G^{-1}:(\chi^{k_{z}}_{I},\chi^{k_{z}}_{II})\mapsto[\chi^{k_{z}}_{I},\chi^{k_{z}}_{II}]\,,$
(79)

it will be convenient to also have a “gauge-fixing” map

$G:[\chi^{k_{z}}_{I},\chi^{k_{z}}_{II}]\rightarrow(\chi^{k_{z}}_{I},\chi^{k_{z}}_{II}),$
(80)

that represents the assignment of signed frame charges with respect to a
chosen gauge at a fixed base point, e.g. in the example of Figure 5 we have
taken $G([\chi^{k_{z},1}_{I}])=G([2])=-2$. We remark that contrary to the
function $G^{-1}$ which is surjective, the gauge-fixing map is multi-valued,
corresponding to all the possible choices of global gauges and locations of
the Dirac sheets.

In particular, the gauge fixing of a 3D $PT$-symmetric phase built from the
transition between two Euler phases provides signed differences of Euler
classes, i.e.

$G(\Delta[\boldsymbol{\chi}])=G([\boldsymbol{\chi}^{\pi}])-G([\boldsymbol{\chi}^{0}])=(\Delta\chi_{I},\Delta\chi_{II})\,,$
(81)

which we will use below to introduce the signed linking numbers of the
adjacent nodal rings.

###  V.4 Euler-class valued linking numbers, or the monopole charges of nodal
rings

In this section we motivate that the linked nodal ring mediating a transition
of Euler phases can be generally characterized by two linking numbers which we
obtain by gauge fixing through

$\displaystyle G(\Delta[\boldsymbol{\chi}])$
$\displaystyle=G([\boldsymbol{\chi}^{\pi}])-G([\boldsymbol{\chi}^{0}])\,,$
(82) $\displaystyle=\left(\Delta\chi_{I},\Delta\chi_{II}\right)\,,$
$\displaystyle=\left(\chi_{I}[S],\chi_{II}[S]\right)\,,$
$\displaystyle=\left(\mathsf{Ln}_{I}[\mathcal{L}_{(I,II)}],\mathsf{Ln}_{II}[\mathcal{L}_{(I,II)}]\right)\,,$
$\displaystyle=\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]\,,$

where $\chi_{I,II}[S]$ are the Euler class-valued monopole charges of the
adjacent nodal ring $\mathcal{L}_{(I,II)}$ wrapped by the surface $S$. We
motive this definition with two examples below.

####  V.4.1 Single Euler class transition

Returning to the example of Figure 5, i.e. a single Euler class transition
with $[\chi^{0}_{I}]=[2]$, $[\chi^{\pi}_{I}]=[0]$ and
$[\chi^{0}_{II}]=[\chi^{\pi}_{II}]=[0]$, the signed difference of Euler class
is $G(\Delta[\chi_{I}])=G([0]-[2])=0-(-2)=+2$, given the gauge choice of
Figure 5. We have noted above that the cancellation of Euler class at
$k_{z}=\pi$ requires that a number $|G(\Delta[\chi_{I}])|=2$ of $I$-th nodes
be braided with an adjacent node. As a consequence, the adjacent nodal ring
$\mathcal{L}_{(I,II)}$ mediating the change of Euler class of the $I$-th
subspace must be linked with two $I$-th nodal braids, those are
$\\{\mathcal{L}_{I,i}\\}_{i=3,4}$ in Figure 5.

This motivates the definition of the linking number Ahn  _et al._ (2018) of
the adjacent nodal ring with the $I$-th nodal braids by the $I$-th monopole
Euler classes, i.e.

$\displaystyle\mathsf{Ln}_{I}[\mathcal{L}_{(I,II)}]$
$\displaystyle=\chi_{I}[S]\,$ (83) $\displaystyle=G(\Delta[\chi_{I}])=+2\;,$

where $S$ is an oriented envelope wrapping $\mathcal{L}_{(I,II)}$, see Fig. 5
in gray. We note that the monopole Euler class $\chi_{I}[S]$ is readily given
by half the number of oriented $I$-th braids crossing the surface $S$, for a
fixed choice of gauge. From Fig. 5, we readily get
$\chi_{I}[S]=(1+1+1+1)/2=2$, i.e. we count $+1$ for each $I$-th nodal braid
oriented outwards the surface $S$. The linking number corresponds to the
number of stable linking among the nodal braids, that is, not counting
possible accidental linking that can be removed adiabatically (i.e. while
preserving the $(I,II)$-gap of the initial and final Euler phases).

(a)
---
(b)
Figure 6:  The same linked nodal structure as in Fig. 5 (where it has the linking numbers $\mathsf{Ln}[\mathcal{L}_{(I,II)}]=(+2,0)$) upon the adiabatic reversal of the Euler class at $k_{z}=0$. (a) The Euler class reversal at $k_{z}=0$ has the effect of introducing one extra adjacent nodal ring (dashed line red) encircling all $I$-th nodal braids (full line green). (b) The resulting linked nodal structure after combining the two adjacent nodal rings, giving the linking numbers $\mathsf{Ln}[\mathcal{L}_{(I,II)}]=(-2,0)$.  (a) | (b) | (c)
---|---|---
|  |
Figure 7:  Nodal braids and linked adjacent nodal ring obtained for the
topological transition between the Euler phases
$[\chi^{0}_{I},\chi^{0}_{II}]=[1,1]$ at $k_{z}=0$ and
$[\chi^{\pi}_{I},\chi^{\pi}_{II}]=[1,-1]$ at $k_{z}=\pi$. By fixing the gauge
globally, we allocate consistent non-Abelian frame charges to every nodal
braid, here representing the charges through open/full circles at
$k_{z}=0,\pi$ and through arrows Tiwari and Bzdušek (2020) at intermediary
$k_{z}$ (see text). We write the gauge fixing as
$G([\boldsymbol{\chi}^{0}])=(-1,-1)$ and
$G([\boldsymbol{\chi}^{\pi}])=(+1,-1)$, with $G$ the “gauge-fixing” map. From
(a) to (c), we show the effect of the adiabatic reversal of Euler class at
$k_{z}=0$ on the linking numbers that are computed in terms of Euler monopole
charges. First, in (a) the adjacent nodal ring encircles the $I$-th nodal
braids (green). Then, the Euler class reversal at $k_{z}=0$ induces the
creation of an extra adjacent nodal ring in (b), which, after combining with
the preexistent adjacent nodal ring, gives rise to an adjacent nodal ring that
now encircles the $II$-th nodal braids. We find different linking numbers in
(a),
$\boldsymbol{\mathsf{Ln}}(\mathcal{L}_{(I,II)})=(\chi_{I}[S],\chi_{II}[S])=(+2,0)$,
and in (c),
$\boldsymbol{\mathsf{Ln}}(\mathcal{L}_{(I,II)})=(\chi_{I}[S^{\prime}],\chi_{II}[S^{\prime}])=(0,-2)$,
where $S$, and $S^{\prime}$ are the surfaces wrapping the adjacent nodal ring
in (a), and in (c), respectively. From the homotopy equivalence of (a) and
(c), the linked nodal structure is hence characterized by the unique homotopy
invariant $\Delta[\boldsymbol{\chi}]=[1,1]-[1,-1]=\\{[2,0],[0,2]\\}$ (see
text).

Combining the above result with the fact that there is no linking with the
$II$-th nodal braids, we conclude that the nodal structure is characterized by
the linking numbers

$\displaystyle\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]$
$\displaystyle=\left(\chi_{I}[S],\chi_{II}[S]\right)\,,$ (84)
$\displaystyle=G([\boldsymbol{\chi}^{\pi}])-G([\boldsymbol{\chi}^{0}])\,,$
$\displaystyle=(0,0)-(-2,0)=(+2,0)\,.$

Let us now address the effect of the (adiabatic) Euler class reversal map on
the linking numbers. On one hand, we note that the chosen gauge can be flipped
globally, in which case the linking numbers become $(-2,0)$. On the other
hand, we can act with the Euler class reversal map on the Euler phases at
$k_{z}=0$ and at $k_{z}=\pi$, independently. We show in Figure 6 the effect of
reversing the Euler classes at $k_{z}=0$. First, an extra adjacent nodal ring
(dashed red) is introduced, which encircles all the nodal braids [Fig. 6(a)].
After recombining the two adjacent nodal rings [Fig. 6(b)], we get the linking
numbers $\mathsf{Ln}[\mathcal{L}_{(I,II)}]=\chi_{I}[S^{\prime}]=(-2,0)$.

While the effects of the Euler class reversal and that of a global change of
gauge appear to be the same, with all the homotopy equivalent phases captured
by $[\,\mathsf{Ln}[\mathcal{L}_{(I,II)}]\,]$, we show below with an other
example that the Euler reversals allows more possibilities leading to a larger
homotopy equivalence class given by Eq. (78).

####  V.4.2 Double Euler class transition

We now generalize the above results to the cases when the Euler classes of
both subspaces (the $I$-th and $II$-th) change. For this, we must again
address all the consequences of the homotopy equivalence
$(\chi_{I},\chi_{II})\sim(-\chi_{I},-\chi_{II})$ on the linking numbers (i.e.
the Euler class-valued monopole charges) of the adjacent nodal ring. To this
end, we consider the example of the transition from an Euler phase
$[\chi^{0}_{I},\chi^{0}_{II}]=[1,1]$ to
$[\chi^{\pi}_{I},\chi^{\pi}_{II}]=[1,-1]$, shown schematically in Figure 7(a)
with the choice of gauge in which $G(\boldsymbol{\chi}^{0})=(-1,-1)$ and
$G(\boldsymbol{\chi}^{\pi})=(+1,-1)$.

Taking into account that the initial and final Euler phases are only defined
up-to-homotopy by an equivalence class, we deduce that the linking nodal
structure is in principle characterized by all the combinatorial differences
of Euler classes
$(\chi^{0}_{I},\chi^{0}_{II})-(\chi^{\pi}_{I},\chi^{\pi}_{II})$, i.e.

$\displaystyle(+1,-1)$ $\displaystyle-(-1,-1)$ $\displaystyle=(+2,0)\,,$
$\displaystyle(+1,-1)$ $\displaystyle-(+1,+1)$ $\displaystyle=(0,-2)\,,$
$\displaystyle(-1,+1)$ $\displaystyle-(-1,-1)$ $\displaystyle=(0,+2)\,,$
$\displaystyle(-1,+1)$ $\displaystyle-(+1,+1)$ $\displaystyle=(-2,0)\,.$

Combining the pairs $(\pm 2,0)$ and $(0,\pm 2)$ into equivalence classes, i.e.
$[2,0]$ and $[0,2]$, we are left with showing the homotopy equivalence
$[2,0]\sim[0,2]$, which is the meaning of the definition of the difference of
two equivalence classes $[1,1]-[1,-1]$ as a pair of equivalence classes, i.e.
$\Delta[\boldsymbol{\chi}]=[1,1]-[1,-1]=\left\\{[2,0],[0,2]\right\\}$.

Using Figure 7, we find that the above algebraic expression gives the right
homotopy invariant of the linked nodal structures obtained from the transition
of Euler phases. Starting from panel (a) obtained for a fixed global choice of
gauge, we see that there is a change of (signed) Euler class of the $I$-th
subspace from $\chi^{0}_{I}=1$ to $\chi^{0}_{I}=-1$, which is mediated by the
presence of an adjacent $(I,II)$-nodal ring, $\mathcal{L}_{(I,II)}$ (red),
linked to the $I$-th nodal braids (green). Wrapping $\mathcal{L}_{(I,II)}$
with the grey sphere in Fig. 7(a), we obtain the linking numbers of the
adjacent nodal ring in terms of signed Euler monopole charges

$\displaystyle\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]$
$\displaystyle=(\chi_{I}[S],\chi_{II}[S])=G\left(\Delta[\boldsymbol{\chi}]\right)=(+2,0)\,,$
(85)

where the map $G$ emphasizes that this holds upon the fixing of the gauge with
respect to a unique base point.

We now proceed with the homotopy equivalence between the a priori different
linking nodal structures corresponding to the linking numbers $(\pm 2,0)$ and
$(0,\pm 2)$. We start with Fig. 7(a), where one adjacent nodal ring encircles
the two $I$-th nodal braids (green), which mediates the charge conversion of
the latter from the plane $k_{z}=0$ to $k_{z}=\pi$. Fig. 7(b,c) shows the
effect of the adiabatic reversal of Euler class of the phase on the
$k_{z}=0$-plane (similarly to Fig. 6). The Euler class-reversal induces the
creation of a new adjacent nodal ring that encircles all the nodal braids,
both from the $I$-th and $II$-th gaps, in Fig. 7(b) (note the reversed non-
Abelian frame charges of the nodal braids at $k_{z}=0$). After combining the
two adjacent nodal rings, we get Fig. 7(c), where the resulting adjacent nodal
ring now encircles the two $II$-th nodal braids (blue). We say that the
deformation is adiabatic because we do not need to close the ($I,II$)-band
gaps at the $k_{z}=0,\pi$-planes, which is the rule of the game here. After
the adiabatic deformation, we define the new oriented wrapping surface
$S^{\prime}$ in Fig. 7(c), with respect to which we obtain the linking numbers
in terms of the signed Euler monopole charges, i.e.

$\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]=(\chi_{I}[S^{\prime}],\chi_{II}[S^{\prime}])=(0,-2)\,,$
(86)

where we assumed the same global choice of gauge as Fig. 7(a).

Since the two configurations Fig. 7(a) and Fig. 7(c) are homotopy equivalent,
the nodal structure resulting from the Euler phase transition
$[1,1]\leftrightarrow[1,-1]$ is allowed to realize all the linking numbers
$\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]\in\\{(2,0),(-2,0),(0,2),(0,-2)\\}$,
where the transition from one pair of linking numbers (given a fixed choice of
gauge) to another is obtained through the adiabatic reversal of Euler classes
at $k_{z}=0,\pi$. We thus conclude that the linked nodal structure is
characterized by the single homotopy invariant

$\Delta[\boldsymbol{\chi}]=[1,1]-[1,-1]=\left\\{[2,0],[0,2]\right\\}\,.$ (87)

##  VI Linked nodal structures in 3D

In this section, we specify the 3D models hosting nodal line structures for
every pair of 2D Euler models presented in Section III. We show that the
transition from one Euler phase to an other, which is homotopically
inequivalent, must be mediated by an adjacent nodal ring, that is linked with
the nodal braids of the $I$-th or $II$-th subspaces. We show the nodal
structures obtained numerically and we characterize them with the concepts
introduced in Section V.

Within this section, we use the general ansatz Eq. (76) for
$H_{PT}^{\Delta[\boldsymbol{\chi}]}(\boldsymbol{k}_{\parallel},k_{z})$, and
hence we refer to the two 2D building blocks
$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k}_{\parallel})$ and
$H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k}_{\parallel})$ chosen among the
models of Section III, where we take
$(k_{1},k_{2})=\boldsymbol{k}_{\parallel}$.

(a) $\Delta[\chi_{I}]=[0]-[2]$ | (b) $\Delta[\chi_{I}]=[2]-[4]$ | (c) $\Delta[\chi_{I}]=[0]-[4]$
---|---|---
|  |
Figure 8:  $PT$-symmetry protected linked nodal structures in 3D generated by
a pair of 2D Euler phases, located at $k_{z}=0$ (cyan plane) and $|k_{z}|=\pi$
(yellow plane), in the three-band case. The nodal braids belonging to the
$I$-th two-band subspace are colored in blue, and the adjacent nodal rings,
belonging to the $(I,II)$-gap are in red. The linking of an adjacent nodal
rings $\mathcal{L}_{(I,II)}$ with the $I$-th nodal braids are required to
mediate the transition between any two inequivalent Euler phases, with an
Euler class-valued monopole charge corresponding its linking number with
respect to the $I$-th subspaces $\mathsf{Ln}_{I}\in 2\mathbb{Z}$ (in the
three-band and orientable case, only even Euler class can be formed). Note
that the signed linking number requires the global fixing of the gauge (see
Section V). (a) Linked nodal structure for the Euler phases
$[\chi^{0}_{I}]=[2]$ and $[\chi^{\pi}_{I}]=[0]$. The linking number of the
adjacent nodal ring is $|\mathcal{Ln}|=2$, matching with the monopole Euler
class computed over a wrapping cylinder (since $\mathcal{L}_{(I,II)}$ is
winding through one non-contractible loop of the Brillouin zone). (b) Linked
nodal structure for the Euler phases $[\chi^{0}_{I}]=[4]$ and
$[\chi^{\pi}_{I}]=[2]$, inducing an adjacent nodal ring with the linking
number $|\mathsf{Ln}_{I}|=2$. (c) Linked nodal ring with a linking number
$|\mathsf{Ln}_{I}|=4$, induced by the Euler phases $[\chi^{0}_{I}]=[4]$ and
$[\chi^{\pi}_{I}]=[0]$.

###  VI.1 Three-band $PT$-models and their linked nodal structure

####  VI.1.1 $\Delta[\chi_{I}]=[0]-[2]$

We take

$H^{[\chi^{0}_{I}]}(\boldsymbol{k})=H^{[2]}_{3B}(\boldsymbol{k})\,,~{}H^{[\chi^{\pi}_{I}]}(\boldsymbol{k})=H^{[0]}_{3B}(\boldsymbol{k})\,,$
(88)

with $(m,t_{1},t_{2},\lambda,\delta)=(1,1,0,1,1/4)$ in Eq. (40) for the phase
$\chi_{I}=[2]$, and for the trivial phase, we take

$H^{[0]}_{3B}(\boldsymbol{k})=H^{A}_{3B,\text{deg}}[5,0,0,1](\boldsymbol{k})-\dfrac{1}{4}\text{diag}(0,1,-1)\,,$
(89)

i.e. that is Eq. (35a) with $(m,t_{1},t_{2},\lambda)=(5,0,0,1)$, and an
additional mass term. We show the linked nodal structure generated by the 3D
model in Fig. 8(a). The blue lines correspond to the $I$-th nodal braids, and
the red line is the adjacent $(I,II)$-nodal ring, $\mathcal{L}_{(I,II)}$. The
cyan plane at $k_{z}=0$ locates the Euler phase $[\chi^{0}_{I}]=[2]$ with four
stable nodal points,i.e. at the intersections of the blue nodal braids, and
the yellow plane at $k_{z}=\pi$ locates the trivial phase with no node. The
adjacent nodal ring is characterized by a linking number
$|\mathsf{Ln}_{I}[\mathcal{L}_{(I,II)}]|=2$. We note that
$\mathcal{L}_{(I,II)}$ is also winding through the $k_{2}$-axis of the
Brillouin zone.

####  VI.1.2 $\Delta[\chi_{I}]=[2]-[4]$

We take

$H^{[\chi^{0}_{I}]}(\boldsymbol{k})=H^{[4]}_{3B}(\boldsymbol{k})\,,~{}H^{[\chi^{\pi}_{I}]}(\boldsymbol{k})=H^{[2]}_{3B}(\boldsymbol{k})\,,$
(90)

where $H^{[2]}_{3B}(\boldsymbol{k})$ is the same as above. For the phase
$[4]$, we take Eq. (43) with $\boldsymbol{h}^{B}$ and
$(m,t_{1},t_{2},\lambda)=(1/2,0,5/2,1/4)$, i.e.

$H^{[4]}_{3B}(\boldsymbol{k})=H^{B}_{3B,\text{deg}}[1/2,0,5/2,1](\boldsymbol{k})-\dfrac{1}{4}\text{diag}(1,0,-1)\,.$
(91)

We show the linked nodal structure in Fig. 8(b) [note the shift of the axes,
when comparing the $k_{z}=\pi$ plane in (b) with the $k_{z}=0$ plane in (a)],
which is now characterized by eight stable nodal points (blue) on the
$k_{z}=0$ plane, and an adjacent nodal ring (red) with a linking number
$|\mathsf{Ln}_{I}(\mathcal{L}_{(I,II)})|=2$.

####  VI.1.3 $\Delta[\chi_{I}]=[0]-[4]$

We take the same model $H^{[0]}_{3B}(\boldsymbol{k})$ for the trivial phase,
and for the $[4]$ phase we take,

$H^{[4]}_{3B}(\boldsymbol{k})=H^{B}_{3B,\text{deg}}[1/2,0,-3/2,1](\boldsymbol{k})-\dfrac{1}{4}\Lambda_{3}\,.$
(92)

We show the linked nodal structure in Fig. 8(c), that again exhibits eight
stable nodal points at $k_{z}=0$, but now with two adjacent nodal rings
leading to a doubled linking number
$|\mathsf{Ln}_{I}(\mathcal{L}_{(I,II)})|=4$.

###  VI.2 Four-band $PT$-models and their linked nodal structure

(a) $\Delta[\boldsymbol{\chi}]=[0,0]-[1,1]$ | (b) $\Delta[\boldsymbol{\chi}]=[1,-1]-[1,1]$
---|---
|
(c) $\Delta[\boldsymbol{\chi}]=[0,0]-[2,2]$ | (d) $\Delta[\boldsymbol{\chi}]=[2,-2]-[2,2]$
|
Figure 9:  $PT$-symmetry protected linked nodal structures in 3D generated by a pair of 2D Euler phases, located at $k_{z}=0$ (cyan plane) and $k_{z}=\pi$ (yellow plane), in the balanced four-band case. The $I$-th ($II$-th) nodal braids are drawn in green (blue), and the adjacent nodal rings $\mathcal{L}_{(I,II)}$ (i.e. within the $(I,II)$-gap) are in red. Similarly to Fig. (8), the transition between two inequivalent Euler phases enforces the linking of the adjacent nodal rings with the $I$-th and $II$-th nodal braids, leading to the linking numbers $\boldsymbol{\boldsymbol{Ln}}=(\mathsf{Ln}_{I},\mathsf{Ln}_{II})$ (note that the signed linking numbers require the global fixing of the gauge, see Section V). (a) $\boldsymbol{\boldsymbol{Ln}}[\mathcal{L}_{(I,II)}]=(1,1)$. (b) $\boldsymbol{\boldsymbol{Ln}}[\mathcal{L}_{(I,II)}]=(1,0)$. (c) $\boldsymbol{\boldsymbol{Ln}}[\mathcal{L}_{(I,II)}]=(2,2)$. (d) $\boldsymbol{\boldsymbol{Ln}}[\mathcal{L}_{(I,II)}]=(0,4)$.  (a) $\Delta[\boldsymbol{\chi}]=[0,0]-[2,0]$ | (b) $\Delta[\boldsymbol{\chi}]=[0,0]-[4,0]$
---|---
|
(c) $\Delta[\boldsymbol{\chi}]=[0,0]-[3,1]$ | (d) $\Delta[\boldsymbol{\chi}]=[3,-1]-[3,1]$
|
Figure 10:  Same data as those presented in Fig. 9 for the imbalanced four-
band case. (a) $\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]=(2,0)$. (b)
$\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]=(4,0)$. (c)
$\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]=(3,1)$. (d)
$\boldsymbol{\mathsf{Ln}}[\mathcal{L}_{(I,II)}]=(0,2)$.

###  VI.3 $\Delta[\boldsymbol{\chi}]=[0,0]-[1,1]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[1,1]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[0,0]}_{4B}(\boldsymbol{k})\,,$
(93)

with $H^{[1,1]}_{4B}(\boldsymbol{k})$ given by Eq. (50) with
$(m,t_{1},\delta)=(1,3/2,1/2)$, and for the trivial phase, we take Eq. (46a)
with $(m,t_{1},t_{2})=(1,0,0,1/2)$ and
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime})=(1,0,0,1/2)$, i.e.

$H^{[0,0]}_{4B}(\boldsymbol{k})=H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&0&0&0\\\
1&0&0&0\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,.$
(94)

We show the linked nodal structure in Fig. 9(a), where the $I$-th nodal braid
is down in green, the $II$-th nodal braid is blue (here visible by
transparency below the green line), and the adjacent linked nodal ring in red.
There is a pair of stable nodes at $k_{z}=0$ in both the $I$-th and $II$-th
subspaces. Each pair is connected via a braid that is lined with the red nodal
ring. The linking numbers of the later are $\boldsymbol{\mathsf{Ln}}=(1,1)$.

###  VI.4 $\Delta[\boldsymbol{\chi}]=[1,-1]-[1,1]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[1,1]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[1,-1]}_{4B}(\boldsymbol{k})\,,$
(95)

with $H^{[1,1]}_{4B}(\boldsymbol{k})$ the same as above, and for the $[1,-1]$
phase, we take Eq. (51) with $(m^{\prime},t_{1}^{\prime},\delta)=(1,0,1/2)$.
We show the linked nodal structure in Fig. 9(b). The adjacent nodal ring is
now linked with the $I$-th nodal braids only, given the linking numbers
$\boldsymbol{\mathsf{Ln}}=(2,0)$.

###  VI.5 $\Delta[\boldsymbol{\chi}]=[0,0]-[2,2]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[2,2]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[0,0]}_{4B}(\boldsymbol{k})\,,$
(96)

with $H^{[2,2]}_{4B}(\boldsymbol{k})$ given by Eq. (52) with
$(m,t_{1},t_{2},\delta)=(1/2,0,-3/2,1/2)$, and
$H^{[0,0]}_{4B}(\boldsymbol{k})$ the same as above. We show the linked nodal
structure in Fig. 9(c), where two adjacent nodal rings are each linked with
both one $I$-th and one $II$-th nodal braid, leading to the linking numbers
$\boldsymbol{\mathsf{Ln}}=(2,2)$.

###  VI.6 $\Delta[\boldsymbol{\chi}]=[2,-2]-[2,2]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[2,2]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[2,-2]}_{4B}(\boldsymbol{k})\,,$
(97)

with $H^{[2,2]}_{4B}(\boldsymbol{k})$ the same as above, and with
$H^{[2,-2]}_{4B}(\boldsymbol{k})$ given by Eq. (53) with
$(m^{\prime},t_{1}^{\prime},t_{2}^{\prime},\delta^{\prime})=(1/2,0,-3/2,1/2)$.
We show the linked nodal structure in Fig. 9(d), where one adjacent nodal ring
is linked with four $II$-th nodal braids (blue), leading to the linking
numbers $\boldsymbol{\mathsf{Ln}}=(0,4)$.

###  VI.7 $\Delta[\boldsymbol{\chi}]=[0,0]-[2,0]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[2,0]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[0,0]}_{4B}(\boldsymbol{k})\,,$
(98)

with $H^{[2,0]}_{4B}(\boldsymbol{k})$ given by Eq. (54), and with

$H^{[0,0]}_{4B}(\boldsymbol{k})=H^{CA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&0&0&1\\\
-1&0&0&1\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{11}\,.$
(99)

We show the linked nodal structure in Fig. 10(a), where one adjacent nodal
ring is linked with two $I$-th nodal braids (green), leading to the linking
numbers $\boldsymbol{\mathsf{Ln}}=(2,0)$.

###  VI.8 $\Delta[\boldsymbol{\chi}]=[0,0]-[4,0]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[4,0]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[0,0]}_{4B}(\boldsymbol{k})\,,$
(100)

with $H^{[4,0]}_{4B}(\boldsymbol{k})$ given by Eq. (56), and with

$H^{[0,0]}_{4B}(\boldsymbol{k})=H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}1&0&0&1\\\
1&0&0&-1\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,.$
(101)

We show the linked nodal structure in Fig. 10(b), where four adjacent nodal
rings are each linked one time with both one $I$-th and one $II$-th nodal
braid. Each adjacent nodal ring that is linked one time with a single $I$-th
braid, must also be linked one time with a $II$-th braid, otherwise its non-
Abelian charge would be ill defined Tiwari and Bzdušek (2020). This has the
consequence that, even though the $II$-th braids (blue) are not stable (since
$\chi^{0}_{II}=0$), they must be linked with the adjacent ring. The
instability of the $II$-th braids tells us that Fig. 10(b) can be deformed
adiabatically such that the $II$-th braids annihilate while leaving the linked
adjacent rings paired two-by-two. The linking numbers here are
$\boldsymbol{\mathsf{Ln}}=(4,0)$.

###  VI.9 $\Delta[\boldsymbol{\chi}]=[0,0]-[3,1]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[3,1]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[0,0]}_{4B}(\boldsymbol{k})\,,$
(102)

with $H^{[3,1]}_{4B}(\boldsymbol{k})$ given by Eq. (58), and with

$H^{[0,0]}_{4B}(\boldsymbol{k})=\dfrac{1}{4}H^{AA}_{\text{4B,deg}}\left[\begin{smallmatrix}-3&0&0&0\\\
-1&0&0&0\end{smallmatrix}\right](\boldsymbol{k})+\dfrac{1}{2}\,\Gamma_{13}\,.$
(103)

We show the linked nodal structure in Fig. 10(c), where three adjacent nodal
rings are each linked one time with both one $I$-th and one $II$-th nodal
braid. Here again, the $II$-th braids (blue) tend to follow the the $I$-th
braids (green), in order to satisfy the consistency of their non-Abelian
charges. However, only one $II$-th braid is stable, implying that the two
others can be removed upon the pairing of two adjacent nodal rings. The
linking numbers are $\boldsymbol{\mathsf{Ln}}=(3,1)$.

###  VI.10 $\Delta[\boldsymbol{\chi}]=[3,-1]-[3,1]$

We take

$H^{[\boldsymbol{\chi}^{0}]}(\boldsymbol{k})=H^{[3,1]}_{4B}(\boldsymbol{k})\,,~{}H^{[\boldsymbol{\chi}^{\pi}]}(\boldsymbol{k})=H^{[3,-1]}_{4B}(\boldsymbol{k})\,,$
(104)

with $H^{[3,1]}_{4B}(\boldsymbol{k})$ the same as above, and with
$H^{[3,-1]}_{4B}(\boldsymbol{k})$ given by Eq. (60). We show the linked nodal
structure in Fig. 10(d), where one adjacent nodal ring is lined with two
$II$-th nodal braids. This is compatible with the linking numbers
$\boldsymbol{\mathsf{Ln}}=(0,2)$.

##  VII 2D Chern phases and 3D chiral phases

By breaking $C_{2}T$ (or by breaking $PT$ and taking $0<|k_{z}|<\pi$, in the
3D context), the stable nodes of the Euler phases may be gapped. When starting
from a phase that hosts $C_{2z}T$ symmetry only, the breaking of this symmetry
readily converts the Euler topology to the Chern topology. In this section, we
first want to identify the minimal terms $H_{\text{ch}}(\boldsymbol{k})$ that
bring every nontrivial Euler phase discussed in Section III to a nontrivial
Chern phase upon the breaking of $C_{2z}T$ perturbatively, i.e. taking

$H^{c_{1}}(\boldsymbol{k})=H^{[\chi]}(\boldsymbol{k})+\lambda_{\text{ch}}H_{\text{ch}}(\boldsymbol{k})$
(105)

with $0<|\lambda_{\text{ch}}|$, and such that the Chern number is directly
determined by the Euler class of the symmetric model. More precisely, the
gaping of the $\nu$-th connected two-band subspace with an Euler class
$\chi_{\nu}$ gives rise to two separated bands, each carrying a finite Chern
number $c_{1,(\nu,1)}=-c_{1,(\nu,2)}$ with $|c_{1,\nu}|=|\chi_{\nu}|$, where
the sign of the Chern number of each band is determined by the sign of
$\lambda_{\text{ch}}$. In the three-band case we find $|c_{1}|=|\chi_{I}|$. In
the four-band case, we find $|c_{1,I}|=|c_{1,II}|=|\chi_{I}|=|\chi_{II}|$ for
the balanced Euler phases, while there is more freedom for the imbalanced
phases with $|c_{1,I(II)}|\in\\{|\chi_{I}|,|\chi_{II}|\\}$. We have listed in
Table 2 all the terms that gap the connected two-band subspaces of every Euler
model of Section III, distinguishing those that lead to the nontrivial Chern
phases. Importantly, the results of Table 2 are conditioned by the additional
symmetries carried by the models of Section III. Below, we discuss separately
and in more details each three-band and four-band phase, together with the
effect of their additional symmetries on the Chern phases.

From the identification of the symmetry-breaking terms
$\lambda_{\text{ch}}H_{\text{ch}}(\boldsymbol{k})$ leading to the nontrivial
Chern phases, we can then formulate a systematic route for the building of 3D
chiral models from pairs of 2D Euler phases. Our choice is to take

$H^{\Delta
c_{1}}_{\text{Weyl}}(\boldsymbol{k}_{\parallel},k_{z})=H^{\Delta[\chi]}_{PT}(\boldsymbol{k}_{\parallel},k_{z})+\\\
\lambda_{\text{ch}}\sin k_{z}\left(\dfrac{1+\cos
k_{z}}{2}H^{0}_{\text{ch}}(\boldsymbol{k}_{\parallel})+\dfrac{1-\cos
k_{z}}{2}H^{\pi}_{\text{ch}}(\boldsymbol{k}_{\parallel})\right)\,,$ (106)

with $H^{\Delta[\chi]}_{PT}(\boldsymbol{k}_{\parallel},k_{z})$ defined in
section V and given explicitly for all pairs of the Euler phases in Section
VI, and where
$\\{H^{k_{z}}_{\text{ch}}(\boldsymbol{k}_{\parallel})\\}_{k_{z}=0,\pi}$ are
minimal terms of Table 2. These 3D chiral phases are characterized by two
$C_{2}T$ planes, at $k_{z}=0$ and $k_{z}=\pi$, on which the Euler topology is
preserved and where the associated stable nodal points are pinned. Then, for
$0<|k_{z}|<\pi$, $C_{2}T$ symmetry is broken ($PT$ symmetry is broken all
together) and the nodes of the $C_{2}T$-symmetric planes become gapped. In
other words, the nodes pinned at $k_{z}=0,\pi$ by $C_{2}T$ constitute Weyl
points in the 3D Brillouin zone. We show below, for the three-band and the
four-band systems, that the transition between two inequivalent Euler phases,
from $k_{z}=0$ to $k_{z}=\pi$, is mediated by the presence of simple Weyl
points between the two symmetric planes (i.e. $0<|k_{z}|<\pi$), the number of
which matches the difference in the Chern numbers of the 2D planes at
$|k_{z}|=\varepsilon$ and at $|k_{z}|=\pi-\varepsilon$, for a small deviation
$\varepsilon$.

In the following, first for the three-band case, then for the four-band case,
we start with a discussion of the terms that give 2D Chern phases upon
breaking $C_{2}T$ symmetry, and we show one example of embedding of a pair of
2D Euler phases into one 3D chiral phase.

$\begin{array}[]{ l | c | c }&|c_{1,I(II)}|=0&|c_{1,I(II)}|\in\\{|\chi_{I}|,|\chi_{II}|\\}\\\ \hline\cr\hline\cr{[}\chi_{I}{]}={[}2{]}&\Lambda_{7}&\sin k_{2}\Lambda_{5}\\\ {[}\chi_{I}{]}={[}4{]}&\Lambda_{2}&\sin k_{2}\Lambda_{7}\\\ \hline\cr[\boldsymbol{\chi}]=[1,1]&\Gamma_{21},\Gamma_{32}&\Gamma_{20}\\\ \hline\cr[\boldsymbol{\chi}]=[1,-1]&\Gamma_{21},\Gamma_{32}&\Gamma_{23}\\\ \hline\cr[\boldsymbol{\chi}]=[2,2]&\Gamma_{21},\Gamma_{32}&\Gamma_{20}\\\ \hline\cr[\boldsymbol{\chi}]=[2,-2]&\Gamma_{21},\Gamma_{32}&\Gamma_{23}\\\ \hline\cr[\boldsymbol{\chi}]=[2,0]&\Gamma_{23},\Gamma_{32}&\sin k_{2}\\{\Gamma_{23},\Gamma_{32}\\}\\\ \hline\cr[\boldsymbol{\chi}]=[4,0]&\Gamma_{21},\Gamma_{32}&\sin k_{1}\Gamma_{21}+\sin k_{2}\Gamma_{32}\\\ \hline\cr[\boldsymbol{\chi}]=[3,1]&\Gamma_{21},\Gamma_{32}&\sin k_{1}\Gamma_{21}+\sin k_{2}\Gamma_{32}\\\ \hline\cr[\boldsymbol{\chi}]=[3,-1]&\Gamma_{21},\Gamma_{32}&\sin k_{1}\Gamma_{21}+\sin k_{2}\Gamma_{32}\end{array}$ Table 2: List of the $C_{2}T$-breaking terms $H_{\text{ch}}(\boldsymbol{k})$ that gap perturbatively (i.e. for $0<|\lambda_{\text{ch}}|$ in Eq. (105)) the connected two-band subspaces of the Euler phases of Section III. The second column lists the terms that gap the two-band subspaces and convert them into nontrivial Chern bands.

###  VII.1 Three-band Chern and chiral phases

Since the three-band models span all the Gell-Mann matrices, see Section
III.1, there is no extra symmetry beyond $C_{2z}T$ (with the caveat of an
accidental symmetry discussed below). Given the representation of $C_{2z}T$ in
the Bloch orbital basis, i.e.
$U_{C_{2z}T}\mathcal{K}=\mathbb{1}_{3}\mathcal{K}$, $C_{2z}T$ is broken
whenever we add a term with a complex Gell-Mann matrix, i.e.

$\begin{array}[]{l}\begin{array}[]{lll}\Lambda_{2}\,=\,\begin{pmatrix}[r]0&-\mathrm{i}&0\\\
\mathrm{i}&0&0\\\
0&0&0\end{pmatrix},&\Lambda_{5}\,=\,\begin{pmatrix}[r]0&0&-\mathrm{i}\\\
0&0&0\\\
\mathrm{i}&0&0\end{pmatrix},&\Lambda_{7}\,=\,\begin{pmatrix}[r]0&0&0\\\
0&0&-\mathrm{i}\\\ 0&\mathrm{i}&0\end{pmatrix}.\end{array}\end{array}$ (107)

We list in Table 2 the minimal terms that open a gap perturbatively (i.e. with
a small prefactor $|\lambda_{\text{ch}}|=\varepsilon>0$), distinguishing the
terms that lead to nontrivial Chern phase (third column). We note that the
models $H^{[2]}_{\text{3B}}(\boldsymbol{k})$ and