\makeatletter
\@ifundefined{HCode}
{\documentclass[CRPHYS,Unicode,screen,biblatex,published]{cedram}
\addbibresource{crphys20260109.bib}
\newenvironment{Table}{\begin{table}}{\end{table}}
\newenvironment{noXML}{}{}
\def\tsup#1{\textsuperscript{#1}}
\def\tsub#1{\textsubscript{#1}}
\def\tminus{$-$}
\def\ndash{\text{--}}
\usepackage[T1]{fontenc}
\def\thead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tabnote#1{\vskip4pt\parbox{.96\linewidth}{#1}}
\def\hyphen{\text{-}}
\def\mathbi#1{\text{\textbf{\textit{#1}}}}
\usepackage[figuresright]{rotating}
\RequirePackage{etoolbox}
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1}/{#2})}
\def\sttfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\def\bsigma{\boldsymbol{\sigma}}
\newenvironment{fxequation}{\begin{equation}}{\end{equation}}
\def\inlinefig#1{\includegraphics{#1}}
\def\jobid{crphys20260109}
%\graphicspath{{/tmp/\jobid_figs/web/}}
\graphicspath{{./figures/}}
\newcounter{runlevel}
\let\MakeYrStrItalic\relax
\csdef{Seqnsplit}{\\}
\def\refinput#1{}
\def\back#1{}
\def\botline{\\\hline}
\def\Lbreak{\newline}
\def\dollar{\$}
\def\ubreak{\break}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\def\xmorerows#1#2{#2}
\def\mn{\phantom{$-$}}
\def\0{\phantom{0}}
\def\figType#1{}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\usepackage{hyperref}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\DOI{10.5802/crphys.287}
\datereceived{2026-02-16}
\daterevised{2026-07-29}
\dateaccepted{2026-07-30}
\ItHasTeXPublished
}
{\documentclass[crphys]{article}
\usepackage[T1]{fontenc}
\def\CDRdoi{10.5802/crphys.287}
\newenvironment{sidewaystable}{\begin{table}}{\end{table}}
\let\refinput\input
\let\ubreak\relax
\makeatletter
\def\href#1#2{\url[#1]{#2}}
\def\tminus{\unient{2212}}
\def\CDRsupplementaryTwotypes#1#2{}
\let\splittabular\relax
\def\sfrac#1#2{{#1}/{#2}}
\def\stfrac#1#2{({#1}/{#2})}
\def\sttfrac#1#2{({#1})/{#2}}
\def\stofrac#1#2{(({#1})/{#2})}
\newcommand\@coi{}
\newcommand\COI[1]{\gdef\@coi{#1}}
\newcommand\printCOI{\ifx\@coi\@empty\else%
\section*{Declaration of interests}
\@coi\fi
}
}
\makeatother

\usepackage{upgreek}

\COI{The authors do not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
have declared no affiliation other than their research organizations.}

%\dateposted{2026-04-23}

\dateposted{2026-08-25}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Spin response of a magnetic monopole and quantum Hall response
in topological lattice models through local invariants and light}

\alttitle{Spin d'un monopole magn\'etique et r\'eponse de Hall 
quantique dans des mod\`eles sur r\'eseau topologiques via des
invariants locaux et la  lumi\`ere}

\author{\firstname{Karyn} \lastname{Le Hur}\CDRorcid{0000-0002-3990-4782}\IsCorresp}
\address{CPHT, CNRS, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau, France}
\email[K. Le Hur]{karyn.le-hur@polytechnique.edu}

\author{\firstname{Andrea} \lastname{Baldanza}\CDRorcid{0009-0005-3546-5558}}
\address{Dipartimento di Fisica, Universit\` a di Roma La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy}

\keywords{\kwd{Topological system}
\kwd{Geometry}
\kwd{Light and Mathematics}
\kwd{Quantized responses}
\kwd{Topological quantum physics}
\kwd{Berry phase}
\kwd{Anomalous Hall effect and Quantum spin Hall effect}}

\altkeywords{\kwd{Syst\`eme topologique}
\kwd{G\'eom\'etrie}
\kwd{Lumi\`ere et Mathematique}
\kwd{R\'eponses quantifi\'ees}
\kwd{Physique quantique topologique}
\kwd{Phase de Berry}
\kwd{Effet Hall quantique anormal et Effet Hall de spin}}

\begin{abstract}
Here, we elaborate on and develop the geometrical approach introduced
in Le Hur, \textit{Phys. Rep.} 1104 (2025) between the magnetic
monopole created from a radial field, quantum physics and topological
lattice models through quantum phase transitions. We introduce an
effective magnetic moment for a monopole when applying an additional
source field along $z$-direction which also mediates the quantum phase
transition. We present its relation with the transverse pumped quantum
Hall current. The magnetic susceptibility can be introduced as a
measure of the topological invariant i.e. it remains quantized within
the topological phase until the transition. We show the relation with
two-dimensional topological lattice models such as a honeycomb Haldane
model in real space. We develop the theory and present a numerical
analysis between local invariants in momentum space introduced from
Dirac points, a real space correspondence and the responses to
circularly polarized light. We develop the formalism for coupled-planes
materials including the possibility of quantum spin Hall effect and
address a relation between the Ramanujan infinite alternating series
and an interface in real space with a topological number one-half. 
\end{abstract}

\begin{altabstract}
Nous \'elaborons et d\'eveloppons une approche  g\'eom\'etrique
introduite dans Le Hur, \textit{Phys. Rep.} 1104 (2025) unifiant  le
monopole magn\'etique d\^u \`a l'application d'un champ (magn\'etique)
radial, la physique  quantique et les mod\`eles sur r\'eseau
topologiques en pr\'esence de transitions de phase  quantiques. Nous
introduisons le moment magn\'etique pour le monopole en r\'eponse \`a 
l'application d'un champ magn\'etique additionnel le long de la
direction $z$, qui induit  aussi la transition de phase quantique
topologique. Nous pr\'esentons une correspondance  avec le courant de
Hall quantique. La susceptibilit\'e magn\'etique associ\'ee peut ainsi 
\^etre introduite comme une mesure de l'invariant topologique qui reste
en effet quantifi\'e  dans la phase topologique jusqu'\`a la
transition. Nous montrons la relation avec le  mod\`ele de Haldane sur
le r\'eseau en nid d'abeille hexagonal. Nous d\'eveloppons la th\'eorie
et pr\'esentons une analyse num\'erique des invariants topologiques
locaux introduits  aux points de Dirac dans l'espace r\'eciproque, une
correspondence dans l'espace r\'eel,  et \'etudions la r\'eponse \`a la
lumi\`ere polaris\'ee circulaire. Nous d\'eveloppons le formalisme 
pour des mat\'eriaux en forme de planches incluant la possibilit\'e
d'un effet Hall quantique  de spin et adressons une relation entre la
s\'erie infinie altern\'ee de Ramanujan et  une interface topologique
dans l'espace r\'eel avec un nombre topologique un demi.
\end{altabstract}

%\input{CR-pagedemetas}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

The theory of magnetic monopoles attracts attention since the works of
Curie~\cite{Curie} and Dirac~\cite{Dirac,Polchinski}.
Detecting such monopoles associated to the planetary system, the
universe, is yet of great interest.  Recently, one of us has introduced
a geometrical approach showing a correspondence between classical
magnetic monopoles and quantum magnetic monopoles from a formulation at
the poles on the sphere~\cite{KLHReview}. The magnetic monopole is
produced from a radial magnetic field. Topological properties or
equivalently the magnetic topological charge are generally measured
through the integral of the magnetic field or through the integral of
the Berry curvature in quantum physics on the surface
area~\cite{Berry}. This approach~\cite{KLHReview} can be precisely
rephrased into an effective cartesian metric in the vicinity of the
poles in terms of a classical vector potential or in terms of the Berry
gauge potential~\cite{Berry} in quantum physics associated to a
coherent gauge (phase) representation for the eigenstates of the
spin-$1/2$ particle on the Bloch sphere.  Equivalently, the information
on the magnetic charge is transported on each side of the equator to
the poles in a thin handle. {Before emphasizing on the goals in this
article,  we remind several practical applications~\cite{KLHReview}.}
This approach allows for a description of the quantum Hall response
when building an analogy to the Newtonian approach. When applying an
electric field along the polar angle an electron goes down from north
to south pole with the induction of a transverse pumped current or a
topological transverse pumped charge which is related to the quantum
Hall conductivity and to the physics at the edges on a cylinder. It
also allows us to show how photo-electric effects can become
topological~\cite{KarynLight}. The responses to circularly polarized
light can measure the topological charge or the quantum Hall
conductivity from the poles which can be directly applied in
two-dimensional (2D) topological lattice models through the Haldane
model~\cite{Haldane} and the quantum anomalous Hall effect
(QAH)~\cite{QiZhang,BernevigHughes} associated to the $\mathbb{Z}$
quantized invariant, where the poles correspond to the two Dirac points
of the honeycomb lattice. From the same light signals, the quantum Hall
conductivity~\cite{TKNN} was first introduced through a summation on
the wavevectors of the whole Brillouin
zone~\cite{NathanPeter,DFG,Japan}.  Our geometrical formulation can
also account for interaction effects in the detection of light in a
quantitative manner through a variational stochastic
approach~\cite{PhilippAdolfoKaryn,KaneMelevariational}. When adding
interaction effects on the lattice in the Haldane model, our
variational stochastic approach is very efficient to reveal the
first-order nature of the quantum phase transition towards the Mott
phase~\cite{PhilippAdolfoKaryn}. This approach is also applicable for
two-dimensional topological insulators, i.e.\ the quantum spin Hall
(QSH) effect and the Kane--Mele model~\cite{KaneMele1,KaneMele2,FuKane},
where circularly polarized light can also detect the $\mathbb{Z}_2$
invariant locally from the Dirac points~\cite{KarynLight}. It is
important to emphasize that through the effect of a radial magnetic
field magnetic monopoles are realized in quantum circuits with
applications to the Haldane model~\cite{Google,Boulder,Tran}. The
quantum metric tensor is also measured associated to phase
transitions~\cite{Tran}. Within our approach, we have shown that
circularly polarized light produces the same effect as a boost of the
azimuthal angle building a correspondence between quantum metric and
responses to circularly polarized light through the introduction of a
geometrical function~\cite{KLHReview}. It is also possible to measure
the geometrical information through the optical conductivity in quantum
materials~\cite{BansilFu}.

{This is the beginning of this work today. We will then develop the
formalism and geometrical responses from the Bloch sphere and the poles
down to  the honeycomb Haldane model in the presence of a topological
quantum phase transition. This will be produced from a longitudinal
fixed magnetic field $M$ on the sphere corresponding to an alternating
potential on the lattice. We will present an algorithm of the method
applicable for topological lattice models mapping the surface of the
sphere onto the rhomboid Brillouin zone. We present a unification
between QAH and QSH effects in materials from coupled planes, leading
to a relation with the Ramanujan alternating infinite series and to a
$\sfrac{1}{2}$ topological proximity effect in the thermodynamic limit,
related to transport and to the response to circularly polarized light
through the Riemann zeta function. 

What are then the precise goals in this article? In recent works, we
have shown how the topological invariant can also be re-written in
terms of the pseudo-spin responses at the poles related to a protected
quantum dynamo effect~\cite{dynamo,KLHReview,FractionalArticle} and
with applications on topological quantum
wires~\cite{FrederickLoicKaryn,FrederickLoicOlesiaKaryn,KarynFanMagali}. In this article, in Equation~(\ref{kappaspin}), we introduce the effective magnetic moment of the monopole as a function of $M$ associated to the longitudinal pseudo-spin response. We show how this is related to transport properties. We present the susceptibility for the monopole
in Equation~(\ref{chitopo}) which remains invariant within the
topological phase until the phase transition and therefore can be
introduced as a topological marker. To the best of our knowledge, this
magnetic susceptibility of the monopole was not mentioned before in the
literature; see e.g.\ Ref.~\cite{Alba} where some pseudo-spin responses
are introduced related to time of flight in cold atoms.} This
topological susceptibility shows then an analogy with orbital magnetism
where the quantum Hall response is also thought of as a topological
orbital susceptibility~\cite{SariahKarynFrederic} corresponding to the
derivative of the orbital magnetic moment with respect to the applied
perpendicular magnetic field.  {The local Berry curvature on the
surface of the sphere is also related to the in-plane pseudo-spin
response.} We will then develop the corresponding theory for the
Haldane model in two dimensions~\cite{Haldane} with the introduction of
a Semenoff mass~\cite{Semenoff} which corresponds e.g.\ to the effect of
a charge density wave substrate~\cite{Eva}. We develop the
correspondence between magnetic moment of the monopole and real-space
geometrical marker for the Haldane model. The local definition of the
topological invariant in terms of Berry gauge potentials and
pseudo-spin responses at Dirac points, within our approach, also
reveals standard definition in terms of sign of mass at each Dirac
point~\cite{Haldane,Nathan,Orsay}. Topological properties describing
the occurrence of an edge mode are generally robust until the phase
transition.  { Finding observables to describe such a transition
towards this trivial phase is yet of great interest related to the
relative populations on the two sublattices~\cite{Alba}. Indeed, above
the transition, particles progressively reside on one site preferably
as a result of the proximity effect referring to a trivial insulator
which does not present an edge mode and is characterized through a zero
index (invariant).} It is then relevant to mention here recent progress
on measuring locally topological or geometrical properties in momentum
space from photo-luminescence related to the Bloch sphere
approach~\cite{C2N}.

{ The theory and the developed algorithm will show how the response to
circularly polarized light at the Dirac points is efficient to reveal
the topological nature of the quantum phase transition i.e.\ only one
Dirac point will resonate with the circularly polarized waves leading
to a $\sfrac{1}{2}$ response compared to the topological and trivial
phases emphasizing on the $\sfrac{1}{2}$ topological nature similar to a
half Skyrmion at the transition. Our geometrical approach then will
have implications on geometrical properties of coupled-planes
materials~\cite{DFG2}, alternating the possibility of QSH and QAH
effects through the Ramanujan alternating infinite series  as a
physical $\sfrac{1}{2}$ topological invariant in real
space~\cite{Review2022}. This will give rise to a topological interface
in real space linked to the $\sfrac{1}{2}$ number that will then relate
to our recent works on fractional topological numbers that we take time
to also remind in the next sentences. Through a model of coupled
spins-$\sfrac{1}{2}$ on the Bloch
sphere~\cite{FractionalArticle,KLHReview} and two-dimensional materials
such as graphene and bilayer
systems~\cite{FractionalArticle,KarynSariah1,KarynSariah2,KLHQSHSemimetal}, 
interacting quantum wires~\cite{FrederickLoicKaryn,FrederickLoicOlesiaKaryn,KarynFanMagali}, 
the $\sfrac{1}{2}$ topological
number gives rise to interesting topological phases such as topological
semimetals in two dimensions which can also be revealed through
circularly polarized light~\cite{FractionalArticle,KarynSariah1,KarynSariah2,KLHQSHSemimetal},
and yet characterizes interacting topological phases in one
dimension~\cite{FrederickLoicKaryn,FrederickLoicOlesiaKaryn,KarynFanMagali}. 
We underline here recent interesting works in the community related to
these efforts~\cite{BoFu1,BoFu2,IkedaRayan,Nascimbene}. This model of
two spheres can be realized in quantum circuits~\cite{Google} and can
be applied as a platform for quantum information through protected
Majorana fermions~\cite{MajoranaHalf}. The half topological invariant
then means that the surface radiating the Berry curvature is halved
producing a half Skyrmion and that the geometrical properties are
locally resolved at one pole as a $\uppi$ Berry phase or $\uppi$ winding
number~\cite{OneHalf}. For the situation of entangled Bloch spheres,
this also builds a parallel with the half flux quantum in the sense of
superconductivity compared to a metallic phase~\cite{OneHalf}.  }

{ The organization of the manuscript is as follows.} In
Section~\ref{Monopole}, we develop the geometrical aspects of the
monopole in quantum physics associated to the quantum phase transition
and we present the evaluation of the spin response and of the effective
magnetic moment as a function of the magnetic field $M$. We also
present the relation with the (mean) quantized transverse Hall current
from a Newtonian approach.  In Section~\ref{maplattice}, we develop the
theory for the honeycomb Haldane model in the presence of a Semenoff
mass producing the quantum phase transition. We present a precise map
from the rhomboid Brillouin zone onto the sphere and derive
analytically and numerically the local responses associated to local
invariants in momentum (reciprocal) space. The spin-$1/2$ on the Bloch
sphere will correspond to the pseudo spin-$1/2$ measuring the population
imbalance between the two sublattices of the honeycomb lattice. We
introduce a real space analysis related to the effective magnetic
moment of the monopole. We elaborate on the local responses from
circularly polarized light associated to the quantum phase transition.
In Section~\ref{coupledplanes}, we develop the theory of coupled-planes
related to the $\sfrac{1}{2}$ topological invariant building an analogy
between a topological interface and the Ramanujan infinite alternating
series. We analyse transport properties and the responses to circularly
light within this correspondence. The coupled-planes model includes the
alternating presence of QAH and QSH effects. In Section~\ref{summary},
we summarize our findings. Appendices are devoted to additional
mathematical proofs. In Appendix~\ref{AppendixA}, we elaborate on the
relation between the geometrical approach and transport, quantum Hall
response with a map onto a cylinder. In Appendix~\ref{AppendixB}, we
present a geometrical justification of the $\sfrac{1}{2}$ topological
invariant for the coupled planes when reaching the infinity limit.

\section{Bloch sphere as a magnetic monopole, quantum Hall response and
topological magnetism} \label{Monopole}

The magnetic monopole in quantum physics is then formed through the
application of a radial magnetic field associated to the Bloch sphere
of a spin-$1/2$, resulting in a hedgehog topological
sphere~\cite{KLHReview}.  We remind here that such an Hamiltonian is
realized in quantum circuits~\cite{Google,Boulder,Tran}. 

In this Section, we study the physical responses as a function of the
parameter $M$ corresponding to the fixed additional magnetic field
along $z$ direction which will drive the quantum phase transition
associated to the Poincar\'e--Hopf theorem. We evaluate the local spin
magnetization responses at and in the vicinity of the poles on the
Bloch sphere as a function of $M$.  We present a simple proof in a
dressed polar angle representation showing how the global topological
number is generally measurable locally through the Berry gauge
potential~\cite{KLHReview,FractionalArticle}  which can then be
re-written in terms of the spin magnetization along $z$ direction
through Eherenfest theorem. We will then introduce the mean (medium) of
the spin magnetization on the surface area and address the relation
with the transverse quantum Hall current or transverse pumped charge in
a Newtonian (gravitational) correspondence when driving a charge from
north to south pole through a longitudinal electric field. This will
give rise to the effective topological magnetic moment $\kappa(M)$ in
Equation~(\ref{kappaspin}), associated to the loops of transverse
currents formed along the equatorial plane on the whole surface. When
$M=0$, as a result of the radial magnetic field, magnetic moments of
the two hemispheres compensate each other and $\kappa(M=0)=0$. This
analysis will also be related to the cylinder
geometry~\cite{KLHReview}.

It should be emphasized that for classical magnetic monopoles produced
e.g.\ by the same radial magnetic field, the magnetic field $M$
corresponds to an additional source to the Gauss theorem that produces
an additional effective zero flux on a spherical surface i.e.\ the
in-coming magnetic flux at south pole is compensated by the out-going
magnetic flux at north pole. In that case, the $M$ term will not
mediate a classical phase transition.

\subsection{Topological Bloch sphere and geometrical approach of the
quantum phase transition through a local spin marker}

We begin with the quantum Hamiltonian for the spin-$1/2$ 
{\begin{equation}
H = - (B\mathbf{e}_r +M\mathbf{e}_z)\cdot {\bsigma}.
\end{equation}}\unskip
Here, the vector ${\bsigma}=(\sigma_x,\sigma_y,\sigma_z)$ is
associated to Pauli matrices. The total magnetic field corresponds to
the addition of a radial magnetic field $\mathbf{B}$ and of a magnetic
field $M$ polarized along $z$ direction. Then, $\mathbf{e}_r$ and
$\mathbf{e}_z$ are two unit vectors associated to these directions. The
Hamiltonian can be equivalently written in spherical coordinates as
{\begin{equation}
H = - \mathbf{B}^*\cdot {\bsigma}
\end{equation}}\unskip
with the magnetic field 
{\begin{equation}
\mathbf{B}^* = B\left(\sin\theta \cos \varphi, 
\sin\theta \sin \varphi, \cos \theta+\frac{M}{B}\right).
\end{equation}}\unskip

Here, $\theta\in [0;\uppi]$ and $\varphi\in [0;2\uppi]$ are the polar and
azimuthal angles respectively in spherical coordinates. In
Figure~\ref{spinresponse}, we present the local variation of the unit
vector $\mathbf{n}=\sfrac{\mathbf{B}^*}{B^*}$ as a function of $M$ on the
unit sphere associated to the angles $\theta$ and $\varphi$. The energy
eigenvalues are
{\begin{equation}
E_{\pm} = \pm \sqrt{(B\cos \theta +M)^2 + B^2 
\sin^2 \theta} = \pm B^* = \pm |\mathbf{B}^*|.
\end{equation}}\unskip
The eigenstate with (lowest) energy $E_-=-B^*$ takes the form
{\begin{equation}
\left|\psi\right\rangle = |\alpha| {\mathrm{e}}^{-\mathrm{i}
\frac{\varphi}{2}}\left|+\right\rangle +  |\beta|
{\mathrm{e}}^{\mathrm{i} \frac{\varphi}{2}}\left|-\right\rangle
\end{equation}}\unskip
where
{\begin{eqnarray}
\begin{array}{rcl}
|\alpha|^2 &=& (E_- - (B\cos \theta+M))/(2 E_-)  \\[6pt]
|\beta|^2 &=& (E_- + (B\cos \theta+M))/(2 E_-).
\end{array}
\end{eqnarray}}\unskip
The Hilbert space is formed with the 2D eigenstates $|{+}\rangle$,
$|{-}\rangle$ associated to the $z$ direction (or Pauli matrix
$\sigma_z$) with spin eigenvalues ${+}1$ and ${-}1$ respectively. The
eigenstates are formulated within a particular gauge choice for the
phase related to the azimuthal angle. The theory presented below will
be gauge invariant.

\begin{figure}
\includegraphics{fig01}
\caption{\label{spinresponse}Local variation of the
$\mathbf{n}=\sfrac{\mathbf{B}^*}{B^*}$ vector associated to the total
magnetic field as a function of $M$ on the unit sphere associated to
the angles $\theta$ and $\varphi$. Phase transition refers here to the
situation of the topological quantum phase transition occurring when
$M=B$ i.e.\  it will be associated to the jump of the topological
invariant from one to zero. Associated to this transition, the
$n_z$-component of this unit vector flips its direction at south pole.}
\end{figure}

In quantum physics the Berry curvature plays the role of the magnetic
field and the equivalent to the vector potential is the local Berry
gauge potential~\cite{Berry}. The topological invariant is measured
from the integral of the Berry curvature on the surface. It
equivalently measures a topological charge in the core of the sphere.
It can be equivalently introduced in spherical coordinates or through a
cartesian metric from the simplification of the $\sin \theta$ function
in the differential area and the $\sfrac{1}{\sin\theta}$ factor stemming
in the definition of the Berry gauge potential component along the
equatorial direction in spherical coordinates~\cite{Google}. In this
way, we introduce the Berry gauge potential as
{\begin{equation}
A_{\varphi} = -\mathrm{i}\langle \psi| \partial_{\varphi} |\psi\rangle = 
-\frac{1}{2}(|\alpha|^2-|\beta|^2) = \frac{1}{2 E_-}(B\cos\theta+M).
\end{equation}}\unskip
We also have that $A_{\theta} = -\mathrm{i} \langle \psi| \partial_{\theta}
|\psi\rangle = 0$. The Berry curvature then takes the form $F_{\theta
\varphi} = \partial_{\theta} A_{\varphi} - \partial_{\varphi}
A_{\theta} = \partial_{\theta} A_{\varphi}=\sfrac{\partial
A_{\varphi}}{\partial \theta}$. We can then introduce the topological
invariant as 
{\begin{equation}
C = \frac{1}{2\uppi}\int_0^{2\uppi} \mathrm{d}\varphi\int_0^{\uppi}
F_{\theta \varphi}  \mathrm{d}\theta = \int_0^{\uppi} \frac{\partial
A_{\varphi}(\theta,M)}{\partial \theta} \mathrm{d}\theta.
\end{equation}}\unskip
The differential associated to $A_{\varphi}$ depends on the parameter
$M$. Therefore to simplify this formula, it is useful to introduce the
{\it dressed angle}
{\begin{equation}
\tan \tilde{\theta} = \frac{\sin \theta}{B \cos \theta + M}.
\end{equation}}\unskip
In principle, the general correspondence should be thought of as
$B^*\cos \tilde{\theta}=B\cos \theta + M$ and we introduce
$B^*\sin\tilde{\theta}=B\sin\theta$. On the sphere described through
the dressed angle $\tilde{\theta}$ and $\varphi$, the magnetic field is
radial $\mathbf{B}^*=B^*\mathbf{e}_r$. The Berry gauge potential and
the Berry curvature will not depend specifically on the precise form of
$B^*$. Therefore, we equivalently introduce the unit radial $\mathbf{n}^*$
vector for this geometrical correspondence in
Figure~\ref{anglecorrespondence}(Right)
{\begin{equation}
\label{effectivedirection}
{\mathbf{n}}^* = (\sin\tilde{\theta}\cos\varphi,\sin\tilde{\theta}
\sin\varphi,\cos\tilde{\theta}).
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig02}
\caption{\label{anglecorrespondence}(Left) Representation of the total
magnetic field ${\mathbf{B}}^*$ on the unit sphere described through
the angles $\theta$ and $\varphi$. (Right) Representation of the
effective topological properties through the (dressed) angle
$\tilde{\theta}(\theta)$. The direction of the total magnetic field
${\mathbf{B}}^*$ is then radial for all the values of $M$ associated to
the unit vector $\mathbf{n}^*$ in Equation~(\ref{effectivedirection}).
We introduce $\mathcal{A}=A_{\varphi}$ in Equation~(\ref{Aphi}). For
all the phases the north pole is identified with $\theta=0$ i.e.\ 
$\tilde{\theta}=0$. Within the topological phase $B<M$, the south pole
corresponds to $\tilde{\theta}(\theta=\uppi)=\uppi$. For $M<B$
topological properties are equivalent to those of a Skyrmion. For
$M>B$, the north and south poles are identical and the equivalent
geometry does not encircle the topological charge at the origin. For
$M=B$, geometrical properties are equivalent to a half sphere
corresponding to a half monopole or a half Skyrmion.}
\end{figure}

In this way, the topological invariant reads
{\begin{eqnarray}
\label{invariant}
C &=& \int_{\tilde{\theta}(\theta=0)}^{\tilde{\theta}(\theta=\uppi)}
\frac{\partial A_{\varphi}(\tilde{\theta})}{\partial
\tilde{\theta}}\frac{\partial \tilde{\theta}}{\partial \theta}
\mathrm{d}{\theta} = 
\int_{\tilde{\theta}(\theta=0)}^{\tilde{\theta}(\theta=\uppi)}
\frac{\partial A_{\varphi}(\tilde{\theta})}{\partial \tilde{\theta}}
\mathrm{d}\tilde{\theta} \nonumber \\
&=& A_{\varphi}(\tilde{\theta}(\theta=\uppi)) -
A_{\varphi}(\tilde{\theta}(\theta=0)),
\end{eqnarray}}\unskip
where
{\begin{equation}
\label{Aphi}
A_{\varphi}(\tilde{\theta}) = - \tfrac{1}{2}\cos\tilde{\theta}.
\end{equation}}\unskip
The last equation is equivalent to
$|\alpha|^2=\cos^2\stfrac{\tilde{\theta}}{2}$ and
$|\beta|^2=\sin^2\stfrac{\tilde{\theta}}{2}$. The topological invariant
is measured for any fixed value of $M$. This local formulation of the
topological invariant from the poles is gauge invariant and it agrees
with general geometrical thoughts { allowing for a correspondence on
the cylinder}~\cite{KLHReview,FractionalArticle} 
(see Appendix~\ref{AppendixA}). All the dependence on the parameter $M$
is then hidden into the definitions of the angles
$\tilde{\theta}(\theta=0)$ and $\tilde{\theta}(\theta=\uppi)$. We
emphasize that the eigenstates are introduced within the same coherent
gauge on the whole surface.

As shown in Figure~\ref{anglecorrespondence},  for all values of $M$, 
the north pole associated to $\theta=0$ is also equivalent to
$\tilde{\theta}=0$ i.e.\ $\tilde{\theta}(\theta=0)=0$. On the other
hand, we do observe a quantum phase transition when $M=B$ from the
angle $\tilde{\theta}$: as long as $M<B$, then $\tilde{\theta}=\uppi$
also corresponds to south pole i.e.\ to $\theta=\uppi$ such that
$\tilde{\theta}(\theta=\uppi)=\uppi$, when $M=B$ the topological properties
become equivalent to a half of a sphere
$\tilde{\theta}(\theta=\uppi)=\sfrac{\uppi}{2}$ and for $M>B$ we also have
the important equality 
$\tilde{\theta}(\theta=\uppi)=\tilde{\theta}(\theta=0)$ implying $C=0$.
The topological number then jumps from one when $M<B$ to zero when
$M>B$. Within this formulation, at the quantum phase transition we also
identify
{\begin{equation}
C_{1/2} =  - A_{\varphi}(\tilde{\theta}(\theta=0)) = +\tfrac{1}{2},
\end{equation}}\unskip
which can be interpreted as a half Skyrmion. It is interesting to
mention that $A_{\varphi}(\tilde{\theta}(\theta=0))=-\sfrac{1}{2}$
corresponds to a topological characterization of a $+\uppi$ Berry phase
encircling the north pole~\cite{FractionalArticle,OneHalf}. The
classical analogue of the local Berry gauge potential i.e.\ the
classical vector potential reads ${\mathcal
A}=A_{\varphi}(M)=-B\stfrac{\cos\theta}{2}-\stfrac{M}{4}\cos(2\theta)$ when
setting $r=1$.  This generalizes the formula introduced in
Equation~(10) in Ref.~\cite{KLHReview} for $M=0$.  We verify that {\it
classically} the induced $M$-component $-\stfrac{M}{4}\cos(2\theta)$
remains identical at the two poles  justifying why the Gauss theorem
only measures the presence of the radial magnetic field for this
situation.

From Ehrenfest's theorem $\langle \psi|\sigma_z |\psi \rangle = \langle
\sigma_z\rangle = \cos\tilde{\theta}=-2A_{\varphi}$. Then, the
topological number can be reformulated through the spin polarizations
at the north pole and at the dressed angle associated to  $\theta=\uppi$:
{\begin{equation}\label{topo}
C = \tfrac{1}{2}(\langle \sigma_z(\tilde{\theta}(\theta=0))\rangle - 
\langle \sigma_z(\tilde{\theta}(\theta=\uppi))\rangle).
\end{equation}}\unskip
This definition agrees with previous results on one sphere when
$M=0$~\cite{dynamo}.  This formula was adapted for the analysis of
topological quantum wires with applications in real space associated to
correlation
functions~\cite{FrederickLoicKaryn,FrederickLoicOlesiaKaryn,KarynFanMagali}.
The profile of the magnetic structure at the poles in
Figure~\ref{spinresponse} clearly reveals the quantum phase transition.
At the topological quantum phase transition, we have the identity
{\begin{equation}
\label{onehalf}
C_{1/2} = \tfrac{1}{2} = \tfrac{1}{2} \langle \sigma_z(\theta=0)\rangle.
\end{equation}}\unskip
For two spheres, it is possible to reach fractional $\sfrac{1}{2}$
numbers on a line or in a phase of the parameters space associated to
the interaction between spins~\cite{KLHReview}. The local formalism
between Berry gauge potential, pseudo-spin physics is also able to
measure the existence of a Bell state or Einstein--Podolsky--Rosen pair
at one pole~\cite{OneHalf}. For $M>B$, the spin magnetizations
associated to $\theta=0$ and $\theta=\uppi$ are the same such that $C=0$
i.e.\ $\tilde{\theta}(\theta=0)=\tilde{\theta}(\theta=\uppi)$.

In superconducting quantum circuits, the topological invariant is
measured from the Berry curvature when driving from north to south
pole~\cite{Google,Boulder}. The topological quantum phase transition
was also revealed in this way. We propose a local alternative
representation of the topological invariant and of the topological
phase transition which is also related to the physical response of the
spin at the poles.

Since the longitudinal and in-plane pseudo-spin responses are related
through a partial derivative with respect to $\tilde{\theta}$,
$|\sigma_x(\tilde{\theta})|$ is then directly related to the local
Berry curvature  $F_{\tilde{\theta}\varphi}=
\partial_{\tilde{\theta}}A_{\varphi}$ such that we also have
{\begin{equation}
C = \frac{1}{2}\int_{\tilde{\theta}(\theta=0)}^{\tilde{\theta}
(\theta=\uppi)} |\sigma_x(\tilde{\theta})| \mathrm{d}\tilde{\theta}.
\end{equation}}\unskip

\subsection{Map of spin response on the area and effective topological
magnetic moment, quantized transverse Hall current from a Newtonian
approach} \label{sphereresponse}

In Appendix~\ref{AppendixA}, we generalize the proof of
Refs.~\cite{KLHReview,FractionalArticle} from the Parseval--Plancherel
theorem showing the induction of a quantized transverse pumped current
in the {\it dressed polar angle} representation on the sphere
{\begin{equation}
J_{\perp}(\tilde{\theta}) = \frac{Q_{\perp}}{T} = 
\frac{1}{2\uppi}\oint J_{\perp}(\tilde{\theta})\mathrm{d}\varphi.
\end{equation}}\unskip
The charge $e$ is the charge of an electron. The pumped charge {\it at
an angle $\tilde{\theta}$} reads
{\begin{equation}
Q_{\perp}(\tilde{\theta}) = \sin^2\frac{\tilde{\theta}}{2}.
\end{equation}}\unskip
In this formulation, in the presence of the radial magnetic field
acting on the spin of an electron and in the presence of the magnetic
field source $M$, we measure the transverse pumped charge associated to
an electric field applied along the polar angle such that
$\hbar\dot{\tilde{\theta}}=e{\mathcal E}$. At time $t=0$, a charge is
present at north pole. The charge goes down similar to the apple and
the gravitational force is a result of the electric field directed
along the polar angle direction i.e.\ it corresponds to a Coulomb force.
The electron moves on the surface of this ball associated to the Bloch
sphere. The final time $T$, corresponding to the time of the measure,
is fixed to be identical for any $M$ such that the angle
$\tilde{\theta}$ reaches $\uppi$ i.e.\ $T=\sfrac{e{\mathcal E}}{\hbar \uppi}
= \sfrac{2e{\mathcal E}}{h}$. Within the topological phase, the
transverse pumped charge at south pole precisely measures the
topological invariant itself $C=1$. See Appendix~\ref{AppendixA}. When
we reach the transition maintaining the same value of the (final) time
$T$ for the measure, associated to Figure~\ref{anglecorrespondence},
this corresponds to maintain the Berry gauge potential fixed to
$A_{\varphi}(\tilde{\theta}=\uppi)=A_{\varphi}(\tilde{\theta}
=\sfrac{\uppi}{2})=0$ 
for dressed angles $\tilde{\theta}\in [\sfrac{\uppi}{2},\uppi]$ such that
there is no Berry curvature effectively present in the south
hemisphere. In that case, the topological invariant becomes halved i.e.\ 
equal to $C_{1/2}$ and the transverse pumped charge is also halved.
Above the transition, the charge remains at the north pole i.e.\ 
$A_{\varphi}(\tilde{\theta}=\uppi)=A_{\varphi}(\tilde{\theta}=0)$ and
there is no transverse pumped current.  In Appendix~\ref{AppendixA}, we
also show the relation with the cylinder geometry. 

In this article, then we introduce the {\it mean} value of the
transverse pumped charge on the original sphere associated to the polar
angle $\theta\in [0;\uppi]$
{\begin{equation}
\bar{Q}_{\perp} = \frac{e}{4\uppi}\int_0^{2\uppi} \mathrm{d}\varphi
\int_0^{\uppi} Q_{\perp}(\tilde{\theta})  \sin \theta\,
\mathrm{d}\theta=\frac{e}{4\uppi}\int_0^{2\uppi} \mathrm{d}\varphi
\int_0^{\uppi} \sin^2\frac{\tilde{\theta}}{2} \sin \theta\,
\mathrm{d}\theta.
\end{equation}}\unskip
The justification is that when $M=0$, we find that this response also
acquires a topological origin
{\begin{equation}
\bar{Q}_{\perp} = \frac{e C(M=0)}{2}.
\end{equation}}\unskip
When $M\neq 0$, we introduce the formula
{\begin{equation}
\bar{Q}_{\perp} = \frac{e C(M=0)}{2} - \frac{e\kappa(M)}{2},
\end{equation}}\unskip
with
{\begin{equation}
\label{kappaspin}
\kappa(M) = \frac{1}{4\uppi}\int_0^{2\uppi} \mathrm{d}\varphi
\int_0^{\uppi}  \langle \sigma_z(\tilde{\theta})\rangle \sin \theta\,
\mathrm{d}\theta = \frac{1}{2} \int_0^{\uppi}  \langle
\sigma_z(\tilde{\theta})\rangle \sin \theta\, \mathrm{d}\theta =
\frac{I}{2}.
\end{equation}}\unskip
Here, $\kappa$ is the mean value of the spin response along $z$
direction or the effective spin response as a function of $M$. When
$M=0$, we emphasize here that from symmetry between the two hemispheres
$\kappa=0$. In response to the radial magnetic field in
Figure~\ref{spinresponse} the two hemispheres produce an effective
moment equal in amplitude but opposite in directions. When
$M\rightarrow +\infty$, $\langle
\sigma_z(\tilde{\theta})\rangle\rightarrow 1$ such that the mean value
of the transverse pumped charge on the whole area is zero.  In that
limit, $\kappa$ is a measure of the unit area. We are precisely
questioning the (topological) properties of $\kappa$ as a function of
$M$ when inserting the equations
{\begin{equation}
\label{sigmaz}
\langle \psi | \sigma_z | \psi\rangle = \cos\tilde{\theta} =
\frac{B\cos\theta+M}{\sqrt{B^2\sin^2 \theta + (B\cos\theta+M)^2}} =
\frac{\partial B^*}{\partial M}=-\frac{\partial E_-}{\partial M}.
\end{equation}}\unskip

To evaluate $\kappa(M)$, we will first write down a correspondence
between spin response and magnetic field.  We introduce the definition
{\begin{equation}
B^* = B\sqrt{1+x}
\end{equation}}\unskip
with
{\begin{equation}
x=\left(2\frac{M}{B}\cos\theta+\frac{M^2}{B^2}\right).
\end{equation}}\unskip
At fixed $M$, $2M \,\mathrm{d}(\cos \theta) = B \,\mathrm{d}x$. 
In this way,
{\begin{equation}
\kappa(M) = -\frac{B}{2}\frac{\partial}{\partial M} 
\left( \frac{1}{2M}\int_{x_{\mathrm{min}}=x(\theta=0)}^{x_{\max}=
x(\theta=\uppi)} B^* \,\mathrm{d}x\right).
\end{equation}}\unskip
This is equivalent to
{\begin{equation}\label{equation}
\kappa(M) = -\frac{B^2}{2}\frac{\partial}{\partial M} 
\left(\frac{1}{2M}
\int_{x_{\mathrm{min}}=x(\theta=0)}^{x_{\max}=x(\theta=\uppi)} 
\sqrt{1+x}\ \mathrm{d}x\right).
\end{equation}}\unskip
Then, we can relate $x_{\mathrm{min}}=2(\sfrac{M}{B})+\stfrac{M^2}{B^2}$ and
$x_{\max}=-2(\sfrac{M}{B})+\stfrac{M^2}{B^2}$ revealing the properties of the
magnetic field $\mathbf{B}^*$ at the poles in Figure~\ref{spinresponse}:
{\begin{equation}
\sqrt{1+x_{\max}} = \frac{B^*(\theta=\uppi)}{B}= \left|1-\frac{M}{B}\right|
\end{equation}}\unskip
{\begin{equation}
\sqrt{1+x_{\mathrm{min}}} = \frac{B^*(\theta=0)}{B}=
\left(1+\frac{M}{B}\right).
\end{equation}}\unskip
This leads to
{\begin{equation}
\label{formulaspin}
\kappa(M) = -\frac{B^2}{2}\frac{\partial}{\partial M} 
\left(\frac{1}{3M} \left|1-\frac{M}{B}\right|^3 - 
\frac{1}{3M}\left(1+\frac{M}{B}\right)^3 \right).
\end{equation}}\unskip
For $M<B$, the only terms which are relevant are the quadratic terms in
the parenthesis which then turn into a linear term when applying the
partial derivative with respect to $M$. For $M<B$, we find
{\begin{equation}
\kappa(M) = \frac{M}{B}\frac{2}{3}.
\end{equation}}\unskip
When $M<B$, this results in a quantized plateau for the susceptibility
response. The linear response regime then remains applicable until the
transition point. We verify this result with a numerical integration of
$\sfrac{\mathrm{d}I}{\mathrm{d}\xi}$ with $I=2\kappa$ and $\xi=\sfrac{M}{B}$ in
Figure~\ref{pseudospinsphere}. A nice result of this article is then to
introduce a {\it topological magnetic susceptibility response} for the
hedgehog sphere (Skyrmion qubit) within the topological phase
{\begin{equation}
\chi = B\frac{\partial \kappa}{\partial M} = \frac{2}{3}.
\end{equation}}\unskip
The susceptibility has indeed a topological origin when $M<B$. To show
this we can re-phrase the result when $M<B$ as 
{\begin{equation}
\kappa = -\frac{1}{6B} \frac{\partial}{\partial M}\left(\left(1-
\frac{M}{B}\right)M\right) + \frac{1}{6B} \frac{\partial}{\partial M}
\left(\left(1+\frac{M}{B}\right)M\right).
\end{equation}}\unskip
This is then equivalent to
{\begin{equation}\label{chitopo}
\chi= \chi_{\mathrm{topo}} = B\frac{\partial \kappa}{\partial M} = 
\frac{2}{3}C(M<B) = -2B\frac{\partial \bar{Q}_{\perp}}{\partial M},
\end{equation}}\unskip
where the topological invariant is the invariant in the topological
phase which can be written as
{\begin{equation}
C(M<B) = \tfrac{1}{2}(\langle \sigma_z(0)\rangle - 
\langle \sigma_z(\uppi)\rangle)=1
\end{equation}}\unskip
from Equation~(\ref{topo}) and we insert the definition in
Equation~(\ref{sigmaz}) for the spin responses.  Within the topological
phase $M=B^-$ the topological character of $\kappa$ can be summarized
as
{\begin{equation}
\kappa = \kappa_{\mathrm{topo}} = -\frac{1}{6B}\left(B^*(\theta=\uppi)-
B^*(\theta=0)\right) + \frac{M}{3B}C(M<B).
\end{equation}}\unskip
Since
{\begin{equation}
\tfrac{1}{2}\left(B^*(\theta=0)-B^*(\theta=\uppi)\right) = MC(M<B)
\end{equation}}\unskip
then, this also leads to
{\begin{equation}
\kappa = \kappa_{\mathrm{topo}} = \frac{2M}{3B}C(M<B).
\end{equation}}\unskip
This formula is valid until $M=B^-$. 
For $M>B$, the form of $\kappa$ in Equation~(\ref{formulaspin}) is
{\begin{equation}
\kappa(M) = 1-\frac{1}{3}\frac{B^2}{M^2},
\end{equation}}\unskip
which leads to a power-law susceptibility response in $M^{-3}$. For
$M=B$, the mean value of the magnetic moment and the susceptibility are
continuous. 

\begin{figure}
\includegraphics{fig03}
\caption{\label{pseudospinsphere}Numerical evaluation of
$\sfrac{\mathrm{d}I}{\mathrm{d}\xi}=2\chi$ as a function of $\xi$ that
reproduces the analytical results, in particular the quantized plateau
for the susceptibility within the topological phase.}
\end{figure}

Then, we present another view of this proof related to the properties
of the topological invariant within the topological phase. We develop
the $\sqrt{1+x}$ function in Equation~(\ref{equation}) as an infinite
series and we develop the response order by order in
$x=2\stfrac{M}{B}\cos\theta+\stfrac{M^2}{B^2}$. To order $\sfrac{x}{2}$,
this leads to
{\begin{equation}
\kappa_{\frac{x}{2}} = \frac{M}{B}C(M=0)
\end{equation}}\unskip
with the topological invariant 
{\begin{equation}
C(M=0) = \frac{1}{2}\int_0^{\uppi} \sin\theta\, \mathrm{d}\theta=
\frac{1}{2}\int_0^{\uppi} \sin\tilde{\theta}\, \mathrm{d}\tilde{\theta}=+1.
\end{equation}}\unskip
The second equality reveals $C(M=0)=C(M<B)$ and imply that we study the
topological phase. The linear term in $M$ in $\kappa$ coming from
$-\stfrac{x^2}{8}$ in the series associated to $\sqrt{1+x}$ then reads
{\begin{equation}
\kappa_{-\frac{x^2}{8}} = -\frac{1}{3}\frac{M}{B}C(M=0).
\end{equation}}\unskip
Therefore, 
{\begin{equation}
\kappa_{\frac{x}{2}} + \kappa_{-\frac{x^2}{8}} = \frac{2}{3}
\frac{M}{B} C(M=0).
\end{equation}}\unskip
Then, we verify that this result is in fact robust when including
higher order terms in the series. This requires a systematic approach
that the numerical method also reproduces. Therefore, we verify in this
way that
{\begin{equation}
\kappa(M<B)=\kappa_{\frac{x}{2}} + \kappa_{-\frac{x^2}{8}} = 
\frac{2}{3} \frac{M}{B} C(M<B) = \frac{2}{3}\frac{M}{B}.
\end{equation}}\unskip

These results can be directly tested in quantum circuits~\cite{Google,
Boulder, Tran}.

\section{Correspondences on a 2D topological lattice model}\label{maplattice}

We discuss below an application for the topological Haldane model on
the honeycomb lattice~\cite{Haldane} that can be realized with
circularly polarized light~\cite{Cavalleri} and may be engineered in
transition metal dichalcogenide (TMD) materials~\cite{Mak}.

On the honeycomb 2D lattice, the analogous quantity to $\kappa(M)$
reads
{\begin{equation}
\langle \sigma_z(\mathbf{R}_i) \rangle = \frac{1}{N}\sum_{\mathbf{k}}
\langle \sigma_z(\mathbf{k})\rangle.
\end{equation}}\unskip
Here, $N=N_A=N_B$ corresponds to the number of $A$ or $B$ inequivalent
sites and $\langle \sigma_z\rangle$ measures the difference of
occupancies on $A$ and $B$ sublattices.  In Section~\ref{realspace}, we
will study the properties of this local marker $\langle
\sigma_z({\mathbf{R}}_i) \rangle$ on the lattice as a function of $M$. 

\subsection{Map from Brillouin zone of Honeycomb lattice onto the
sphere and local topological responses for the Haldane model}
\label{Map}

The monopole formalism~\cite{KLHReview} can be developed to analyse
properties of lattice models such as the Haldane model on the honeycomb
lattice~\cite{Haldane}. 

We develop the theory from the Brillouin zone onto the sphere and
introduce a numerical algorithm for the evaluation of physical
properties in momentum (reciprocal) space and in real space. The
Hamiltonian can be written as a $2\times 2$ matrix in the spinor
representation associated to sublattices $A$ and $B$, $\{|\mathbf{k},
A\rangle, |\mathbf{k}, B\rangle\}$, such that $H=\sum_{\mathbf{k}}
H(\mathbf{k})$ where $H(\mathbf{k})=-\mathbf{d}\cdot
{\bsigma}$ and $\mathbf{d}=(d_x,d_y,d_z+M)$ with
{\begin{eqnarray}\label{dvector}
\begin{array}{rcl}
\displaystyle d_x &=& \displaystyle 
t\left(1+2\cos\left(\frac{3a}{2}k_x\right)\cos
\left(\frac{\sqrt{3}a}{2}k_y\right)\right) \\[8pt]
\displaystyle  d_y &=&\displaystyle  2t \sin
\left(\frac{3a}{2}k_x\right) \cos\left(\frac{\sqrt{3}a}{2}k_y\right) 
\\[8pt]
\displaystyle  d_z &=&\displaystyle  2t_2\left(\sin \left(\sqrt{3}a
k_y\right) -
2\cos\left(\frac{3a}{2}k_x\right)\sin\left(\frac{\sqrt{3}a}{2}
k_y\right)\right).
\end{array}
\end{eqnarray}}\unskip
The term $t$ corresponds to the hopping of electrons on
nearest-neighboring sites and $t_2$ is the Haldane hopping term between
second nearest neighbors with a complex phase fixed to
$\sfrac{\uppi}{2}$~\cite{Haldane}.  Here, $a$ is the lattice spacing.
The two Dirac points have the same component $K_x=\sfrac{2\uppi}{3a}$
such that $\cos(\sfrac{3a}{2}K_x)=-1$. We also have
$K_y=\sfrac{2\uppi}{(a3\sqrt{3})}$ and
$K'_y=-\sfrac{2\uppi}{(a3\sqrt{3})}$ such that close to the Dirac
points we have the identifications~\cite{Haldane}
{\begin{equation}\label{dz}
d_z(\mathbf{K}^{\zeta}) = \pm 3\sqrt{3}t_2 = \zeta 3\sqrt{3}t_2.
\end{equation}}\unskip
The symbol $\zeta$ will be fixed such that $\zeta=+1$ at the $K$ Dirac
point and $\zeta=-1$ at the $K'$ Dirac point which can be viewed as the
mass inversion effect at the two Dirac points.  The parameter $M$,
favoring the occupancy of a particle (electron) on one sublattice, is
called a Semenoff mass term related to the Dirac
equation~\cite{Semenoff}.

It is judicious to introduce the definitions 
{\begin{eqnarray}
\begin{array}{rcl}
d_z &=& -\gamma(\mathbf{k})  \\[6pt]
d_x+\mathrm{i} d_y &=& g(\mathbf{k}).
\end{array}
\end{eqnarray}}\unskip
The energy eigenvalues then can be written as
{\begin{eqnarray}
E_{\pm}(\mathbf{k}) = \pm \sqrt{(M-\gamma(\mathbf{k}))^2 + 
|g(\mathbf{k})|^2}.
\end{eqnarray}}\unskip
For completeness, we illustrate the map from the Brillouin zone,
equivalently drawn as a rhomboid, onto the sphere in
Figure~\ref{honeycomb} that also clarifies the quantum phase transition
from a simple geometrical view, i.e.\ as an equivalent geometry on the
sphere encircling or not the origin of the Brillouin zone. Within the
topological phase the $K$ Dirac point in red is at north pole at
$\theta=0$ and the $K'$ Dirac point in red is at south pole at
$\theta=\uppi$. 

\begin{figure}
\includegraphics{fig04}
\caption{\label{honeycomb}(Top) Honeycomb lattice and Brillouin zone.
The rhomboid Brillouin zone is equivalent to the hexagonal one. Within
the correspondence for $t_2=0.1$ in units of $t$, we introduce the unit
vector $\mathbf{n}_d=\sfrac{\mathbf{d}}{|\mathbf{d}|}$ in
Equation~(\ref{dvector}). When the $\mathbf{d}$ vector covers the
entire sphere then this corresponds to a topological number equal to
one e.g.\ for $M=0$. In this case, Dirac points are located at
different poles on the sphere. Within the trivial phase $M=2M_c$, both
Dirac points then map onto the same pole. In that case, the magnetic
field (Berry curvature) does not wrap around the origin completely.
This leads to a trivial winding number. (Bottom) Path with fixed
azimuthal angle $\varphi$ and parameters $M=0$, $t_2=0.1$.}
\end{figure}

Close to the Dirac points, for the analytical evaluations it is useful
to introduce~\cite{KLHReview}
{\begin{eqnarray}\label{dvectormap}
\begin{array}{rcl}
d_x &=& -\hbar v_F |\mathbf{p}| \cos(\phi_p)  \\[6pt]
d_y &=& -\hbar v_F |\mathbf{p}| \sin(\zeta\phi_p)  \\[6pt]
d_z &=& \zeta 3\sqrt{3} t_2 +M.
\end{array}
\end{eqnarray}}\unskip
For the analytical evaluations, we assume a parabolic dispersion close
to $K$ and $K'$ which is satisfied for not too large values of $t_2$
(e.g.\ $t_2\sim 0.1t$)~\cite{StephanKaryn}. Here, $\phi_p$ represents
the polar angle around each Dirac point which satisfies
{\begin{equation}\label{azimuthal}
\varphi = \zeta \phi_p \pm \uppi,
\end{equation}}\unskip
where $\varphi$ is the azimuthal angle on the sphere. Around the two
Dirac points in the Brillouin zone, compared to the sphere definition, 
the polar angles are then introduced to rotate in different directions
i.e.\ $\zeta=+1$ at the $K$ Dirac point and $\zeta=-1$ at the $K'$ Dirac
point. Close to the Dirac points, the components $d_x$ and $d_y$
precisely give rise to the massless Dirac equation through the
Hamiltonian
{\begin{equation}
H^{\zeta}(\mathbf{p}) = \hbar v_F (p_x \sigma_x + \zeta p_y \sigma_y),
\end{equation}}\unskip
with $v_F=\stfrac{3}{2\hbar}ta$ the Fermi velocity of
graphene~\cite{Wallace}. We also have the correspondence between the
dressed polar angle $\tilde{\theta}$ on the sphere and the deviation
from each Dirac point through the wavevector $\mathbf{p}$:
{\begin{equation}\label{thetatilde}
\tan\tilde{\theta} = \frac{\hbar v_F|\mathbf{p}|}{\zeta 3\sqrt{3} t_2+M}.
\end{equation}}\unskip

The many-body ground state corresponds to the lowest-energy band being
filled 
{\begin{equation}
\left|\Psi\right\rangle = \prod_{\mathbf{k}\in \mathrm{FBZ}} (\alpha(\mathbf{k})c^{\dagger}_{\mathbf{k},A} + 
\beta(\mathbf{k}) c^{\dagger}_{\mathbf{k},B}) \left|0\right\rangle
\end{equation}}\unskip
where FBZ refers to the first Brillouin zone and the electron
operators act on $A$ and $B$ sublattices respectively. It is associated
to the state $\left|\psi\right\rangle$ on the sphere.

Related to Figures~\ref{anglecorrespondence} and \ref{honeycomb}, the
$K$ Dirac point will correspond to the polar angle
$\tilde{\theta}(\theta=0)=0$ and the $K'$ Dirac point will then
correspond to the polar angle $\tilde{\theta}(\theta=\uppi)$. The global
topological invariant can then be defined locally
as~\cite{KLHReview,FractionalArticle}
{\begin{equation}\label{localmarkerlattice}
C = A_{\varphi}(\mathbf{K}') - A_{\varphi}(\mathbf{K}),
\end{equation}}\unskip
with 
{\begin{equation}
A_{\varphi}(\mathbf{K}^{\zeta}) = -\mathrm{i}\zeta \langle \psi_+| 
\partial_{\phi_p}|\psi_+\rangle.
\end{equation}}\unskip
It corresponds to the addition of Berry phases around each Dirac point
where the eigenstates around these two points associated to the
lowest-energy band are introduced with the same $\varphi$-coherent
gauge according to Equation~(\ref{azimuthal}). The local gauge
potential can be efficiently addressed numerically at the Dirac points
through the Taylor formula
{\begin{equation}
A_{\varphi}(\mathbf{K}^{\zeta}) = -\mathrm{i} \zeta 
\lim_{\tilde{\theta}\rightarrow 0 \text{ or }
\tilde{\theta}(\theta=\uppi)} \lim_{\Delta \phi_p\rightarrow 0} 
\frac{\langle \psi_+(\tilde{\theta},\phi_p)|\psi_+(\tilde{\theta},
\phi_p +\Delta \phi_p)\rangle -1}{\Delta \phi_p},
\end{equation}}\unskip
where $\zeta$ is $\pm$ at the $K$ and $K'$ Dirac points respectively.
The formula (\ref{localmarkerlattice}) is very efficient numerically
and it reproduces well the quantum phase transition at $M=M_c=3\sqrt{3}
t_2$ through the jump of the topological invariant locally resolved at
the Dirac points. As illustrated in Figure~\ref{Berrygraphs}, this
formula works well even when the Berry curvatures become very small at
$K$ and $K'$. 

\begin{figure}
\includegraphics{fig05}
\caption{\label{Berrygraphs}Berry curvatures for different values of
$M$ and $t_2$. (Top Left) This corresponds to the physical situation of
graphene with a charge density wave substrate leading to
Equation~(\ref{Berryfunction})~\cite{Eva}. (Top Right) We are within
the topological phase with $C=1$ and the numbers agree with
Equation~(\ref{Berryfunction}) and also with numerical results in the
literature~\cite{Anton}. (Bottom Left) Results within the
non-topological phase with $C=0$. (Bottom Right) When increasing $t_2$
within the topological phase, there is a deviation from the Dirac
approximation for the $d_x$ and $d_y$ components and the local Berry
curvatures become (very) small.  Interestingly the formula $C =
A_{\varphi}(\mathbf{K}') - A_{\varphi}(\mathbf{K})=+1$ from the Dirac
points which is exact yet works well.}
\end{figure}

At the topological quantum phase transition, the gap is closing at one
Dirac point and the half topological invariant can be seen as a $\uppi$
Berry phase or $\uppi$ winding number related to the massive Dirac point.
From the discussion on the sphere in Section~\ref{Monopole}, this
formula $C = A_{\varphi}(\mathbf{K}') - A_{\varphi}(\mathbf{K})$ is
gauge invariant. We emphasize here that the situation of half
topological numbers can be then generalized  to a region of the
parameters space (e.g.\ a line) associated to the quantum anomalous Hall
semimetal~\cite{FractionalArticle,KarynSariah1,KarynSariah2}. This
formalism builds a correspondence with the physics of surface states of
three-dimensional insulators~\cite{FuKane,SekineNomura,Zhang} which
also reveal a massive topological Dirac point with a $\uppi$ winding
number or half topological invariant (see also Section~4.6 in
Ref.~\cite{KLHReview}). We mention here recent efforts to observe a
half-quantized Hall conductance in semimagnetic topological insulator
bilayers~\cite{Mogi} { and in ultra-cold atoms~\cite{Nascimbene}}
related to the $C=1/2$ parity anomaly~\cite{Haldane}. A recent work
also reports a half-quantized chiral edge current~\cite{Zhuo}.

We also show another useful aspect of this local representation of
topological properties through the Berry curvature itself. Swapping
from the sphere to the Brillouin zone, it is then possible to evaluate
analytically the Berry curvature assuming the linear Dirac
approximation for the $d_x$ and $d_y$ components~\cite{KarynLight}.
Including the presence of the term $M$ identical to a Semenoff mass, we
generalize the formula found in Ref.~\cite{KarynLight} by one of us as
{\begin{eqnarray}
\label{Berryfunction}
F_{p_x,p_y}^{\zeta} &=& -\frac{1}{2(M + \zeta  3\sqrt{3}t_2)^2}
\hbox{Im} ( \langle \psi_+|\partial_{p_x}  H^{\zeta} |\psi_-\rangle
\langle \psi_-|\partial_{p_y} H^{\zeta} |\psi_+\rangle) \nonumber \\
&=& \zeta \frac{\hbar^2 v_F^2}{2(M + \zeta
3\sqrt{3}t_2)^2}\cos\tilde{\theta},
\end{eqnarray}}\unskip
where 
{\begin{equation}
\cos\tilde{\theta} = \langle \psi_+| \sigma_z | \psi_+\rangle = -2
A_{\varphi}(\tilde{\theta}).
\end{equation}}\unskip
We emphasize that $\left| \psi_+ \right\rangle$ (called $ \left|\psi
 \right\rangle$ in the preceding Section) and
$ \left|\psi_- \right\rangle$ correspond to the {\it lowest-energy} and
{\it upper-energy} eigenstates of the spin-$1/2$ particle 
{\begin{equation}
\left|\psi_-\right\rangle = -|\beta| {\mathrm{e}}^{-\mathrm{i}
\frac{\varphi}{2}}\left|+\right\rangle +  |\alpha|
{\mathrm{e}}^{\mathrm{i} \frac{\varphi}{2}}\left|-\right\rangle.
\end{equation}}\unskip
These two eigenstates are then related to the lowest and upper energy
bands in the Haldane model. A similar relation between Berry curvature
and pseudo-spin response was derived in relation to the Thomas
precession and Dirac equation in 1991 and
1994~\cite{Mathur,MathurShankar}.  This leads to the results of
Figure~\ref{Berrygraphs}. We verify the height of Berry curvature
surrounding each Dirac point with the topological phase e.g.\ for $M=0$
{\begin{eqnarray}
F_{p_x,p_y}^{\zeta} = -\zeta \frac{\hbar^2}{24 t_2^2}(t a)^2
(2A_{\varphi}(\tilde{\theta})).
\end{eqnarray}}\unskip
When summing the Berry curvature at the two Dirac points then this
equally defines the topological invariant. In this sense, the quantum
Hall conductivity written in terms of Berry curvatures~\cite{TKNN} can
also be resolved  locally from the Dirac points modulo a prefactor that
is also useful to obtain an estimate of the ratio $\sfrac{t_2}{t}$. We
assume here that $t_2\neq 0$ because Equation~(\ref{Aphi}) is
applicable in the presence of a radial magnetic field on the sphere
implying then a term $t_2$ on the lattice.

In Section~\ref{light}, the response to circularly polarized light,
i.e.\ through the photo-induced currents, will precisely allow us to
resolve this information from the Dirac
points~\cite{PhilippAdolfoKaryn}. We will illustrate an application
including the mass  term $M$. 

\subsection{Populations and Pseudo-spin response through a local
marker in momentum space}

It is then useful to build a precise correspondence between populations
on sublattices $A$ and $B$ and the pseudo-spin response along $z$
direction within the ground state. From the correspondence on the
sphere, 
{\begin{equation}
C = \frac{\cos\tilde{\theta}(\mathbf{K}) - 
\cos\tilde{\theta}(\mathbf{K}')}{2} = 
\frac{\langle \sigma_z(\mathbf{K}) \rangle - \langle 
\sigma_z(\mathbf{K}') \rangle}{2}.
\end{equation}}\unskip
We illustrate the usefulness of this local marker in momentum space
associated to the pseudo-spin response in Figure~\ref{populations}. The
populations on $A$ and $B$ sublattices with e.g.\ 
$n_{\mathbf{k},A}=c^{\dagger}_{\mathbf{k},A} c_{\mathbf{k},A}$ take the
forms
{\begin{eqnarray}\label{density}
\begin{array}{rcl}
\langle n_{\mathbf{k},A}\rangle &=&\displaystyle  \langle \Psi
|n_{\mathbf{k},A}|\Psi\rangle = |\alpha(\mathbf{k})|^2 = \frac{1}{2} -
\frac{\gamma(\mathbf{k})-M}{2|\mathbf{d}(\mathbf{k})|} \\[8pt] 
\langle n_{\mathbf{k},B}\rangle &=&\displaystyle  \langle
\Psi|n_{\mathbf{k},B}|\Psi\rangle = |\beta(\mathbf{k})|^2 = \frac{1}{2}
+ \frac{\gamma(\mathbf{k})-M}{2|\mathbf{d}(\mathbf{k})|}.
\end{array}
\end{eqnarray}}\unskip
Close to the Dirac points, since $g(\mathbf{k})\rightarrow 0$, then we
obtain the simple general form 
{\begin{equation}
\langle \sigma_z(\mathbf{K}^{\zeta})\rangle = 
|\alpha(\mathbf{K}^{\zeta})|^2-|\beta(\mathbf{K}^{\zeta})|^2 = 
-\frac{\gamma(\mathbf{K}^{\zeta}) - M}{|\gamma(\mathbf{K}^{\zeta})-M|}=
\mathrm{sgn}(M+\zeta 3\sqrt{3} t_2).
\end{equation}}\unskip
It is also possible to re-interpret the $sgn$ function at each Dirac
point through a mass sign in Equation~(\ref{dz}) which is then dressed
with the $M$ term~\cite{Haldane,Orsay,Nathan}. If we relate the
population in momentum space with the photoluminescence intensity in
Ref.~\cite{C2N}, then Equations~(\ref{density}) may be resolved through
the Stokes parameters. In that article, the authors then introduce the
valley Chern number~\cite{Haldane}. Information about pseudo-spin
physics of Skyrmions may also be accessed through time of flight in
cold atoms~\cite{Alba}.

\begin{figure}
\includegraphics{fig06}
\caption{\label{populations}Pseudo-spin response at the two Dirac
points and illustration of the phase transition through the flip of the
pseudo-spin at one pole. $\langle \sigma_z(\mathbi{k})\rangle$
evaluated at the Dirac points as a function of $\sfrac{M}{(3\sqrt 3
t_2)}$.  The top figure shows the five energy diagrams along the
Brillouin zone axis. The black line response is relative to the
$K$-point, which is mapped to the north pole of the sphere, while the
grey one is relative to the $K'$-point, which corresponds to the south
pole. As long as $|M|<3\sqrt 3 t_2$, we are in the topological phase
and so the pseudospin responses at $K$ and $K'$ are in opposite
directions. In the trivial phase the pseudospins point in the same
directions. This corresponds to the topological band inversion.}
\end{figure}

We also emphasize on the fact that the in-plane pseudo spin observable
in momentum space acquires an important representation on the sphere
related to the local Berry curvature $F_{\tilde{\theta}\varphi}$:
{\begin{equation}
|\langle \psi| \sigma_x |\psi\rangle| =
|\alpha^*(\mathbf{k})\beta(\mathbf{k}) +
\alpha(\mathbf{k})\beta^*(\mathbf{k})| = \sin\tilde{\theta}.
\end{equation}}\unskip
The in-plane pseudo-spin response for the Haldane model was recently
addressed in Ref.~\cite{IkedaRayan} associated to sublattice coherence
effects.

\subsection{Real-space analysis and comparison with the effective
magnetic moment of the monopole} \label{realspace}

From Parseval--Plancherel theorem, the real space pseudo-spin response
reads
{\begin{equation}
\langle \sigma_z(\mathbf{R}_i)\rangle = -\frac{1}{N} 
\sum_{\mathbf{k}} \frac{\gamma(\mathbf{k})-M}{|\mathbf{d}(\mathbf{k})|},
\end{equation}}\unskip
for $M=0$, the function $\gamma$ is odd under the transformation
$k_y\rightarrow -k_y$ such that $\langle
\sigma_z(\mathbf{R}_i)\rangle=0$. 

In Figure~\ref{pseudospin}, we analyse the properties of this local
marker in real space comparatively to $\kappa$ in
Equation~(\ref{kappaspin}). When varying the parameter $M$,
corresponding e.g.\ to describe the effects of a charge density wave
substrate~\cite{Eva}, we do reveal a linear response behavior for the
pseudo-spin locally on the lattice until the phase transition similar
to the analysis of the response of the monopole. The first derivative
in {\it yellow} of the pseudo-spin is also fixed until the topological
phase transition. We report a similar behavior as the susceptibility
$\chi=B\stfrac{\partial \kappa}{\partial M}$ in Equation~(\ref{chitopo}).
The first derivative from the real space analysis on the lattice in
Figure~\ref{pseudospin} is around ${\sim} 0.33\ldots,$ which is
approximately half of $\chi$.  The susceptibility in yellow in
Figure~\ref{pseudospin} clearly distinguishes the position (location)
of the transition. In Figure~\ref{pseudospin}, we show in {\it red} the
second derivative  of  $\langle \sigma_z(\mathbf{R}_i)\rangle$ as a
function of $\sfrac{M}{M_c}$.

\begin{figure}
\includegraphics{fig07}
\caption{\label{pseudospin}Numerical study of the populations in real
space and of $\langle \sigma_z(\mathbf{R}_i)\rangle$ as a function of
$M$ with $t_2=0.1t$ as in Figure~\ref{honeycomb}. First and second
derivatives as a function of $M$. Here, we remind that
$M_c=3\sqrt{3}t_2$ which is analogous to $B$ for the monopole. The
calculation is done by doing a discrete Fourier transform in a $100
\times 100$ finite size lattice in Periodic Boundary Conditions.}
\end{figure}

The comparison between the local marker in real space $\langle
\sigma_z(\mathbf{R}_i)\rangle$ and the theory of the monopole's spin
response is interesting since in general for any observable a careful
correspondence between quantities evaluated on the lattice and on the
sphere is required. For the quantum Hall response it is possible to
re-write the global topological invariant in the plane as an evaluation
of the Berry curvature on the sphere through $F_{\theta\varphi}$ with a
measure of the area integration as 
$\mathrm{d}\theta \,\mathrm{d}\varphi$~\cite{KLHReview}.
There is also a simple understanding of this mapping from the Kubo
formula for the quantum Hall response~\cite{KLHReview}. For the
pseudo-spin response, this is evaluated on the spherical surface and we
have verified that making a flat-space simplification does not
reproduce the same results in particular at the transition. The
Jacobian of a transformation necessitates some thoughts in general. As
long as Equations~(\ref{dvectormap}) are satisfied i.e.\ with an energy
spectrum of the form $\pm \sqrt{(\hbar v_F |\mathbf{p}|)^2 + (\zeta
3\sqrt{3}t_2+M)^2}$ around each Dirac point we can justify the results
as follows. On the lattice, we evaluate 
{\begin{equation}
\frac{1}{(2\uppi)^2}\iint \langle \sigma_z(\mathbf{k})\rangle
\,\mathrm{d}k_x\, \mathrm{d}k_y.
\end{equation}}\unskip
In the vicinity of the north pole or of the $K$ Dirac point, from
Equation~(\ref{thetatilde}), we can develop  $\mathrm{d}k_x\,
\mathrm{d}k_y\rightarrow (2\uppi) |\mathbf{p}|
\,\mathrm{d}|\mathbf{p}|$ such that $(\hbar v_F)^2
\,\mathrm{d}|\mathbf{p}||\mathbf{p}|=\mathrm{d}\theta  \sin \theta
M_c^2$ with $M_c=3\sqrt{3}t_2$. We develop the area of the rhomboid
Brillouin zone from a disk centered around the $K$ Dirac point. In this
correspondence the radius of the sphere is $M_c$ whereas the radius of
the disk around the $K$ Dirac point is $\hbar v_F$.  To reproduce the
lattice  result from the sphere which shows a linear relation between
$\theta$ and  $|\mathbf{p}|$ around north pole then this requires e.g.\
to rescale  $\hbar v_F=1$ and $M_c=1$ corresponding to $t_2\sim 0.19$
and $t=\sfrac{2}{3}$ when setting the lattice spacing to unity. In this
way,
{\begin{equation}
\frac{1}{(2\uppi)^2}\iint \langle \sigma_z(\mathbf{k})\rangle 
\,\mathrm{d}k_x\, \mathrm{d}k_y \sim \frac{1}{\uppi}\kappa \sim
0.21\frac{M}{M_c}.
\end{equation}}\unskip
This approximation tends to provide a good justification of why
$\langle \sigma_z(\mathbf{R}_i)\rangle$ is related to $\kappa$ such
that it also shows a plateau within the topological phase, as observed
numerically  in Figure~\ref{pseudospin}. For larger values of $t_2$,
numerically we observe some curvature effects on the plateau for
$\langle \sigma_z(\mathbf{R}_i)\rangle$ which can be associated to
deviations from the parabolic band approximation and to the flattening
of the bands~\cite{StephanKaryn}. The topological phase transition
remains visible. Yet, the topological marker introduced on the sphere
in Figure~\ref{spinresponse} remains applicable for all values of $B$
or $t_2$ on the lattice.

It is then useful to present the numerical analysis in real space,
resolved in each sublattice, associated to the Fourier transforms of
$\langle n_{\mathbf{k},A}\rangle$ and $\langle
n_{\mathbf{k},B}\rangle$:
{\begin{eqnarray}
\begin{array}{rcl}
\displaystyle \langle n_{\mathbf{k},A} \rangle = |\alpha(\mathbf{k})|^2
&=&\displaystyle  \sum_{\mathbf{R}} {\mathrm{e}}^{\mathrm{i}
\mathbf{k}\cdot \mathbf{R}} C(\mathbf{R},A) \\[8pt]
\displaystyle \langle n_{\mathbf{k},B}\rangle =|\beta(\mathbf{k})|^2
&=&\displaystyle  \sum_{\mathbf{R}} {\mathrm{e}}^{\mathrm{i}
\mathbf{k}\cdot \mathbf{R}} C(\mathbf{R},B).
\end{array}
\end{eqnarray}}\unskip
In this way, the topological invariant takes the equivalent form 
{\begin{equation}
\label{invariantcorrelation}
C = \frac{1}{2} \sum_{\mathbf{R}} (C(\mathbf{R},A) - C(\mathbf{R},B))
\left({\mathrm{e}}^{\mathrm{i}\mathbf{K}\cdot \mathbf{R}} -
{\mathrm{e}}^{\mathrm{i}\mathbf{K}'\cdot \mathbf{R}}\right).
\end{equation}}\unskip
This can also be re-written as
{\begin{equation}
C = \mathrm{i}\sum_{\mathbf{R}}(C(\mathbf{R},A) - C(\mathbf{R},B))
{\mathrm{e}}^{\mathrm{i} K_x R_x} \sin(K_y R_y).
\end{equation}}\unskip
For one-dimensional topological systems, it is in general possible to
evaluate the summation
analytically~\cite{FrederickLoicKaryn,FrederickLoicOlesiaKaryn,KarynFanMagali}.
It is then useful to introduce the correlation function in real space
associated to the topological invariant
{\begin{equation}\label{f}
f(\mathbf{R}) = C(\mathbf{R},A) - C(\mathbf{R},B) =
\frac{1}{N}\sum_{\mathbf{q}} {\mathrm{e}}^{-\mathrm{i}\mathbf{q}\cdot
\mathbf{R}} \langle
\sigma_z(\mathbf{q})\rangle = \frac{1}{N}\sum_{\mathbf{q}}
{\mathrm{e}}^{-\mathrm{i}\mathbf{q}\cdot \mathbf{R}}
\left(\frac{-\gamma(\mathbf{q})+M}{\sqrt{(\gamma(\mathbf{q})-M)^2 +
|g(\mathbf{q})|^2}}\right).
\end{equation}}\unskip
When $M\gg M_c=3\sqrt{3}t_2$, $\langle \sigma_z(\mathbf{q})\rangle
\rightarrow +1$ and therefore correlation functions are very
short-range such that $f(\mathbf{R})\rightarrow \delta(\mathbf{R})$
traducing indeed an {\it insulating phase} such that $C=0$ due to the
presence of the sine function in the sum. Within the topological phase,
the function ${\mathrm{e}}^{\mathrm{i}K_x R_x}$ 
is real for nearest neighbors of the same
flavour. Therefore, this requires to have longer-range of correlations.
Yet, the phase shows a gap such that a relatively short decay also
occurs. Numerically, we identify an exponential decay for
$f(\mathbf{R})$ at short distances for length scales up to 10 units of
lattice vectors. Correlation functions also decay in a fast way for the
Kane--Mele model~\cite{StephanKaryn}. Within the {\it topological phase}
e.g.\ for $M=0$, if we insert Equation~(\ref{f}) into
Equation~(\ref{invariantcorrelation}) we can verify that $C=1$. For the
p-wave Kitaev superconductor, at half-filling and for $t=\Delta$, the
nearest-neighbor correlator reproduces the topological
invariant~\cite{FrederickLoicOlesiaKaryn}.  At the transition, from
this analysis, the correlation length diverges from the trivial phase.

\subsection{Relation to local responses from circularly polarized light}
\label{light}

Here, we show how the pseudo-spin response and the Berry curvature are
locally measured through the responses to circularly polarized light
and develop the theory of the quantum phase transition.

It is useful to remind that for a two-level system the light matter
interaction takes the form
{\begin{equation}
\delta H_{\pm} = A_0 {\mathrm{e}}^{\pm \mathrm{i}\omega t}
\left|+\right\rangle \left\langle -\right|  +\mbox{h.c.} = A_0
{\mathrm{e}}^{\pm \mathrm{i}\omega t}\sigma^+ +\mbox{h.c.}
\end{equation}}\unskip
The classical vector potential is described through its components
$A_x=A_0\cos \omega t$ and $A_y=\mp A_0 \sin \omega t$. This is
equivalent as a circularly polarized vector potential $\mathbf{A}_{\pm}
= A_0 {\mathrm{e}}^{-\mathrm{i}\omega t} (\mathbf{e}_x \mp \mathrm{i} 
\mathbf{e}_y)= A_0 {\mathrm{e}}^{-\mathrm{i}\omega
t} (\mp \mathbf{e}_{\varphi})$. It is useful to introduce the
inter-band transition probabilities associated to each
polarization~\cite{KarynLight,KLHReview}
{\begin{equation}
\Gamma_{\pm} = \frac{2\uppi}{\hbar} |\langle \psi_+| 
\delta H_{\pm} |\psi_-\rangle|^2 \delta (E_b-E_a \mp \hbar \omega).
\end{equation}}\unskip
The meaning of right $(+)$ and left-handed $(-)$ polarizations can be
understood from the {\it resonance} which is obtained through the
transformation  $\left|-\right\rangle \rightarrow {\mathrm{e}}^{\mp
\sttfrac{\mathrm{i}\omega t}{2}}\left|-\right\rangle$ and
$\left|+\right\rangle \rightarrow {\mathrm{e}}^{\pm
\sttfrac{\mathrm{i}\omega t}{2}}\left|+\right\rangle$.  On the lattice
$\left|+\right\rangle$ refers to the  $\left|A\right\rangle$ sublattice
polarization and $\left|-\right\rangle$ refers to the $|B\rangle$
sublattice polarization.  In this way, the time-dependent phases can be
interpreted as an energy shift  $E_b - E_a = \pm \hbar\omega$. 

From the form of the eigenstates
$\left|\psi_+(\tilde{\theta},\varphi)\right\rangle
=\left|\psi_+\right\rangle$ and
$\left|\psi_-(\tilde{\theta},\varphi)\right\rangle
=\left|\psi_-\right\rangle$ in the presence
of the mass $M$, we can then generalize the geometrical function
$\alpha(\theta)$ introduced in previous
articles~\cite{KarynLight,KLHReview} as:
{\begin{equation}
\alpha(\tilde{\theta}) =  |\langle \psi_+| \sigma_x|\psi_-\rangle|^2 +  
|\langle \psi_+| \sigma_y|\psi_-\rangle|^2 = 
\sin^4\frac{\tilde{\theta}}{2} + \cos^4\frac{\tilde{\theta}}{2}
= \alpha(\uppi-\tilde{\theta}).
\end{equation}}\unskip
For $M=0$, one of us has shown that this function at the two Dirac
points is precisely related to the square of the topological
invariant~\cite{KarynLight,KLHReview}. Close to $\tilde{\theta}=0$ we
can equivalently write
{\begin{equation}
\sin^4\frac{\tilde{\theta}}{2} + \cos^4\frac{\tilde{\theta}}{2} =
\langle \sigma_z(\theta=0)\rangle 
\end{equation}}\unskip
and close to $\tilde{\theta}(\theta=\uppi)$, from definitions in 
Appendix~\ref{AppendixA}, we also have
{\begin{equation}
\sin^4\frac{\tilde{\theta}}{2} + \cos^4\frac{\tilde{\theta}}{2} =
\langle \sigma_z(\tilde{\theta}(\theta=\uppi))\rangle +2C^2.
\end{equation}}\unskip
As long as we are within the topological phase, the points
$\tilde{\theta}=0$ and $\tilde{\theta}=\uppi$ correspond respectively to
the points $\theta=0$ and $\theta=\uppi$.  The functions
$\sin^4\stfrac{\tilde{\theta}}{2}$ and $\cos^4\stfrac{\tilde{\theta}}{2}$
can also be interpreted in terms of the topological responses
$(A_{\varphi}'(\theta_c^-))^2$ and $(A_{\varphi}'(\theta_c^+))^2$; see
Appendix~\ref{AppendixA}. The transition probabilities at $\theta=0$
from the right-handed light source ($\Gamma_{+}$) and at $\theta=\uppi$
from the left-handed light source ($\Gamma_{-}$) reveal quantized
responses  $\Gamma_{+}(0) = \Gamma_{-}(\uppi)$. The quantized light
information at resonance then reveals the definition of the topological
invariant resolved at the Dirac points. Fixing $\tilde{\theta}$, when
varying $\omega$ we can then resolve the form factor associated to each
light response. When crossing the phase transition, the two poles
become equivalent from the eigenstates structure. In that case,
$\Gamma_{+}(0)=\Gamma_{+}(\uppi)$ from the energy conservation. The light
responses also reveal the pseudo-spin state at the two Dirac  points. 

Related to the left- and right-handed waves on the sphere, in the plane
this turns into
{\begin{equation}
\zeta \phi_p \pm \uppi = \pm \omega t = \varphi
\end{equation}}\unskip
Compared to the definitions on the sphere, this implies
$\Gamma_{+}(\uppi)\rightarrow \Gamma_{LP}(\mathbf{K}')$ and
$\Gamma_{-}(\uppi)\rightarrow \Gamma_{RP}(\mathbf{K}')$ where $LP$ and
$RP$ designate circular light and right polarizations in the plane.
Then, we verify the topological structure
${\Gamma}_{RP}(\mathbf{K})={\Gamma}_{RP}(\mathbf{K}')$ within the
topological phase and
${\Gamma}_{RP}(\mathbf{K})={\Gamma}_{LP}(\mathbf{K}')$ within the
trivial phase. For these responses,  we have
$A_0=\sfrac{\mathcal{E}}{\Delta(\mathbf{k})}$ where $\Delta(\mathbf{k})$
corresponds to the resonance energy gap for a wavevector $\mathbf{k}$
and  $\mathcal{E}$ is the electric field.  At the transition, the gap
is closing at the $K'$ point which means effectively that
${\Gamma}_{RP}(\mathbf{K}')={\Gamma}_{LP}(\mathbf{K}')=0$ if we assume
a light source with $\omega\neq 0$.

We emphasize on the fact that the detection of the topological
invariant through circularly polarized light attracts some attention in
the community related to the quantum Hall
conductivity~\cite{NathanPeter,DFG,Japan}. The light signal is then
related to the integral of the Berry curvature when summing the
photo-induced currents on all momenta in the Brillouin zone. In
Ref.~\cite{PhilippAdolfoKaryn}, we have suggested that the same
photo-induced response written in terms of Berry curvatures may be
revealed from the Dirac points. Here, we elaborate on this fact and
show that the method from the Dirac points is very efficient
quantitatively to detect the quantum phase transition. The
photo-induced response for the {\it currents} takes the
form~\cite{NathanPeter,PhilippAdolfoKaryn},
{\begin{equation}
\Gamma_{\lambda}(\omega,\mathbf{k}) = \frac{2\uppi}{\hbar}
\left(\frac{{\mathcal E}}{\Delta(\mathbf{k})}\right)^2 
\left|\langle u_{\mathbf{k}}| \left(\lambda \mathrm{i} 
\frac{\partial H}{\partial p_x}+\frac{\partial H}{\partial p_y}\right)
|l_{\mathbf{k}}\rangle\right|^2 
\delta(\hbar\omega-\Delta(\mathbf{k})).
\end{equation}}\unskip
In this formula the lowest and upper energy eigenstates are introduced
as $\left|l_{\mathbf{k}}\right\rangle$ and 
$\left|u_{\mathbf{k}}\right\rangle$, $\lambda$
refers to a right or left circular polarization corresponding to
$\lambda=\pm$ respectively. In Section~\ref{Map}, $p_x$ and $p_y$ are
introduced as wavevector components measured from each Dirac point. 
These responses are in accordance with the transition probabilities
introduced on the sphere and the selection rules mentioned above.  It
is then useful e.g.\ to integrate the responses in frequency. In
experiments, for $t_2\sim 0.1t$, this is equivalent to fix
$\hbar\omega=\Delta(\mathbf{K})$ and integrate on momenta (see
Figure~\ref{LightResponse}) and similarly for the response at the other
Dirac point obtained when fixing the resonance
$\hbar\omega=\Delta(\mathbf{K}')$. We can then verify the precise
result
{\begin{equation}
{\Gamma}_{RP}(\mathbf{K})-{\Gamma}_{LP}(\mathbf{K})=
\frac{2\uppi}{\hbar}\frac{{\mathcal E}^2}{\Delta(\mathbf{K})^2} \langle
u_{\mathbf{K}} | \mathrm{i} \partial_{p_x} H | l_{\mathbf{K}}\rangle
\langle l_{\mathbf{K}} | \partial_{p_y} H | u_{\mathbf{K}}\rangle =
\frac{2\uppi}{\hbar} {\mathcal E}^2 F_{p_x p_y}(\mathbf{K})
\end{equation}}\unskip
with $F_{p_x p_y}$ the Berry curvatures evaluated in Section~\ref{Map}
in the presence of $M$. We also have
{\begin{equation}
{\Gamma}_{RP}(\mathbf{K}')-{\Gamma}_{LP}(\mathbf{K}')=
\frac{2\uppi}{\hbar}\frac{\mathcal{E}^2}{\Delta(\mathbf{K}')^2} \langle
u_{\mathbf{K}' }| \mathrm{i} \partial_{p_x} H | l_{\mathbf{K}'}\rangle
\langle l_{\mathbf{K}'} | \partial_{p_y} H | u_{\mathbf{K}'}\rangle =
\frac{2\uppi}{\hbar} {\mathcal E}^2 F_{p_x p_y}(\mathbf{K}').
\end{equation}}\unskip
If we add (sum) these two equations multiplying by
$\Delta(\mathbf{k})^2=4E_-(\tilde{\theta})^2$ resulting then in the
responses $\tilde{\Gamma}$ locally then  this measures the topological
invariant $C$. It is then relevant to introduce
{\begin{equation}
\label{kappa}
\kappa'=\frac{\tilde{\Gamma}_{RP}(\mathbf{K}) +
\tilde{\Gamma}_{RP}(\mathbf{K}') - \tilde{\Gamma}_{LP}(\mathbf{K}) - 
\tilde{\Gamma}_{LP}(\mathbf{K}')}{\tilde{\Gamma}_{RP}(\mathbf{K}) + 
\tilde{\Gamma}_{RP}(\mathbf{K}') +\tilde{\Gamma}_{LP}(\mathbf{K}) +
\tilde{\Gamma}_{LP}(\mathbf{K}')}=C,
\end{equation}}\unskip
with
{\begin{equation}
\tilde{\Gamma}_{RP}(\mathbf{K}) + \tilde{\Gamma}_{RP}(\mathbf{K}') 
+\tilde{\Gamma}_{LP}(\mathbf{K}) + \tilde{\Gamma}_{LP}(\mathbf{K}') = 
\frac{2\uppi}{\hbar} {\mathcal E}^2 (\hbar v_F)^2.
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig08}
\caption{\label{LightResponse}Light responses for the right- and left
circular polarizations within the topological phase and the trivial
phase for $t_2=0,1$ with $t=1$.  We evaluate numerically the
photo-induced currents and verify the analytical formulas for each
circular drive $\tilde{\Gamma}_{RP}$ and $\tilde{\Gamma}_{LP}$ at the
two Dirac points. Within the topological phase the two Dirac points
interact with the right-handed circular drive and we verify that the
height of the signal peaks remains invariant when varying $M$. We
verify the value of $\kappa'$ in the two phases. At the phase
transition, the gap is closing at the $K'$ Dirac point, i.e.\ the light
response with frequency $\omega\neq 0$ occurs only at the $K$ Dirac
point. The $\sfrac{1}{2}$ light response can then define the
$\sfrac{1}{2}$ invariant at the topological phase transition. Above the
transition, the $K'$ Dirac point interacts with the left-handed
circular drive and the $K$ Dirac point with the right-handed circular
wave.}
\end{figure}

We summarize the results in Figure~\ref{LightResponse}. Within the
topological phase, the right-handed circular drive induces two peaks at
$K$ and $K'$ of height $18\uppi\approx 56,55$ if we set $t_2=0.1$,
$t=1=\mathcal{E}$ and similarly for the lattice spacing. We emphasize
on the fact that the heights of these peaks remain identical within the
topological phase. When crossing the phase transition, we see clearly
that the $K'$ Dirac point now interacts with the left-handed circular
drive and the peak height remains the same. From the peak heights
resolved at the two Dirac points it is now possible to read if the
phase is topological or not. It also encodes quantitative information
on local Berry curvature and pseudo-spin response. 

Locating the transition with the topological invariant
$C_{1/2}=\sfrac{1}{2}$ then corresponds to detect half of the signal
from the two Dirac points compared to the topological phase (i.e.\ only
one peak with the same height or intensity  located at the $K$ Dirac
point), with the right-handed circularly polarized light. A half signal
is also measurable for the quantum anomalous Hall
semimetal~\cite{KarynSariah2} and quantum spin Hall
semimetal~\cite{KLHQSHSemimetal} in this way.

This analysis can be generalized to the quantum spin Hall
effect~\cite{KarynLight}. When coupling a plane of graphene with a
topological thin material described through a Haldane
model~\cite{Haldane}, we have shown the induction of a $\mathbb{Z}_2$
topological state where through proximity effect  the graphene system
acquires a topological number different in sign compared to the one in
the Haldane model~\cite{DFG2}. This produces a $\mathbb{Z}_2$
topological number  $C_h-C_g=\pm 2$ with e.g.\ $C_h=1$ and $C_g=-1$
referring to the topological numbers in the two planes. The proximity
effect is induced from a AA-BB stacking where we introduce $r$ as the
inter-planes hopping term. This generalizes the quantum spin Hall
effect of Kane and Mele~\cite{KaneMele1,KaneMele2} to the situation
with different $d_z$ components in the two planes. The graphene is then
described through a $\mathbf{d}_g$-vector of the form
{\begin{equation}
\mathbf{d}_g =
\left(d_x,d_y,-\frac{|r|^2}{|d_z^h(\mathbf{k})|}d_z^h(\mathbf{k})\right),
\end{equation}}\unskip
with $d_z^h$ as in Equation~(\ref{dvectormap}) with $M=0$. This
analysis shows that the quantum anomalous Hall effect with $\mathbb{Z}$
topological order~\cite{Haldane} and the quantum spin Hall effect with
$\mathbb{Z}_2$ topological number~\cite{KaneMele1,KaneMele2,Sheng} are
both present in coupled-planes systems when going from one to two
dimensions. The $\mathbb{Z}_2$ invariant is then measurable through
circularly polarized light~\cite{KarynLight}: in the thin material
described through a Haldane model the two Dirac points interact with
the right-handed circularly polarized light whereas in the topological
graphene plane the two Dirac points interact with the left-handed
circularly { polarized} light, following the precise protocol described
above. Inverting the sign of the invariant is equivalent to invert the
roles of $A$ and $B$ sublattices in the eigenstates at the two Dirac
points corresponding then to a modification of the direction of the
circular drive.

\section{Application to coupled planes materials through the Ramanujan
alternating infinite series} \label{coupledplanes}

\subsection{Alternating $\mathbb{Z}$ and $\mathbb{Z}_2$ topological
states in coupled planes models}

Here, we generalize the analysis to multi-layered systems and we are
questioning the thermodynamical situation in the vertical $z$ direction
perpendicular to the plane(s). We are then addressing a situation where
successive planes will show an alternating topological invariant of the
form $(-1)^j$ with $j=0,1,\ldots\,$. 

For three planes, this can be realized with a topological layer
described through the Haldane model in between two layers of graphene.
The protocol can be generalized adding further thin layers. In
principle, it is also possible to find a material where the Berry
curvature or topological invariant would alternate in sign in the $z$
direction. The system of two layers discussed above and in
Ref.~\cite{DFG2} was motivated from an analysis in ultra-cold atoms in
optical lattices that may be generalized for multi-layered systems with
alernating topological invariants. The model can also be realized when
assembling successively QAH thin materials and QSH thin materials.  The
inter-layer hopping term is smaller than the smallest energy gap in the
system such that topological properties are well defined in each plane.
For a finite number of planes, we can introduce two invariants, the
quantum Hall conductivity
{\begin{equation}
\sigma_{xy} = \frac{e^2}{h}\sum_{j=0}^N (-1)^j,
\end{equation}}\unskip
and from the divergence theorem a $\mathbb{Z}_2$ invariant that
describes the physics on top and bottom surfaces. In physics, the
divergence theorem was introduced associated to fluid mechanics by 
J.~L.~Lagrange in 1762 and was then addressed by C.~F.~Gauss and 
M.~Ostrogradsky. In mathematics, it is associated to the Green's theorem.
We can then rephrase the divergence theorem in the present situation as
{\begin{equation}
C_{\mathbb{Z}_2}=C_N - C_{N=0} = C_{N=0}((-1)^N -1) = 
\frac{1}{2\uppi}\iiint \frac{\partial F_z=j}{\partial z} \mathrm{d}z\,
\mathrm{d}k_x\, \mathrm{d}k_y,
\end{equation}}\unskip
such that the topological invariant in each plane reads
{\begin{equation}
C_j = \frac{1}{2\uppi}\oiint F_j\,\mathrm{d}k_x\, \mathrm{d}k_y  = (-1)^j 
\end{equation}}\unskip
with
{\begin{equation}
F_j =(-1)^j F_{k_x k_y} \delta_{zj} = F_{z=j}.
\end{equation}}\unskip
Here, $F_{k_x k_y}$ is the Berry curvature in one plane that
corresponds to $F_{p_x p_y}^{\zeta}$ close to each Dirac point, as
described in the preceding Sections. When $N$ is even with ($N+1$)
planes, the situation is similar to the $\mathbb{Z}$ quantum Hall
effect i.e.\ $C_{\mathbb{Z}_2}=0$ and $\sigma_{xy}=\sfrac{e^2}{h}$. When
$N$ is odd with ($N+1$) planes, then the situation is yet comparable to
the quantum spin Hall effect i.e.\ $C_{\mathbb{Z}_2}=+2$ and
$\sigma_{xy}=0$. For $N$ finite, we have a {\it even-odd} effect
alternating QAH and QSH states successively.

A question then comes from the infinite series of Ramanujan
{\begin{equation}
{\mathcal S} = \sum_{j=0}^{\mathcal R} (-1)^j = \frac{1}{2}.
\end{equation}}\unskip
The symbol ${\mathcal R}$ refers to the Ramanujan way of performing the
infinite series. We are then wondering what is the physical meaning of
this $\sfrac{1}{2}$ related to the material i.e.\ how should we think
about the thermodynamical limit then? From the physical point of view,
one may argue that an interface between a QAH state and the vacuum or a
non-trivial material allows for an interface with a jump of the
topological invariant corresponding within our formulation to a
situation with $M=M_c$ implying a $\sfrac{1}{2}$ topological number of
one massive Dirac point characterized through a $\uppi$ Berry
phase~\cite{FractionalArticle,OneHalf}. In addition, surface states of
three-dimensional topological insulators  with one Dirac point on the
upper or bottom surface are also described through a similar QAH
state~\cite{FuKane,SekineNomura,Zhang}. Such surfaces then give rise to
a massive Dirac point with a half-quantized quantum Hall conductance
which is measured in layered systems~\cite{Mogi}. We propose then to
unify these two ways of thinking through the infinite alternating
series related to observables such as the quantum Hall response and the
response to circularly polarized light. { The geometrical analysis
presented in Appendix~\ref{AppendixB}, from the divergence theorem, yet
shows the possibility of a $\sfrac{1}{2}$ topological number  associated
to the thermodynamical limit, which may also have practical
applications for classical physics through the occurrence  of a
modulated electric or magnetic field alternating its sign in $z$
direction.} It is relevant to emphasize here the recent interest in
relating Ramanujan infinite series with topological quantum Hall
physics~\cite{Montambaux}.

\subsection{Ramanujan alternating infinite series: application in
materials and $\sfrac{1}{2}$ invariant at a topological phase transition
in real space through quantum Hall response and light}

To acquire a physical understanding of the Ramanujan series, we can
regularize the series in the sense of Abel~\cite{Candelpergher}
{\begin{equation}
{\mathcal S} = \lim_{\epsilon\rightarrow 0}\left(\sum_{j=0}^{\mathcal
A} (-1)^j(1-\epsilon)^j =  \frac{1}{1+(1-\epsilon)} =
\sum_{j=0}^{\mathcal A} (-1)^j(1-j\epsilon)\right).
\end{equation}}\unskip
Suppose we apply an electric field ${\mathcal E}$ in each plane where a
plane also corresponds to a sphere.  In this way we measure the quantum
Hall conductivity.  Then, the factor $(1-\epsilon j)$ is equivalent to
say that the electric field in each plane is effectively ${\mathcal
E}(1-\epsilon j)$. When $j\rightarrow +\infty$ there exists a limit for
which the effective electric field is zero or equivalently the pumped
transverse charge (Hall current) is zero. Equivalently, the charge
remains at the north pole on each sphere associated to each plane on
the right of the interface. The charge cannot be negative  because in
the view of Newton in Appendix~\ref{AppendixA} when reaching the north
pole a particle stays there.  Therefore, when we meet the plane such
that the quantum Hall current is zero then above this limit this is
similar as if we insert an infinity of layers which are topologically
trivial (e.g.\ an ordinary insulator of variable width) corresponding
then to add an infinite number of pairs of $(1-1)$ in the Ramanujan
series. This implies the presence of an interface that we can model
through a topological phase transition in real space through the
parameter $M$ introduced in preceding Sections: for planes below the
interface each plane may be described with a Semenoff mass $M=0$ (or
any $M$ smaller than $M_c$), then at the interface $M=M_c$ and above
the interface the material is trivial i.e.\ $M>M_c$.  In this sense, we
re-interpret the Ramanujan series which is equivalent to
{\begin{equation}\label{S}
{\mathcal S} = 1-{\mathcal S} = \tfrac{1}{2}
\end{equation}}\unskip
in a physical way as two equivalent infinite series
{\begin{equation}\label{S1}
{\mathcal S}_1 = \left(1-1+1-1 + 
%\tfrac{1}{2}
\raisebox{-4pt}{\inlinefig{fx01}}
\right) + (1-1) + (1-1) + (1-1)+\cdots = \tfrac{1}{2}
\end{equation}}\unskip
{\begin{equation}
\label{S2}
{\mathcal S}_2 = \left(1-1+1-1 +1 -
%\tfrac{1}{2}
\raisebox{-4pt}{\inlinefig{fx02}}
\right) + (1-1) + (1-1) + (1-1)+\cdots = {\mathcal S}_1 = \tfrac{1}{2}.
\end{equation}}\unskip
In the first summation, the gap closes on the layer 5 at the $K'$ Dirac
point corresponding to a topological invariant $C=+\sfrac{1}{2}$
(mentioned in blue in the equation above) within our analysis.  This is
precisely equivalent to add a Semenoff mass on the fifth layer such
that $M=M_c$. The plane 5 is similar to the first plane with $j=0$ and
has a topological invariant $+1$ with the definition of the series. The
trivial region then may be thought of as planes with $M>M_c$. In the
second summation, the gap closes on the layer 6 at the $K$ Dirac point
corresponding then to a topological invariant $C=-\sfrac{1}{2}$. In this
way, when saying ${\mathcal S}_1={\mathcal S}_2$ the infinity is indeed
reached when we meet the interface because the situations with a even
and a odd number of planes in the topological region become identical.
Multiplying  ${\mathcal S}_1={\mathcal S}_2$ by $\sfrac{e^2}{h}$ then
this is also equivalent to measure the quantum Hall response of the
material. This situation can also describe the situation where a
material would have a different height in different regions of the
sample, i.e.\ on one region the system will be physically described
through the series ${\mathcal S}_1$ and on one region the system will
be physically described through the series ${\mathcal S}_2$.

Below, we show how the (physical) regularization of infinite series can
also measure the response to circularly polarized light in the
material. First, we evaluate the response to circularly polarized light
for the mathematical situation of an infinite number of planes with
alternating topological numbers. We will then show that this response
is equivalent to the materials associated to Equations~(\ref{S1}) and
(\ref{S2}). The planes associated to an index $j$ which is even will
interact with the right-handed circularly polarized light and the
planes described with $j$ odd will interact with the left-handed
circularly polarized light. If we measure $\kappa'$ in
Equation~(\ref{kappa}) for each plane i.e.\ the topological invariant,
then the total response for the situation of the right-handed
circularly polarized light will be
{\begin{equation}\label{serieslight}
C_{j=0} + \sum_{j=1}^{+\infty} C_{2j} \frac{1}{j^s}.
\end{equation}}\unskip
The factor $\sfrac{1}{j^{\sfrac{s}{2}}}$ refers to the light propagation in
each plane through a power-law attenuation factor for the in-coming
vector potential in Section~\ref{light}. This series can be re-summed
as
{\begin{equation}\label{zeta}
C_{j=0}(1+\zeta(s)),
\end{equation}}\unskip
with $\zeta(s)$ the Riemann zeta function. { We suppose that each plane
is characterized through a resonance with light. Assuming a power-law
decay of the light signal in the bulk this would allow to reveal
Equation~(\ref{zeta}), such that the re-summation of the infinite
number of planes in the neighborhood of the plane at $j=0$ will then
reveal the Riemann zeta function $\zeta(s)$.} The interesting surprise
then is that if we extrapolate $s\rightarrow 0$ i.e.\ light can reach
the infinity in the vertical direction such that the result of the
measure will be
{\begin{equation}
C_{j=0}(1+\zeta(0)) = C_{j=0}\left(1-\tfrac{1}{2}B(0)\right) = 
\tfrac{1}{2}C_{j=0} = {\mathcal S}.
\end{equation}}\unskip
Here, $B(0)=+1$ is the Bernoulli number. {The limit $s\rightarrow 0$
may be reached when trying various forms of power-law attenuations from
extrapolation analysis.} This is similar to say that compared to the
result of one plane, the effect of the (many) other planes is then to
compensate for an additional ${-}\sfrac{1}{2}$. This is similar to the
result obtained from the geometrical analysis in
Appendix~\ref{AppendixB}. We obtain the same result if we apply a
left-handed circularly polarized light resonating with the planes with
odd $j$ indices and measure the heights of the red peaks located at the
two Dirac points. In that case, from the definition of $\kappa'$ the
result is $\stfrac{1}{2}C_{j=1}=-{\mathcal S}$. If the number of planes
is infinite, this is equivalent as if we modify $C_{j=0}\rightarrow
-C_{j=0}=C_{j=1}$ in Equation~(\ref{serieslight}). If we measure
circular dichroism in this way corresponding to measure the half
difference of the two signals (i.e.\ of the left and right  light
polarizations) then this measures ${\mathcal S}$. 

Now, we build a correspondence with the measures of the light responses
in the materials associated to the series ${\mathcal S}_1$ and
${\mathcal S}_2$. For the material associated to the series ${\mathcal
S}_1$, if we { collect} the information on the right-handed and
left-handed circularly polarized lights, i.e.\ we measure $\kappa'$ in
each plane through the peaks associated to the light responses,  then
two successive planes described through topological numbers $(+1 -1)$
or $(-1+1)$ will reveal zero. In this way, from the interface we will
measure physically a one-half response: the light signals will reveal
only one peak from the $K$ Dirac point. For the material associated to
the series ${\mathcal S}_2$, if we { collect} the information on the
right-handed and left-handed circularly polarized lights from the
planes five and six, then this will also reveal a one-half response
i.e.\ the response is also equivalent to the response of one Dirac point
within the topological phase. This analysis then supports that the
infinity is reached when meeting the interface. 

It is also important to mention that in the model of interacting
spheres revealing fractional topological
numbers~\cite{FractionalArticle}, when taking the thermodynamical
limit, we can also reach a $\sfrac{1}{2}$ topological number for each 
sphere. 

\section{Conclusion}\label{summary}

We have developed the formalism of local topological invariants
(markers) from the magnetic monopole to lattice models with an emphasis
on the Haldane model and on its topological quantum phase transition.
The numerical analysis presented in this work through the various
probes shows how this geometrical approach of local character in
momentum space is efficient, simple and useful. The local invariants
are associated to physical observables such as the quantum Hall
response and local response in reciprocal space to circularly polarized
light.  We introduce an effective magnetic moment for the monopole such
that the susceptibility response with respect to the additional
magnetic field along $z$ direction reveals the topological phase
transition. Through a proximity effect in coupled-planes systems, the
QAH effect and the QSH effect are alternatively related through a
even-odd effect. When taking the thermodynamical limit, we formulate a
correspondence (analogy) between a $\sfrac{1}{2}$ invariant at a
topological interface and the Ramanujan infinite alternating series.
Many challenging questions remain  to be addressed within this
geometrical approach that we hope will be useful to the community,
related to practical applications and to recent results on
photoluminescence results locally resolved in momentum
space~\cite{C2N}. Topological properties are also measurable in circuit
quantum electrodynamics lattices through a local pump probe in real
space~\cite{JulianKaryn}.

\section*{Acknowledgments}
KLH acknowledges interesting discussions at the Conference on
Quantum physics in curved spacetimes at Tours in Le Studium, Institute
for Advanced Studies, Loire Valley. AB is thankful to
Universities Paris-Saclay, Porto and Roma through the  Erasmus Mundus
Quarmen Master program, and to Ecole Polytechnique for the support in
his Master Thesis. 

\printCOI

\back{}
\appendix{}

\def\appendixlabel{Appendix \Alph{section}}

%\def\theequation{\thesection.\arabic{equation}}
%\setcounter{equation}{0}


\section{Relation to transport and quantum Hall response,  map onto the
cylinder} \label{AppendixA}

Here, we formulate an analogy with a charge $e$ navigating from north
to south in an electric field oriented along the unit vector related to
the dressed polar angle. The second Newton  equation is $\hbar
\dot{\tilde{\theta}}= e{\mathcal E}$. From quantum mechanics and
Parseval--Plancherel theorem, it is then possible to  show the existence
of a {\it transverse pumped current} of the
form~\cite{KLHReview,FractionalArticle}
{\begin{equation}
J_{\perp}(\tilde{\theta}) = \frac{Q_{\perp}}{T} = 
\frac{e A'_{\varphi}(\tilde{\theta}=\tilde{\theta}_c^-)}{T}.
\end{equation}}\unskip
The charge is measured at angle $\tilde{\theta}_c$ at time $T$ such
$\hbar \tilde{\theta}_c= e {\mathcal E} T$. This form of transverse
current can also be verified from many-body physics  related to the
quantum Hall response ~\cite{KLHReview,FractionalArticle} and can be
interpreted as a Karplus--Luttinger velocity~\cite{KarplusLuttinger}.
The definition of the $A'_{\varphi}$-variable is obtained from Stokes
theorem applied with two domains meeting at angle
$\tilde{\theta}_c$~\cite{KLHReview,FractionalArticle}. To validate
Stokes theorem and geometrical definitions for a radial magnetic field,
this is equivalent to say that this requires two reference points for
the vector potential~\cite{Nakahara} or for the Berry gauge potential
such that these two reference points may be the two poles belonging to
different domains or more generally the points defined as
$\tilde{\theta}(\theta=0)$ and $\tilde{\theta}(\theta=\uppi)$. On the
domain related to the north pole, we can introduce the 
definition~\cite{KLHReview,FractionalArticle}
{\begin{equation}
A'_{\varphi}(\tilde{\theta}<\tilde{\theta}_c) = 
A'_{\varphi}(\tilde{\theta}=\tilde{\theta}_c^-)= 
A_{\varphi}(\tilde{\theta}=\tilde{\theta}_c)-
A_{\varphi}(\tilde{\theta}(\theta=0))
\end{equation}}\unskip
and on the domain related to  $\tilde{\theta}(\theta=\uppi)$ we can then
introduce
{\begin{equation}
A'_{\varphi}(\tilde{\theta}>\tilde{\theta}_c) = 
A'_{\varphi}(\tilde{\theta}=\tilde{\theta}_c^+)= 
A_{\varphi}(\tilde{\theta}=\tilde{\theta}_c)-
A_{\varphi}(\tilde{\theta}(\theta=\uppi)).
\end{equation}}\unskip
These definitions can be viewed as transporting information from each
pole on a thin cylinder (similar to a candle or Dirac string) on each
side of the equator. The important point is that the function
$A_{\varphi}(\tilde{\theta})$ is continuous and differentiable on the
whole surface and the Berry functions are then introduced with the same
$\varphi$ coherent gauge, such that the topological invariant
equivalent reads
{\begin{equation}
C = A'_{\varphi}(\tilde{\theta}<\tilde{\theta}_c)  - 
A'_{\varphi}(\tilde{\theta}>\tilde{\theta}_c).
\end{equation}}\unskip
In the trivial phase,
$A'_{\varphi}(\tilde{\theta}<\tilde{\theta}_c)=A'_{\varphi}(\tilde{\theta}>
\tilde{\theta}_c)$ which is another way to say we have only one domain.
To obtain a quantized transverse topological response within the
topological phase we can e.g.\ measure the total pumped transverse
charge when $\theta=\uppi$ i.e.\ when
$\tilde{\theta}_c=\tilde{\theta}(\theta=\uppi)=\uppi$.  We measure
$A'_{\varphi}(\tilde{\theta}<\tilde{\theta}_c)=C$ such that
$A'_{\varphi}(\tilde{\theta}>\tilde{\theta}_c)=0$. For $M<B$ this
corresponds to a navigation from north to south pole on the effective
sphere associated to the dressed angle. The total transverse pumped
charge then measures the topological invariant $C$. 

Since the topological invariant $C$ can be measured from the poles we
can then progressively transform the geometry e.g.\ onto an ellipse or
onto a cylinder such that information at the poles remains invariant.
We assume a corresponding cylinder of fixed height $H=2$ such that the
total surface area of the cylinder of unit radius i.e.\ $2\uppi H$ is
equal to the surface area on the sphere. On the top and bottom disks of
the cylinder then we maintain the Berry gauge potentials to
$A_{\varphi}(\tilde{\theta}(\theta=0))$ and
$A_{\varphi}(\tilde{\theta}(\theta=\uppi))$ respectively. The variable
$u$ associated to the vertical axis of the cylinder is
$u=\cos\tilde{\theta}$ such that within the topological phase
$A_{\varphi}(u)=-u/2$.  Within the topological phase, the Berry
curvature is $F_{\varphi u}=-\partial_u A_{\varphi}=\sfrac{1}{2}$ on
the whole surface area of the cylinder~\cite{KLHReview} and at the
transition this is equivalent to modify $F_{\varphi u}=0$ for
$\tilde{\theta}>\sfrac{\uppi}{2}$ corresponding to maintain the Berry
gauge potential fixed to $A_{\varphi}(\tilde{\theta}(\theta=\uppi))=0$
in the region $\tilde{\theta}\in
\left]\sfrac{\uppi}{2};\uppi\right]$. The total transverse
pumped current then is not modified in the region
$\tilde{\theta}\in \left]\sfrac{\uppi}{2};\uppi\right]$.
We can now measure the conductance located at the two edges associated
with the vertical region of the surface area of the cylinder,
introducing a difference of potential $H\cdot E=(V_b-V_t)$ between bottom
and top of the cylinder.  We measure the transverse pumped current at
the final time to reach $\tilde{\theta}=\uppi$.  On the cylinder
geometry, the transverse pumped current then takes the form
{\begin{equation}
|J_{\perp}| = \frac{e^2}{h}2EC = \frac{e^2}{h} C(V_b-V_t) = I_b-I_t.
\end{equation}}\unskip
Within the topological phase with $C=1$, we identify edge currents
moving in different directions $I_b$ and $I_t$ at $z=-1$ and $z=+1$
respectively satisfying the Landauer-B\" uttiker
formula~\cite{Buttiker,Halperin}
{\begin{equation}
G = \frac{\mathrm{d}I}{\mathrm{d}V} = \frac{q^2}{h}C = \frac{e^2}{h}.
\end{equation}}\unskip
We can then reveal the quantized conductance of an edge state. When we
reach the transition $C=\sfrac{1}{2}$, within this formulation this
corresponds to a halved current on each edge.  For $B>M$, the
transverse pumped current is zero which corresponds effectively to a
charge staying at the north pole or top disk of the cylinder with
$I_b-I_t=0$.

\section{Divergence theorem for the coupled planes when reaching the
infinity limit} \label{AppendixB}

When we reach the dense limit of planes (planks), the Berry curvature
takes the form
{\begin{equation}
F_z = (-1)^z F_{k_x k_y} \theta(z) = {\mathrm{e}}^{\pm \mathrm{i}\uppi
z} F_{k_x k_y} \theta(z)
\end{equation}}\unskip
with the function $F_{k_x k_y}$ which does not depend on $z$. Here,
$\theta(z)$ is the Heaviside step function. This leads to
{\begin{equation}
\frac{\partial F_z}{\partial z} = \pm \mathrm{i}\uppi {\mathrm{e}}^{\pm
\mathrm{i}\uppi z} F_{k_x k_y}\theta(z) + {\mathrm{e}}^{\pm
\mathrm{i}\uppi z} F_{k_x k_y} \delta(z).
\end{equation}}\unskip
The divergence theorem requires us to evaluate
{\begin{eqnarray}
\frac{1}{2\uppi}\int_0^{+\infty} \mathrm{d}z \iint \mathrm{d}k_x\,
\mathrm{d}k_y  \left(\pm \mathrm{i}\uppi {\mathrm{e}}^{\pm
\mathrm{i}\uppi z} F_{k_x k_y}\theta(z) + {\mathrm{e}}^{\pm \mathrm{i}
\uppi z} F_{k_x k_y} \delta(z)\right).
\end{eqnarray}}\unskip
Integrating the second term on the variable $z$ from zero to $+\infty$
gives $\sfrac{C_{N=0}}{2}$. The first term is zero because we can
introduce the identity $(-1)^z = \sttfrac{{\mathrm{e}}^{\mathrm{i}\uppi z}
+ {\mathrm{e}}^{-\mathrm{i}\uppi z}}{2}$ which corresponds to add the
two terms with different signs above. From the divergence theorem, we
can then interpret the term $\sfrac{C_{N=0}}{2}$ as the effective result
on one surface when summing the effect of the infinite number of thin
planes. This surface  may be seen as the bottom plane from the
$\delta(z)$ function corresponding then to slightly transport this
topological charge from the interface to the first plane at $j=0$.

\printbibliography
\refinput{crphys20260109-reference.tex}

\end{document}
