# Chameleon Mechanism in Scalar Nonmetricity Cosmology

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

We investigate the dynamics and the phase-space evolution for the scalar
nonmetricity cosmology with a Chameleon mechanism. In particular, we consider
a spatially flat Friedmann–Lemaître–Robertson–Walker geometry and within the
framework of scalar nonmetricity theory, we consider a generalization of the
Brans-Dicke theory in nonmetricity gravity. Introducing a pressureless gas as
the matter source, we also incorporate a coupling function responsible for the
interaction. Our findings reveal that the choice of connection in nonmetricity
gravity significantly impacts the interaction between the scalar field and the
matter source. For one particular connection, we discover the absence of
asymptotic solutions with a nontrivial interacting component. More precisely,
in this scenario, the matter source does not directly interact with the scalar
field; however, there is an interaction with the dynamical degrees of freedom
provided by the connection.

Symmetric teleparallel; nonmetricity gravity; scalar field; Chameleon
mechanism; Interaction

##  I Introduction

Scalar nonmetricity theory sc1  is a generalization of Symmetric Teleparallel
Equivalent General Relativity (STEGR) Nester:1998mp , where a scalar field
nonminimally coupled to gravity is introduced in the gravitational Action
Integral. This scalar field interacts with the nonmetricity scalar through a
coupling function. The scalar nonmetricity theory serves as the analogue of
scalar tensor theory and scalar torsion theory in symmetric teleparallel
theory gravity, see the discussion in trinity . While STEGR is equivalent to
General Relativity, the presence of a minimally coupled scalar field maintains
this equivalence. However, in the case of a nonminimally coupled scalar field,
there is no equivalence between scalar nonmetricity and scalar tensor
theories.

In the $f\left(R\right)$ generalization of General Relativity Buda ; BOtt ,
the new geometrical degrees of freedom can be attributed to a scalar field,
and the gravitational theory is equivalent to a Brans-Dicke theory Brans  with
zero Brans-Dicke parameter, also known as the O’Hanlon theory Sotiriou ; odin1
; ff1 . In this work, we are interested in the Brans-Dicke extension in
nonmetricity theory anpal01 . This theory has the property of attributing the
geometrical degree of freedom provided by the $f\left(Q\right)$ generalization
of STEGR Koivisto2 ; Koivisto3 ; rev10 ; fq1 ; fq2 . While the scalar tensor
theories are mathematically related through conformal transformations, this is
not the case for scalar nonmetricity theories. Conformal transformations in
symmetric teleparallel gravity lead to gravitational theories where a boundary
term coupled to the scalar field contributes to the gravitational dynamics.
This implies that a nonminimally coupled scalar field in nonmetricity gravity
is not conformally equivalent to a minimally coupled scalar field in STEGR gg1
. The effects of conformal transformations on the physical properties in
scalar nonmetricity theory have been studied recently in pal2 . It was found
that there exists a unique correspondence for the asymptotic solutions under
conformal transformations, and exact solutions describing cosmic acceleration
remain invariant under the transformation that relates these conformally
equivalent theories.

Constraints of $f\left(Q\right)$-gravity with cosmological observations reveal
that the alternative theory challenge the standard $\Lambda$-cosmology fot1 ,
a similar result was found in fot2 ; fot3 . For an extended analysis we refer
the reader to fot4  where the inertia effects was considered in the analysis
of the observational data.

Within this study, we postulate an interaction mechanism between the
nonminimally coupled scalar field and the matter component int1 ; int2 ; int3
; int4 ; int5 ; int6 ; int7 . This interaction is because of a Chameleon
component in the gravitational Action Integral ch1 ; ch2 . The matter source
Lagrangian is coupled to the scalar field through an interacting functions.
Because of this function there is mass transfer between the scalar field and
the matter source. Hence, the mass of the scalar field depends on the density
of the matter source, which means that the mass of the scalar field is
different in open space and in compact objects, which gives a Chameleon
behaviour to the scalar field. The Chameleon mechanism in General Relativity
can have geometric origin and specifically it is related to the Weyl
Integrable Spacetime theory salim96 . The Chameleon mechanism is a particular
case of the general interaction theory and it has been widely applied in
gravitational physics and cosmology va5 ; va6 ; hot ; hot2 ; ame1 ; ame2 ;
pavon .

The cosmological Chameleon mechanism can be observable in strong gravitational
effects, specifically the cosmological chameleon may affects the orbits in the
solar system sol2 , see also the discussion in sol3 . Parameters of the
Chameleon mechanism have been constrained by the analysis of the data provided
by CAST sol4a  and XENON1T data sol4b . There are various tests which have
been proposed in the literature for the investigation of the chameleon
mechanism, we refer the reader to sol5 ; sol6  and references therein.
Therefore the study of a Chameleon mechanism in the modified theories of
gravities and specifically in the extension of the STEGR with a scalar field
is of special interest.

In this study we focus on the cosmological framework of a spatially flat
Friedmann–Lemaître–Robertson–Walker (FLRW) geometry, and we investigate the
dynamics and the asymptotic solutions in scalar nonmetricity theory with a
Chameleon component. For the matter source, we consider a pressureless dust
fluid source which represents the dark matter component of the universe. In
STEGR, the definition of the gravitational Lagrangian depends on the choice of
the connection, which differs from the Levi-Civita connection rev10 .
Specifically, the connection is considered to be flat and symmetric, leading
to a non-unique definition for the connection. In STEGR, the selection of the
connection does not impact the gravitational theory. However, this is not the
case in extensions of STEGR.

We will observe that for the cosmological model under consideration, the
choice of the connection plays a crucial role in the presence of the Chameleon
mechanism in cosmic evolution. Indeed, within the framework of scalar
nonmetricity theory with matter component coupled to the scalar field in the
gravitational Action, it follows that the Chameleon mechanism contributes to
the mass of the scalar field only for specific selection of the connection.
This result is different from that provided by the gravitational theories
where the Levi-Civita connection, i.e. extensions of General Relativity va5 ,
or the vierbein fields, i.e. teleparallelism, are applied va001 . That is an
interesting property of symmetric teleparallel theory.

For our study, we employ the method of dynamical system analysis. We utilize
the Hubble normalization approach to determine the existence of asymptotic
solutions in the phase-space. This approach has been extensively studied in
cosmology and has yielded many important results. It has been instrumental in
deriving selection rules for the viability of gravitational models and
understanding the initial condition problem dn1 ; dn2 ; dn3 ; dn4 ; dn5 .
Regarding nonmetricity theory, dynamical system analysis has also been widely
applied, as seen in df1 ; df2 ; df3 ; df4 ; df5  and references therein. The
structure of the paper is outlined as follows.

In Section II, we introduce the basic framework of STEGR. The Scalar
nonmetricity theory is presented in Section III, where we focus on the Brans-
Dicke analogue in nonmetricity theory. The Chameleon mechanism, which
describes the interaction between the scalar field and the matter source, is
introduced in Section IV. The cosmological application of the Chameleon
mechanism in scalar nonmetricity theory is discussed in Section V, where we
consider the background geometry to be the spatially flat FLRW spacetime. We
consider the three different families $\Gamma_{1},~{}\Gamma_{2}$ and
$\Gamma_{3}$ for the connection given by STEGR, and we present three different
sets of field equations. We find that only for connections $\Gamma_{1}$ and
$\Gamma_{3}$ does the mass of the scalar field depend on the Chameleon
mechanism, while for connection $\Gamma_{2}$, the Chameleon mechanism
introduces an interaction between the matter source and the geometrical
dynamical degrees of the connection. For connection $\Gamma_{2}$, there is no
dynamical interaction between the scalar field and the matter source.

In Section VI, we conduct a detailed analysis of the phase-space. We apply the
Hubble normalization approach and express the field equations in an equivalent
form of an algebraic-differential system. We determine the stationary points
and investigate the physical properties of the asymptotic solutions at these
points. Furthermore, we discuss the stability properties of these asymptotic
solutions. Specifically, for connection $\Gamma_{2}$, we find no asymptotic
solutions with nontrivial Chameleon mechanism. However, for connections
$\Gamma_{1}$ and $\Gamma_{3}$, we identify stationary points where the
Chameleon mechanism significantly impacts the universe’s evolution. These
points can act as sources or attractors, indicating that the Chameleon
mechanism in scalar nonmetricity cosmology can play an important role in both
early-time and late-time stages of the universe. The case of arbitrary
potential is discussed in Section VII. Finally, in Section VIII, we present
our conclusions.

##  II Symmetric teleparallel Gravity

We consider a four-dimensional manifold which is described by the metric
tensor $g_{\mu\nu}$ and the symmetric and flat connection
$\Gamma_{\mu\nu}^{\kappa}$. Therefore, the curvature and the torsion tensors
defined for the connection $\Gamma_{\mu\nu}^{\kappa}$ are always zero, that
is, trinity

$\displaystyle R_{\;\lambda\mu\nu}^{\kappa}$ $\displaystyle\equiv
2\left(\Gamma_{\;\lambda\left[\nu,\mu\right]}^{\kappa}+\Gamma_{\;\lambda[\nu}^{\sigma}\Gamma_{\;\mu]\sigma}^{\kappa}\right)=0,$
(1) $\displaystyle\mathrm{T}_{\mu\nu}^{\lambda}$ $\displaystyle\equiv
2\Gamma_{\;\left[\mu\nu\right]}^{\lambda}=0.$ (2)

However, the nonmetricity scalar defined

$Q_{\lambda\mu\nu}\equiv
g_{\mu\nu,\lambda}-\Gamma_{\;\lambda\mu}^{\sigma}g_{\sigma\nu}-\Gamma_{\;\lambda\nu}^{\sigma}g_{\mu\sigma},$
(3)

plays an important role in nonmetricity gravity.

We define the non-metricity conjugate tensor Koivisto2

$P_{\;\mu\nu}^{\lambda}=-\frac{1}{4}Q_{\;\mu\nu}^{\lambda}+\frac{1}{2}Q_{(\mu\phantom{\lambda}\nu)}^{\phantom{(\mu}\lambda\phantom{\nu)}}+\frac{1}{4}\left(Q^{\lambda}-\bar{Q}^{\lambda}\right)g_{\mu\nu}-\frac{1}{4}\delta_{\;(\mu}^{\lambda}Q_{\nu)}$
(4)

in which $Q_{\lambda}=Q_{\lambda\mu}^{\phantom{\lambda\mu}\mu}$,
$\bar{Q}_{\lambda}=Q_{\phantom{\mu}\lambda\mu}^{\mu}$. The nonmetricity
$Q=Q_{\lambda\mu\nu}P^{\lambda\mu\nu}~{}$is the fundamental scalar of STEGR.
The gravitational Action in STEGR is defined as

$S_{Q}=\int d^{4}x\sqrt{-g}Q\,.$ (5)

However, the nonmetricity scalar $Q$ is related to the Ricci scalar
$\tilde{R}$ defined by the Levi-Civita connection
$\tilde{\Gamma}_{\mu\nu}^{\kappa}$ of the metric tensor, with the relation
$\tilde{R}-Q=B_{Q},$where $B_{Q}$ is defined by the expression gg1
$B_{Q}=-\tilde{\nabla}_{\mu}\left(Q^{\mu}-\bar{Q}^{\mu}\right)$ and it is a
boundary term. Therefore, the Action Integral for STEGR reads

$\int d^{4}x\sqrt{-g}Q\simeq\int d^{4}x\sqrt{-g}\tilde{R}+\text{boundary
terms.}$ (6)

##  III Scalar nonmetricity Gravity

In scalar nonmetricity theory we assume the existence of a scalar field
$\phi$, coupled to the gravity where the gravitational Action Integral reads
sc1

$S_{ST\varphi}=\int
d^{4}x\sqrt{-g}\left(\frac{F\left(\varphi\right)}{2}Q+\frac{\omega\left(\varphi\right)}{2}g^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}+V\left(\varphi\right)\right),$
(7)

$F\left(\varphi\right)$ is the coupling function of the scalar field with the
nonmetricity scalar, and $V\left(\varphi\right)$ is the scalar field
potential. Function $\omega\left(\phi\right)$ defines only the “coordinate”
where the scalar field is defined, and it can be removed from the system with
the introduction of the new scalar field
$d\Phi=\sqrt{\omega\left(\varphi\right)}d\varphi~{}$anpal01 .

Variation with respect to the metric tensor in (7) gives the modified field
equations jjd1

$F\left(\varphi\right)G_{\mu\nu}+2F_{,\phi}\varphi_{,\lambda}P_{~{}~{}\mu\nu}^{\lambda}+g_{\mu\nu}V\left(\varphi\right)+\frac{\omega\left(\varphi\right)}{2}\left(g_{\mu\nu}g^{\lambda\kappa}\varphi_{,\lambda}\varphi_{,\kappa}-\varphi_{,\mu}\varphi_{,\nu}\right)=0.$
(8)

where $G_{\mu\nu}$ is the Einstein tensor of general relativity. Moreover, the
Klein-Gordon equation for the scalar field reads

$\frac{\omega\left(\varphi\right)}{\sqrt{-g}}g^{\mu\nu}\partial_{\mu}\left(\sqrt{-g}\partial_{\nu}\varphi\right)+\frac{\omega_{,\varphi}}{2}g^{\lambda\kappa}\varphi_{,\lambda}\varphi_{,\kappa}+\frac{1}{2}F_{,\varphi}Q-V_{,\varphi}=0.$
(9)

Finally, variation of (7) with respect to the
connection$~{}\Gamma_{\mu\nu}^{\kappa}$ gives the equation of motion

$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}F\left(\varphi\right)P_{\phantom{\mu\nu}\sigma}^{\mu\nu}\right)=0.$
(10)

When the latter equation is trivially satisfied, we shall say that the
connection $\Gamma_{\mu\nu}^{\kappa}$ is defined in the coincidence gauge.

In the following we consider the $F\left(\varphi\right)=\varphi$ and
$\omega\left(\varphi\right)=\frac{\omega_{0}}{\varphi}$, where the Action (7)

$S_{BD\varphi}=\int
d^{4}x\sqrt{-g}\left(\frac{\varphi}{2}Q+\frac{\omega}{2\varphi}g^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}+V\left(\varphi\right)\right),$
(11)

and takes the form of the Brans-Dicke analogue in nonmetricity theory. When
parameter $\omega=0$, the gravitational theory (11) is equivalent to the
symmetry teleparallel $f\left(Q\right)$-gravity, where the scalar field
$\varphi$ attributes the new dynamical degrees of freedom provided by the
nonlinear function $f\left(Q\right)$. Specifically, the scalar field is
introduced as $\varphi=f_{,Q}\left(Q\right)$ and
$V\left(\varphi\right)=f_{,Q}\left(Q\right)Q-f\left(Q\right)$.

We employ the change of variable $\varphi=e^{\phi}$ and the Brans-Dicke
(-like) Action Integral (10) takes the form of the dilaton analogue in
nonmetricity theory, that is, anpal01

$S_{D}=\int
d^{4}x\sqrt{-g}e^{\phi}\left(\frac{Q}{2}+\frac{\omega}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}+\hat{V}\left(\phi\right)\right)~{}\text{\
with }\hat{V}\left(\phi\right)=V\left(\phi\right)e^{-\phi}.$ (12)

for the dilaton field the gravitational field equations are

$G_{\mu\nu}+2\phi_{,\lambda}P_{~{}~{}\mu\nu}^{\lambda}+g_{\mu\nu}V\left(\phi\right)e^{-\phi}+\frac{\omega}{2}\left(g_{\mu\nu}g^{\lambda\kappa}\phi_{,\lambda}\phi_{,\kappa}-\phi_{,\mu}\phi_{,\nu}\right)=0.$
(13) $\omega\nabla_{\mu}\nabla^{\mu}\phi+\frac{1}{2}Q-V_{,\phi}e^{-\phi}=0.$
(14)
$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}e^{\phi}P_{\phantom{\mu\nu}\sigma}^{\mu\nu}\right)=0.$
(15)

Conformal transformations of the cosmological nonmetricity Brans-Dicke theory
have been investigated in pal2 . It was found that there exists a unique one-
to-one correspondence for asymptotic solutions between different frames, the
correspondence is independent on the selection of the connection. That result
differs from that of metric Brans-Dicke theory, which means that nonmetricity
theory opens new directions on the discussion of the physical frame.

The nonmetricity scalar $Q$ is related to the Ricci scalar $R$ for the Levi-
Civita connection of the metric tensor $g_{\mu\nu}$, with a boundary term,
that is $Q=R+B~{}$fbb . Consequently, by replacing in (7) it follows that the
theory deviates from the usual scalar-tensor model.

##  IV Chameleon Mechanism

Let us now introduce a matter component in the gravitational Action Integral
for the dilaton field (12) such that

$S=S_{D}+\int\sqrt{-g}g\left(\phi\right)L_{m}$ (16)

where $g\left(\phi\right)$ is the coupling function between the matter source
and the scalar field $\phi$. A nonconstant function $g\left(\phi\right)$
indicates that there exists energy transfer between the matter source and the
scalar field $\phi$. Lagrangian $L_{m}$ describes the matter term.

The gravitational field equations read

$G_{\mu\nu}=g\left(\phi\right)T_{\mu\nu}^{m}+T_{\mu\nu}^{\phi}~{},~{}$ (17)
$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}e^{\phi}P_{\phantom{\mu\nu}\sigma}^{\mu\nu}\right)=0.$
(18)

where tensor $T_{\mu\nu}^{\phi}$ attributes the scalar field dynamical terms
and $T_{\mu\nu}^{m}$ is the energy momentum tensor for the matter source. For
the comoving observer $u_{\mu}$ with $g_{\mu\nu}u^{\mu}u^{\nu}=-1$, the energy
momentum tensor for a perfect fluid is

$T_{\mu\nu}^{m}=\left(\rho_{m}+p_{m}\right)u_{\mu}u_{\nu}+p_{m}g_{\mu\nu},$
(19)

where $\rho_{m}$ is the energy density and $p_{m}$ is the pressure component.

The equation of motion for the matter source reads

$\tilde{\nabla}_{\nu}\left(g\left(\phi\right)T^{m~{}\mu\nu}+T^{\phi~{}\mu\nu}\right)=0$
(20)

where $\tilde{\nabla}_{\nu}$ is defined by the Levi-Civita connection. We
remark that we have assumed no interaction of the matter source with the
connection $\Gamma_{\mu\nu}^{\kappa}$. The latter equation of motion reads

$\tilde{\nabla}_{\nu}\left(f\left(\phi\right)T^{m~{}\mu\nu}\right)=f_{,\phi}\rho_{m}~{},~{}\tilde{\nabla}_{\nu}\left(T^{\phi~{}\mu\nu}\right)=-f_{,\phi}.$
(21)

Therefore, for the perfect fluid and the scalar field we find the modified
equations of motions

$\tilde{\nabla}_{\mu}\left(\rho_{m}\right)u^{\mu}+\left(\rho_{m}+p_{m}\right)\tilde{\nabla}_{\mu}u^{\mu}-\rho_{m}\tilde{\nabla}_{\mu}\ln\left(g\left(\phi\right)\right)u^{\mu}=0,$
(22)
$\tilde{\nabla}^{\nu}\left(2e^{\phi}\phi_{,\lambda}P_{~{}~{}\mu\nu}^{\lambda}+g_{\mu\nu}V\left(\phi\right)+\frac{\omega}{2}e^{\phi}\left(g_{\mu\nu}g^{\lambda\kappa}\phi_{,\lambda}\phi_{,\kappa}-\phi_{,\mu}\phi_{,\nu}\right)\right)+e^{\phi}\phi^{,\nu}G_{\mu\nu}+2\rho_{m}\tilde{\nabla}_{\mu}g\left(\phi\right)=0.$
(23)

In the following we consider the exponential coupling function

$g\left(\phi\right)=g_{0}e^{g\phi}.$ (24)

For this model, the latter equations of motion are simplified as follows

$\tilde{\nabla}_{\mu}\left(\rho_{m}\right)u^{\mu}+\left(\rho_{m}+p_{m}\right)\tilde{\nabla}_{\mu}u^{\mu}-\lambda\rho_{m}\phi_{,\mu}u^{\mu}=0,$
(25)
$\tilde{\nabla}^{\nu}\left(2e^{\phi}\phi_{,\lambda}P_{~{}~{}\mu\nu}^{\lambda}+g_{\mu\nu}V\left(\phi\right)+\frac{\omega}{2}e^{\phi}\left(g_{\mu\nu}g^{\lambda\kappa}\phi_{,\lambda}\phi_{,\kappa}-\phi_{,\mu}\phi_{,\nu}\right)\right)+e^{\phi}\phi^{,\nu}G_{\mu\nu}+2\rho_{m}\lambda
e^{\lambda\phi}\phi_{,\mu}=0.$ (26)

The exponential coupling function describes the simplest interacting scenario
and it is the toy model which has been applied before in the case of General
Relativity ch1 ; ch2 .

##  V Interaction in Nonmetricity Cosmology

In large scales the universe is considered to be homogeneous and isotropic
described by the spatially flat FLRW spacetime with line element

$ds^{2}=-N(t)^{2}dt^{2}+a(t)^{2}\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right],$ (27)

where $a\left(t\right)$ is the scale factor denotes the radius of the universe
and $N\left(t\right)$ is the lapse function. The Hubble function is defined as
$H=\frac{1}{N}\frac{\dot{a}}{a}$, where dot means total derivative with
respect to the $t~{}$parameter, i.e. $\dot{a}=\frac{da}{at}$. The FLRW
geometry for arbitrary scale factor $a\left(t\right)$ admits a sixth-
dimensional Killing algebra consisted by the six isometries of the three-
dimensional Euclidean space, three translations and three rotations.

We introduce a symmetric connection $\Gamma_{\mu\nu}^{\kappa}$, and the
requirements the connection to admit the symmetries of the FLRW spacetime and
the curvature tensor to be zero, leads to the following three families of
connections with nonzero coefficients Hohmann

$\Gamma_{1}:\Gamma_{\;tt}^{t}=\gamma(t),$
$\Gamma_{2}:\Gamma_{\;tt}^{t}=\frac{\ddot{\psi}(t)}{\dot{\psi}(t)}+\dot{\psi}(t),\quad\Gamma_{\;tr}^{r}=\Gamma_{\;rt}^{r}=\Gamma_{\;t\theta}^{\theta}=\Gamma_{\;\theta
t}^{\theta}=\Gamma_{\;t\varphi}^{\varphi}=\Gamma_{\;\varphi
t}^{\varphi}=\dot{\psi}(t),$

and

$\Gamma_{3}:\Gamma_{\;tt}^{t}=\ln\left(\dot{\Psi}\left(t\right)\right)^{\cdot},\quad\Gamma_{\;rr}^{t}=\frac{1}{\dot{\Psi}\left(t\right)},\quad\Gamma_{\;\theta\theta}^{t}=\frac{r^{2}}{\dot{\Psi}\left(t\right)},\quad\Gamma_{\;\varphi\varphi}^{t}=\frac{r^{2}\sin^{2}\theta,}{\dot{\Psi}\left(t\right)}$

and common coefficients
$\Gamma_{\theta\theta}^{r}=-r~{},~{}\Gamma_{\varphi\varphi}^{r}=-r\sin^{2}\theta$,
$\Gamma_{\varphi\varphi}^{\theta}=-\sin\theta\cos\theta~{},~{}\Gamma_{\theta\varphi}^{\varphi}=\Gamma_{\varphi\theta}^{\varphi}=\cot\theta$,
$\Gamma_{\;r\theta}^{\theta}=\Gamma_{\;\theta
r}^{\theta}=\Gamma_{\;r\varphi}^{\varphi}=\Gamma_{\;\varphi
r}^{\varphi}=\frac{1}{r}$. Functions
$\gamma\left(t\right),~{}\psi\left(t\right)$ and $\Psi\left(t\right)~{}$are
dynamical variables Hohmann .

Since the three connections are flat, it means that there exist coordinate
transformations in order to be written as $\Gamma_{\mu\nu}^{\kappa}=0$.
However, when we introduce the metric tensor (27) we make the selection for a
coordinate system, such that connection $\Gamma_{1}$ to be defined in the
coincidence gauge while connections $\Gamma_{2}$ and $\Gamma_{3}$ to be
defined in the noncoincidence gauge. Consequently, new dynamical degrees of
freedom are introduced in the gravitational field equations from these two
connections.

For the matter source we assume an ideal gas with constant equation of state
parameter, i.e. $p_{m}=w_{m}\rho_{m}$, then for the exponential coupling
function the equation of motion for the matter source (25) reads

$\dot{\rho}_{m}+3\left(1+w_{m}\right)H\rho_{m}-\lambda\dot{\phi}\rho_{m}=0$
(28)

from where it follows
$\rho_{m}\left(t\right)=\rho_{m0}a^{-3\left(1+w_{m}\right)}e^{\lambda\phi}$.
In the following we consider that the matter source describes the pressureless
dark matter, that is, $w_{m}=0$.

As far as the modified gravitational field equations (17) is concerned, in the
following lines we present the field equations for the dilaton field (12) for
the three different connections which describe the spatially flat FLRW
geometry.

Connection $\Gamma_{1}$ is defined in the coincidence gauge and the
corresponding field equations are

$\displaystyle
e^{\phi}\left(3H^{2}+\frac{\omega}{2}\dot{\phi}^{2}+V\left(\phi\right)e^{-\phi}\right)-g_{0}e^{\lambda\phi}\rho_{m}$
$\displaystyle=0,$ (29) $\displaystyle
e^{\phi}\left(2\dot{H}+3H^{2}+2H\dot{\phi}-\frac{\omega}{2}\dot{\phi}^{2}+V\left(\phi\right)e^{-\phi}\right)$
$\displaystyle=0.$ (30)

while the Klein-Gordon equation (26) is expressed as

$e^{\phi}\left(2\omega\left(\ddot{\phi}+3H\dot{\phi}\right)-6H^{2}+\omega\dot{\phi}^{2}+e^{-\phi}V_{,\phi}\right)+4g_{0}\lambda
e^{\lambda\phi}\rho_{m}=0.$ (31)

The field equations for the latter system is equivalent with that of the
scalar-torsion model in teleparallelism with a Chameleon mechanism va001 .

For connection $\Gamma_{2}$, the modified Friedmann’s equations read

$\displaystyle
e^{\phi}\left(3H^{2}+\frac{\omega}{2}\dot{\phi}^{2}+\frac{3}{2}\dot{\phi}\dot{\psi}+V\left(\phi\right)e^{-\phi}\right)-g_{0}e^{\lambda\phi}\rho_{m}$
$\displaystyle=0,$ (32) $\displaystyle
e^{\phi}\left(2\dot{H}+3H^{2}+2H\dot{\phi}-\frac{\omega}{2}\dot{\phi}^{2}-\frac{3}{2}\dot{\phi}\dot{\psi}+V\left(\phi\right)e^{-\phi}\right)$
$\displaystyle=0,$ (33)

while the equation of motion for the connection and the scalar field are

$\displaystyle
e^{\phi}\left(3\ddot{\psi}+2\omega\left(\ddot{\phi}+3H\dot{\phi}\right)+3\left(\ddot{\psi}+3H\dot{\psi}\right)-6H^{2}+\omega\dot{\phi}^{2}+2e^{-\phi}V_{,\phi}\right)+4g_{0}\lambda
e^{\lambda\phi}\rho_{m}$ $\displaystyle=0,$ (34)
$\displaystyle\ddot{\phi}+\dot{\phi}^{2}+3H\dot{\phi}$ $\displaystyle=0.$ (35)

From equation (35) it follows that $e^{\phi}=\int\frac{\phi_{0}}{a^{3}}dt$,
which means that the mass of the scalar field does not depend on the Chameleon
mechanism and there is not interaction between the scalar field and the fluid
source. However, equation (34) indicates that the scalar field $\psi$ which is
provided by the connection in the dynamical system interacts with the
Chameleon mechanism.

Finally, for the third connection, $\Gamma_{3}$, the system of the
cosmological field equations is expressed as follows

$\displaystyle\left(3H^{2}+\frac{\omega}{2}\dot{\phi}^{2}-\frac{3}{2a^{2}}\frac{\dot{\phi}}{\dot{\Psi}}+V\left(\phi\right)\right)-g_{0}e^{\lambda\phi}\rho_{m}$
$\displaystyle=0,$ (36) $\displaystyle
e^{\phi}\left(2\dot{H}+3H^{2}+2H\dot{\phi}-\frac{\omega}{2}\dot{\phi}^{2}-\frac{1}{2a^{2}}\frac{\dot{\phi}}{\dot{\Psi}}+V_{,\phi}e^{-\phi}\right)$
$\displaystyle=0.$ (37)

The equations of motion for the scalar field and the connection are

$\displaystyle
e^{\phi}\left(-\frac{3}{a^{2}}\left(\ddot{\Psi}-H\dot{\Psi}\right)+\dot{\Psi}^{2}\left(-6H^{2}+\omega\dot{\phi}^{2}+2\omega\left(\ddot{\phi}+3H\dot{\phi}\right)\right)+2V_{,\phi}e^{-\phi}\dot{\Psi}^{2}\right)+4g_{0}\lambda
e^{\lambda\phi}\rho_{m}$ $\displaystyle=0,$ (38)
$\displaystyle\dot{\Psi}\left(\dot{\phi}\left(H+\dot{\phi}\right)+\ddot{\phi}\right)-2\dot{\phi}\ddot{\Psi}$
$\displaystyle=0.$ (39)

From the modified Klein-Gordon equations (31), (35) and (39) we observe that
the contribution of the Chameleon mechanism in the mass of the scalar field
depends on the selection of the connection for the definition of the
nonmetricity scalar. For the coincidence gauge, and from the Klein-Gordon
equation (31) the term provided by the Chameleon mechanism in the Klein-Gordon
equation is similar to that of the metric theory. Nevertheless, for connection
$\Gamma_{2}$, and from the Klein-Gordon equation (35) it follows that the mass
of the scalar field is always zero, while the Chameleon mechanism interacts
with the connection. Finally, for the third connection $\Gamma_{3}$ and
equation (39) we observe that the scalar field is given by the expression
$e^{\phi}=\int\frac{\phi_{0}\dot{\Psi}^{2}}{a}d\not t$, which means that the
scalar field depends on the dynamical variable of the connection; however the
scalar field interacts indirectly with the Chameleon mechanism through the
field $\Psi\left(t\right)$.

The effects of the Chameleon mechanism in the scalar field mass depends on the
selection of the connection. For the coincidence gauge the Chameleon mechanism
has similarities with the metric theories, while for the noncoincidence gauge
the Chameleon mechanism interact with the scalar field indirectly.

##  VI Phase-space analysis

In this Section we investigate the asymptotic solutions and the general
evolution for the cosmological model of scalar nonmetricity theory with
Chameleon mechanism. For the different cosmological theories related to the
three different connections of nonmetricity theory, we define dimensionless
variables in the Hubble normalization cope  in order to write the field
equations as a set of algebraic-differential system.

The stationary points of the latter system describe asymptotic cosmological
solutions. From the stability properties of the stationary points we can
construct the global evolution for the dynamical and physical parameters of
the model and to understand the effects of the Chameleon mechanism in the
cosmological evolution. The new set of dependent and independent variables in
the Hubble normalization are

$x=\frac{\dot{\phi}}{2\sqrt{6}H}~{},~{}y=\frac{e^{-\phi}V\left(\phi\right)}{3H^{2}}~{},~{}\Omega_{m}=g_{0}e^{\left(\lambda-1\right)\phi}\frac{\rho_{m}}{3H^{2}}~{},$
(40)
$\kappa=\frac{V_{,\phi}}{V}~{},~{}v=\sqrt{\frac{3}{2}}\frac{\dot{\psi}}{H}~{},~{}\frac{1}{\bar{z}}=\sqrt{\frac{3}{2}}a^{2}\dot{\Psi}H~{},~{}\tau=\ln\left(a\right).$
(41)

Let $\tau$ be the new independent variable, then for each connection
$\Gamma_{1},~{}\Gamma_{2}$ and $\Gamma_{3}$ we can write the corresponding
reduced dynamical system. For the potential function $V\left(\phi\right)$ we
introduce the exponential function $V\left(\phi\right)=V_{0}e^{\kappa\phi}$,
in such a way that the parameter $\kappa$ to be always a constant. The
analysis for the exponential potential includes the derivation of the
stationary points for every potential function which can be described in the
limit by the exponential potential such that parameter $\kappa$ remains a
constant.

We are interested in the asymptotic limit and the exponential function
provides the asymptotics where the scalar field dominates in the field
equations and also it the case where it does not contribute in the evolution
of the universe. This has been widely discussed before for instance in ans11
and references therein.

In this case the dimension of the dynamical system is reduced by one.

For connection $\Gamma_{1}$, we find the dynamical system $DS_{I}$

$\displaystyle\Omega_{m}$ $\displaystyle=1+\omega x^{2}+y~{},$ (42)
$\displaystyle\frac{dx}{d\tau}$
$\displaystyle=x^{2}\left(\sqrt{6}\left(1+2\lambda\right)-3\omega
x\right)+3x\left(y-1\right)+\frac{\sqrt{6}}{\omega}\left(1+2\lambda-\left(\kappa-2\lambda\right)y\right),$
(43) $\displaystyle\frac{dy}{d\tau}$
$\displaystyle=y\left(\sqrt{6}\left(1+\kappa\right)x-3\left(\omega
x^{2}-\left(1+y\right)\right)\right).$ (44)

The second connection $\Gamma_{2}$ provides the dynamical system $DS_{II}$

$\displaystyle\Omega_{m}$ $\displaystyle=1+\omega x^{2}+y+xz~{},$ (45)
$\displaystyle\frac{dx}{d\tau}$
$\displaystyle=-\frac{3}{2}x\left(1-y+x\left(\omega x+z\right)\right),$ (46)
$\displaystyle\frac{dy}{d\tau}$
$\displaystyle=y\left(3\left(1+y\right)+x\left(\sqrt{6}\left(1+\kappa\right)-3\left(\omega
x+z\right)\right)\right),$ (47) $\displaystyle\frac{dz}{d\tau}$
$\displaystyle=\left(\sqrt{6}\left(1+2\lambda\right)-\frac{3}{2}z\right)\left(1+\omega
x^{2}+xz\right)+y\left(\frac{3}{2}z-\sqrt{6}\left(\kappa-2\lambda\right)\right).$
(48)

Finally, for connection $\Gamma_{2}$ the corresponding dynamical system
$DS_{III}~{}$in the Hubble normalization is determined to be

$\displaystyle\Omega_{m}$ $\displaystyle=1+\omega x^{2}+y+xv,$ (49)
$\displaystyle\frac{\left(4\omega x+v\right)}{x}\frac{dx}{d\tau}$
$\displaystyle=\left(2\sqrt{6}\left(1+2\lambda\right)+\frac{v}{2}-6\omega
x\right)\left(1+\omega x^{2}\right)+2v~{}...$
$\displaystyle~{}~{}~{}~{}~{}...~{}+y\left(\frac{3}{2}z-2\sqrt{6}\left(\kappa-2\lambda\right)\right)+x\left(6\omega
y+\left(4\sqrt{6}\lambda+\frac{v}{2}\right)v\right),$ (50)
$\displaystyle\frac{dy}{d\tau}$
$\displaystyle=y\left(3\left(1+y\right)+x\left(\sqrt{6}\left(1+\kappa\right)-3\omega
x+v\right)\right),$ (51) $\displaystyle\frac{3\left(4\omega
x+v\right)}{v}\frac{dv}{d\tau}$
$\displaystyle=2\sqrt{6}\left(1+2\lambda\right)+\omega x^{2}\left(12\omega
x-z+2\sqrt{6}\left(2\lambda-3\right)\right)+3z~{}...$
$\displaystyle...~{}-x\left(4\omega\left(1+3y\right)+z\left(2\sqrt{6}\left(1-2\lambda\right)+z\right)\right)-3y\left(3z+2\sqrt{6}\left(\kappa-2\lambda\right)\right).$
(52)

For the above dynamical systems the dynamical variables are not constraint
$\left\\{\Omega_{m},x,y,z,v\right\\}$. For instance the effective energy
density for the matter source $~{}\Omega_{m}$ can be expressed as
$\Omega_{m}=g_{0}e^{\left(\lambda-1\right)\phi}\hat{\Omega}_{m}$,
$\hat{\Omega}_{m}=\frac{\rho_{m}}{3H^{2}}$; nevertheless
$0\leq\hat{\Omega}_{m}\leq 1$, the range of the dynamical variable
$\Omega_{m}$ depend on the value of $g_{0}$, and of the scalar
$\left(\lambda-1\right)\phi$. Below we focus in the case where the dynamical
variables take values in the finite regime.

###  VI.1 Dynamical system $DS_{I}$

We consider connection $\Gamma_{1}$ where the physical parameters are
described by the dynamical system $DS_{I}\,$. In the following we study the
phase-space of system $DS_{I}$.

The stationary points $A=\left(x\left(A\right),y\left(A\right)\right)$ for the
dynamical system (42)-(44) describe asymptotic solutions where the effective
cosmological fluid has the equation of state parameter
$w_{eff}^{\Gamma_{1}}\left(x,y\right)=\left(2\sqrt{\frac{2}{3}}-\omega
x\right)x+y$.

Thus, from the dynamical system (42)-(44) we derive the points:

  * •

$A_{1}=\left(\sqrt{\frac{2}{3}}\frac{\left(1+2\lambda\right)}{\omega},0\right),$

with
$\Omega_{m}\left(A_{1}\right)=1+\frac{2\left(1+2\lambda\right)^{2}}{3\omega}$.
The stationary point exist for $\omega\neq 0$. It describes a universe where
the scalar field and the matter source contribute in the cosmological
evolution and
$w_{eff}^{\Gamma_{1}}\left(A_{1}\right)=\frac{2-8\lambda^{2}}{\omega}$. The
physical properties of the asymptotic solution depend on the coupling
parameter $\omega$, and on the parameter $\lambda$ of the interaction term,
thus this asymptotic solution can describe the cosmic acceleration for
specific values of the free parameters. The eigenvalues of the linearized
system (43), (44) around the stationary point are
$e_{1}\left(A_{1}\right)=-\frac{2+8\lambda\left(1+\lambda\right)+3\omega}{\omega}$,
$e_{2}\left(A_{1}\right)=3+\frac{2\left(\kappa-2\lambda\right)\left(1+2\lambda\right)}{\omega}$,
from where we conclude that $A_{1}$ is an attractor when
$\kappa<\frac{4\lambda\left(1+2\lambda\right)-3\omega}{2\left(1+2\lambda\right)}$
with$~{}\left\\{\lambda<-\frac{1}{2},\omega<-\frac{2}{3}\left(1+2\lambda\right)^{2}\right\\}$,
$\left\\{\lambda>-\frac{1}{2},\omega>0\right\\}$ or
$\kappa>\frac{4\lambda\left(1+2\lambda\right)-3\omega}{2\left(1+2\lambda\right)}$
with $\left\\{\lambda<-\frac{1}{2},\omega>0\right\\}$ or
$\left\\{\lambda>-\frac{1}{2},\omega<-\frac{2}{3}\left(1+2\lambda\right)^{2}\right\\}$.

  * •

$A_{2}=\left(\frac{1+\kappa}{\sqrt{6}\omega},-1-\frac{\left(1+\kappa\right)^{2}}{6\omega}\right),$

with $\Omega_{m}\left(A_{2}\right)=0$,
$w_{eff}^{\Gamma_{1}}\left(A_{2}\right)=-1+\frac{1-\kappa^{2}}{3\omega}$ and
eigenvalues
$e_{1}\left(A_{2}\right)=-3-\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)}{\omega}$,
$e_{2}\left(A_{2}\right)=-3-\frac{\left(1+\kappa\right)^{2}}{2\omega}$. In
this asymptotic solution only the scalar field contributes in the universe,
and acceleration is occurred for specific values of the free parameters
$\omega$ and $\kappa$. As far as the stability of the point is concerned, for
$\kappa=-1$, the point is always an attractor, for $\kappa<-1$ the point is an
attractor when
$\left\\{\omega<-\frac{\left(1+\kappa\right)^{2}}{6},\lambda<\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$
or
$\left\\{\omega>0,\lambda>\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$
while for $\kappa>-1$, the point is an attractor for
$\left\\{\omega<-\frac{\left(1+\kappa\right)^{2}}{6},\lambda>\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$
or
$\left\\{\omega>0,\lambda<\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$.

  * •

$A_{3}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa-2\lambda},\frac{2\left(\kappa-2\lambda\right)\left(1+2\lambda\right)+3\omega}{2\left(\kappa-2\lambda\right)^{2}}\right),$

with
$\Omega_{m}\left(A_{3}\right)=\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)+3\omega}{\left(\kappa-2\lambda\right)^{2}}$,
$w_{eff}^{\Gamma 1}\left(A_{3}\right)=\frac{2\lambda-1}{\kappa-2\lambda}$ and
eigenvalues
$e_{1,2}\left(A_{3}\right)=\frac{3\left(1-\kappa+4\lambda\right)}{4\left(\kappa-2\lambda\right)}\pm\sqrt{\frac{9\left(1-7\kappa^{2}+8\lambda\left(7+10\lambda\right)-\kappa\left(26+24\lambda\right)\right)}{16\left(\kappa-2\lambda\right)^{2}}-\frac{3\left(1+\kappa\right)\left(1+2\lambda\right)}{\omega}-\frac{27\omega}{2\left(\kappa-2\lambda\right)^{2}}}$.
The point exist for $\kappa\neq 2\lambda$ and it describes an asymptotic
solution where the two fluids contribute in the cosmological evolution. The
asymptotic solution describes an acceleration universe for specific values of
the free parameters $\kappa$ and $\lambda$. The stationary point is an
attractor when the real components of the two eigenvalues are negative, that
is, when $\lambda\neq-\frac{1}{2}$, and
$\kappa\left(1+\kappa\right)+3\omega<2\lambda\left(1+\kappa\right)~{},~{}\sqrt{\left(1-\kappa+4\lambda\right)^{3}\left(1-49\kappa+100\lambda\right)}+\kappa\left(26+7\kappa+24\lambda\right)+48\omega>1+8\lambda\left(7+10\lambda\right)$
or
$1+\sqrt{\left(1-\kappa+4\lambda\right)^{3}\left(1-49\kappa+100\lambda\right)}+8\lambda\left(7+10\lambda\right)<\kappa\left(26+7\kappa+24\lambda\right)+48\omega$,
$2\left(\kappa-2\lambda\right)\left(1+2\lambda\right)+3\omega>0$. In the
special case where $\kappa=-1$, the stationary point is attractor when
$\frac{2}{3}<\frac{\omega}{\left(1+2\lambda\right)^{2}}<\frac{5}{6}$.

  * •

$A_{4}^{\pm}=\left(\pm\frac{i}{\sqrt{\omega}},0\right),$

with $\Omega_{m}\left(A_{4}^{\pm}\right)=0$,
$w_{eff}^{\Gamma_{1}}\left(A_{4}^{\pm}\right)=1\pm\frac{2i}{3\sqrt{\omega}}\sqrt{\frac{2}{3}}$
and eigenvalues $e_{1}\left(A_{4}^{\pm}\right)=3\pm
i\left(1+2\lambda\right)\sqrt{\frac{6}{\omega}}$,
$e_{2}\left(A_{4}^{\pm}\right)=6\pm
i\left(1+\kappa\right)\sqrt{\frac{6}{\omega}}$. The stationary points are real
and physically accepted when $\omega<0$. They describe asymptotic solutions
where only the scalar field contributes in the universe and acceleration is
occurred only at the stationary point $A_{4}^{-}$ for $-\frac{8}{3}<\omega<0$.
Point $A_{4}^{+}$ is an attractor when
$\left\\{\left(1+2\lambda\right)<-3\sqrt{\frac{\left|\omega\right|}{6}},~{}\left(1+\kappa\right)<-6\sqrt{\frac{\left|\omega\right|}{6}}\right\\}$
while point $A_{4}^{-}$ is an attractor for
$\left\\{\left(1+2\lambda\right)<3\sqrt{\frac{\left|\omega\right|}{6}},~{}\left(1+\kappa\right)<6\sqrt{\frac{\left|\omega\right|}{6}}\right\\}$.

The above results are summarized in table 1.

Table 1: Stationary points and physical parameters for dynamical system $DS_{I}$. Point | Existence | Interaction | $\mathbf{\Omega}_{m}$ | $\mathbf{w}_{eff}^{\Gamma_{1}}$ | Can be Attractor?
---|---|---|---|---|---
$A_{1}$ | $\omega\neq 0$ | Yes | $1+\frac{2\left(1+2\lambda\right)^{2}}{3\omega}$ | $\frac{2-8\lambda^{2}}{\omega}$ | Yes
$A_{2}$ | $\omega\neq 0$ | No | $0$ | $-1+\frac{1-\kappa^{2}}{3\omega}$ | Yes
$A_{3}$ | $\kappa\neq 2\lambda$ | Yes | $\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)+3\omega}{\left(\kappa-2\lambda\right)^{2}}$ | $\frac{2\lambda-1}{\kappa-2\lambda}$ | Yes
$A_{4}^{\pm}$ | $\omega<0$ | No | $0$ | $1\pm\frac{2i}{3\sqrt{\omega}}\sqrt{\frac{2}{3}}$ | Yes

###  VI.2 Dynamical system $DS_{II}$

For the second connection, namely $\Gamma_{2}$, the evolution of the physical
variables is given by the dynamical system $DS_{II}$.

The stationary points
$B=\left(x\left(B\right),y\left(B\right),z\left(B\right)\right)$ for the
dynamical system (45)-(48) describe asymptotic solutions with equation of
state parameter for the cosmological fluid
$w_{eff}^{\Gamma_{2}}\left(x,y,z\right)=y-x\left(\omega
x+z-\frac{2\sqrt{6}}{3}\right)$.

We calculate the stationary points:

  * •

$B_{1}=\left(x_{1},0,-\left(\frac{1}{x_{1}}+\omega x_{1}\right)\right),$

with $\Omega_{m}\left(B_{1}\right)=0$ and
$w_{eff}^{\Gamma_{2}}\left(B_{2}\right)=1+2\sqrt{\frac{2}{3}}x_{1}$. Parameter
$x_{1}$ is arbitrary, which means that $B_{1}$ describes a family of points.
The eigenvalues of the linearized system are derived
$e_{1}\left(B_{1}\right)=0$,
$e_{2}\left(B_{1}\right)=6+\sqrt{6}\left(1+\kappa\right)x_{1}~{}$and
$e_{3}\left(B_{3}\right)=3+\sqrt{6}\left(1+2\lambda\right)x_{1}$. The
asymptotic solutions given by the family of points $B_{1}$ describe cosmic
acceleration when $x_{1}<-\sqrt{\frac{2}{3}}$. These points have been derived
before in pal2 , and the application of the Center Manifold Theorem indicates
that there is not any stable submanifold where the points $B_{1}$ are stable.

  * •

$B_{2}=\left(0,-1,\sqrt{\frac{2}{3}}\left(1+\kappa\right)\right),$

where $\Omega_{m}\left(B_{2}\right)=0$,
$w_{eff}^{\Gamma_{2}}\left(B_{2}\right)=-1$, and eigenvalues
$e_{1}\left(B_{2}\right)=-3$, $e_{2}\left(B_{2}\right)=-3$ and
$e_{3}\left(B_{3}\right)=-3$. Point $B_{2}$ describe the de Sitter universe
and it is always an attractor.

  * •

$B_{3}=\left(0,0,2\sqrt{\frac{2}{3}}\left(1+2\lambda\right)\right),$

where $\Omega_{m}\left(B_{3}\right)=1$,
$w_{eff}^{\Gamma_{3}}\left(B_{2}\right)=0$, and eigenvalues
$e_{1}\left(B_{3}\right)=3$, $e_{2}\left(B_{3}\right)=-\frac{3}{2}$ and
$e_{3}\left(B_{3}\right)=-\frac{3}{2}$. The asymptotic solution at point
$B_{3}$ describe a universe dominated by the dust fluid. $B_{3}$ is always a
saddle point which means that the asymptotic solution is unstable.

The above results are summarized in table 2.

Table 2: Stationary points and physical parameters for dynamical system $DS_{II}$. Point | Existence | Interaction | $\mathbf{\Omega}_{m}$ | $\mathbf{w}_{eff}^{\Gamma_{1}}$ | Can be Attractor?
---|---|---|---|---|---
$B_{1}$ | Always | No | $0$ | $1+2\sqrt{\frac{2}{3}}x_{1}$ | No
$B_{2}$ | Always | No | $0$ | $-1$ | Yes
$B_{3}$ | Always | No | $1$ | $0$ | No

###  VI.3 Dynamical system $DS_{III}$

For the dynamical system $DS_{III}$ of connection $\Gamma_{3}$, the stationary
points $C=\left(x\left(C\right),y\left(C\right),v\left(C\right)\right)$
describe asymptotic solutions with
$w_{eff}^{\Gamma_{3}}\left(x,y,v\right)=y+x\left(\frac{2}{3}\sqrt{6}-\omega
x+\frac{1}{3}z\right)$.

The stationary points of the dynamical system (49)-(52) are:

  * •

$C_{1}=\left(0,-1,-\sqrt{\frac{2}{3}}\left(1+\kappa\right)\right),$

from where we calculate $\Omega_{m}\left(C_{1}\right)=0$,
$w_{eff}^{\Gamma_{3}}\left(C_{1}\right)=-1$ and the three eigenvalues
$e_{1}\left(C_{1}\right)=-5$,
$e_{2}\left(C_{1}\right)=-3$,$~{}e_{3}\left(C_{1}\right)=2$. Point $C_{1}$
describes a de Sitter universe where the scalar field plays the role of the
cosmological constant. From the eigenvalues we infer that the point is always
a saddle point.

  * •

$C_{2}=\left(0,0,-2\sqrt{\frac{2}{3}}\left(1+2\lambda\right)\right),$

describes a universe dominated by the dust fluid,
$\Omega_{m}\left(C_{2}\right)=1$ and
$w_{eff}^{\Gamma_{3}}\left(C_{2}\right)=0$. The eigenvalues of the linearized
system are $e_{1}\left(C_{2}\right)=3$,
$e_{2}\left(C_{2}\right)=3$,$~{}e_{3}\left(C_{2}\right)=-\frac{1}{2}$ from
where infer that $C_{2}$ is always a saddle point.

  * •

$C_{3}=\left(\frac{1+\kappa}{\sqrt{6}\omega},-1-\frac{\left(1+\kappa\right)^{2}}{6\omega},0\right),$

with physical variables $\Omega_{m}\left(C_{3}\right)=0$ and
$w_{eff}^{\Gamma_{3}}\left(C_{3}\right)=-1+\frac{1-\kappa^{2}}{3\omega}$. The
asymptotic solution describes a universe dominated by the scalar field, while
acceleration can be occurred for specific values of the free parameters. The
eigenvalues of the linearized system are
$e_{1}\left(C_{3}\right)=-3-\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)}{\omega}$,
$e_{2}\left(C_{3}\right)=-3-\frac{\left(1+\kappa\right)^{2}}{2\omega}$ and
$e_{3}\left(C_{3}\right)=\frac{\kappa\left(2+3\kappa\right)-1+10\omega}{6\omega}$.
Hence, point $C_{3}$ is an attractor when
$\left\\{\kappa<-1,~{}\frac{\left(1+\kappa\right)\left(3\kappa-1\right)}{10}<\omega<-\frac{\left(1+\kappa\right)^{2}}{6},~{}\lambda<\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$
or
$\left\\{-1<\kappa<-\frac{1}{3},~{}0<\omega<\frac{\left(1+\kappa\right)\left(3\kappa-1\right)}{10}\right\\}$
or
$\left\\{\kappa>2,~{}\frac{\left(1+\kappa\right)\left(3\kappa-1\right)}{10}<\omega<-\frac{\left(1+\kappa\right)^{2}}{6},~{}\lambda>\frac{\kappa\left(1+\kappa\right)+3\omega}{2\left(1+\kappa\right)}\right\\}$
.

  * •

$C_{4}=\left(\sqrt{\frac{2}{3}}\frac{1+2\lambda}{\omega},0,0\right),$

where
$\Omega_{m}\left(C_{4}\right)=1+\frac{2+8\lambda\left(1+\lambda\right)}{3\omega}$,
$w_{eff}^{\Gamma_{3}}\left(C_{4}\right)=\frac{2\left(1-4\lambda^{2}\right)}{3\omega}$
and eigenvalues
$e_{1}\left(C_{4}\right)=3+\frac{2\left(\kappa-2\lambda\right)\left(1+2\lambda\right)}{\omega}$,
$e_{2}\left(C_{4}\right)=\frac{1}{6}+\frac{4\lambda\left(1+3\lambda\right)-1}{3}$,
$e_{3}\left(C_{4}\right)=-\frac{3}{2}-\frac{4\lambda\left(1+\lambda\right)+1}{\omega}$.
This point describes a solution where the scalar field and the dust fluid
contributes in the cosmological solution. Point $C_{4}$ is an attractor if the
free parameters are constraint as follows
$\left\\{\lambda<-\frac{1}{2},\frac{\left(1-6\lambda\right)}{1+2\lambda}<\frac{\omega}{\left(1+2\lambda\right)^{2}}<-\frac{2}{3},\kappa<\frac{4\lambda\left(1+2\lambda\right)-3\omega}{2\left(1+2\lambda\right)}\right\\}$
or
$\left\\{-\frac{1}{2}<\lambda<\frac{1}{6},0<\frac{\omega}{\left(1+2\lambda\right)^{2}}<2\frac{\left(1-6\lambda\right)}{\left(1+2\lambda\right)},\kappa<\frac{4\lambda\left(1+2\lambda\right)-3\omega}{2\left(1+2\lambda\right)}\right\\}$
or
$\left\\{\lambda>\frac{1}{4},\frac{\left(1-6\lambda\right)}{1+2\lambda}<\frac{\omega}{\left(1+2\lambda\right)^{2}}<-\frac{2}{3},\kappa>\frac{4\lambda\left(1+2\lambda\right)-3\omega}{2\left(1+2\lambda\right)}\right\\}$.

  * •

$C_{5}^{\pm}=\left(\pm\frac{i}{\sqrt{\omega}},0,0\right),$

exist when $\omega<0$. The physical properties of the asymptotic solutions are
similar to that of points $A_{4}^{\pm}$, that is,
$\Omega_{m}\left(C_{5}^{\pm}\right)=0$ and
$w_{eff}^{\Gamma_{3}}\left(C_{5}^{\pm}\right)=1\pm\frac{2i}{3\sqrt{\omega}}\sqrt{\frac{2}{3}}$.
We calculate the eigenvalues of the linearized system
$e_{1}\left(C_{5}^{\pm}\right)=3\pm
i\left(1+2\lambda\right)\sqrt{\frac{6}{\omega}}$,
$e_{2}\left(C_{5}^{\pm}\right)=6\pm
i\left(1+\kappa\right)\sqrt{\frac{6}{\omega}}$ and
$e_{3}\left(C_{5}^{\pm}\right)=-\left(\frac{4}{3}\pm
2i\sqrt{\frac{2}{3\omega}}\right)$. Thus, point $C_{5}^{+}$ is attractor for
$\left(1+2\lambda\right)<-3\sqrt{\frac{\left|\omega\right|}{6}},~{}\left(1+\kappa\right)<-6\sqrt{\frac{\left|\omega\right|}{6}}$,
while point $C_{5}^{-}$ is attractor for
$\left\\{\left(1+2\lambda\right)<3\sqrt{\frac{\left|\omega\right|}{6}},~{}\left(1+\kappa\right)<6\sqrt{\frac{\left|\omega\right|}{6}},~{}\omega<-\frac{3}{2}\right\\}$.

  * •

$C_{6}^{\pm}=\left(\frac{1\pm\sqrt{1-2\omega}}{\sqrt{6}\omega},0,\sqrt{\frac{2}{3}}\left(-2\pm\sqrt{1-2\omega}\right)\right),$

where $\Omega_{m}\left(C_{6}^{\pm}\right)=0$ and
$w_{eff}^{\Gamma_{3}}\left(C_{6}^{\pm}\right)=\frac{1}{9}+\frac{2\left(1\pm\sqrt{1-2\omega}\right)}{\omega}$.
The points are real when $\omega\leq\frac{1}{2}$, where easy we can infer that
only the asymptotic solution at point $C_{6}^{+}$ can describe cosmic
acceleration for $-\frac{3}{2}<\omega<0$. The eigenvalues are studied
numerically. We find that point $C_{6}^{-}$ can not be an attractor, while
point $C_{6}^{+}$ is attractor when
$\left\\{\omega<0,~{}\kappa>-\frac{4}{3}+\frac{5}{3}\sqrt{1-2\omega}\right\\}$
or
$\left\\{0<\omega<\frac{1}{2},~{}\kappa<-\frac{4}{3}+\frac{5}{3}\sqrt{1-2\omega}\right\\}$
and $\lambda$ is constraint as given in Fig. 1.

  * •

$C_{7}=\left(\frac{5}{1-3\kappa}\sqrt{\frac{2}{3}},\frac{2\left(9\kappa\left(8+3\kappa\right)+50\omega-9\right)}{3\left(1-3\kappa\right)^{2}},\frac{\sqrt{6}\left(1-\kappa\left(2+3\kappa\right)+10\omega\right)}{2\left(1-3\kappa\right)}\right),$

with $\Omega_{m}\left(C_{7}\right)=0$ and
$w_{eff}^{\Gamma_{3}}\left(C_{7}\right)=\frac{\kappa-7}{3\left(3\kappa-1\right)}$.
The point exist for $\kappa\neq\frac{1}{3}$ and describes an accelerated
universe for $\frac{1}{3}<\kappa<2$. The eigenvalues are determined
numerically. In Fig. 2 we present region plots in the space of the free
parameters $\left\\{\kappa,\lambda,\omega\right\\}$ where the stationary point
is attractor.

  * •

$C_{8}=\left(-\frac{1}{\kappa-2\lambda}\sqrt{\frac{2}{3}},\frac{2\left(\kappa-2\lambda\right)\left(1+2\lambda\right)+3\omega}{2\left(\kappa-2\lambda\right)^{2}},0\right),$

from where we calculate
$\Omega_{m}\left(C_{8}\right)=\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)+3\omega}{\left(\kappa-2\lambda\right)}$
and
$w_{eff}^{\Gamma_{3}}\left(C_{8}\right)=\frac{2\lambda-1}{\kappa-2\lambda}$.
The stationary point exist for $\kappa-2\lambda\neq 0$ and it can describe
cosmic acceleration for specific values of the free parameters
$\kappa,\lambda$. The eigenvalues are studied numerically. In Fig. 3 we
present region plots in the space of the free parameters
$\left\\{\kappa,\lambda,\omega\right\\}$ where the stationary point $C_{8}$ is
attractor.

  * •

$C_{9}=\left(\frac{1}{\sqrt{6}\left(1-6\lambda\right)},0,\frac{\sqrt{6}\left(\omega-2+8\lambda\left(1+3\lambda\right)\right)}{2\left(1-6\lambda\right)}\right),$

with
$\Omega_{m}\left(C_{9}\right)=\frac{2\left(12\lambda\left(6\lambda-1\right)+\omega\right)}{3\left(1-6\lambda\right)^{2}}$
and
$w_{eff}^{\Gamma_{3}}\left(C_{9}\right)=\frac{1-2\lambda}{3\left(1-6\lambda\right)}$.
The stationary point exist for $\lambda\neq\frac{1}{6}$ and it describes an
accelerated universe for $\frac{1}{6}<\lambda<\frac{1}{4}$. The eigenvalues
are studied numerically. In Fig. 4 we present region plots in the space of the
free parameters $\left\\{\kappa,\lambda,\omega\right\\}$ where the stationary
point $C_{9}$ is attractor.

The above results are summarized in table 3.

Figure 1: Region plot in the two-dimensional plane $\left\\{\lambda,\omega\right\\}$, where point $C_{6}^{+}$ is attractor. Figure 2: Region plots in the space of the free parameters $\left\\{\kappa,\lambda,\omega\right\\}$ where the three eigenvalues of the dynamical system $DS_{III}$ around the stationary point $C_{7}$ have negative real parts and $C_{7}$ is attractor. Figure 3: Region plots in the space of the free parameters $\left\\{\kappa,\lambda,\omega\right\\}$ where the three eigenvalues of the dynamical system $DS_{III}$ around the stationary point $C_{8}$ have negative real parts and $C_{8}$ is attractor. Figure 4: Region plots in the space of the free parameters $\left\\{\kappa,\lambda,\omega\right\\}$ where the three eigenvalues of the dynamical system $DS_{III}$ around the stationary point $C_{8}$ have negative real parts and $C_{8}$ is attractor. Table 3: Stationary points and physical parameters for dynamical system $DS_{III}$. Point | Existence | Interaction | $\mathbf{\Omega}_{m}$ | $\mathbf{w}_{eff}^{\Gamma_{1}}$ | Can be Attractor?
---|---|---|---|---|---
$C_{1}$ | Always | No | $0$ | $-1$ | No
$C_{2}$ | Always | No | $1$ | $0$ | No
$C_{3}$ | $\omega\neq 0$ | No | $0$ | $-1+\frac{1-\kappa^{2}}{3\omega}$ | Yes
$C_{4}$ | $\omega\neq 0$ | Yes | $1+\frac{2+8\lambda\left(1+\lambda\right)}{3\omega}$ | $\frac{2\left(1-4\lambda^{2}\right)}{3\omega}$ | Yes
$C_{5}$ | $\omega<0$ | No | $0$ | $1\pm\frac{2i}{3\sqrt{\omega}}\sqrt{\frac{2}{3}}$ | Yes
$C_{6}^{\pm}$ | $\omega\leq\frac{1}{2}$ | No | $0$ | $\frac{1}{9}+\frac{2\left(1\pm\sqrt{1-2\omega}\right)}{\omega}$ |  $C_{6}^{+}$ Yes
$C_{7}$ | $\kappa\neq\frac{1}{3}$ | No | $0$ | $\frac{\kappa-7}{3\left(3\kappa-1\right)}$ | Yes
$C_{8}$ | $\kappa\neq 2\lambda$ | Yes | $\frac{\left(1+\kappa\right)\left(\kappa-2\lambda\right)+3\omega}{\left(\kappa-2\lambda\right)}$ | $\frac{2\lambda-1}{\kappa-2\lambda}$ | Yes
$C_{9}$ | $\lambda\neq\frac{1}{6}$ | Yes | $\frac{2\left(12\lambda\left(6\lambda-1\right)+\omega\right)}{3\left(1-6\lambda\right)^{2}}$ | $\frac{1-2\lambda}{3\left(1-6\lambda\right)}$ | Yes

##  VII Arbitrary potential

Let us now focus in the case of the exponential coupling function (24) with
the arbitrary scalar field potential $V\left(\phi\right)$ where now parameter
$\kappa$ is not a constant. Specifically, the evolution of parameter $\kappa$
is given by the first-order differential equation

$\frac{d\kappa}{d\tau}=\sqrt{6}x\kappa^{2}\left(\Lambda\left(\kappa\right)-1\right)~{},~{}\Lambda\left(\kappa\right)=\frac{V_{,\phi\phi}V}{\left(V_{,\phi}\right)^{2}}.$
(53)

Hence, for an arbitrary potential $V\left(\phi\right)$, the dimension of the
dynamical systems $DS_{I}$, $DS_{II}$ and $DS_{III}$ is increased by one,
where the new equation is (53). The stationary points of the dynamical systems
$DS_{I}$, $DS_{II}$ and $DS_{III}$ should satisfy the additional constraint
$\sqrt{6}x\kappa^{2}\left(\Lambda\left(\kappa\right)-1\right)=0$. Thus, if
$\kappa_{0}$ is a root of the algebraic equation
$\kappa_{0}^{2}\left(\Lambda\left(\kappa_{0}\right)-1\right)=0$, then the
previous stationary points for the exponential potential are recovered.
Additional, the new stationary point which is introduced in the dynamical
system is that with $x=0$. For the dynamical system $DS_{I}$ stationary points
with $x=0$, exist only when $\kappa=-1$ or $\lambda=-\frac{1}{2}$. For the
dynamical system $DS_{II}$, we find only the stationary points $B_{2}$ and
$B_{3}$, while for the dynamical system $DS_{III}$ the stationary points
$C_{1}$ and $C_{2}$ exists. Hence, the physical properties of the asymptotic
solutions remain the same with the exponential potential. However, because of
the new dynamical variable, the stability properties are affected by the
definition of function $\Lambda\left(\kappa\right)$.

If we consider the general coupling function $f\left(\phi\right)$, then,
$\lambda$ is a dynamical variables and satisfies the equation

$\frac{d\lambda}{d\tau}=\sqrt{6}x\lambda^{2}\left(\hat{\Lambda}\left(\lambda\right)-1\right)~{},~{}\hat{\Lambda}\left(\lambda\right)=\frac{f_{,\phi\phi}f}{\left(f_{,\phi}\right)^{2}}\text{.}$
(54)

When the scalar field potential $V\left(\phi\right)$ is the exponential
function, such that $\kappa$ is constant, then the stationary points for the
extended dynamical systems $DS_{I}$, $DS_{II}$ and $DS_{III}$ are similar with
that which discussed before. Nevertheless, when $\kappa~{}$and $\lambda$ are
not constants, parameters $\lambda$ and $\kappa$ are not independent, and we
can express $\lambda=\lambda\left(\kappa\right)$. However, the families of the
stationary points are these which we have already discussed. Hence, our main
conclusions holds and for other functional forms for the scalar field
potential and the coupling function.

##  VIII Conclusions

In this study, we considered an interaction within the components of the dark
sector of the universe within the framework of gravitational theory with
nonmetricity. Specifically, we employed a pressureless fluid source to
describe the dark matter component of the universe, coupled with a scalar
field that is nonminimally coupled to the nonmetricity tensor to account for
the universe’s acceleration. To model this interaction, we introduced a
coupling function in the Action Integral, which gives rise to a Chameleon
mechanism.

For this gravitational model, we conducted a comprehensive analysis of the
dynamics governing the cosmological field equations within a spatially flat
FLRW geometry. We found that the cosmological history depends on the selection
of the connection which defines the nonmetricity scalar. We remark that the
definition of a flat and symmetric connection in FLRW geometry is not unique,
and for the spatially flat geometry there are three families of connections.

Connection $\Gamma_{1}$ provides the cosmological model with the fewest number
of dynamical variables. Utilizing the Hubble normalization approach, we
computed the stationary points for this connection. The field equations
corresponding to connection $\Gamma_{1}$ yield four families of stationary
points as summarized in Table 1. Two of these families describe universes with
a nontrivial interaction term, while the rest two points when they exist
describe scaling solutions, which can correspond to the asymptotic behaviour
of the early or time acceleration phases of the universe.

Moving to the second connection, $\Gamma_{2}$, we found that the field
equations admit three families of stationary points as presented in Table 2.
However, none of these points describe asymptotic solutions with an
interaction between the matter components of the universe. Lastly, the field
equations of connection $\Gamma_{3}$ allow for the maximum number of
stationary points, with nine families of such points identified with physical
properties are given in Table 3. There are three stationary points with
nonzero interaction term, one stationary point where the matter source
dominates in the cosmic evolution and five stationary points where the scalar
field dominates in the universe.

Among these, three families support the Chameleon mechanism. It is important
to note that our analysis was performed within a finite regime. Although the
dynamical variables are not constrained and can take values at infinity, we
omitted the analysis at infinity due to our finding that the stationary points
of the vacuum scalar nonmetricity model are recovered in this regime.

The findings from this study can serve as selection rules for the connection
in nonmetricity gravity when interaction is required in the gravitational
model. For connection $\Gamma_{2}$ we did not find any stationary point with
nonzero contribution of the Chameleon mechanism. From the Klein-Gordon
equation (35) of the second scalar field, it follows that the scalar field has
an exact solution, and there is not any contribution of the matter source to
the mass of the scalar field. It is clear that either in the perturbation
level that will not change. On the other hand, for the rest two connections,
namely, connections while for connections $\Gamma_{1}$ and $\Gamma_{3}$, the
admitted stationary points can describe the major eras of the cosmological
evolution, while they provide asymptotic solutions with interacting fluids.

Indeed, coexistence and nonzero interaction in the dark sector of the universe
is provided only by the connections $\Gamma_{1}$ and $\Gamma_{3}$. Thus, this
study can be used to define criteria for the selection of the connection in
nonmetricity theory. Indeed, if interaction is essential in the dark sector,
then connection $\Gamma_{2}$ is excluded from being physically accepted.

At this stage it is important to mention that because of the introduction of
the Chameleon mechanism, that is, of the nonzero parameter $\lambda$, new
stationary points of the cosmological field equations exist, with different
physical properties and stability behaviour for the majority of the stationary
points, either for that which does not describe coexistence and they do not
support a Chameleon term.

Recent studies per1 ; per2  on the analysis of the cosmological perturbations
in $f\left(Q\right)$-gravity indicate the pathological character of the theory
because of the presence of ghost or of the strong coupling problem. Although
the symmetric teleparallel Brans-Dicke model (11) is equivalent to
$f\left(Q\right)$-gravity when $\omega=0$, in the cosmological perturbation
theory, for $\omega\neq 0$, the scalar field perturbations provides dynamical
degrees of freedom, which may overpass the pathologies of
$f\left(Q\right)$-theory. On the other hand, the existence of the Chameleon
mechanism can affect the dynamical evolution for the cosmological
perturbations mot1 , such that to solve the strong coupling problem. However,
such analysis is out of the scopus of this work and will be studied elsewhere.

###### Acknowledgements.

AP thanks the support of VRIDT through Resolución VRIDT No. 096/2022 and
Resolución VRIDT No. 098/2022.

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# Single-photon scattering and bound states in a one-dimensional waveguide
with topological giant atom

Wei Zhao  Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China  Tian Tian  School of Science, Changchun
University, Changchun 130022, China  Zhihai Wang<EMAIL_ADDRESS>Center
for Quantum Sciences and School of Physics, Northeast Normal University,
Changchun 130024, China

###### Abstract

We investigate the single photon scattering and bound states in a coupled
resonator waveguide (CRW) which couples to a topological giant atom (TGA) via
two distant sites. Here, the TGA is constructed by a one dimensional Su-
Schrieffer-Heeger chain with finite length. By modulating the topological
phase of the TGA, the incident photon in the CRW can be completely reflected
or transmitted, and is therefore beneficial to design the coherent photonic
device. Meanwhile, we also achieve two pairs of bound states locating
respectively above and blow the continuum. Whether the gap is open or closed
depends on the boundary condition of the TGA. Therefore, the combination of
the topology and the interference provides us an exciting opportunity to
manipulate the photonic state in the context of waveguide QED.

##  I INTRODUCTION

The light-matter interaction plays a crucial role in the fundamental sciences
[1, 2], underpinning the rapid development of quantum technology. In recent
years, the light-matter interaction in waveguide structures has attracted a
lot of attentions, which have brought a lot of theoretical [3] and
experimental studies in the waveguide QED [4] community, such as dressed or
bound states [5, 6, 7, 8, 9, 10, 11], phase transitions [12, 13], single-
photon devices [14, 15, 16], exotic topological and chiral phenomena [17, 18,
19, 20, 21, 22]. At the same time, the dynamical control of single-photon
transmission is a hot topic in constructing quantum networks [23]. Since
photons provide a reliable output of quantum information and the single-photon
is considered to be one of the most suitable carriers of quantum information.
In quantum devices and quantum networks [24, 25, 26], one-dimensional (1D)
waveguides [27, 28] are essential light-matter interfaces, and the
controllable single-photon transport with linear and nonlinear dispersion
relationships has been extensively studied [11, 45, 54, 55, 56].

In the traditional treatment, the light-matter interaction is often modeled by
the dipole approximation, where the atoms are treated as point-like dipoles
[29]. However, in recent years, the study of the interaction in atom-waveguide
systems has been extended to the interaction between the photon and a giant
atom, where the non-local light-matter interaction occurs with multiple
points. In the giant atom community [30, 31, 32, 33, 34, 35, 36, 37], since
the interactions between the waveguide photons and the giant atoms include
multi-path quantum interferences, many new phenomena those do not exist in the
traditional small atom systems have been predicted, such as tunable bound
state [38, 39, 40, 41], decoherence free subspaces [42], electromagnetically
induced transparency (EIT) [43, 44], Autler-Townes splitting (ATS) phenomena
[45], chiral physics [46, 47, 48, 49, 50], and many others [51, 52, 53, 54,
55, 56]. The physical basis behind these phenomena is the interference and
delay effects in the propagation of the photon/phonon between different
coupling points.

On the other hand, Haldane and Raghu [57, 58] proposed to manipulate the
photon transport via topological structure in 2008, which paves the way for
the development of topological photonics [59, 60, 61, 62]. In the simplest one
dimensional case, the topological waveguide can be constructed [63, 64]
through the Su-Schrieffer-Heeger (SSH) [65] chain, which is characterized by
nonzero winding number or zero-mode edge state(s) in the topological
nontrivial phase with periodical or open boundary condition, respectively.

Based on the above achievements, it is meaningful to combine the topological
nature of the SSH chain and the interference effect from the giant atom, to
manipulate the single photon state in the context of the waveguide QED. To
this end, we construct a topological giant atom (TGA) by the SSH chain with
finite length, which couples to the coupled resonator waveguide (CRW) via two
distant sites. Our findings imply that the single photon scattering can be
modulated by the topological phase of the TGA and the bound state is
controlled by the boundary condition.

##  II MODEL AND HAMILTONIAN

As schematically shown in Fig. 1, the system we consider is composed of two
arrays of CRW. The upper one is a $N+1$ sites [i.e., $(N+1)/2$ pairs for odd
$N$] SSH chain and the lower one is an infinite CRW with uniform hopping
strength. Since the upper one supports a finite of discrete energy levels, we
name it as the topological giant atom (TGA), whose topological nature is
discussed below. The TGA couples to the CRW non-locally via two separate sites
of $j=0$ and $j=N$. The Hamiltonian $H$ of the system can be divided into
three parts, i.e., $H=H_{c}+H_{S}+H_{I}$.

Figure 1: Schematic diagram of coupling of a TGA to an infinite CRW via two
coupling point.

The first part $H_{c}$ represents the free Hamiltonian of the CRW, and is
expressed as

$H_{c}=\omega_{c}\sum_{j}a_{j}^{\dagger}a_{j}-\xi\sum_{j=-\infty}^{+\infty}\left(a_{j+1}^{\dagger}a_{j}+a_{j}^{\dagger}a_{j+1}\right),$
(1)

where $\omega_{c}$ is bare frequency of the resonators and $a_{j}^{\dagger}$
($a_{j}$) is the bosonic creation (annihilation) operator on site $j$, $\xi$
is the hopping strength between the nearest resonators.

The second part $H_{S}$ of the Hamiltonian $H$ represents the free Hamiltonian
of the TGA [consisting of $L$ unit cells, $L=\left(N+1\right)/2$ for odd $N$],
which can be written as

$\begin{split}H_{S}&=\sum_{l=1}^{L}\omega_{e}\left(C_{A,l}^{\dagger}C_{A,l}+C_{B,l}^{\dagger}C_{B,l}\right)\\\
&+t_{1}\sum_{l=1}^{L}\left(C_{A,l}^{\dagger}C_{B,l}+{\rm H.c.}\right)\\\
&+t_{2}\sum_{l=1}^{L-1}\left(C_{A,l+1}^{\dagger}C_{B,l}+{\rm H.c.}\right)\\\
&+t_{3}\left(C_{A,1}^{\dagger}C_{B,L}+{\rm H.c.}\right),\end{split}$ (2)

In the TGA, each unit cell hosts $A$ and $B$ two resonators with the identical
frequency $\omega_{e}$, and $C_{A,l}\left(C_{B,l}\right)$ is the annihilation
operators at the $A(B)$ sublattices of the $l$th unit cell (see Fig. 1). The
intra- and inter-cell hopping amplitudes are $t_{1}$ and $t_{2}$,
respectively. Here, we use the value of $t_{3}$ to distinguish the boundary
condition of the TGA. For the periodical boundary condition, we set
$t_{3}=t_{2}$. In this case, the TGA is topologically nontrivial with a
winding number of $w=1$ for $\left|t_{1}\right|<\left|t_{2}\right|$ and
topologically trivial with $w=0$ for $\left|t_{1}\right|>\left|t_{2}\right|$
[66]. For the open boundary condition, we set $t_{3}=0$, and the TGA supports
the boundary state in the topologically nontrivial phase, but not in the
trivial phase.

The third part $H_{I}$ of the Hamiltonian describes the coupling between the
TGA and the waveguide via the $0$th and $N$th resonators with the same
coupling strength $J$. Under the rotating wave approximation, the Hamiltonian
$H_{I}$ can be written as

$H_{I}=J\left(a_{0}^{\dagger}C_{A,1}+a_{N}^{\dagger}C_{B,L}+{\rm
H.c.}\right).$ (3)

##  III SINGLE-PHOTON SCATTERING

In this section, we will discuss the behavior of the single-photon scattering
by an open TGA ($t_{3}=0$). We consider that a single-photon with wave vector
$k$ is incident from the left side of the waveguide. Since the excitation
number in the system is conserved, the eigenstate in the single-excitation
subspace can be written as

$\left|E_{k}\right\rangle=\sum_{j}U_{j}a_{j}^{\dagger}\left|G\right\rangle+\sum_{l}X_{l}C_{A,l}^{\dagger}\left|G\right\rangle+\sum_{l}Y_{l}C_{B,l}^{\dagger}\left|G\right\rangle,$
(4)

where $\left|G\right\rangle$ represents that all of the resonators in the
waveguide are in the vacuum states, while the TGA is in the ground state
$\left|g\right\rangle$. $U_{j}$ is the probability amplitude for finding a
photonic excitation in the resonator $j$, and $X_{l}(Y_{l})$ describe the
excitation amplitudes in the site $A_{l}(B_{l})$ of the TGA. In the regimes of
$j<0$, $0\leqslant j\leqslant N$ and $j>N$, the amplitude $U_{j}$ can be
written in the form

$\begin{split}U_{j}=\begin{cases}e^{ikj}+re^{-ikj},&j<0\\\
Ae^{ikj}+Be^{-ikj},&0\leqslant j\leqslant N\\\
te^{ikj},&j>N,\end{cases}\end{split}$ (5)

where $r$ and $t$ are respectively the single-photon reflection and
transmission amplitudes. Furthermore, the second line in the above equation
implies that the incident photon can be transmitted (reflected) by the
left(right) leg of the TGA giant atom with amplitude $A$ ($B$). The
transmitted photon then propagates back and forth in the spatial regime
covered by the giant atom with $0\leqslant j\leqslant N$.

Solving the Schrödinger equation
$H\left|E_{k}\right\rangle=E\left|E_{k}\right\rangle$ in the region of $j\neq
0,N$, it yields a dispersion relationship of $E_{k}=\omega_{c}-2\xi\cos k$.
Furthermore, the continuity conditions at $j=0$ and $j=N$ give $1+r=A+B$ and
$Ae^{ikN}+Be^{-ikN}=te^{ikN}$, respectively.

Figure 2: (a) Schematic diagram of the model when $N=1$. (b) Schematic diagram
of TGA where only one of the two sites is couple to CRW via one resonator. (c)
The reflection rate $R$ as a function of detuning $\Delta_{2}$ for $N=1$. The
parameters are set as $t_{1}=0.5\xi$, $\omega_{e}=\omega_{c}=20\xi$.

We begin with the simplest case with $N=1$, in which the TGA is composed of
one $A$ site and one $B$ site as shown in Fig. 2 (a). An analytical expression
for the reflection amplitude $r$ can be obtained as

$\begin{split}r&=\frac{i\xi\left(\Delta_{2}+t_{1}\right)\sin
k}{\left[\Delta_{1}+\xi\left(e^{ik}-1\right)\right]\left(\Delta_{2}+t_{1}\right)-J^{2}}\\\
&+\frac{i\xi\left(\Delta_{2}-t_{1}\right)\sin
k}{\left[\Delta_{1}+\xi\left(e^{ik}+1\right)\right]\left(\Delta_{2}-t_{1}\right)-J^{2}}-1,\\\
\end{split}$ (6)

where $\Delta_{1}=E_{k}-\omega_{c}$ and $\Delta_{2}=E_{k}-\omega_{e}$
represent the detuning between the propagating photon in the waveguide and the
bare resonators in the CRW and the TGA, respectively. Since the spontaneous
radiation of the resonators and TGA are ignored, it satisfies that $R+T=1$,
where $R=\left|r\right|^{2}$ is the reflection rate and $T=\left|t\right|^{2}$
is the transmission rate.

In Fig. 2 (c), we demonstrate the reflection rate $R$ as a function of the
photon-atom detuning $\Delta_{2}$ in the resonance condition that is,
$\omega_{c}=\omega_{e}$. The result for the scheme of the single cell [$N=1$
shown in Fig. 2 (a)] is illustrated by the blue solid curve. The two complete
reflection peaks ($R=1$) characterize the inter coupling inside the TGA.
Besides, when only one of the two sites in the TGA couple to the one resonator
in the waveguide [shown in Fig. 2 (b)], the single photon reflection is
characterized by the Rabi splitting behavior. As shown in the figure, the
peaks of the orange dashed line locate exactly at the frequency
$\Delta_{2}=\pm t_{1}$. The deviation from the Rabi splitting for the two
sites coupling originates from the photonic interference effect when it
propagates back and forth, and this interference is peculiar for the giant
atom setup.

Figure 3: The reflection rate $R$ as a function of detuning $\Delta_{2}$ for
$N=5$, and (a) $t_{2}=0.1\xi$, (b) $t_{1}=0.1\xi$. The parameters are set as
$J=0.9\xi$, $\omega_{e}=\omega_{c}=20\xi$.

Next, let us move to the case of $N>1$, in which the topological properties of
the TGA take effect. In Fig. 3, we demonstrate the reflection rate $R$ for
$N=5$ as a function of the detuning $\Delta_{2}$. As shown in the figures, the
reflection rate shows a complicated behavior in the two bands regime of
$-|t_{1}+t_{2}|<\Delta_{2}<-|t_{1}-t_{2}|$ and
$|t_{1}-t_{2}|<\Delta_{2}<|t_{1}+t_{2}|$. In the gapped regime with
$-|t_{1}-t_{2}|<\Delta_{2}<|t_{1}-t_{2}|$, the reflection rate behaves
differently when the TGA undergoes the topological phase transition. For
example, in the topological trivial phase ($t_{1}>t_{2}$), we illustrate the
results for $t_{2}=0.1\xi$ in the cases of $t_{1}=0.2\xi$ and $t_{1}=0.5\xi$
respectively in Fig. 3(a). For small $t_{1}$ with $t_{1}=0.2\xi$, there exists
a small peak at the zero detuning $\Delta_{2}=0$. As the increases of $t_{1}$,
the peak is smoothed and it forms a relatively wide band for $R=0$ nearby the
regime of $\Delta_{2}=0$. It implies that the TGA in the topological trivial
phase can be used to realized the quantum cloaking [67, 68], that is, the TGA
does not affect the propagation of the photon in the waveguide. On the other
hand, when the TGA works in the topological non-trivial phase, the complete
reflection window ($R=1$) is demonstrated in Fig. 3(b) nearby the resonant
regime of $\Delta_{2}=0$. The results in Fig. 3 indicate that, we can use the
TGA to construct the coherent single-photon device. When the value of
$t_{1(2)}$ be is achieved by the order of $100$ MHz experimentally [69], we
can realize the quantum cloaking or complete reflection with the bandwidth of
about $100-200$ MHz, which is in the same order of $t_{1}$ and $t_{2}$.

##  IV Single photon bound state

In the above section, we have studied the single photon scattering states who
locate inside the propagating band
$E_{k}\in\left[\omega_{c}-2\xi,\omega_{c}+2\xi\right]$. Due to the coupling
between the TGA and the waveguide, the translational symmetry of the system is
broken, leading to the photonic bound states which lay outside the propagating
band. Here, we resort to the numerical diagonalization of the Hamiltonian to
find these bound states when the TGA works in the topologically non-trivial
phase ($t_{1}<t_{2}$).

Figure 4: The energy spectrum for $N=29$ in (a) $t_{3}=t_{2}$, and
(b)$t_{3}=0$. The parameters are set as $t_{1}=0.1\xi$, $t_{2}=0.2\xi$,
$\omega_{e}=\omega_{c}=20\xi$.

In Figs. 4 (a) and (b), we plot the energy spectrum by considering that the
TGA works in the periodical boundary condition $(t_{3}=t_{2})$ and open
boundary condition $(t_{3}=0)$, respectively. As expected, we observe the
continual band which supports the propagating modes in both of the cases and
the difference arises from the bound states outside the continuum. In the
periodical boundary condition, there exist two pairs of non-degenerate bound
states, which locate symmetrically above and below the continuum and are
denoted by $E_{1},E_{2},E_{3}$ and $E_{4}$, respectively as shown in Fig. 4
(a). Within the available experimental parameter $\xi=100$ MHz, the gap
between the gap state $\Delta_{E}=E_{1}-E_{2}=E_{3}-E_{4}$ is in the order of
$10$MHz in the condition of $J=3\xi$. As a comparison, when the TGA works in
the open boundary condition, the pairs of the bound states become degenerate
as shown in Fig. 4 (b). Moreover, it satisfies $E_{5}=(E_{1}+E_{2})/2$ and
$E_{6}=(E_{3}+E_{4})/2$.

Figure 5: The evolution of the population in the excited state for the probing
atom for $t_{2}=t_{3}$ (blue solid line), and $t_{3}=0$ (orange dashed line).
The parameters are set as $N=29$, $\omega_{e}=\omega_{c}=20\xi$,
$\omega_{p}=E_{6}=16.65\xi$, $t_{1}=0.1\xi$, $t_{2}=0.2\xi$, $J=3\xi$,
$f=2\times 10^{-3}\xi$, $\gamma=2\times 10^{-4}\xi$.

The above transition from the degenerate bound states to non-degenerate ones
provide us an effective approach to detect the boundary condition of the TGA.
To this end, we introduce an auxiliary probing atom whose resonant frequency
satisfies $\omega_{p}=E_{6}$ and observe its excitation evolution. Then, the
probing Hamiltonian is expressed as

$H_{p}=H+(\omega_{p}-i\gamma)\left|e\right\rangle_{p}\left\langle
e\right|+f\left(\tau_{+}a_{j}+{\rm H.c.}\right)$ (7)

which demonstrates that the probing atom is coupled to the $j$th cavity with
the coupling strength $f$, $\gamma$ and
$\tau_{+}=\left|e\right\rangle_{p}\left\langle g\right|$ are the spontaneous
emission and raising operator of the probing atom.

Preparing the probing atom in the excited state and the TGA-waveguide system
in the ground state initially, we plot the excitation dynamics of the probing
atom in Fig. 5 by taking $J=3\xi$. Here the probing atom is located in the
resonator of $j=0$, that is, the left leg of the TGA. Since the probing atom
is large detuned from both of the bound states in the condition of
$f\ll|\omega-E_{3(4)}|$, the probing atom is effectively decoupled from the
TGA-waveguide system in the open boundary condition. As a result, the dynamics
of $P_{e}(t)=\langle|e\rangle_{p}\langle e|\rangle$ shows an exponential
decay, which is determined by the spontaneous dissipation rate $\gamma$ as
shown by the solid line in Fig. 5. In the periodical boundary condition, the
pair of non-degenerate bound states emerge into the degenerate ones, which are
resonate with the probing atom. So that, the dynamics exhibits a Rabi
oscillation character as shown by the dashed line. In this sense, the boundary
condition of the TGA can be detected in a coherent manner.

##  V REMARKS and CONCLUSIONS

In this paper, we have constructed a TGA via a finite SSH chain which couples
to the CRW. In the single excitation subspace, we discuss the scattering and
bound states, respectively. We find that the photonic scattering behavior can
be modulated by the topological nature of the TGA. Together with the
interference effect during the propagation of the photon in the waveguide, we
can design the wide band cloaking or reflection photonic device when the TGA
works in the topological trivial or non-trivial phases, respectively. As for
the bound states which locate outside the continuum, the energy gap can be
closed or opened, depending on the boundary condition of the TGA.

In the current experimental availability, the proposed model can be realized
in the superconducting circuits.In 2014, the first experimental realized the
coupling between a superconducting transmon quantum qubit and the surface
acoustic waves (SAWs) waveguide was realized. Here, the qubit is $20$ times
larger than the wavelength of the SAWs in size and it therefore reached the
giant atom regime [70]. This ratio between the size of the giant atom and the
wavelength was then developed to $100$ and the light-matter interaction was
achieved by tens of MHz [34]. Additionally, the giant atom was also achieved
by coupling artificial atoms created with Josephson junctions to
superconducting circuits through the capacitance or inductance. In Ref. [37],
the giant atom was experimentally achieved by and couple it to a waveguide at
multiple, yet well separated, discrete locations. The distance between two
coupling points could reach $20.54$ mm [44]. Recently, the topology of two
nested giant spin ensembles(GSE) has been experimentally demonstrated, the
distance between the two coupling points of the inner(outer) is designed to be
$8.3\,(16.6)$cm [71]. It provides a new platform for “giant atom” physics.
Moreover,the CRW constructed with $9$ superconducting qubits is also realized
in the experiment, where the nearest-neighbor coupling strength is
$\xi/2\pi=50$ MHz [72]. In a recent experiment, Painter’s group expanded the
CRW consist of a $42$ unit-cell array of capacitively coupled lumped-element
microwave resonators [73], and the SSH topological structure was also
constructed with the hopping strength $J(1\pm\delta)$ where $J/2\pi=368$ MHz,
$\delta=0.282$. Furthermore, the qubit-resonator coupling strength in such
structure was achieved by $g/2\pi=124.6$ MHz [69]

The establishment of TGA in our work provides an unconventional approach to
regulate the transmission and distribution of the single photon in the CRW.
Here, the interference effect for the propagation of the photon in the regime
of the giant atom will be applicable in quantum control and quantum
information processing. We hope that our work of controlling photons through
the giant atom systems combined with topologies will stimulate new research of
the hybrid system and broaden the range of application of artificial giant
atom.

###### Acknowledgements.

This work is supported by the funding from Jilin Province (Grant Nos.
20230101357JC and 20220502002GH) and the National Science Foundation of China
(Grants No. 12105026 and No. 12375010).

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