# Generalized coherent states of exceptional Scarf-I potential: Their spatio-
temporal and statistical properties

T. Shreecharan<EMAIL_ADDRESS>Department of Physics, Faculty of
Science and Technology,
ICFAI Foundation for Higher Education,
Dontanapally, Hyderabad, Telangana, India: 501 203.  S. Sree Ranjani
<EMAIL_ADDRESS>Department of Physics, Faculty of Science and
Technology,
ICFAI Foundation for Higher Education,
Dontanapally, Hyderabad, Telangana, India: 501 203.

###### Abstract

Abstract

We construct generalized coherent states for the rationally extended Scarf-I
potential. Statistical and geometrical properties of these states are
investigated. Special emphasis is given to the study of spatio-temporal
properties of the coherent states via the quantum carpet structure and the
auto-correlation function. Through this study, we aim to find the signature of
the “rationalisation” of the conventional potentials and the classical
orthogonal polynomials.

MSC2020: 81Q60, 81R30.
Keywords: Exceptional Orthogonal Polynomials, Coherent states, Quantum
carpets, Statistical and geometrical properties.

##  I Introduction

The discovery of exceptional orthogonal polynomials (EOPs) Kamran1 ; Kamran2
has had a significant impact on the areas of Sturm-Liouville theory and
exactly solvable quantum mechanics Sasaki1 . The latter has seen a revival, as
the existence of these polynomials has led to the contruction of new exactly
solvable potentials Quesne1 ; Sasaki2 . Many groups worked in tandem using
various methods such as Darboux transformations, Darboux-Crum, supersymmetry
(SUSY), group theory etc., to construct these new polynomials and potentials
kam_dar ; Sasaki3 ; Dutta ; Ramos ; Grandati ; Mandal ; Chaitanya ;
Sreeranjani1 . Another interesting development that has occurred recently is
the construction of multi-indexed EOPs leading to generalised families of new
potentials Sasaki4 ; Quesne2 ; multi1 ; multi2 ; Grandati-Quesne ;
Sreeranjani2 .

These potentials are rational extensions of the conventional exactly solvable
potentials wherein additional rational terms get added, hence the new
potentials are also called as the rational potentials. Interestingly the old
and the corresponding new potentials are isospectral to each other. While the
wavefunctions of the rational potentials involve EOPs, the conventional
potentials have the classical orthogonal polynomials (COPs) as their
eigenfunctions. Like the potentials, the EOPs are also rational extensions of
the COPs.

Various aspects of the conventional potentials have been extensively explored.
One such aspect is the construction of the coherent states (CS). Ever since
the introduction of coherent states, as the most classically behaving quantum
states, by Schrödinger, this concept has evolved over the years. At present
there are three universally accepted definitions of CS. The first is that
these states be the eigenstates of the annihilation operator. The second
definition states that the coherent state must be generated by a unitary
displacement operator. The third definition is that the CS must saturate the
uncertainty relation. It is well known all three definitions coincide only for
the harmonic oscillator potential, which obeys the Heisenberg-Weyl algebra.
Other exactly solvable potentials have either the $su(2)$ or the $su(1,1)$
algebra as their underlying algebra, therefore the CS corresponding to a given
potential are distinct and do not coincide. This makes the study of these
states and their properties quite fruitful and exciting.

Though the Lie algebraic techniques are very powerful and elegant, it is
sometimes quite difficult to establish the generators of the algebra in terms
of the potential parameters. This makes the construction of the CS difficult
and therefore we have to resort to other methods. In this context the shape
invariance symmetry or the discrete reparametrisation symmetry of these
potentials has played a pivotal role Gedenstein ; Aizawa1 ; Balantekin1 ;
Balantekin2 . Since the rational potentials also respect reparametrisation
symmetry, it is natural that we exploit this to construct the CS. Similar
studies for the new rational potentials and the underlying dynamical symmetry
groups have been done Aizawa2 . Recently CS for the rationally extended
oscillator have been constructed and their statistical and dynamical
properties have been studied hoffman1 ; hoffman2 ; hoffman3 .

We construct the generalised CS (GCS) as defined in Ref. Balantekin2 , for the
rational Scarf-I potential, which has $X_{1}$ EOPs as solutions. In addition
to studying their statistical and geometrical aspects, we also study the
quantum carpet structures. These are interesting patterns generated by the
spatio-temporal evolution of the probability density function of the CS. We
also study the auto-correlation functions of the rational CS. All the results
obtained for the rational Scarf-I potentials are compared with those of the
conventional Scarf-I potential, so as to obtain the signature of
rationalization of the potential and the wave functions.

The paper is organized as follows. In section II, we summarize the rational
and the conventional Scarf-I potentials and their solutions and briefly
summarise supersymmetry (SUSY) and shape invariance (SI). In section III, we
present the four types of GCS for this potential. This is followed by the
quantum carpets structures of the old and the new potentials and their
comparisons in section IV. Section V, deals with the auto-correlation
function, while section VI deals with the statistical and geometrical aspects
of the GCS. In section VII, we present our conclusions.

##  II The rational trigonometric Scarf-I potential

As is well known the shape invariant potentials (SIPs) are classified into six
categories based on the Infeld and Hull factorization scheme Infeld . This
classification is based on the translational transformation properties of the
potential parameters. The trigonometric Scarf-I potential belongs to the type
$\mathcal{A}$ of SIPs as per this scheme. The rational Scarf-I has the same
reparametrisation symmetry as the conventional potential therefore it can also
be classified as a type $\mathcal{A}$ potential.

In this section, we present the Scarf-I and the rational Scarf-I potentials
and their solutions. These will be used in our subsequent construction of GCS
and their properties. The conventional supersymmetric trigonometric Scarf-I
potential with $(\hbar=2m=1)$ is given by

$V^{-}(x)=[\alpha(\alpha-1)+\beta^{2}]\sec^{2}x-\beta(2\alpha-1)\sec x\tan x,$
(1)

whose eigenfunctions and eigenvalues are

$\psi^{-}_{n}(x)=N_{n}(1-\sin x)^{\frac{(\alpha-\beta)}{2}}(1+\sin
x)^{\frac{(\alpha+\beta)}{2}}P_{n}^{(\alpha-\beta-\frac{1}{2}\,,\,\alpha+\beta-\frac{1}{2})}(\sin
x),$ (2)

and

$E^{-}_{n}=(n+\alpha)^{2},\qquad n=0,1,2,\dots$ (3)

respectively. Here
$P_{n}^{(\alpha-\beta-\frac{1}{2}\,,\,\alpha+\beta-\frac{1}{2})}(\sin x)$ are
the classical Jacobi polynomials and the normalization constant in Eq. (2) is

$\hskip-14.22636ptN_{n}=\sqrt{\frac{n!(2n+\alpha)\Gamma(n+2\alpha)}{2^{2\alpha}\Gamma(\alpha-\beta+n+\frac{1}{2})\Gamma(\alpha+\beta+n+\frac{1}{2})}}$
(4)

The corresponding superpotential associated with this potential is given by

$W(x)=\alpha\tan x-\beta\sec x\ .$ (5)

The rational trigonometric Scarf-I potential is given by

$\displaystyle\widetilde{V}^{-}(x)$ $\displaystyle=$
$\displaystyle[\alpha(\alpha-1)+\beta^{2}]\sec^{2}x-\beta(2\alpha-1)\sec x\tan
x$ (6) $\displaystyle+$ $\displaystyle\frac{2(2\alpha-1)}{2\alpha-1-2\beta\sin
x}-\frac{2[(2\alpha-1)^{2}-4\beta^{2}]}{(2\alpha-1-2\beta\sin x)^{2}},$

where $-\frac{\pi}{2}<x<\frac{\pi}{2}$ and $0<\beta<\alpha-1$. The bound state
solutions are

$\widetilde{\psi}^{-}_{n}(x)=N_{n}\frac{(1-\sin
x)^{\frac{1}{2}(\alpha-\beta)}(1+\sin
x)^{\frac{1}{2}(\alpha+\beta)}}{2\alpha-1-2\beta\sin x}\\\
\widetilde{P}_{n+1}^{(\alpha-\beta-\frac{1}{2},\alpha+\beta-\frac{1}{2})}(\sin
x)\ ,$ (7)

where

$\displaystyle\widetilde{P}_{n+1}^{(\alpha-\beta-\frac{1}{2},\alpha+\beta-\frac{1}{2})}(\sin
x)=-\frac{1}{2}\left[\sin
x-\frac{2\alpha-1}{2\beta}\right]P_{n}^{(\alpha-\beta-\frac{1}{2},\alpha+\beta-\frac{1}{2})}(\sin
x)$
$\displaystyle+\frac{1}{2\alpha-1+2n}\Bigg{[}\frac{2\alpha-1}{2\beta}P_{n}^{(\alpha-\beta-\frac{1}{2},\alpha+\beta-\frac{1}{2})}(\sin
x)-P_{n-1}^{(\alpha-\beta-\frac{1}{2},\alpha+\beta-\frac{1}{2})}(\sin
x)\Bigg{]}$

are the $X_{1}$ exceptional Jacobi polynomials and

$\displaystyle\widetilde{N}_{n}=\frac{\beta}{2^{\alpha-2}}\Bigg{[}\frac{n!(2n+2\alpha)\Gamma(n+2\alpha)}{(n+\alpha-\beta+\frac{1}{2})(n+\alpha+\beta+\frac{1}{2})}$
$\displaystyle\frac{1}{\Gamma(n+\alpha-\beta-\frac{1}{2})\Gamma(n+\alpha+\beta-\frac{1}{2})}\Bigg{]}^{\frac{1}{2}}$
(8)

is the normalization constant. The corresponding energy eigenfunctions are
given by

$\widetilde{E}^{-}_{n}=(n+\alpha)^{2},\quad\quad n=0,1,2,\dots$ (9)

The superpotential associated with this potential is given by

$\displaystyle W(x)=\alpha\tan x-\beta\sec x$ $\displaystyle-2\beta\cos
x\Bigg{[}\frac{1}{2\alpha-1-2\beta\sin x}-\frac{1}{2\alpha+1-2\beta\sin
x}\Bigg{]}.$ (10)

It is clear from the above expressions, as to why the new potentials and the
polynomials are called the rational extensions of their conventional and
classical counterparts respectively. For both the supersymmetric potentials we
can construct their isospectral partners $V^{+}(x)$ and
$\widetilde{V}^{+}(x)$, using the given superpotentials respectively. Similar
to the Scarf-I, the rational Scarf-I potential satisfies the shape invariance
condition

$\widetilde{V}^{+}(x,a_{1})=\widetilde{V}^{-}(x,a_{2})+R(a_{1}),$ (11)

where $R$ is a function of $a_{1}$ with $a_{1}=\alpha$ and $a_{2}=\alpha+1$
for the Scarf-I potential. For further details on supersymmetric quantum
mechanics, we refer to Khare-book .

In figure (1), we give the plots of these potentials for certain parameter
values.

Figure 1: The potential with $\alpha=12$ and $\beta=10.9$. The dashed line is
for the Scarf-I potential and the continuous curve depicts the rational
Scarf-I potential.

As can be seen, the rational terms in the potential lead to a small dip at one
end of the potential well. As will be seen later in the paper this dip has an
effect on the quantum carpet structure.

##  III The coherent state construction

For any $j$th potential in the hierarchy of SIPs, the GCS, as defined in Ref.
Balantekin2 , is

$\big{|}\zeta;a_{j}\big{\rangle}=\frac{1}{\sqrt{N\big{(}|\zeta|^{2};a_{j}\big{)}}}\sum_{n=0}^{\infty}\frac{\zeta^{n}}{h_{n}(a_{j})}\
\big{|}\psi_{n}\big{\rangle}\ .$ (12)

In the above $h_{n}(a_{j})$ are the expansion coefficients defined below

$h_{n}(a_{j})=\sqrt{\prod_{k=1}^{n}\Big{[}\sum_{s=k}^{n}R(a_{s})\Big{]}}\Bigg{/}\prod_{k=1}^{n-1}\mathcal{Z}_{j+k}$
(13)

for $n\geq 1$. We have used the notation
$(a_{j})\equiv[R(a_{1}),R(a_{2}),R(a_{3}),\cdots,R(a_{n});a_{j},a_{j+1},\cdots
a_{j+n-1}]$ and $h_{0}(a_{j})=1$. The index $j$ is the family index in the
hierarchy of shape invariant potentials. These potentials are related to each
other by a translational shift of the potential parameter.The normalization
constant is given by

$N\big{(}|\zeta|^{2};a_{j}\big{)}=1\Bigg{/}\sqrt{\sum_{n=0}^{\infty}\frac{|\zeta|^{2n}}{|h_{n}(a_{j})|^{2}}}\
.$ (14)

Note that $\big{|}\psi_{n}\big{\rangle}$ is the eigenstate of the SIP and the
eigenvalues are given by $E_{n}=\sum_{k=1}^{n}R(a_{k})$ Khare-book . In the
above equation
$\mathcal{Z}_{j+k}\equiv\mathcal{Z}(a_{1},a_{2},\cdots,a_{j+k})$ is an
arbitrary functional of the potential parameters. It plays a crucial role in
the construction of the CS. It is worth mentioning that $Z_{j}$ is not
completely arbitrary, but is constrained by the shape invariance condition.
$\mathcal{Z}_{j+k}$ and $\mathcal{Z}_{j}$ are related via a similarity
transformation which is given by
$\mathcal{Z}_{j+k}=\\{\hat{T}(a_{1})\\}^{k}\mathcal{Z}_{j}\\{\hat{T}^{\dagger}(a_{1})\\}^{k}$.

A few comments are in order here about the operator $\hat{T}(a_{1})$: (1) It
is unitary (2) Its action on an eigenstate $|\psi_{n}(a_{j})\rangle$
corresponding to a SIP is given by $\hat{T}(a_{1})|\psi_{n}(a_{j+1})\rangle$
(3) Its action on an operator $O(a_{j})$ is via a similarity transformation
$\hat{T}(a_{1})\ O(a_{j})\ \hat{T}^{\dagger}(a_{1})=O(a_{j+1})$. For more
details about the operator $\hat{T}(a_{1})$ and the details of the
construction of CS, the readers should see Ref Aizawa1 ; Balantekin1 ;
Balantekin2 .

Four different types of CS are constructed, for the Scarf-I potential and its
rational extension, by suitably defining $Z_{j}$ in terms of an auxiliary
function $g(c,d,a_{j})$. Here $c$ and $d$ are constants and
$a_{1}=\alpha,a_{2}=\alpha+1,\cdots,a_{j}=\alpha+j-1$. Below we present the
expressions for the four different CS. Note in what follows for the sake of
notational simplicity we suppress the $a_{j}$ dependency in the expressions of
the CS.

GCS 1:

$\mathcal{Z}_{j}=C$ (constant) leads to the coherent state constructed by
Aizawa et. al. Aizawa2

$\displaystyle\big{|}\zeta\big{\rangle}=\frac{1}{\sqrt{N\big{(}|\zeta|^{2}\big{)}}}\sum_{n=0}^{\infty}\sqrt{\frac{\Gamma(2\alpha+n)}{\Gamma(n+1)\Gamma(2\alpha+2n)}}\
\zeta^{n}\ \big{|}\psi_{n}\big{\rangle}\ .$ (15)

Here,

$\frac{1}{\sqrt{N\big{(}|\zeta|^{2}\big{)}}}=\sum_{n=0}^{\infty}\frac{\Gamma(2\alpha+n)|\zeta|^{2n}}{\Gamma(n+1)\Gamma(2\alpha+2n)},$
(16)

which can be summed to obtain

$N(|\zeta|^{2})={{}_{1}}F_{2}\big{(}2\alpha;\alpha,\alpha+\frac{1}{2};\frac{|\zeta|^{2}}{4}\big{)}\
.$ (17)

For the other three CS the auxiliary function $g(c,d,a_{j})$ is non trivial
and of the form

$g(a_{j};c,d)=ca_{j}+d,$ (18)

such that

$\prod_{k=0}^{n-1}g(a_{j+k};c,d)=\frac{c^{n}\Gamma(\alpha+j+n+\frac{d}{c}-1)}{\Gamma(\alpha+j+\frac{d}{c}-1)}$
(19)

GCS 2:

By defining

$\mathcal{Z}_{1}=\sqrt{g(\alpha;2,1)\ g(\alpha;2,2)}\
e^{-i\tilde{\alpha}R(a_{1})}$ (20)

and using Eq. (19), we are led to the second type of GCS

$\displaystyle\big{|}\zeta\big{\rangle}=\frac{1}{\sqrt{N\big{(}|\zeta|^{2}\big{)}}}\sum_{n=0}^{\infty}\sqrt{\frac{\Gamma(2\alpha+n)\Gamma(2\alpha+2n+1)}{\Gamma(n+1)\Gamma(2\alpha+2n)\Gamma(2\alpha+1)}}$
$\displaystyle
e^{-i\tilde{\alpha}E_{n}}\zeta^{n}\big{|}\psi_{n}\big{\rangle},$ (21)

which is the Perelomov coherent state for the Scarf-I potential. Here
$\tilde{\alpha}$ is a real constant. The normalization constant for this case
turns out to be

$N\big{(}|\zeta|^{2}\big{)}={{}_{2}}F_{1}\big{(}2\alpha;\alpha+1,\alpha;|\zeta|^{2}\big{)}\
.$ (22)

GCS 3:

The third GCS, which happens to be a Barut-Girardello coherent state, is
obtained by defining

$\mathcal{Z}_{1}=\frac{\sqrt{g(\alpha;2,1)g(\alpha;2,2)}}{g(\alpha;1,1+\alpha)}\
e^{-i\tilde{\alpha}R(a_{1})}\ .$ (23)

The expression for the coherent state turns out to be

$\displaystyle\big{|}\zeta\big{\rangle}=\frac{1}{\sqrt{N\big{(}|\zeta|^{2})}}\sum_{n=0}^{\infty}\sqrt{\frac{\Gamma(2\alpha+2n+1)\Gamma(2\alpha+n)\Gamma(2\alpha+1)}{\Gamma(2\alpha+2n)\Gamma(2\alpha+n+2)\Gamma(n+1)}}\
e^{-i\tilde{\alpha}E_{n}}\zeta^{n}\big{|}\psi_{n}\big{\rangle},$ (24)

where

$\displaystyle
N\big{(}|\zeta|^{2}\big{)}=\frac{\Gamma(2\alpha+1)^{2}}{\Gamma(2\alpha+2)}\
{{}_{2}}F_{2}\big{(}2\alpha,\alpha+1;\alpha,2\alpha+2;|\zeta|^{2}\big{)}\ .$
(25)

GCS 4:

The fourth coherent state is obtained by defining

$\displaystyle\mathcal{Z}_{j}=\sqrt{\frac{g(a_{j};1,0)\,g(a_{j};1,1/2)\,g(a_{j+2};2,-2\alpha-2\sigma)}{g(a_{j};1,\alpha)\,g(a_{j+2};1,-\alpha)}}\
e^{-i\tilde{\alpha}R(a_{1})}.$ (26)

Setting $j=1$ in the above equation we get

$\displaystyle\mathcal{Z}_{1}=\sqrt{\frac{g(\alpha;1,0)\,g(\alpha;1,1/2)\,g(\alpha+2;2,-2\alpha-2\sigma)}{g(\alpha;1,\alpha)\,g(\alpha+2;1,-\alpha)}}\
e^{-i\tilde{\alpha}R(a_{1})},$ (27)

which leads to

$\displaystyle\big{|}\zeta\big{\rangle}=\frac{1}{\sqrt{N\big{(}|\zeta|^{2}\big{)}}}\sum_{n=0}^{\infty}\sqrt{\frac{\Gamma(n+2-\sigma)}{\Gamma(n+2)\Gamma(n+1)}}\
e^{-i\tilde{\alpha}E_{n}}\zeta^{n}\big{|}\psi_{n}\big{\rangle},$ (28)

where $\sigma$ is a real number dependent on $d$ via equations (18) and (19),
whose explicit form is $\sigma=-\alpha-d/2$. The normalization constant is

$N\big{(}|\zeta|^{2}\big{)}=\Gamma(2-\sigma)\
{{}_{1}}F_{1}(2-\sigma;2;|\zeta|^{2})\ .$ (29)

As already mentioned $\big{|}\psi_{n}\big{\rangle}$ are the normalized wave
functions, Eq. (2), of the Scarf-I potential. Replacing these by
$\widetilde{\psi}_{n}(x)$ from Eq. (7) in the above expressions, will yield
the corresponding GCS for the rational potential.

##  IV Quantum Carpet Structure

In the section we study the spatio-temporal behaviour of the GCS of the
conventional Scarf-I potential and its rational extension. By evolving the
probability density of the CS over space-time leads to quantum carpet
structures Berry ; Rost ; Marzoli ; Loinaz ; Hall ; Shree . These patterns are
generated because of the revival dynamics exhibited by the wavepacket. This
idea of the revival dynamics was first put forth in the work of Ref. Averbukh
. A revival is said to take place when a wavepacket under time evolution
returns to its initial form. Fractional revival is said to occur when a time
evolving wavepacket breaks up into sub-packets at different locations, but
with a shape that closely matches the initial form. Since the eigenvalues of
the Scarf-I and the rational Scarf-I potential are quadratic, there will be
revivals as well as fractional revivals. Potentials that have linear energy
spectrum do not exhibit such a behaviour. For mathematical details concerning
the revival dynamics we refer to the exhaustive review article of Robinett
Robinettphysrep .

Figure 2: Carpet structure for GCS1, Scarf-I.

Figure 3: Carpet structure for GCS1, rational Scarf-I.

Figure 4: Carpet structure for GCS2, Scarf-I.

Figure 5: Carpet structure for GCS2, rational Scarf-I.

Figure 6: Carpet structure for GCS3, Scarf-I.

Figure 7: Carpet structure for GCS3, rational Scarf-I.

Figure 8: Carpet structure for GCS4, Scarf-I.

Figure 9: Carpet structure for GCS4, rational Scarf-I.

The quantum carpet structures corresponding to the GCS (1-4), for the
conventional Scarf-I Potential, are shown in Figures (3), (5), (7), and (9)
respectively. In parallel, the four GCS for the rational Scarf-I potential
lead to the quantum carpet structures that are shown in the Figures (3), (5),
(7), and (9) respectively. The parameter values taken for the above carpets
are $\alpha=12,\beta=10.9$ and $n=20$. For the GCS4 case $\sigma=-\alpha-d/2$
with $d=0$.

In all the carpets the white regions indicate a very high probability of
finding the particle as compared to other areas. From the plot of the
potentials, it is clear that the wells are asymmetric about their center. This
leads to the localisation of the particle to the right end of the well as seen
in all the quantum carpet figures.

The additional terms in the rational Scarf-I potential lead to a dip near the
right wall leading to an increase in the asymmetry as compared to the
conventional potential. This has a profound effect on the quantum carpet
structure of the rational potential. The high probability regions of the
carpet which for the conventional potential were thick white bands, exhibits
intricate substructures for the rational potential.

These intricate patterns in our carpet structures, corresponding to the
rational Scarf-I, are because of the wavepacket splitting caused by two
reasons. First because of a dip due to the rational terms in the potential and
the second due to the the revival and fractional revival phenomena occurring
as a result of the quadratic energy structure. The dip leads to increased wave
packet collisions not only with the walls of the potential but also with each
other. These revival aspects are studied in the next section using the auto-
correlation function. It is worth mentioning that in Ref. hoffman1  the
authors have shown a similar wavepacket breaking in the rationally extended
oscillator potential due to the dip in the center that this potential has as
compared to the usual parabolic potential of an ordinary oscillator.

##  V Auto-correlation function/Fidelity

The autocorrelation function or fidelity is defined as the overlap between the
state at $t=0$ and at a later time $t$:

$A(t)=\big{\langle}\zeta;a_{r};t\big{|}\zeta;a_{r};0\big{\rangle}\ .$ (30)

This quantity provides further evidence of the revival dynamics. Numerically
$|A(t)|^{2}$ lies between 0 and 1. Zero means no overlap while one indicates a
perfect overlap, pointing to a complete revival of the CS. The fidelity is
very sensitive to the weighting factors of the CS. This is evident from the
different crest and trough structure seen in the plots of $|A(t)|^{2}$ of all
the four GCS. These are plotted in the Figures (11), (11), (13), and (11)
corresponding to the GCS 1, 2, 3, and 4 respectively of the the rational
Scarf-I potential. It must be pointed out that the autocorrelation plots for
the Scarf-I potential will be the same.

Figure 10: Auto correlation function for GCS1.

Figure 11: Auto correlation function for GCS2.

Figure 12: Auto correlation function for GCS3.

Figure 13: Auto correlation function for GCS3.

##  VI Statistical and Geometrical aspects

###  VI.1 Statistical Property

Various statistical properties of the GCS that have been constructed, will be
presented in this section. The study of these properties gives us insight into
various interesting phenomena observed in quantum optics, quantum electronics,
and some foundational aspects of quantum mechanics. The classical nature of
the GCS is encoded in the Poissonian distribution and any deviation from it
indicates non-classical behaviour. Note that, all the figures that follow
below have been plotted for the same values of the potential parameter
$\alpha$. In every figure corresponding to GCS1, 2, and 3 the blue dotted line
corresponds to $\alpha=2$, the red dot-dashed line corresponds to $\alpha=5$,
and the purple dashed line is for $\alpha=10$ whereas for the GCS4 we have
chosen $\sigma=1.5,-1,-9$ respectively.

The deviation from classicality can be measured using the second order
correlation factor $g^{(2)}$. Three possibilities exist depending on the
values that $g^{(2)}$ takes. If $g^{(2)}=1$ then the states are classical. If
$g^{(2)}>1$ it indicates bunching effect and $g^{(2)}<1$ antibunching. In
other words whenever $g^{(2)}\neq 1$, it signifies non-classsical behaviour.
The second order correlation function is given by

$g^{(2)}=\frac{N^{\prime\prime}(z)\ N(z)}{N^{\prime 2}(z)}\ ,$ (31)

where $z=|\zeta|^{2}$ and the primes denote derivatives with respect to $z$.
$N(z)$ is the normalization constant corresponding to the different GCS as
given in equations (17), (22), (25), and (29) respectively.

Figure 14: Intensity correlation for GCS1.

Figure 15: Intensity correlation for GCS2.

Figure 16: Intensity correlation for GCS3.

Figure 17: Intensity correlation for GCS4.

The intensity correlation plot for the GCS1, shown in Fig. (15), is less than
one indicating antibunching effect; whereas Fig. (15) is the plot for GCS2
which show bunching behaviour. This behaviour is seen for all values of the
potential parameter considered. Figure (17) and Figure (17) show nonclassical
for small values of $z$ but tend towards classical behaviour asymptotically.

Mandel parameter ($\mathcal{Q}$) Mandel  is another way to characterise
nonclassical states. If the value of $\mathcal{Q}<0$ the statistics is sub-
Poissonian whereas, for positive values of $\mathcal{Q}$ the statistics is
super-Poissonian. The Mandel parameter can be defined in terms of the
normalisation constant of the CS as

$\mathcal{Q}=z\
\left(\frac{N^{\prime\prime}(z)}{N^{\prime}(z)}-\frac{N^{\prime}(z)}{N(z)}\right)\
,$ (32)

Figure 18: Mandel parameter for GCS1.

Figure 19: Mandel parameter for GCS2.

Figure 20: Mandel parameter for GCS3.

Figure 21: Mandel parameter for GCS4.

The plots of the Mandel parameter for GCS1 and GCS2 are shown in Figures (19)
and (19) respectively. The former shows sub-Poissonian behaviour while the
latter super-Poissonian. Interestingly the GCS2 has the same $Q$ values for
all values of $\alpha$ and $z$. Figures (21) and (21) coresponding to GCS3 and
GCS4 respectively, yield a similar information as the intensity correlation
graphs i.e., the CS approach classicality for large values of $z$.

In what follows, we plot the weighting distributions, defined as
$P=|\langle\zeta;a_{r}|\zeta;a_{r}\rangle|^{2}$, for the GCS 1-4 in Figures
(23), (23), (25), and (25) respectively.

Figure 22: Photon distribution GCS1.

Figure 23: Photon distribution GCS2.

Figure 24: Photon distribution GCS3.

Figure 25: Photon distribution GCS4.

Similarly the mean photon number which can be defined as

$\mathcal{N}=z\ \frac{N^{\prime}(z)}{N(z)}\ ,$ (33)

have been plotted in Figures (27), (27), (29), and (29).

Figure 26: Mean photon number for GCS1.

Figure 27: Mean photon number for GCS2.

Figure 28: Case 3 mean photon number.

Figure 29: Case 4 mean photon number.

###  VI.2 Geometrical Property

The GCS constructed above have not only interesting statistical behaviour they
also have fascinating geometric structure. Perelomov Perelomov  and Gilmore’s
Gilmore1 ; Gilmore2  construction of the CS has a group manifold
interpretation. This group manifold exposition of the CS runs into trouble
when the Barut-Girardello CS is constructed. This is because it is not based
on the group theoretical aspect. This does not mean that the situation is
hopeless as far as geometry is concerned. One can resort to something called
the metric factor $\omega$, first introduced by Klauder et. al. Klauder , and
still gain an understanding of the geometric nature of the GCS.

It is a well known fact that CS provide a mapping between the set of complex
numbers and the rays in Hilbert space. This projective Hilbert space can be
endowed with Fubini-Study metric. This metric can be written via the line
element in terms of the CS as
$ds^{2}:=||d|\zeta\rangle||^{2}-|\langle\zeta|d|\zeta\rangle|^{2}$ Klauder .

The metric factor surprisingly can be expressed entirely in terms of the
normalisation constant of the CS:

$\omega(z)=\frac{N^{\prime}(z)}{N(z)}+z\
\left(\frac{N^{\prime\prime}(z)}{N(z)}-\frac{N^{\prime
2}(z)}{N^{2}(z)}\right).$ (34)

If $\omega(z)=1$ describes a flat two dimensional surface and when
$\omega(z)\neq 1$ it is nonflat. For more details we refer to Klauder .

The graphs corresponding to the GCS 1-4 are shown in Figures (31), (31), (33),
and (33). From the Fig (31) which corresponds to GCS1, we conclude that the
surface of the CS is non-flat. Fig (31), which happens to be a Perelomov CS,
we can see that it starts out as a flat surface and becomes non flat as $z$
increases. In figures (33) and (33) the CS surfaces become flat
asymptotically.

Figure 30: Metric factor for GCS1.

Figure 31: Metric factor for GCS2.

Figure 32: Metric factor for GCS3.

Figure 33: Metric factor for GCS4.

##  VII Conclusions

In this paper we have constructed the GCS for the rational Scarf-I and the
Scarf-I potential. We looked at their quantum carpet structures and have made
a comparative study. It is noticed that the quantum carpets for the rational
potential show intricate sub-structures as compared to the carpets of the
conventional potential. We were able to trace this to the increased asymmetry
in the potential, due to the presence of the rational terms in the exceptional
Scarf-I potential. This shows that these new potentials have interesting
revival dynamics. The auto-correlation studies also confirm the same. The
statistical aspects have been studied via the intensity correlation, Mandel
parameter, mean photon number, and the probability distribution. The
geometrical aspects were investigated using the metric factor. A more detailed
analysis along the lines of Lane  needs to be done and we hope to return to it
in future.

Acknowledgements TS is supported by SERB, India Grant: ECR/2015/000081. SSR is
supported by SERB, India Grant: EMR/2016/005002.

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# The physical connection between central stellar surface density and stellar
spin in SAMI and MaNGA nearby galaxies

L. Cortese,1,2 A. Fraser-McKelvie,1,2 J. Woo,3 B. Catinella,1,2 K.
Harborne,1,2 J. van de Sande,2,4J. Bland-Hawthorn,2,4 S. Brough,2,5 J. J.
Bryant2,4,6 S. Croom,2,4 S. Sweet2,7
1International Centre for Radio Astronomy Research, The University of Western
Australia, 35 Stirling Hw, 6009 Crawley, Australia
2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia
3Department of Physics, Simon Fraser University, 8888 University Drive,
Burnaby, BC V5A 1S6, Canada
4Sydney Institute for Astronomy (SIfA), School of Physics, The University of
Sydney, NSW 2006, Australia
5School of Physics, University of New South Wales, NSW 2052, Australia
6Australian Astronomical Optics, AAO-USydney, School of Physics, University of
Sydney, NSW 2006, Australia
7School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia E-mail<EMAIL_ADDRESS>(LC)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The stellar surface density within the inner 1 kpc ($\Sigma_{1}$) has become a
popular tool for understanding the growth of galaxies and its connection with
the quenching of star formation. The emerging picture suggests that building a
central dense core is a necessary condition for quenching. However, it is not
clear whether changes in $\Sigma_{1}$ trace changes in stellar kinematics and
the growth of dispersion-dominated bulges. In this paper, we combine imaging
from the Sloan Digital Sky Survey with stellar kinematics from the Sydney-AAO
Multi-object Integral-field unit (SAMI) and Mapping Nearby Galaxies at Apache
Point Observatory (MaNGA) surveys to quantify the correlation between
$\Sigma_{1}$ and the proxy for stellar spin parameter within one effective
radius ($\lambda_{re}$) for 1599 nearby galaxies. We show that, on the star-
forming main sequence and at fixed stellar mass, changes in $\Sigma_{1}$ are
mirrored by changes in $\lambda_{re}$. While forming stars, main sequence
galaxies remain rotationally-dominated systems, with their $\Sigma_{1}$
increasing but their stellar spin staying either constant or slightly
increasing. The picture changes below the main sequence, where $\Sigma_{1}$
and $\lambda_{re}$ are no longer correlated. Passive systems show a narrower
range of $\Sigma_{1}$, but a wider range of $\lambda_{re}$ compared to star-
forming galaxies. Our results indicate that, from a structural point of view,
passive galaxies are a more heterogeneous population than star-forming
systems, and may have followed a variety of evolutionary paths. This also
suggests that, if dispersion-dominated bulges still grow significantly at
$z\sim$0, this generally takes place during, or after, the quenching phase.

###### keywords:

galaxies: evolution – galaxies: disc – galaxies: bulges – galaxies: kinematics
and dynamics – galaxies: photometry

††pubyear: 2015††pagerange: The physical connection between central stellar
surface density and stellar spin in SAMI and MaNGA nearby galaxies–C

##  1 Introduction

One hundred years since its conception, understanding the physical origin of
the Hubble morphological sequence of galaxies (Hubble, 1926) remains one of
the major challenges for extragalactic astronomy. While relatively simple in
its original definition, the Hubble ‘tuning fork’ simultaneously traces
changes in star formation activity (e.g., prominence of spiral arms) and
stellar structure (e.g., presence of stellar bulges and/or bars), making its
physical interpretation not always straightforward. Thus, it is not surprising
that, throughout the decades, various re-incarnations of the Hubble tuning
fork have been proposed (e.g., de Vaucouleurs, 1959; Cappellari et al., 2011b;
Kormendy & Bender, 2012), with an effort to provide more physical basis to the
initial visual criteria used to separate different morphological types (see
also Romanowsky & Fall, 2012; Cortese et al., 2016; Hardwick et al., 2022).

Figure 1: The distributions of (from left to right) redshift ($z$), stellar
mass ($M_{*}$), specific star formation rate ($sSFR\equiv SFR/M_{*}$) and
stellar spin parameter within one effective radius ($\lambda_{re}^{c}$) for
our final sample (filled orange histograms). The empty black histograms show
the distribution of our parent sample extracted from the intersection of the
SAMI+MaNGA sample of Fraser-McKelvie et al. (2021) and the $\Sigma_{1}$ SDSS
catalogue of Woo & Ellison (2019).

Even when the focus is on stellar structure alone, the complexity of the
galaxy ecosystem can make it difficult to link visual morphology to the actual
orbital distribution of stars (e.g., Zhu et al., 2018). This is showcased, for
example, by the degeneracy in the use of the term ‘bulge’ in the context of
galaxy evolution. From a physical point of view, it is generally assumed that
this should be a structural component different from a disk, where most (if
not all) the dynamical support comes from non circular orbits (now-a-days
referred to as ‘classical bulge’). In reality, most works use this term to
simply identify deviations in central light surface brightness/mass surface
density from what is expected from an exponential disk (e.g. Kent, 1985; Peng
et al., 2002; Simard et al., 2011; Lange et al., 2016). This culminates with
the use of the term ‘pseudo-bulge’ to indicate features with a disc-like 2D
structure at the centre of galaxies (Kormendy & Kennicutt, 2004; Erwin et al.,
2015). In addition, the exact quantification of the properties of the ‘bulge’
heavily depends on the technique used to separate this component from a disk
(Kormendy & Bender, 2012; Meert et al., 2015; Cook et al., 2019; Tabor et al.,
2019; Oh et al., 2020), potentially hampering our ability to unveil the
evolutionary history of the inner parts of galaxies.

An alternative approach is to refrain from trying to separate the ‘bulge’ from
the disc using 2D imaging observations and instead quantify how much, and how
fast, the central core of galaxies grows. Specifically, in recent years, parts
of the community have focused on the central stellar surface density of
galaxies, generally measured within the central 1 kpc ($\Sigma_{1}$, Cheung et
al., 2012). Being a non-parametric quantity, $\Sigma_{1}$ is independent of
the actual nature of the central core of galaxies and, compared to the
historically more popular stellar surface density within one effective radius,
is less sensitive to changes in disk properties at larger radii, and supposed
to trace the growth of the central core of galaxies (Fang et al., 2013; Barro
et al., 2017; Walters et al., 2021).

The popularity of $\Sigma_{1}$ has further increased due to the fact that it
appears to be one of the best structural parameters able to discriminate
between active and passive galaxies. As initially shown by Cheung et al.
(2012) and Barro et al. (2017), and then confirmed by several others (e.g.,
Tacchella et al., 2015; Whitaker et al., 2017; Mosleh et al., 2017; Woo et
al., 2015; Woo et al., 2017; Woo & Ellison, 2019; Wang et al., 2018; Chen et
al., 2020; Suess et al., 2021), there appears to be a clear threshold in the
value of $\Sigma_{1}$ above which galaxies are almost always quiescent. This
has provided indirect support to a scenario in which the growth of the central
stellar core in galaxies may be physically linked to the process driving
quenching (e.g., the growth of the central supermassive black hole and the
onset of feedback from active galactic nuclei, AGN), a process sometimes
referred to as ‘compaction’ (Zolotov et al., 2015; Tacchella et al., 2015).
While, from a theoretical point of view, compaction is expected to be more
important at high redshift, recent works have suggested that it may also be an
effective mechanism in the local galaxy population (e.g., Wang et al., 2018).

While the quantification of the correlation between $\Sigma_{1}$, stellar
mass, star formation rate (SFR) and other galaxy properties, as well as their
dependence on redshift, has now been well established, the interpretation of
the correlations in the framework of galaxy transformation, and specifically
compaction, remains challenging. This is due, once again, to the fact that a
higher value of $\Sigma_{1}$ at fixed other galaxy properties may be blindly
assumed as evidence for the build-up of a dispersion-supported structure. This
would require invoking specific physical processes able to alter the orbital
structure of stars in the disc (e.g., mergers or violent disc instability,
Noguchi, 1999). However, it has never been demonstrated whether or not
$\Sigma_{1}$ is a good proxy for the presence of classical bulges in galaxies.
Could it be that $\Sigma_{1}$ simply traces the stellar mass growth of discs?
Is the link between $\Sigma_{1}$ and star formation quenching truly a by-
product of the build-up of dispersion-supported structures in the core of
galaxies?

There are very good reasons to expect that $\Sigma_{1}$ may not always trace
the presence of classical bulges. Indeed, we already know that, at high
stellar mass and stellar surface density, galaxies show a wide range of
kinematic properties, as highlighted by the ATLAS 3D survey (Cappellari et
al., 2011a), and confirmed by more recent integral field spectroscopic (IFS)
surveys (Cortese et al., 2016; van de Sande et al., 2018; Falcón-Barroso et
al., 2019; Wang et al., 2020; Guo et al., 2020). With the advent of large IFS
surveys it is paramount that we start to include kinematic information in our
characterisation of galaxy structure. In particular, stellar kinematics are
critical for discriminating between classical bulges and disc-like structures
at the centre of galaxies.

A wide range of stellar kinematic properties at fixed $\Sigma_{1}$ would
impact on the use of the mass-SFR-$\Sigma_{1}$ plane to constrain the various
paths to quiescence followed by galaxies. Indeed, the narrow range of
$\Sigma_{1}$ observed in passive systems has sometimes been used to suggest a
very narrow range of evolutionary paths (and thus physical processes) leading
to quiescence (Barro et al., 2017; Woo & Ellison, 2019; Suess et al., 2021).
Of course, if this limited range was just the result of the inability of
$\Sigma_{1}$ to discriminate between discs and bulges in the inner 1 kpc of
galaxies, this would have major implications on our view of galaxy evolution.

To address this issue, in this paper we present a detailed analysis of the
correlation between central stellar surface density $\Sigma_{1}$ and stellar
spin parameter for a sample of galaxies extracted from the overlap of the
Sloan Digital Sky Survey (SDSS, York et al., 2000), and the Mapping Nearby
Galaxies at Apache Point Observatory (MaNGA, Bundy et al., 2015) and SAMI
(Bryant et al., 2015; Croom et al., 2021a) IFS surveys. Our study extends on
previous works that looked at the correlation between $\Sigma_{1}$ and central
stellar velocity dispersion (e.g., Fang et al., 2013; Chen et al., 2020) by
exploiting the resolved stellar kinematics maps provided by SAMI and MaNGA.

This paper is organised as follows. In Section 2, we describe our sample
selection and estimates of $\Sigma_{1}$ and stellar spin, as well the
ancillary data used in this work. In Section 3, we look at the correlation
between $\Sigma_{1}$ and stellar spin for star-forming and passive galaxies,
separately. This section includes the main results of this work. Lastly, we
discuss the implications of our results in Section 4 and summarise in Section
5. Throughout this paper, we use a flat $\Lambda$ cold dark matter concordance
cosmology: $H_{0}$= 70 km s-1 Mpc-1, $\Omega_{M}$=0.3 and
$\Omega_{\Lambda}$=0.7.

Figure 2: The SFR-M∗ plane for our sample color-coded by the median value of
$\Sigma_{1}$ (top panel) and $\lambda_{re}^{c}$ (bottom panel). Medians are
calculated in steps of 0.1 dex including all galaxies within 0.2 dex in both
stellar mass and SFR. The solid line shows the best-fit to the star-forming
main sequence of Fraser-McKelvie et al. (2021), with the dashed line
indicating the threshold used here to separate star-forming and passive
galaxies. Only bins with five or more galaxies are shown.

##  2 Sample selection

Our parent sample is taken from Fraser-McKelvie et al. (2021), who combined
stellar kinematic measurements from both the SAMI and MaNGA surveys for a
sample of 3289 nearby galaxies over a stellar mass range
9.5<log($M_{*}/M_{\odot}$)<12 and up to at least one effective radius.
Briefly, galaxies are extracted from the overlap of the two surveys with the
GALEX-SDSS-WISE Legacy Catalog (GSWLC, version 2; Salim et al., 2016) to
guarantee homogeneous estimates of global stellar masses and current SFRs
across the redshift range and stellar mass regimes covered by both surveys. We
start from the stellar line-of-sight velocities and velocity dispersion maps
produced by the SAMI (van de Sande et al., 2017) and MaNGA (Westfall et al.,
2019) teams, which have been extracted from data cubes adaptively binned to a
signal-to-noise of 10 using the Voronoi binning code of Cappellari & Copin
(2003). The stellar flux-weighted spin parameter proxy within one effective
radius ($\lambda_{re}$) was estimated from the line-of-sight and velocity
dispersion maps following the definition presented in Emsellem et al. (2007),
with Petrosian $r$-band effective radii obtained from the NASA-Sloan Atlas
(Blanton et al., 2011). In order to correct for the effect of both beam
smearing and inclination we proceeded as follows. First, $\lambda_{re}$
estimates were corrected for beam smearing following the empirical recipes
presented in Harborne et al. (2020), which are based on seeing and Sersic
index. Beam-smearing corrected spin values were then corrected for inclination
using the $r$-band ellipticity value as a proxy for inclination following del
Moral-Castro et al. (2020). We point the reader to Fraser-McKelvie et al.
(2021) and van de Sande et al. (2021) for an extensive discussion of these
corrections. In the following, we use the corrected value $\lambda_{re}^{c}$
as a proxy for the stellar spin parameter, but we note that the main
conclusions of this paper do not qualitatively change if observed (i.e., not
corrected for either beam smearing or inclination) values are used, as
discussed in Appendix A of this paper.

Figure 3: The correlation between $\Sigma_{1}$ and $\lambda_{re}^{c}$ on the
star-forming main sequence. Left column. The $\Sigma_{1}$-$M_{*}$ (top) and
$\lambda_{re}^{c}$-$M_{*}$ (bottom) for galaxies on the star-forming main
sequence. Point size and color-coding reflect the probability density
distributions of galaxies across each parameter space, with yellow/lighter
colors indicating higher density. The black line shows the median relation for
the whole sample, with dashed lines showing the 25% and 75% percentiles (see
also Table 1). Middle column. Same as left column, with coloured lines now
showing the running medians for galaxies with $\lambda_{re}^{c}$ (top) or
$\Sigma_{1}$ (bottom) higher/lower than the observed median value at fixed
stellar mass. Right. Variation in $\lambda_{re}^{c}$ as a function of
variations in $\Sigma_{1}$ at fixed stellar mass. It is clear that, once we
control for mass, an increase in central stellar surface density traces a
decrease in stellar spin, and vice-versa.

We cross-correlated the SAMI+MaNGA sample with the measurements of central
stellar surface density ($\Sigma_{1}$) presented in Woo & Ellison (2019).
These have been obtained by converting the SDSS surface brightness profiles in
$g$ and $i$ band into stellar surface density profiles using the stellar mass-
to-light color relation presented in Fang et al. (2013). $\Sigma_{1}$ is then
obtained by interpolating the stellar mass profiles and measuring the total
stellar mass within 1 kpc. While $\sim$95% of the galaxies in our IFS sample
are included in the parent sample of Woo & Ellison (2019) (3102 out of 3289),
for a good fraction of them 1 kpc is smaller than the point-spread function
(PSF) of SDSS observations. Thus, to remove objects for which the estimate of
$\Sigma_{1}$ could be unreliable and/or biased, we focus on galaxies with
$z<$0.07, for which the PSF of SDSS images corresponds to less than 1 kpc, and
exclude highly inclined systems (i.e., minor-to-major axis ratio $b/a<$0.5).
Residual seeing effects do not affect our conclusions, as discussed in
Appendix B. These cuts reduce the overlap with the IFS sample to 1599. While
the sample decreases by a factor of $\sim$2 in size, we note that our final
sample still spans the same parameter space in stellar mass, specific SFR and
stellar spin parameter as the whole sample, as clearly shown in Fig. 1. As
expected, the redshift cut primarily affects our ability to sample the very
high stellar mass regime (M${}_{*}\gtrsim$1011.5 M⊙).

Throughout this paper, we investigate galaxies on and below the star-forming
main sequence (SFMS) separately. We use the best fit to the SFMS presented by
Fraser-McKelvie et al. (2021). This was calculated using all galaxies in SAMI
and MaNGA with SFRs included in the GSWLC catalog:

$\log(SFR_{MS})=0.256-\log(1+\frac{10^{10.064}}{M_{*}})$ (1)

Our active population includes all galaxies whose SFR is higher than
$SFR_{MS}-$0.5 dex (see Fig. 2, 720 galaxies), whereas the passive population
includes all galaxies with SFR below this threshold (879 galaxies).

##  3 The connection between stellar central surface density and spin

We start by showing in Fig. 2 the SFR-M∗ plane covered by our sample, color-
coded by the median value of $\Sigma_{1}$ (top panel) and $\lambda_{re}^{c}$
(bottom panel)111In this paper, we treat $\lambda_{re}^{c}$ as a log-normally
distributed quantity and plot it in log scale. While this may seem at odds
with what is generally done in the literature, where stellar spin is treated
as a linear quantity, from a theoretical point of view the stellar spin is
expected to follow a log-normal distribution (e.g., Mo et al., 1998; Bullock
et al., 2001). . We divided this plane into a grid with pixels 0.1 dex wide.
For each pixel, we show the median value for all galaxies with stellar mass
and SFR within 0.2 dex from the centre of each pixel. This means that we are
oversampling the distribution of galaxies in this plane.

The comparison between the two panels of Fig. 2 already indicates that, while
at fixed stellar mass passive galaxies have generally higher $\Sigma_{1}$ and
lower $\lambda_{re}^{c}$ than active systems, once we focus on either star-
forming or quiescent populations alone changes in $\Sigma_{1}$ may not always
be followed by changes $\lambda_{re}^{c}$, and vice-versa. To better quantify
the differences between the two structural indicators, in the following we
examine star-forming and quiescent galaxies, separately.

###  3.1 The star-forming main sequence

In Fig. 3, we show the $\Sigma_{1}$-$M_{*}$ (top left) and
$\lambda_{re}^{c}$-$M_{*}$ (bottom left) relations for all star-forming
galaxies in our sample. Color-coding and symbol size scale with the
probability density distributions of galaxies across both parameter spaces.
The black solid line shows the running medians for both relationships, with
dashed lines indicating the 25% and 75% interquartile ranges (see also Table
1).

Figure 4: The correlation between $\Sigma_{1}$ and $\lambda_{re}^{c}$ below
the SFMS. Colors and symbols are as in Fig. 3. Contrary to what observed in
star-forming galaxies, for passive systems we find no correlation between
$\Sigma_{1}$ and $\lambda_{re}^{c}$ at fixed stellar mass.

As already shown by several authors in the last decade (e.g., Cheung et al.,
2012; Fang et al., 2013; Suess et al., 2021), on the SFMS, $\Sigma_{1}$
monotonically increases with stellar mass. The slope of the correlation is
clearly shallower than 1 implying that, if galaxies grow in mass moving along
this relation (or following even shallower tracks as suggested by cosmological
simulations, see e.g., Tacchella et al., 2016; Chen et al., 2020; Walters et
al., 2021), most of the mass growth would take place in the outer parts of the
disk (i.e., at radii larger than 1 kpc). In other words, galaxies would not
become more compact (or more centrally concentrated) while on the SFMS. This
simple speculation becomes even more interesting when we look at the
distribution of stellar spin parameter as a function of stellar mass. Star-
forming galaxies show a narrow range of stellar spins, with $\lambda_{re}^{c}$
slightly increasing with mass (i.e., $\lesssim$0.1 dex across 1 dex in M∗; see
Table 1), as already shown by Wang et al. (2020) and Fraser-McKelvie et al.
(2021). From a structural point of view, SFMS galaxies appear to form a disc
main sequence.

If we combine the top and bottom left panels of Fig. 3, it appears that, on
the SFMS, the fact that more massive galaxies have a higher value of
$\Sigma_{1}$ does not necessarily imply that they have a more prominent
dispersion-supported central component (i.e., a classical bulge). On the
contrary, despite having higher central surface densities than lower mass
systems, high mass galaxies are still rotationally-dominated systems from a
kinematic perspective (e.g., $\lambda_{re}^{c}$>0.5). Intriguingly, this would
be in line with the idea that most of the growth of mass on the SFMS at
$z\sim$0 takes place beyond 1 kpc, as already hinted from the slope of the
$\Sigma_{1}-M_{*}$ relation.

Now that we have established how both $\Sigma_{1}$ and $\lambda_{re}^{c}$ vary
with stellar mass, it is important to investigate whether or not these two
structural parameters are correlated at fixed mass. After all, both scaling
relations have a significant scatter (i.e., average 25% to 75% inter-quartile
range IQR $\sim$0.3 dex and 0.13 dex for the $\Sigma_{1}-M_{*}$ and
$\lambda_{re}^{c}$-$M_{*}$ relationships, respectively; see Table 1) and our
goal is to establish whether higher central surface densities imply lower
spins, and vice-versa. To do so, we start by quantifying the offset in
$\Sigma_{1}$ and $\lambda_{re}^{c}$ for every galaxy in our sample with
respect to the median value observed for SFMS galaxies at fixed stellar mass.
Specifically, for each galaxy, we define $\Delta(\Sigma_{1})$ and
$\Delta(\lambda_{re}^{c})$ as the logarithmic difference between the observed
value and the median value of $\Sigma_{1}$ or $\lambda_{re}^{c}$ for all SFMS
galaxies within 0.1 dex in stellar mass. In the top right panel of Fig. 3, we
plot $\Delta(\lambda_{re}^{c})$ vs. $\Delta(\Sigma_{1})$ to show that the two
quantities are clearly correlated (Spearman correlation coefficient
$\rho\sim-$0.30, with a standard error from bootstrapping of 0.03). At fixed
stellar mass, galaxies with higher $\Sigma_{1}$ show lower stellar spins, and
vice-versa. This indicates that, once we control for stellar mass, an increase
in central stellar surface density traces an increase in the importance of
random motions in the gravitational support of galaxies, and thus most likely
tracing the presence of small classical bulges (or potentially just the
presence of a significantly thicker stellar disc). However, it is important to
remind the reader that all these galaxies are rotationally-dominated, and that
the decrease in spin is minimal, consistent with structural differences
between those pure discs and galaxies with small bulge components (e.g., the
equivalent of moving from Sd to Sb morphological types, e.g., Cortese et al.,
2016; van de Sande et al., 2018; Falcón-Barroso et al., 2019).

To show how the population of high/low $\Sigma_{1}$/$\lambda_{re}^{c}$
populates the $\Sigma_{1}-M_{*}$ and $\lambda_{re}^{c}$-$M_{*}$ scaling
relations, respectively, in the middle column of Fig. 3 we show the running
medians for galaxies with $\Delta(\lambda_{re}^{c})>0$ and
$\Delta(\lambda_{re}^{c})<0$ on the $\Sigma_{1}-M_{*}$ relation, and for
galaxies with $\Delta(\Sigma_{1})>0$ and $\Delta(\Sigma_{1})<0$ on the
$\Sigma_{1}-M_{*}$ relation. In both cases, galaxies with higher spin/ lower
central surface densities (and vice-versa) map each other very well, and they
follow parallel relations as a function of stellar mass. This confirms that
the two observables are tracing the same structural changes at fixed stellar
mass.

###  3.2 The passive population

The natural next step is to investigate whether or not the correlation between
$\Sigma_{1}$ and stellar spin holds for galaxies that are no longer on the
SFMS. A rapid glimpse at Fig. 4, which provides the same parametrisation
presented in Fig. 3 but for passive galaxies, immediately shows that here the
situation is dramatically different. The picture put forward in the subsection
above may no longer be valid below the SFMS.

Starting from the $\Sigma_{1}$-$M_{*}$ (top left) and
$\lambda_{re}^{c}$-$M_{*}$ (bottom left) scaling relations, we see significant
differences. First, as already known (e.g., Fang et al., 2013), at fixed
stellar mass the range of $\Sigma_{1}$ observed in passive galaxies is
significantly narrower (IQR$\sim$0.19 dex) than that of star-forming systems.
Second, and most importantly, the opposite happens for the stellar spin. The
spread in $\lambda_{re}^{c}$ at fixed mass increases by more than a factor of
2 (IQR increasing with mass from $\sim$0.2 to 0.57 dex, see Table 1) and, if
any trend is present, now spin clearly decreases with increasing stellar mass.
However, this should not be interpreted as passive galaxies simultaneously
growing their central surface density while losing their degree of rotational
support. Indeed, as it is unlikely that passive galaxies grow significantly in
mass via star formation, their mass increase must happen via mergers. The
chaotic nature of these processes, in particular in the case of major mergers,
makes it unlikely that the scaling relations shown in the left column of Fig.
4 trace evolutionary tracks for passive galaxies.

The lack of a correlation between central surface density and spin becomes
even more striking if we compare $\Delta(\Sigma_{1})$ to
$\Delta(\lambda_{re}^{c})$. The only difference here is that the offsets are
measured from the median values of the passive population at fixed mass. As
shown in the right panel of Fig. 4, $\Delta(\Sigma_{1})$ and
$\Delta(\lambda_{re}^{c})$ are no longer correlated
($\rho\sim-$0.04$\pm$0.03). Thus, there is no difference in average spin
between galaxies with high and low stellar central surface density, and no
difference in $\Sigma_{1}$ for galaxies with low and high spin at fixed mass.
In other words, in the passive population, central surface density is not a
good proxy for the kinematic state of the stellar population and,
specifically, to gauge the balance between rotationally- and dispersion-
supported structures within one effective radius. This result is not only
driven by the presence of slow rotators, but it still holds if we focus on
galaxies with $\lambda_{re}^{c}>$0.25 ($\rho\sim-$0.09$\pm$0.04). Only once we
focus our attention on galaxies with $\lambda_{re}^{c}>$0.4, hints of a
correlation between $\Sigma_{1}$ and $\lambda_{re}^{c}$ start to re-emerge,
but still with lower statistical significance than that observed for SFMS
galaxies ($\rho\sim-$0.22$\pm$0.03).

In retrospect, and as already hinted in the introduction, this result should
not come as a complete surprise. Already from a visual morphology point of
view we know that, while star-forming galaxies are primarily disks, passive
systems include pure discs, discs plus ‘bulges’ and pure ‘bulges’, with
‘bulges’ including both classical and pseudo-bulge structures. However, this
morphological spread is not encapsulated by parameters that are sensitive to
the 2D projected distribution of stars in galaxies, while they are clearly
traced by stellar kinematic information such as the stellar spin parameter
(e.g., Cappellari et al., 2011a; Krajnović et al., 2013; Cortese et al.,
2016).

##  4 Discussion

The main result of this work is the lack of a direct connection between
central stellar surface density and stellar spin parameter across the entire
SFR-M∗ parameter space covered by nearby galaxies. For star-forming galaxies,
differences in $\Sigma_{1}$, at fixed mass trace changes in stellar
kinematics. This correlation disappears below the SFMS. Here, $\Sigma_{1}$
cannot be used to infer the presence (and the importance) of classical bulges
in passive galaxies. We can use these findings, combined with previous works,
to improve our current understanding of how galaxies evolve from a structural
point of view across the SFR-M∗ plane.

Starting from Fig. 4, the striking difference in dynamic range between
$\Sigma_{1}$ and $\lambda_{re}^{c}$ at fixed stellar mass and the wide range
(0.2-0.57 dex) in observed stellar spin imply that the passive population
spans a variety of structural properties (and hence morphologies),
significantly larger than what is observed in the star-forming galaxy
population. Assuming that at higher redshift SFMS galaxies still show a small
scatter in stellar spin at fixed stellar mass, our finding implies that during
(or after) the quenching phase galaxies can experience a wide range of
structural transformation, from remaining rotationally-dominated with
relatively small classical bulges at their centre, up to becoming slow-
rotators fully dominated by random motions, with this extreme case becoming
relevant only at high stellar masses ($>$1010.5 M⊙, e.g., Guo et al., 2020;
van de Sande et al., 2021).

It would be tempting to use our findings to question the notion of a single
(or small number) of evolutionary paths towards quiescence for galaxies at
$z\sim 0$ (e.g., Woo & Ellison, 2019; Suess et al., 2021), and instead support
a more complex picture with a diversity of paths leading to quenching and
structural transformation (Cortese & Hughes, 2009; Fraser-McKelvie et al.,
2018; Janowiecki et al., 2020; Cortese et al., 2021; Saintonge & Catinella,
2022), which would also be in line with the diversity observed in stellar
population properties of passive galaxies both in the local Universe and at
higher redshift (e.g., Thomas et al., 2005; Graves et al., 2009; Tacchella et
al., 2022). However, we cannot exclude that just mergers alone (i.e., minor
and major mergers combined) may still explain the stellar kinematics
properties of galaxies, with the wide range of spin simply due to the huge
variety of, e.g., minor mergers that a galaxy can experience (see also Grand
et al., 2017; Garrison-Kimmel et al., 2018), combined with the inevitable mass
loss and associated adiabatic expansion (e.g., van Dokkum et al., 2014).
Nevertheless, our results clearly demonstrate that the $\Sigma_{1}$-$M_{*}$
parameter space alone cannot be used to fully unveil the structural evolution
of galaxies, and at least confirm that major/disruptive mergers become a
significant pathway for galaxy transformation only in the very high stellar
mass regime, where a statistically significant population of slow rotating
galaxies starts to emerge.

Whether the structural transformation happens during or after quenching
(Cortese et al., 2019), and if the same physical process is responsible for
both, cannot be established from this analysis. However, a comparison between
the range of $\Sigma_{1}$ and $\lambda_{re}^{c}$ covered by the active and
passive population, as shown in Figs. 3 and 4, suggests that the bulk of
dispersion-supported cores in our sample do not form on the SFMS. In other
words, we are not seeing major classical bulge growth on the SFMS, before
quenching. Of course, there is a small fraction of star-forming galaxies that
shows spin parameters consistent with a high degree of random motion (i.e.,
$\sim$10% with $\lambda_{re}^{c}$<0.4) but, as already shown by Fraser-
McKelvie et al. (2021), these are primarily interacting systems and cannot,
alone, explain the observed difference between the active and passive
populations. Similarly, while the scatter in stellar spin observed in our SFMS
sample indicates that galaxies may be growing dispersion-supported central
structures in the SFMS, these remain a relatively small fraction of the total
galaxy mass, and correspond to the typical population of early-type star-
forming spirals observed in the local Universe (e.g., Cook et al., 2019;
Falcón-Barroso et al., 2019). This is even more the case if we take into
account the fact that small changes in $\lambda_{re}^{c}$ in the SFMS may also
trace the presence of a thick disc component, and not of a central dispersion-
supported bulge, suggesting that our findings provide some conservative upper-
limits on the importance of dispersion-supported bulges in the SFMS.

As discussed by Croom et al. (2021b) and Cortese et al. (2019), part of the
differences in stellar spin between the active and passive populations could
be due to a combination of size evolution and progenitor bias, as most of the
quiescent systems in our sample left the SFMS a few billion years ago, at the
very least. However, even ignoring this, we do not really see a large family
of galaxies with kinematic properties consistent with the presence of
prominent central massive classical bulges (e.g., $\lambda_{re}^{c}<$0.5) on
the SFMS (see also Morselli et al., 2017; Cook et al., 2020). Either most of
classical bulges were formed at earlier epochs, via processes that have
gradually become less efficient in recent times, or their mass growth takes
place when (or after) galaxies have started leaving the SFMS. This is a key
difference between using $\Sigma_{1}$ and $\lambda_{re}^{c}$ as an indicator
of structural transformation: i.e., while, when observed in two dimensions,
all passive systems have similar central stellar surface densities consistent
with that expected for classical bulge-dominated systems, from a stellar
kinematic point of view they are far from being a homogeneous family, in
particular for stellar masses greater than $\sim$1010.5 M⊙. An increase in
central stellar surface density cannot be blindly interpreted as evidence for
an increase in the importance of random motions on the gravitational support
of galaxies. Indeed, as shown in Fig. 3, on the SFMS, galaxies grow in both
mass and central stellar surface density (Walters et al., 2021), but their
stellar spin either remains constant or even slightly increases.

Excitingly, the potential disconnect between quenching and morphological
transformation as well as the large variety of merger-driven kinematic changes
in galaxies suggested here appear in line with predictions from large hydro-
dynamical cosmological simulations such as EAGLE (e.g., Lagos et al., 2018,
2022) IllustrisTNG (e.g., Tacchella et al., 2019; Park et al., 2021). However,
it is important to note that some of these findings may be affected by
spurious collisional heating (Ludlow et al., 2021) and next generation/higher
resolution runs are needed to fully confirm these scenarios and allow us to
perform a quantitative comparison between observations and simulations (see
also van de Sande et al., 2019).

The main limitation of our work is that we are not comparing stellar surface
density and stellar spin for the same regions. While $\Sigma_{1}$ traces the
inner 1 kpc, $\lambda_{re}^{c}$ is measured within one effective radius, thus
encapsulating a larger area (on average a factor of $\sim$3 larger in radius
for passive systems and $\sim$4 for star-forming ones) and potentially more
prone to trace changes in the outer parts of galaxies. Unfortunately, with
current facilities, it is practically impossible to obtain higher spatial
resolution IFS observations for $\sim$1500 galaxies to quantify
$\lambda_{re}^{c}$ within 1 kpc. Thus, it is important to discuss if, and how,
our conclusions may be affected by this issue.

On the SFMS, it is possible that the correlation between spin and central
surface density becomes even stronger, but the main conclusion of star-forming
galaxies remaining rotationally-dominated systems and having only small
classical bulges would hold. Below the SFMS, it is difficult to think of a
scenario in which the lack of correlation between $\Sigma_{1}$ and
$\lambda_{re}^{c}$ would only be due to the different apertures used to
estimate the two quantities. This could only happen if the bulk of passive
rotationally-dominated galaxies becomes consistent with being a slow rotator
within 1 kpc. While a smaller aperture does imply a smaller rotational
velocity and thus a narrower range of $\lambda$ (assuming a flat velocity
dispersion profile), it is unlikely that the scatter in spin at fixed mass
decreases so much to become consistent, or even smaller, than that observed in
the SFMS. Indeed, from previous studies of the radial cumulative distribution
of spin parameter in early type galaxies, we know that the variety in spin
parameter values is already present in the very inner parts of galaxies (e.g.,
Emsellem et al., 2011; Arnold et al., 2014; van de Sande et al., 2017).

What we have been able to test is whether our results hold if we use the
stellar surface density within one effective radius ($\Sigma_{e}$ instead of
$\Sigma_{1}$ as a proxy for 2D structure). While, as already known, the
scatter in the $\Sigma_{e}$-M∗ relation is larger than what observed for the
$\Sigma_{1}$-M∗ trend, we still find that $\Sigma_{e}$ and $\lambda_{re}^{c}$
trace each other on the SFMS while they are uncorrelated in the passive
population.

Thus, despite the fact that the different apertures used in this work may
affect the detailed quantification of the correlation between $\Sigma_{1}$ and
$\lambda_{re}^{c}$, we expect that our main conclusion should qualitatively
hold once measuring the spin parameter within 1 kpc for large statistical
samples will become possible.

##  5 Summary

In this paper, we have combined estimates of the 2D stellar surface density
($\Sigma_{1}$) of galaxies within their central 1 kpc, with measurements of
their stellar spin parameter within one effective radius ($\lambda_{re}^{c}$),
to determine if changes in $\Sigma_{1}$ map variations in stellar kinematics.
We showed that:

  * •

On the SFMS, at fixed stellar mass, galaxies with higher $\Sigma_{1}$ have
lower spin, and vice-versa, but the vast majority of star-forming galaxies are
rotationally-dominated (i.e., discs) systems. While $\Sigma_{1}$ clearly grows
with increasing stellar mass, the stellar spin either remains constant or
increases by only $\lesssim$0.1 dex over 1 dex in stellar mass.

  * •

Below the SFMS, $\Sigma_{1}$ and $\lambda_{re}^{c}$ are no longer correlated.
Passive galaxies show a narrow range of $\Sigma_{1}$ but a wide range of
$\lambda_{re}^{c}$ values. Here, while $\Sigma_{1}$ still increases with
increasing mass, $\lambda_{re}^{c}$ decreases with mass, although with a large
scatter.

Together, our findings imply that, in passive galaxies, the central stellar
surface density cannot be used to disentangle rotationally- from dispersion-
supported structures. In the context of galaxy evolution and structural
transformation at $z\sim$0, we interpret our results as follows:

  * •

Only small classical bulges are able to grow while galaxies are on the SFMS.
Major compaction episodes where dispersion-supported structures grow and
create bulge-dominated star-forming systems are extremely rare at $z\sim$0.

  * •

If classical bulge formation is still efficient at $z\sim$0, this takes place
primarily during (or after) galaxies have started their quenching phase.

  * •

Passive galaxies are, from a structural point of view, an heterogeneous
population. While this may still be consistent with a two-way path towards
quiescence, at this stage it cannot be excluded that a multitude of
evolutionary paths (and potentially physical processes) are responsible for
the transformation of active discs into quiescent systems.

## Acknowledgements

We thank the anonymous referee for useful comments which improved the quality
of this manuscript. This research was conducted by the Australian Research
Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO
3D), through project number CE170100013. LC acknowledges support from the
Australian Research Council Discovery Project and Future Fellowship funding
schemes (DP210100337, FT180100066). JvdS acknowledges support of an Australian
Research Council Discovery Early Career Research Award (project number
DE200100461) funded by the Australian Government. JBH is supported by an ARC
Laureate Fellowship FL140100278. JJB acknowledges support of an Australian
Research Council Future Fellowship (FT180100231). SMS acknowledges funding
from the Australian Research Council (DE220100003). Parts of this work were
performed on the OzSTAR national facility at Swinburne University of
Technology. The OzSTAR program receives funding in part from the Astronomy
National Collaborative Research Infrastructure Strategy (NCRIS) allocation
provided by the Australian Government.

The SAMI Galaxy Survey is based on observations made at the Anglo-Australian
Telescope. The Sydney-AAO Multi-object Integral field spectrograph (SAMI) was
developed jointly by the University of Sydney and the Australian Astronomical
Observatory. The SAMI input catalogue is based on data taken from the Sloan
Digital Sky Survey, the GAMA Survey and the VST ATLAS Survey. The SAMI Galaxy
Survey is supported by the Australian Research Council Centre of Excellence
for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013, the Australian Research Council Centre of Excellence for All-sky
Astrophysics (CAASTRO), through project number CE110001020, and other
participating institutions. The SAMI Galaxy Survey website is http://sami-
survey.org/

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## Data Availability

The data used for this work has been presented in Fraser-McKelvie et al.
(2021) and Woo & Ellison (2019). The SAMI data cubes and value-added products
used in this paper are available from Astronomical Optics’ Data Central
service at: https://datacentral.org.au/. The MaNGA data products are available
at: https://www.sdss.org/dr15/manga/manga-data/data-access/.

## Author Contribution Statement

LC devised the project and drafted the paper. AMF and JW performed the
measurements of stellar kinematics and stellar surface density, respectively.
LC, AMF, JW, BC and KH contributed to the initial data analysis and
preliminary interpretation of the results. JvdS, JBH, JJB and SC provided key
support to all the activities of the SAMI Galaxy Survey (’builder status’).
All authors discussed the results and commented on the manuscript.

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##  Appendix A Effect of beam-smearing and inclination corrections on stellar
spin

In the main body of this paper we have used $\lambda_{re}$ values corrected
for both inclination and beam smearing as a proxy for stellar spin parameter
($\lambda_{re}^{c}$). As extensively discussed in the literature (e.g., Graham
et al., 2018; Harborne et al., 2020; Fraser-McKelvie et al., 2021; van de
Sande et al., 2021), for IFS surveys such as SAMI and MaNGA these corrections
combined can easily change the value of $\lambda_{re}$ by $\sim$50% or more.
Thus, it is important to make sure that the main conclusions of this paper are
not driven by potential systematic effects in the correction themselves, but
are still visible in the original measurements. With this aim, in Fig. 5 and 6
we reproduce Fig. 3 and 4 by using the observed value of $\lambda_{re}$
instead of $\lambda_{re}^{c}$. As expected, values of spin parameter are lower
and the scatter in the $\lambda_{re}$-M∗ relations for both active and passive
galaxies increase (average IQR $\sim$0.23 and $\sim$0.45 dex, respectively).
This is due to the large variety of inclinations and observing conditions for
the two samples. However, the lack of correlation between $\lambda_{re}$ and
$\Sigma_{1}$ ($\rho\sim-$0.09$\pm$0.04) and larger scatter in the
$\lambda_{re}$-M∗ relation for passive galaxies with respect to galaxies on
the SFMS still remains. As such, we can conclude that our results are not
qualitatively affected by the beam smearing and inclination corrections
applied in this work.

Figure 5: Same as Fig. 3, but with the observed $\lambda_{re}$ plotted instead
of the value corrected for both inclination and beam smearing. Figure 6: Same
as Fig. 4, but with the observed $\lambda_{re}$ plotted instead of the value
corrected for both inclination and beam smearing.

##  Appendix B Effect of seeing on the estimate of $\Sigma_{1}$

$\Sigma_{1}$ can be affected by seeing and, while our sample selection ensures
that the half-width-at-half-maximum (HWHM) of the SDSS observations is always
smaller than 1 kpc, it is important to briefly discuss if this might affect
our main conclusions. Indeed, Barro et al. (2017) have shown that, even when
the nominal resolution of the data is 0.6-0.7 kpc, $\Sigma_{1}$ could be
systematically underestimated and that this effect is stronger for galaxies
with higher Sersic index. Thus, the lack of correlation between
$\lambda_{re}^{c}$ and $\Sigma_{1}$ in passive galaxies could just be the by-
product of a larger effect of the atmospheric smearing in these galaxies.
Unfortunately, empirical corrections for the effect of seeing applicable to
our sample are not available. So, we cannot perform the same test presented in
Appendix A. Nevertheless, we can use the work of Barro et al. (2017), combined
with the properties of our sample, to discuss the potential role of seeing in
our findings. In Fig. 7 we show the cumulative distributions of HWHM for the
SDSS data used to estimate $\Sigma_{1}$ in physical scales (left) and of
Sersic index (right) for our SFMS (blue) and passive population (red). Passive
galaxies have the same HWHM distribution as SFMS objects and, if anything,
they are marginally better resolved. In addition, as expected, passive systems
cover a significantly narrower range of Sersic indices than active galaxies.

According to Barro et al. (2017, see their Fig. 14), at fixed HWHM, the
effects of seeing on $\Sigma_{1}$ correlate strongly with Sersic index, so
that a population covering a narrow range of Sersic indices will be all
affected in a similar way, whereas for a sample spanning a wide range of
Sersic indices the changes in $\Sigma_{1}$ may vary significantly (up to
$\sim$0.3 dex in the most extreme cases) from galaxy to galaxy. As such, given
that our SFMS spans a significantly larger range of Sersic indices than the
passive population, we should expect this family to be potentially more
affected by atmospheric blurring and any secondary trends to be washed away.
Conversely, we still find secondary trends for the SFMS population, making it
unlikely that the lack of correlation between $\Sigma_{1}$ and
$\lambda_{re}^{c}$ for passive galaxies is just an artefact of the limited
spatial resolution of our data (see also the discussion of seeing effects in
Woo et al., 2015). Lastly, we note that we have also tested our results by
focusing on a sub-sample with HWHM$<$0.5 kpc and our results are unaffected
but, due to the significantly smaller number statistics, the quantification of
trends as a function of mass and $\Delta(\Sigma_{1})$ or
$\Delta(\lambda_{re}^{c})$ becomes highly uncertain, with only two bins of
stellar mass having enough galaxies. This is why, contrary to what done in
Appendix A, we do not show the equivalent of Fig. 3 and 4 for this subsample.

Figure 7: Cumulative distributions of seeing half-width-at-half-maximum (HWHM)
in physical units (left) and Sersic indices (right) for SFMS (blue) and
passive galaxies (red).

##  Appendix C The $\Sigma_{1}$-$M_{*}$ and $\lambda_{re}^{c}$-$M_{*}$
relations for star-forming main sequence and passive galaxies

In Table 1 we present the median values (as well as 25% and 75% percentiles)
for $\Sigma_{1}$ and $\lambda_{re}^{c}$ per bin of stellar mass. Each bin
contains 100 galaxies. As discussed in § 2, we use Eq. 1 to separate star-
forming from passive galaxies, labelling as passive all systems below $-$0.5
dex from the locus of the SFMS.

Table 1: The median $\Sigma_{1}$ and $\lambda_{re}^{c}$ per bin of stellar
mass for SFMS and passive galaxies. Uncertainties indicate the 75% and 25%
percentiles. $\begin{array}[]{ccc}\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\log(M_{*}/\rm
M_{\odot})&\log(\Sigma_{1}/\rm M_{\odot}~{}kpc^{-2})&\log(\lambda_{re}^{c})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\lx@intercol\hfil\rm Star-
forming~{}main~{}sequence~{}galaxies\hfil\lx@intercol\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr
9.89&8.74^{+0.16}_{-0.17}&-0.20^{+0.05}_{-0.09}\\\\[2.84544pt]
10.21&8.99^{+0.13}_{-0.20}&-0.17^{+0.06}_{-0.09}\\\\[2.84544pt]
10.39&9.17^{+0.13}_{-0.21}&-0.15^{+0.04}_{-0.08}\\\\[2.84544pt]
10.59&9.28^{+0.15}_{-0.10}&-0.14^{+0.05}_{-0.05}\\\\[2.84544pt]
10.74&9.45^{+0.10}_{-0.15}&-0.14^{+0.05}_{-0.09}\\\\[2.84544pt]
10.97&9.54^{+0.19}_{-0.10}&-0.15^{+0.05}_{-0.06}\\\\[2.84544pt]
11.22&9.69^{+0.13}_{-0.14}&-0.10^{+0.03}_{-0.08}\\\\[2.84544pt] \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\hline\cr\lx@intercol\hfil\rm
Passive~{}galaxies\hfil\lx@intercol\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr
10.23&9.23^{+0.11}_{-0.14}&-0.24^{+0.07}_{-0.14}\\\\[2.84544pt]
10.45&9.38^{+0.10}_{-0.13}&-0.29^{+0.10}_{-0.18}\\\\[2.84544pt]
10.58&9.49^{+0.11}_{-0.09}&-0.28^{+0.07}_{-0.09}\\\\[2.84544pt]
10.70&9.61^{+0.09}_{-0.10}&-0.25^{+0.07}_{-0.10}\\\\[2.84544pt]
10.82&9.67^{+0.07}_{-0.10}&-0.30^{+0.13}_{-0.15}\\\\[2.84544pt]
10.94&9.72^{+0.07}_{-0.06}&-0.34^{+0.12}_{-0.39}\\\\[2.84544pt]
11.08&9.80^{+0.09}_{-0.09}&-0.47^{+0.24}_{-0.24}\\\\[2.84544pt]
11.28&9.88^{+0.06}_{-0.09}&-0.60^{+0.25}_{-0.32}\\\\[2.84544pt] \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\end{array}$