# Effects of orbital selective dynamical correlation on the spin
susceptibility and superconducting symmetries in Sr2RuO4

Chang-Youn Moon<EMAIL_ADDRESS>Advanced Instrumentation Institute, Korea
Research Institute of Standards and Science, Yuseong, Daejeon 305-340,
Republic of Korea

###### Abstract

We investigate the connection between the local electron correlation and the
momentum dependence of the spin susceptibility and the superconducting gap
functions in Sr2RuO4, using density-functional theory combined with dynamical
mean-field theory. Adopting frequency-dependent two-particle vertex moves the
zero energy spin susceptibility peaks towards the Brillouin zone center,
compared with random-phase approximation which basically retains the peak
positions closer to the Brillouin zone boundary as determined by the Fermi-
surface nesting. We find that $d_{xy}$ orbital plays a central role here via
its enhanced correlation strength. Solving the linearized Eliashberg equation
from this spin susceptibility, prime candidates of the superconducting gap
symmetry are a $s$-wave, along with a nearly degenerate $d$-wave solution, all
in spin singlet. Furthermore, another set of degenerate spin singlet gap
functions emerges, odd with respect to $k$-point as well as orbital exchanges.
We show that the stability of these gap functions are strongly dependent on
the peak position of the spin susceptibility in the Brillouin zone.

Sr2RuO4 remains as one of most intriguing superconductors with the possibility
of highly exotic and unconventional superconductivity, even after three
decades since its discovery Maeno . Noticing the analogy with superfluid 3He,
earlier studies kindled discussions on the $p_{x}+ip_{y}$ pairing
MackenzieMaeno ; RiceSigrist ; Sigrist ; MaenoKittaka ; Kallin . This order
parameter can be characterized by two essentially independent properties; the
chirality, implying non-zero orbital magnetic moment, and the spin-triplet
pairing. Spin-triplet pairing scenario, however, has been put into a serious
question after recent spin susceptibility measurements using nuclear magnetic
resonance (NMR) Knight shift Pustogow ; IshidaManago  and polarized neutron
scattering Petsch . They show that the susceptibility is suppressed below
$T_{c}$ more consistently with the spin-singlet pairing, overruling an earlier
contradicting Knight shift measurement IshidaMukuda . The chirality, or time-
reversal symmetry breaking (TRSB) more generally, has been supported by a
recent zero-field muon spin relaxation (ZF-$\mu$SR) measurement under uniaxial
stress resulting in split superconducting and TRSB transitions Grinenko ,
together with earlier works using ZF-$\mu$SR Luke  and non-zero Kerr rotation
Xia ; Kapitulnik . Evidences of the two-component order parameter Benhabib ;
Ghosh  also have appeared, which can serve as a broader constraint on the
nature of pairing in which the possibility of the chirality is included.
Combinations among even parity order parameters of such as $s$, $d$, and $g$
symmetries are now generally considered plausible candidates.

The local electronic configuration near the Fermi energy ($E_{F}$) comprises
four electrons in $t_{2g}$ levels split from empty $e_{g}$ levels within Ru
$d$ states, and hence Sr2RuO4 is a multi-orbital system where Hund’s coupling
$J_{H}$ plays a significant role in local Coulomb correlations. Consequently
its normal-state behaviors, such as the crossover from high-temperature
incoherent phase to low-temperature Fermi-liquid MackenzieMaeno ; Katsufuji ;
Imai ; Hussey ; Valla ; Wang ; Kidd  are understood in the context of Hund’s
metals MravljeAichhorn ; Medici ; MravljeGeorges ; Kugler . General features
of Hund’s metals include strong local spin fluctuation and orbital
differentiation HauleNJP ; Moon , as manifested in Sr2RuO4 with much enhanced
effective mass of $d_{xy}$-derived Fermi surface (FS) sheet compared with
$d_{xz/yz}$ derived ones MackenzieMaeno ; MravljeAichhorn . Then it is an
important question that how the local correlation that governs the normal
state works on the superconductivity. Assuming the spin-fluctuation-mediated
superconductivity, the two-particle vertex, which forms fully interacting spin
susceptibility ($\chi$) when combined with the polarization bubble
($\chi^{0}$), is one of major channels through which the local correlation
effect is incorporated in electron pairing. While some of previous theoretical
studies adopted the random-phase-approximation (RPA) for the vertex keeping
the lowest order scattering process RomerScherer ; RomerKreisel ;
RomerHirschfeld ; RomerMaier ; Gingras1 ; Gingras2 , others found that using
frequency dependent vertex produces qualitative differences in susceptibility
and superconductivity Strand ; EPL ; Kaser . However, distinctive roles played
by the orbital selectivity and its dynamic nature in the susceptibility
evaluation remain elusive especially with respect to the features critically
connected to the pairing symmetry, such as the peak position of the
susceptibility.

In this study, we demonstrate how the dynamic local correlations shape the
susceptibility in the $k$-space and the pairing symmetry in Sr2RuO4 within the
framework of density-functional theory combined with dynamical mean-field
theory (DFT+DMFT). Adopting the dynamic vertex results in the peak position of
spin susceptibility closer to the Brillouin zone (BZ) center compared with RPA
case, in better agreement with experimental measurements. We argue it
originates from the stronger frequency dependence of the vertex within the
$d_{xy}$ orbital than the other $t_{2g}$ components, relocating the peak off
the position determined by simple FS nesting as in the RPA case. When the
linearized-Eliashberg equation is constructed from the susceptibility and
solved, the most probable gap function turns out to be of a $s$-wave symmetry
with nodes followed by a $d$-wave solution slightly less stable, suggesting a
possible two-component order parameter with an accidental degeneracy. Another
plausible solutions are found to be symmetry-protected doubly degenerate ones
which are odd with respect to both $k$ and orbital exchanges. We show that the
relative stability of these gap functions changes drastically between the
respective susceptibility peak positions from the dynamical vertex and RPA,
emphasizing on the importance of the accurate evaluation of the susceptibility
in $k$-space and of the orbital selective and dynamic local correlation
effects which enable it.

We use the modern implementation of DFT+DMFT method within all electron
embedded DMFT approach DMFT  which is based on Wien2k wien2k , without
downfolding or other approximations. The code is freely available on the web
Haulecode . We employ LDA exchange-correlation functional LDA1 ; LDA2 , and
the quantum impurity model is solved by the continuous time quantum Monte
Carlo (CTQMC) impurity solver CTQMC . Here sampled quantities are expanded in
the basis function obtained by the singular value decomposition of the kernel
for analytic continuation, ensuring reduced high-frequency noise Shinaoka .
Internal atomic positions are optimized, and $U=4.5$ eV and $J=1.0$ eV are
adopted consistently with a previous study on this material DengHaule
employing the same computational method. We use the Slater parametrization of
the Coulomb interaction in this study, and our $U$ and $J$ parameters are
defined with respect to the three Slater parameters in such a way that
$F^{0}=U$, $F^{2}=112/13~{}J$, and $F^{4}=70/13~{}J$. BZ integration is done
on the 30$\times$30$\times$30 k-point mesh for the body-centered tetragonal
primitive unitcell in converging charge density and self energy, while
40$\times$40$\times$2 mesh in the BZ for the conventional tetragonal unitcell
is used for the susceptibility and gap function calculations. All calculations
are done in 116 K. Spin-orbit coupling (SOC) can have non-negligible effects
on the electronic structure mainly near the region where different sheets of
FS intersect, by strong orbital mixing ZhangGorelov ; Kim ; Tamai . As
evaluating the two-particle vertex including SOC is still not available, and
also considering susceptibilities are less affected by SOC than one-particle
spectra Gingras1 , we neglect SOC in this work.

Figure 1: (a) Fermi surface shown on the $k_{z}=0$ plane. Orbital character is
denoted by the depth of the color as well as the thickness of the line. (b)
$[\Gamma\chi\Gamma]^{s}_{\alpha\alpha;\alpha\alpha}(i\nu,i\nu)$ where
$\alpha=d_{xy}$ and $i\nu\approx 0$ on the $k_{z}=0$ plane. The momentum
transfer of one of the four equivalent peaks inside the first Brillouin zone
is marked by $Q_{\omega}$. (c) Infinite series expansion of
$[\Gamma\chi\Gamma]^{s}_{\alpha\alpha^{\prime};\delta\delta^{\prime}}(i\nu,i\nu)$
by the Bethe-Salpeter equation. Here the polarization bubble $\chi^{0}$ is
assumed to be diagonal in orbital indices (see the main text). (d) Same with
(b) but using the frequency-averaged constant two-particle vertex instead of
the dynamic vertex. The peak is denoted by $Q_{\rm RPA}$.

Figure 1(a) shows our calculated FS in the body-centered tetragonal BZ. Two
one-dimensional (1D) FS sheets ($\alpha$ and $\beta$) are almost purely
$d_{xz}$ and $d_{yz}$-derived, respectively, while two-dimensional (2D) FS
($\gamma$) is from $d_{xy}$, consistently with previous studies. All local
correlation effects as contained in the dynamical self energy are included.
Then we evaluated the susceptibility $\chi$, the polarization bubble
$\chi^{0}$, and the two-particle vertex $\Gamma$ which are related by the
Bethe-Salpeter equation for a given bosonic Matsubara frequency $\Omega$ which
is fixed to zero in this work:
$\chi^{s/c}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(i\nu,i\nu^{\prime},q)=((\chi^{0}(q))^{-1}-\Gamma^{s/c})^{-1}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(i\nu,i\nu^{\prime})$.
All the two-particle quantities are defined in the particle-hole channel if
not explicitly indicated otherwise, and $s/c$ stands for spin/charge.
$\Gamma^{s/c}$ is a dynamic local quantity containing all two-particle-
irreducible diagrams and is obtained in the impurity solver Hyowon ; Yin2014 ,
as is the self energy which is the one-particle counterpart of the vertex. We
display $\Gamma^{s}\chi^{s}\Gamma^{s}\equiv[\Gamma\chi\Gamma]^{s}$, which acts
as an effective pairing potential in the Eliashberg equation, for the $d_{xy}$
orbital channel at the lowest fermionic Matsubara frequency
($i\nu=i\nu^{\prime}\approx$ 0) Matsubara  in Fig. 1(b). There is a strong
peak at $q=(0.3,0.3)\equiv Q_{\omega}$ where X point is at $(0.5,0.5)$,
reproducing the peak position of $\chi$ from measurement Steffens  and
previous calculation Strand . Note that $[\Gamma\chi\Gamma]^{s}$ has basically
the same peak structure with $\chi^{s}$ in BZ, since the matrix multiplication
between $\Gamma^{s}$ and $\chi^{s}$ corresponds to the weighted summation over
different components of $\chi^{s}$ which have similar momentum structure with
one another. To highlight the effect of frequency dependence of the vertex, we
evaluate $[\Gamma\chi\Gamma]^{s}$ in Fig. 1(d) using the frequency-averaged
(static) vertex corresponding to the equal-time component, which is equivalent
to RPA. The peak is now at $q=(0.35,0.35)\equiv Q_{\rm RPA}$ closer to X
compared with $Q_{\omega}$ also consistently with the previous calculation
Strand , suggesting that the momentum structure of the susceptibility alters
with local correlations having different dynamic structures. $d_{xz/yz}$
components have the same peak position with $d_{xy}$, but with a distinctively
reduced amplitude, both for dynamic and static vertices SM  where the orbital
selectivity is inscribed. The charge counterpart, $[\Gamma\chi\Gamma]^{c}$, is
found to be negligible compared with $[\Gamma\chi\Gamma]^{s}$ for the dynamic
vertex, while they are comparable for the static vertex SM .

Figure 2: With inter-orbital components of the vertex $\Gamma$ set to zero,
$[\Gamma\chi\Gamma]^{s}_{\alpha\alpha;\alpha\alpha}(i\nu\approx 0,i\nu\approx
0)$ is calculated for $\alpha=d_{xz/yz/xy}$ in (a)/(b)/(c). (e)/(f)/(g) is the
same with (a)/(b)/(c), but using the frequency-averaged constant two-particle
vertex instead of the dynamic vertex. (d) is obtained by multiplying (a), (b),
and (c), while (h) is from multiplying (e), (f), and (g). The peak position is
denoted by $Q^{\prime}_{\omega}$/$Q^{\prime}_{\rm RPA}$ in (d)/(h).

Now we are ready to reveal the mechanism of how the orbital selectivity
encoded in the local vertex determines the peak position of the susceptibility
in the $k$-space. To facilitate isolating the role of each orbital in its
respective dynamic structure, we calculate $[\Gamma\chi\Gamma]^{s}$ with
inter-orbital components of the vertex turned off, i.e.,
$\Gamma^{s}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}=\Gamma^{s}_{\alpha\alpha^{\prime};\alpha\alpha^{\prime}}\delta_{\alpha,\beta}\delta_{\alpha^{\prime},\beta^{\prime}}$,
and display it in Fig. 2. Since the FSs are almost completely decoupled in
orbitals as seen in Fig. 1(a), the polarization bubble, defined as the product
of two counter-propagating one-particle Green’s functions, is also nearly
diagonal:
$\chi^{0}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(q)\approx\chi^{0}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(q)\delta_{\alpha,\beta}\delta_{\alpha^{\prime},\beta^{\prime}}\equiv\chi^{0}_{\alpha\alpha^{\prime}}(q)$.
Then
$[\Gamma\chi\Gamma]^{s}_{\alpha\alpha;\alpha\alpha}(i\nu,i\nu^{\prime},q)$ is
mostly composed of $\chi^{0}_{\alpha\alpha}(i\nu^{\prime\prime},q)$ with the
weight of
$\Gamma^{s}_{\alpha\alpha;\alpha\alpha}(i\nu^{\prime\prime},i\nu^{\prime\prime\prime})$
for the intermediate frequencies $i\nu^{\prime\prime}$ and
$i\nu^{\prime\prime\prime}$, but without incorporating orbital component other
than $\alpha$ in the infinite series as illustrated in Fig. 1(c). The diagonal
component of $[\Gamma\chi\Gamma]^{s}$ in $d_{xz}$ orbital for
$i\nu=i\nu^{\prime}\approx 0$ is shown in Fig. 2(a). The stripe pattern along
the $x$ direction is from the FS nesting in the $d_{xz}$-originated 1D FS
sheet, and the same is true for the $d_{yz}$ component displayed in Fig. 2(b)
with $\pi$/2 rotation. The fact that the $k$-dependence of
$[\Gamma\chi\Gamma]^{s}$ is determined by the simple FS nesting suggests that
the vertex depends only weakly on the frequency, decoupling the polarization
bubble from the vertex in the frequency domain for a given order of the
infinite series in Fig. 1(c). Then the summation of the intermediate frequency
produces $\sum_{\nu}\chi^{0}(i\nu)=\chi^{0}(\Omega=0)$, the polarization
bubble in zero bosonic frequency which is characterized by the FS nesting.
Indeed, these results are almost identical with those from frequency-averaged
static vertex as displayed in Fig. 2(e) and Fig. 2(f). In case of the $d_{xy}$
channel, $[\Gamma\chi\Gamma]^{s}$ exhibits a four-fold rotation symmetry with
weights around X points in the BZ reflecting the symmetry of the orbital as
shown in both Fig. 2(c) and Fig. 2(g) for the dynamic and frequency-averaged
static vertices, respectively. On the contrary to the $d_{xz/yz}$ channels,
however, there is a noticeable difference between the two vertices. Weights
are distributed farther away from the $\Gamma$ point for the dynamic vertex,
which results from the frequency-dependent coefficient (i.e., vertex) of
$\chi^{0}(i\nu)$ in the frequency ($\nu$) summation SM .

As the next step, orbital-separate contributions shown in Fig.
2(a)-(c)/(e)-(g) can be effectively coupled again to re-introduce inter-
orbital components in $[\Gamma\chi\Gamma]^{s}$, by simply multiplying the
contributions from each orbital channel. This is because inter-orbital terms
consisting of the product among $\chi^{0}$’s with different orbital indices,
such as $\chi^{0}_{xy~{}xy}\chi^{0}_{xz~{}xz}$, are restored by the
multiplication in the infinite series expansion of $[\Gamma\chi\Gamma]^{s}$ as
displayed in Fig. 1(c). Figure 2(d)/(h) is obtained by multiplying Fig.
2(a)/(e), (b)/(f), and (c)/(g), and indeed reproduces the strong peak
structure as seen in the original full vertex calculation result in Fig.
1(b)/(d) which includes both intra- and inter-orbital components. Moreover,
the peak position from the dynamic vertex in Fig. 2(d) at
$Q_{\omega}^{\prime}\approx(0.32,0.32)$ is closer to the zone center than that
from the static vertex at $Q_{\rm RPA}^{\prime}\approx(0.35,0.35)$, also in
accordance with the full vertex results. Having demonstrated that
$[\Gamma\chi\Gamma]^{s}$ can be effectively reconstructed from the product of
orbital-separate components, it is evident that the difference of the peak
position between by using the dynamic and static vertices mainly originates
from the contrasting weight distributions of the $d_{xy}$ contribution: weight
is distributed closer to $\Gamma/X$ point in Fig. 2(c)/(g), resulting in the
peak closer to $\Gamma/X$ point in Fig. 2(d)/(h). Therefore, we can conclude
it is the $d_{xy}$ component of the vertex that is responsible for the shift
of the peak position from $Q_{\rm RPA}=(0.35,0.35)$, as determined by the FS
nesting, to $Q_{\omega}=(0.3,0.3)$, via its strong frequency dependence. This
is true not only for $\alpha=d_{xy}$ but also for $\alpha=d_{xz/yz}$ in
$[\Gamma\chi\Gamma]^{s}_{\alpha\alpha;\alpha\alpha}$ SM  which also
incorporates $\chi^{0}_{xy\,xy}$ through the inter-orbital vertex. The
pronounced dynamic nature of the $d_{xy}$ component of the vertex is
consistent with $d_{xy}$ orbital’s larger mass enhancement factor pointing to
the larger correlation effect than $d_{xz/yz}$, which was suggested to
originate from the proximity of the van Hove singularity of $d_{xy}$ band to
$E_{F}$ MravljeAichhorn .

Having calculated $[\Gamma\chi\Gamma]^{s/c}$, we can solve the linearized
Eliashberg equation which also has been adopted in previous works Gingras1 ;
Gingras2 ; Kaser ; Acharya , derived from the divergence condition of the
susceptibility in the particle-particle channel:

$-k_{B}T\sum_{k^{\prime}\nu^{\prime}\alpha^{\prime}\beta^{\prime}\gamma\delta}\Gamma^{pp,s/t}_{\alpha\beta;\alpha^{\prime}\beta^{\prime}}(k\nu;k^{\prime}\nu^{\prime})\chi^{0,pp}_{\alpha^{\prime}\beta^{\prime}\gamma\delta}(k^{\prime}\nu^{\prime})\Delta_{\gamma\delta}(k^{\prime}\nu^{\prime})=\lambda\Delta_{\alpha\beta}(k\nu)$

where the eigenvalue $\lambda$ and the eigenfunction $\Delta$ can be
interpreted as the pairing strength and the gap function, respectively.
$\Gamma^{pp,s/t}$ is the irreducible vertex in the particle-particle channel
and consists of different combinations of $[\Gamma\chi\Gamma]^{s/c}$ in
interchanged momentum and orbital indices depending on spin singlet/triplet
(s/t) pairing. Details of the formalism adopted in this study are found in
Ref. Yin2014 . Here we assume the constant frequency dependence of
$\Gamma^{pp}$ so that $\Gamma^{pp}(i\nu,i\nu^{\prime})=\Gamma^{pp}(i\nu\approx
0,i\nu^{\prime}\approx 0)$, then the resultant gap function $\Delta(k)$ does
not depend on the frequency either as in the BCS approximation where only
even-frequency solutions can be captured. Nevertheless, it should be noted
that the internal frequency summations are performed without any approximation
which is found to be essential to obtain the correct peak position of
$[\Gamma\chi\Gamma]^{s}$ as demonstrated above. The gap function so obtained
is distributed over entire BZ in the orbital basis and in principle at zero
energy, and hence need to be projected to the FS where spectral function is to
be evaluated at zero energy:

$\Delta_{ij}(k)=\sum_{\alpha\beta,i^{\prime}j^{\prime}}\Psi^{L}_{k,ii^{\prime}}U_{k,i^{\prime}\alpha}\Delta_{\alpha\beta}(k)U^{\dagger}_{k,\beta
j^{\prime}}\Psi^{R}_{k,j^{\prime}j}$

. Here $\Psi^{L/R}_{k,ii^{\prime}}$ is the $i$th left/right eigenvector in
$i^{\prime}$th DFT band basis, of the complex DMFT eigenvalue equation at zero
energy, and $U_{k,i^{\prime}\alpha}$ is the projector between orbital $\alpha$
and $i^{\prime}$th DFT band at $k$ Yin2014 ; DMFT . This gives us more
familiar gap functions similar with those defined in the band-based mean-field
formalism instead of Green’s functions.

Figure 3: Superconducting gap functions with three largest eigenvalues from
the linearized Eliashberg equation, projected to the DMFT band basis on the
FS, all in spin singlet. Gap functions in (a) and (b) are both from the
$d_{xy}$ orbital with $\lambda=0.0111$ and 0.0101, and with $s$\- and $d$-wave
symmetries, respectively. $\lambda=0.0088$ solutions are doubly degenerate,
where one has the inter-orbital gap function between $d_{xz}$ and $d_{xy}$ as
shown in (c), and the other is between $d_{yz}$ and $d_{xy}$ with the gap
function $\pi$/2 rotated from (c). These degenerate gap functions have odd-
orbital and odd-parity $f$-wave symmetry. (d) Eigenvalue of each gap symmetry
as a function of the peak position $(q,q)$ in the spin susceptibility from
$Q_{\omega}$ ($q$=0.30) to $Q_{\rm RPA}$ ($q$=0.35). The peak of the spin
susceptibility in an arbitrary position is approximated by a gaussian function
centered at the position. Eigenvalues are rescaled so that they match the
values from the original full calculation at $Q_{\omega}$.

The gap functions for three largest eigenvalues are displayed in Fig.
3(a)-(c), all of which are in spin singlet. The most probable solution with
$\lambda=0.0111$ is from the $d_{xy}$ band and has a $s$-wave symmetry with
nodes (”nodal s-wave”, $A_{1g}$ irreducible representation), followed by a
slightly less stable solution ($\lambda=0.0101$) with also $d_{xy}$ orbital
character but in the $d_{x^{2}-y^{2}}$ symmetry ($B_{1g}$ irreducible
representation). Besides the symmetry-imposed nodes at the intersections
between 2D and 1D FS sheets where the gap function changes the sign, there are
also regions with depleted weights around the van Hove point on the 2D FS for
the $d_{x^{2}-y^{2}}$ solution. The proximity of the eigenvalues or the
accidental (near) degeneracy between the $s$\- and $d$-wave solutions points
to the possibility of the TRSB $s+id$ order parameter suggested earlier
RomerScherer ; RomerKreisel ; RomerHirschfeld ; RomerMaier , although it is
$s+id_{xy}$ rather than $s+id_{x^{2}-y^{2}}$ that is more consistent with
recent experiments Benhabib ; Ghosh ; RomerHirschfeld . Meanwhile, the third
largest eigenvalue ($\lambda=0.0088$) is associated with doubly degenerate
solutions which are protected by the lattice symmetry. Unlike $s$\- and
$d$-wave solutions which represent intra-orbital (band) pairings within the
$d_{xy}$ orbital, the gap functions are inter-orbital between $d_{xy}$ and
$d_{xz/yz}$. One can see that the sign of the gap function is the opposite
between $d_{xy}$ and $d_{xz}$ FSs where they meet as in Fig. 3(c), indicating
it is odd with respect to the orbital (band) interchange. Moreover, it is also
odd in $k$-space ($f$-wave, $E_{u}$ irreducible representation), satisfying
the antisymmetric fermion exchange rule :
$\hat{S}\hat{P}\hat{O}\hat{T}=-1\times-1\times-1\times 1=-1$. Odd-parity
pairing, usually combined with spin-triplet, here associates with spin-singlet
which is only possible with the extra degree of freedom of orbitals, and hence
is characteristic of the multi-orbital superconductivity. The degenerate pair
of gap functions, one from $d_{xy}$ and $d_{xz}$ and another from $d_{xy}$ and
$d_{yz}$ (Fig. 3(c) and its $\pi$/2 rotation, respectively), can form a TRSB
chiral order parameter $f+if$.

Our $s$-wave solution changes sign at every $\pi$/4, hence is effectively
stabilized by the repulsive pairing potential peaked at the smaller momentum
transfer $Q_{\omega}$, compared with longer $Q_{\rm RPA}$ determined by the FS
nesting. $Q_{\omega}$ is rather short to connect the van Hove regions having
higher density of states, hence these regions are depleted in weight for the
$d$-wave gap making it a relatively less stable solution. Indeed, Mazin et al.
adopted susceptibility with a nesting-induced peak at $Q\approx(1/3,1/3)$ to
obtain a $d$-wave gap with maximum weights at the van Hove point in their
early work Mazin , while an $s$-wave gap has been suggested by using the peak
position determined by inelastic neutron scattering at $Q\approx(0.3,0.3)$
Steffens  consistently with our result. To explicitly study the relative
stability of each solution with respect to the susceptibility peak position,
we employ the spin susceptibility ($[\Gamma\chi\Gamma]^{s}$) consisting of
simple gaussian peaks at any desired position in the BZ which should be a
reasonable approximation seeing the simple peak structures of the original
spin susceptibility shown in Fig. 1(b). Solving the gap equation from this
susceptibility consisting of gaussian peaks, we display the result in Fig.
3(d). For $s$\- and $d$-wave solutions, we adopt $[\Gamma\chi\Gamma]^{s}$ with
only $d_{xy}$ diagonal component, while for the $f$-wave gap only the inter-
orbital components between $d_{xy}$ and $d_{xz/yz}$ are set non-zero. The
eigenvalue $\lambda$ of each solution as a function of the peak position is
rescaled so that it retains its original value from the full calculation at
$Q_{\omega}$. As expected, the $s$-wave gap grows unstable with increasing
momentum transfer, while the $d$-wave gap is more stable at $Q_{\rm RPA}$. The
stability of the $f$-wave gap also decreases from $Q_{\omega}$ to $Q_{\rm
RPA}$, so that the $d$-wave is dominant over other gap symmetries at $Q_{\rm
RPA}$. Therefore, we can conclude that TRSB $s+id$ and chiral $f+if$ gap
symmetries are stabilized by the antiferromagnetic spin fluctuation near
$Q_{\omega}=(0.3,0.3)$, which is only available by using the dynamical vertex
function.

Low energy excitations and the presence of (vertical line) gap nodes are
another requirement for the feasible gap symmetries imposed by experiments
node1 ; node2 ; node3 ; node4 ; node5 ; node6 . Our $s$\- and $d$-wave
solutions are from the $d_{xy}$ FS, so the 1D FSs from $d_{xz/yz}$ are
basically gapless for each of the two solutions (see Fig. 3(a) and (b)) and
also for their TRSB complex combination $s+id$. When SOC is included, the
$d_{xy}$ component would be incorporated on the 1D FSs due to the orbital
mixing, but it is likely that some part of 1D FSs are still gapless. On the
other hand, the condition for the existence of gap nodes is tighter for the
chiral $f+if$ order parameter, since both 2D and 1D FSs participate in this
inter-orbital electron pairing (Fig. 3(c) and its $\pi$/2 rotation).
Nevertheless, the van Hove points in the 2D FS would have the least weight in
the complex combination between a sign-changing node and the relatively
smaller weight on the $\pi$/2 rotated point. Further studies might be required
to elaborate on the compatibility of our suggested gap symmetries with
experimental observations, as well as the possible effect of SOC Gingras2 ;
Kaser .

In conclusion, we investigate the role of the dynamic local correlation in the
spin susceptibility and the superconducting symmetry in Sr2RuO4 within the
DFT+DMFT framework. The two-particle vertex is found to weakly depend on the
frequency within $d_{xz/yz}$ orbitals while its $d_{xy}$ component is strongly
frequency-dependent, locating the peak of the spin susceptibility closer to
the BZ center compared with the FS nesting-driven peak position. We find that
the relatively smaller momentum transfer in the spin susceptibility stabilizes
a $s$-wave gap with many sign-changing nodes while it is against the stability
of the $d$-wave, resulting in an accidental near degeneracy of the two gap
symmetries both in spin singlet. Odd-parity and odd-orbital $f$-wave spin
singlet gap functions are also found with symmetry-protected double
degeneracy, which can lead to the realization of a chiral superconductivity.
Our work demonstrates how the local correlation and orbital selectivity in a
Hund’s metal affect the non-local electronic structure in the two-particle
level and the superconductivity.

###### Acknowledgements.

The author thanks Y. Bang, I. I. Mazin, E.-G. Moon, and K. Haule for helpful
discussions.

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Nevertheless, the simulation results roughly reproduce the observational
trends at any given epoch, with the primary difference between the two
simulations discussed by Remus et al. (2017) being their treatment of black
hole feedback.

We can use simple scaling relations to better understand galaxies’ expected
evolution in Figure 14 given various assumptions. Following Figure 12 we adopt
a simple model for size growth during mergers such that $\Delta
r_{\mathrm{e}}\propto\Delta M_{\ast}^{\alpha}$, with $\alpha=1$ or 2 for major
and minor mergers, respectively. We additionally assume that the enclosed dark
matter mass scales with the total (virial) mass of the halo and radius as
$M_{\mathrm{DM}}(r)\propto M_{\mathrm{vir}}r^{2}$, and that the stellar and
halo mass grow at the same rate ($\Delta\log M_{\mathrm{vir}}=\Delta\log
M_{\ast}$). Arrows in the right-hand panel of Figure 14 show the predicted
evolution for a galaxy doubling its stellar mass either through a single major
merger, or successive 10:1 (minor) mergers. As expected, the efficient size
growth associated with minor mergers in our toy model drives rapid evolution
in both $\Sigma_{\mathrm{e}}$ and $f_{\mathrm{DM}}[<r_{\mathrm{e}}]$. However,
while minor mergers can explain the observed decrease in $\Sigma_{\mathrm{e}}$
from high to low redshift, our toy model over-predicts the increase of
$f_{\mathrm{DM}}[<r_{\mathrm{e}}]$. Tortora et al. (2018) report a similar
result for galaxies at $z\approx 0.7$, and suggest that allowing for variation
in the stellar-to-halo mass ratio of accreted galaxies (i.e. $\Delta\log
M_{\mathrm{vir}}\neq\Delta\log M_{\ast}$) can help to lessen the tension
between predicted and observed evolution in $f_{\mathrm{DM}}$. This is
particularly likely for the massive quiescent galaxies in our high-redshift
sample, which are expected to trace the most massive halos at their respective
redshifts (e.g. Lin et al., 2019).

###  5.2 Evolution of the IMF at fixed $\sigma_{\mathrm{e}}$

Figure 15: The IMF offset parameter, $\alpha$, as a function of stellar
velocity dispersion for different samples. In the top panel, filled circles
show at $1.4<z<2.1$ assuming that mass-follows-light (MFL) and a uniform prior
on the intrinsic axis ratio, $q_{\mathrm{int}}$. In the bottom panel we show
the same data, but adopting a prior such that $q_{\mathrm{int}}$$=0.25$. The
error ellipse in the bottom right corner shows the typical uncertainty for
individual galaxies, while the hatched regions illustrates the 16/84th
percentile confidence interval based on a linear fit to the high-redshift
data. Large squares represent the median $\alpha$ derived for GAMA/SDSS data
matched in either evolving number density (filled squares) or stellar velocity
dispersion (open squares) as described in Section 5. Error bars indicate the
16/84th percentile uncertainties on the binned values. Shaded bands show the
$\alpha$-$\sigma_{\mathrm{e}}$ correlation based on a joint analysis of
ATLAS3D and SLACS data by Posacki et al. (2015), with dark and light light
regions indicating the 1 and 2$\sigma$ bounds. There is good agreement between
the matched GAMA/SDSS data and derived best-fit from Posacki et al., both of
which show evidence for a positive correlation between $\alpha$ and
$\sigma_{\mathrm{e}}$. High-redshift galaxies show evidence for a similar
correlation, albeit with a steeper slope such that objects with
$\log\sigma_{\mathrm{e}}\gtrsim 2.45$ have $\alpha$ and $\sigma_{\mathrm{e}}$
comparable to low-redshift galaxies, while at lower $\sigma_{\mathrm{e}}$
further evolution is required to reproduce the low-redshift correlation.

In Section 4.2 we showed that the kinematics of massive quiescent galaxies at
$z>1.5$ are consistent with a MW-like IMF on average. In contrast, a number of
studies based on stellar absorption features, lensing, and dynamical modelling
have shown that a Salpeter or even heavier IMF may be more typical in the in
the inner 0.1–0.2 $R_{\mathrm{e}}$ of present-day early-type galaxies (e.g.
Thomas et al., 2011; Conroy & van Dokkum, 2012; Cappellari et al., 2012,
2013b; Spiniello et al., 2014; van Dokkum et al., 2017; Conroy et al., 2017,
but see also Smith et al., 2015), suggesting some tension between our high-
redshift results and those in the nearby Universe.

Figure 15 shows a comparison of the IMF offset parameter, $\alpha$, as a
function of stellar velocity dispersion, $\sigma_{\mathrm{e}}$, at different
redshifts. We include in this figure constraints derived from low-redshift
galaxies in the ATLAS3D and SLACS samples by Posacki et al. (2015), as well as
binned results for the matched GAMA/SDSS samples described in Section 5. This
comparison shows that the apparent offset in mean $\alpha$ between high and
low redshift depends on $\sigma_{\mathrm{e}}$, such that galaxies with the
highest velocity dispersions generally have $\alpha$ values consistent with
observations at low redshift, while lower velocity dispersion galaxies are
offset towards lower $\alpha$—that is, towards a “lighter” IMF—at fixed
stellar velocity dispersion. Assuming an intrinsically flat axis ratio for
high-redshift galaxies (e.g. $q_{\mathrm{int}}$$=0.25$; bottom panel of Figure
15) reduces the apparent offset of $\alpha$ at low $\sigma_{\mathrm{e}}$,
though the qualitative trend remains unchanged. We note that this offset is
unlikely to be the result of (correlated) uncertainties in
$\sigma_{\mathrm{e}}$ and $\alpha$, shown by the error ellipse in the bottom-
right corner of Figure 15, which tend to scatter galaxies along the low-
redshift $\alpha$–$\sigma_{\mathrm{e}}$ correlation rather than away from it.
It may be that the differential evolution seen between high- and
low-$\sigma_{\mathrm{e}}$ galaxies is tied to the overall buildup of the
velocity dispersion function (VDF) over time. Bezanson et al. (2012) show that
galaxies with $\log\sigma_{\mathrm{e}}\gtrsim 2.4$ form early and their number
density has changed little since at least $z\approx 1.5$, while the population
at lower $\sigma_{\mathrm{e}}$ evolves significantly due to the addition of
newly-quenched galaxies. A more detailed dissection of the
$\alpha$–$\sigma_{\mathrm{e}}$ relation in the context of galaxy star-
formation histories will be the subject of future work.

An interesting implication of Figure 15 is that the
$\alpha$-$\sigma_{\mathrm{e}}$ relation found at low redshift is established
early on, and any scenario invoked to explain quiescent galaxies’ subsequent
evolution in $r_{\mathrm{e}}^{\mathrm{sma}}$, $\sigma_{\mathrm{e}}$, and
$M_{\mathrm{SPS}}$ (e.g. Figure 12) should largely preserve the underlying
correlation; this is especially true at the highest stellar velocity
dispersions. Using high signal-to-noise long-slit spectra of 6 nearby ETGs,
van Dokkum et al. (2017) found evidence for strong radial gradients in
$\alpha$ which increased from MW-like at $R>0.4R_{\mathrm{e}}$ to Salpeter or
heavier at $0.1R_{\mathrm{e}}$ (see also Martín-Navarro et al., 2015b;
Lyubenova et al., 2016). Under the assumption that gas-poor mergers primarily
contribute to the buildup of an extended stellar envelope (e.g. Hopkins et
al., 2009a; van Dokkum et al., 2010; Karademir et al., 2019), then we would
expect the remnants of our high-redshift sample should survive in the cores of
massive nearby ellipticals. Indeed, the typical half-light size of galaxies in
our sample (1–1.5 kpc) are comparable to the physical extent over which the
IMF is found to vary significantly in van Dokkum et al. (2017).

Finally, although we have tried to be comprehensive in our measurement of
stellar masses, we cannot rule out that at least some part of the evolutionary
trends implied by Figure 15 could be the result of redshift-dependent
systematic uncertainties in our derivation of $M_{\mathrm{SPS}}$. These could
be, for example, due to our adopted star-formation history or stellar
population synthesis models (e.g. Pforr et al., 2012; Leja et al., 2019).
Based on the analysis of mock galaxy spectra generated from semi-analytic
models, Pforr et al. (2012) showed that stellar masses for passive galaxies
can be recovered to better than $\sim$0.05 dex for a wide range of star-
formation histories, suggesting that our results are unlikely to be due
_entirely_ to the details of our SFH modelling. Furthermore, while Pforr et
al., 2012 show that mismatches in metallicity can lead to systematic offsets
of up 0.2–0.3 dex, direct observations at $z>1.5$ support our adoption of a
solar metallicity template library (e.g. Onodera et al., 2015; Kriek et al.,
2016).

##  6 Conclusions

We present an analysis of 58 massive quiescent galaxies at $1.4<z<2.1$ with
measured stellar velocity dispersions and deep near-infrared HST imaging. We
use these data to study the evolution of dynamical masses and the dynamical-
to-stellar mass ratio, which are sensitive to the central dark matter fraction
and normalization of the stellar IMF. We find that:

  * i)

The median dynamical-to-stellar mass ratio of quiescent galaxies is lower by
$\sim$0.2 dex at $1.4<z<2.1$ compared to low redshift. In Figures 6 and 7 we
showed that this offset appears to be independent of the method used to derive
dynamical masses (e.g. Jeans models vs. virial mass estimates). The observed
evolution is consistent with an decrease in the fraction (by mass) of dark
matter within the galaxy effective radius, $f_{\mathrm{DM}}[<r_{\mathrm{e}}]$,
which is lower by a factor of roughly two in our high-redshift sample compared
to nearby galaxies in the SDSS/GAMA surveys
($f_{\mathrm{DM}}[<r_{\mathrm{e}}]$$=6.6\pm 1.0$% at $z\approx 1.8$ vs.
$16.3\pm 0.3$% at $z=0$) at fixed stellar mass. These measurements appear
consistent with recent results based on the rotation curves of high-redshift
star-forming galaxies. Based on the matching of progenitor and descendant
populations, we argue in Section 5.1.1 that the evolution of _individual_
galaxies is likely even larger, with $f_{\mathrm{DM}}[<r_{\mathrm{e}}]$
increasing by a factor of 4–5 from high redshift to the present day.

  * ii)

Under the assumption that central dark matter fractions are intrinsically low
in high-redshift galaxies, the dynamical-to-stellar mass ratio can be used as
a probe of the stellar IMF. For mass-follows-light models, we find that high-
redshift data are consistent with a Kroupa-like IMF on average, while models
including an explicit NFW dark matter halo are consistent with a Chabrier IMF
(see Figure 10). We find a correlation between stellar velocity dispersion and
the IMF offset parameter, $\alpha$, at high redshift that is consistent with
low-redshift data, suggesting that any subsequent evolution should act to
preserve this underlying correlation (see Figure 15). We argue that minor
mergers are the most likely drivers of galaxies’ growth in
$r_{\mathrm{e}}^{\mathrm{sma}}$ and $M_{\mathrm{SPS}}$, as they primarily add
material at large radii while preserving the stellar populations in the inner
regions.

  * iii)

Simple models for the contraction of dark matter haloes in response to
baryonic collapse predict high central dark matter fractions. In the most
extreme case of pure adiabatic contraction, such models require IMF
normalizations a factor of $\sim$3 _lighter_ than Chabrier to explain the
observed kinematics (see Figure 11). Significant contraction of the dark
matter halo is difficult to accommodate given current observational
constraints _unless_ other baryonic process (e.g. outflows or AGN feedback)
act to reduce the central dark matter content after collapse.

  * iv)

A comparison of kinematics and structural properties between high-redshift
galaxies and their likely descendants at low-redshift supports minor merging
as the dominant evolutionary pathway after quenching; however those galaxies
with the highest stellar masses and/or stellar velocity dispersions appear to
evolve relatively more than lower mass/dispersion objects. This separation is
apparent for descendants matched both on evolving number density as well as at
fixed $\sigma_{\mathrm{e}}$ (see Figure 12).

  * v)

In the two dimensional parameter space of size and dynamical mass—the so-
called “mass plane” (Figure 13)—high redshift galaxies are both smaller and
have lower dynamical masses than their low-redshift descendants. Separating
the galaxy population in terms of total mass-to-light ratio ($M/L$), which we
interpret here as a proxy for stellar population age, galaxies with the
highest $M/L$ follow lines of roughly constant $\sigma_{\mathrm{e}}$, while
those with lower $M/L$ follow more closely lines of constant stellar surface
density. We interpret these differences as being driven by two separate phases
of passive galaxy formation, whereby galaxies first quench as a strong
function of their stellar mass surface density and their subsequent evolution
on the mass plane is driven by minor merging.

Taken together, our results point towards an evolutionary scenario for massive
quiescent galaxies that sees their formation occurring rapidly at $z>2$.
Subsequent quenching of star formation appears to preserve the disc-like
structural and kinematic signatures associated with massive star-forming
galaxies at those redshifts (e.g. Wuyts et al., 2011; Wisnioski et al., 2015,
2019) as well as their dark matter properties (e.g. Wuyts et al., 2016; Lang
et al., 2017; Genzel et al., 2017, 2020). Subsequent evolution through
continued merging is then required to transform both their structural and
kinematics properties to reproduce the massive, predominantly slowly-rotating
galaxies that constitute their likely descendants at $z=0$ (e.g. Veale et al.,
2017). Based on galaxies’ evolution in size, stellar mass, stellar velocity
dispersion, and central dark matter fraction it appears that the most likely
channel for this evolution is through the accretion of relatively lower-mass
galaxies (i.e. minor mergers). While these results are based on the best
currently available kinematic and photometric data, future spectroscopic
observations with the _James Webb Space Telescope_ and 30+ meter ground-based
observatories will enable systematic surveys of high-redshift stellar
kinematics.

We thank the referee for constructive feedback that helped to improve the
overall quality of this manuscript, as well as Sandesh Kulkarni and Kaushi
Bandara for their help with KMOS observations for the VIRIAL survey. JTM
acknowledges the support of the Australian Research Council Centre of
Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through
project number CE170100013. DJW and MF acknowledge the support of the Deutsche
Forschungsgemeinschaft via Project IDs 3871/1-1 and 3871/1-2. MF has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No 757535).
This work is based on observations taken by the CANDELS Multi-Cycle Treasury
Program (GO 12060 and 12099) and 3D-HST Treasury Program (GO 12177 and 12328)
with the NASA/ESA HST, which is operated by the Association of Universities
for Research in Astronomy, Inc., under NASA contract NAS5-26555. It also
includes data obtained from the Hubble Legacy Archive, which is a
collaboration between the Space Telescope Science Institute (STScI/NASA), the
Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian
Astronomy Data Centre (CADC/NRC/CSA). GAMA is a joint European-Australasian
project based around a spectroscopic campaign using the Anglo-Australian
Telescope. The GAMA input catalogue is based on data taken from the Sloan
Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary
imaging of the GAMA regions is being obtained by a number of independent
survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-
ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the
STFC (UK), the ARC (Australia), the AAO, and the participating institutions.
The VISTA VIKING data used here were taken using ESO Telescopes at the La
Silla Paranal Observatory under programme ID 179.A-2004. The GAMA website is
http://www.gama-survey.org/ . Funding for the SDSS and SDSS-II has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max
Planck Society, and the Higher Education Funding Council for England. The SDSS
Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical
Research Consortium for the Participating Institutions. The Participating
Institutions are the American Museum of Natural History, Astrophysical
Institute Potsdam, University of Basel, University of Cambridge, Case Western
Reserve University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns Hopkins
University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute
for Particle Astrophysics and Cosmology, the Korean Scientist Group, the
Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-
Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), New Mexico State University, Ohio State University,
University of Pittsburgh, University of Portsmouth, Princeton University, the
United States Naval Observatory, and the University of Washington.

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##  Appendix A Additional tests of velocity dispersion measurements

Figure 16: The effects of changing template metallicity on the derived stellar
velocity dispersion. In this case we consider the effects of changing the
assumed metallicity by a factor of $\sim$3, from [Z/H]=-0.3 to 0.2 dex. The
overall impact is small, with the mean dispersion increasing by less that 2%
when going from low to high metallicity. Figure 17: The impact of using
“optimal” vs. parametric template models on the derived stellar velocity
dispersion. The optimal templates are derived from a linear combination of
single stellar population models, while the parametric models assume an
explicit description of SFR as a function of cosmic time as given by Eqn. 2

In this Appendix we explore possible systematics in our derived velocity
dispersions stemming from assumptions made in our spectrophotometric modelling
approach. There are two effects in particular that we are concerned with: the
impact of assuming a fixed solar metallicity for our SED template library, and
the effect of our adopted (parametric) star-formation history.

###  A.1 Metallicity effects

While the majority of low-redshift massive early-type galaxies are consistent
with solar metallicity or higher (e.g. Gallazzi et al., 2006; Thomas et al.,
2010), the picture at high redshift is still uncertain, with different results
suggesting variations in total metallicity of up to 0.8 dex (e.g. Onodera et
al., 2015; Lonoce et al., 2015; Kriek et al., 2016; Morishita et al., 2018).
Given our adoption of a fixed solar metallicity for our template grid, it is
therefore worthwhile investigating the possible impact of this assumption on
our derived velocity dispersions. In Figure 16 we show a comparison of stellar
velocity dispersions obtained with templates a factor of $\sim$2 higher or
lower in metallicity. While there is a clear systematic shift in the resulting
values of $\sigma_{\ast}$,the offsets are of order a few per cent,
significantly smaller than the typical 20% uncertainties on our measurements
of $\sigma_{\ast}$, and are therefore unlikely to bias our results.

###  A.2 Star-formation history effects

While the adoption of a parametric star-formation history to describe the
observed photometry is common practice, the use of such models in performing
kinematic measurements is less common, and bears further investigation. In
brief, we modified the MCMC fitting code described in Section 2.1.1 to
construct the best-fit template from a linear combination of SSP templates, as
opposed to the parametric star-formation histories adopted previously. This
approach mimics the internal fitting procedure adopted in well-known fitting
codes such as pPXF (Cappellari, 2002), while at the same time providing
improved handling of low signal-to-noise data thanks to the MCMC sampling of
the LOSVD. We show the results of this re-fitting in Figure 17.

There is significant scatter between the two velocity dispersion estimates,
but in nearly all cases the derived $\sigma_{\ast}$ values are consistent
within uncertainties. The values derived in our default modelling are larger
by $\sim$0.06 dex on average, however this does not significantly affect our
results.

##  Appendix B Comparisons with previous structural parameter measurements

In this Section we present a comparison of the galaxy structural parameters
derived here with those available in the literature (where available).

###  B.1 High redshift

The comparison of apparent magnitude, Sérsic index, and size for galaxies in
our high-redshift sample is shown in Figure 18. For most galaxies these
quantities were taken from van der Wel et al. (2014), which are based on the
same CANDELS/3D-HST WFC3/F160W imaging. As discussed in Section 2.2, several
galaxies from Bezanson et al. (2013a) and van de Sande et al. (2013) fall
outside the CANDELS/3D-HST footprint, and there our measurements are based on
pipeline-processed data retrieved from the HLA. In most cases the agreement
between different measurements is excellent.

Figure 18: Comparison of total magnitude $m_{r}$, Sérsic index $n$, and semi-
major axis size $r_{\mathrm{e}}^{\mathrm{sma}}$ for galaxies in our high-
redshift sample from different sources.

###  B.2 Low redshift

Figures 19 and 20 show a comparison of apparent $r$-band magnitude, Sérsic
index, and size for galaxies in our low-redshift SDSS/GAMA sample with
measurements from Meert et al. (2015) and Simard et al. (2011), respectively.
While there is generally good agreement between measurements in the different
structural catalogs, there is significant scatter driven by differences in
approach used to mask/model neighboring objects, the size of fitted images,
and the method for measuring the sky background. In most cases any systematic
biases are relatively small, $\lesssim 20$%, and do not effect the conclusions
of this work.

Figure 19: Comparison of total magnitude $m_{r}$, Sérsic index $n$, and semi-
major axis size $r_{\mathrm{e}}^{\mathrm{sma}}$ for low-redshift galaxies
derived in this work vs. Meert et al. (2015). Figure 20: Comparison of total
magnitude $m_{r}$, Sérsic index $n$, and semi-major axis size
$r_{\mathrm{e}}^{\mathrm{sma}}$ for low-redshift galaxies derived in this work
vs. Simard et al. (2011).

##  Appendix C Photometric fits for high-redshift galaxies

In Figure 21 we show the Sersic and MGE fits derived for each galaxy in the
high redshift sample.

Figure 21: Individual galaxy images and fits, following the format of Figure
5. From left to right, panels show the observed HST WFC3/F160W image, the
best-fit Sérsic model derived using galfit, residual maps, and an overlay of
MGE contours on the observed galaxy image. Images are plotted in surface
brightness units, and contours are evenly spaced in steps of 0.5 mag arcsec-2.
Figure 21: (Continued) Figure 21: (Continued) Figure 21: (Continued) Figure
21: (Continued)