# Dynamics of Carrollian Scalar Fields

Luca Ciambelli  Perimeter Institute for Theoretical Physics, 31 Caroline St.
N., Waterloo ON, Canada, N2L 2Y5
<EMAIL_ADDRESS>
###### Abstract

Adopting an intrinsic Carrollian viewpoint, we show that the generic
Carrollian scalar field action is a combination of electric and magnetic
actions, found in the literature by taking the Carrollian limit of the
relativistic scalar field. This leads to non-trivial dynamics: even a single
particle with non-vanishing energy can move in Carrollian physics.

##  I Introduction

Since its discovery by Lévy-Leblond Levy1965  and Sen Gupta Gupta1966 , the
Carrollian universe – found taking the $c\to 0$ contraction of relativistic
physics – keeps surprising us with unexpected and unusual features. By now,
there have been numerous avenues of investigation in this field, such as its
relationship to the BMS group Duval:2014lpa ; Duval:2014uva ; Duval:2014uoa ;
Ciambelli:2019lap , the fluid/gravity correspondence and hydrodynamics
deBoer:2017ing ; Ciambelli:2018xat ; Ciambelli:2018ojf ; Campoleoni:2018ltl ;
Ciambelli:2020eba ; Ciambelli:2020ftk ; Freidel:2022bai ; Petkou:2022bmz ,
flat holography and Carrollian field theory Bagchi:2016bcd ; Ciambelli:2018wre
; Bagchi:2019xfx ; Bagchi:2019clu ; Gupta:2020dtl ; Banerjee:2020qjj ;
Figueroa-OFarrill:2021sxz ; Chen:2021xkw ; Herfray:2021qmp ; Donnay:2022aba ;
Bagchi:2022eav ; Bekaert:2022oeh ; Bagchi:2022emh ; Rivera-Betancour:2022lkc ;
Baiguera:2022lsw ; Campoleoni:2022wmf ; Donnay:2022wvx ; Dutta:2022vkg ;
Mittal:2022ywl ; Chen:2023pqf ; Mehra:2023rmm ; Campoleoni:2023fug ;
Salzer:2023jqv ; Bagchi:2023fbj ; Saha:2023hsl ; Banerjee:2023jpi , black hole
horizon and generic null hypersurfaces Penna:2015gza ; Penna:2018gfx ;
Donnay:2019jiz ; Redondo-Yuste:2022czg ; Freidel:2022vjq ; Gray:2022svz ;
Ciambelli:2023mir ; Ciambelli:2023mvj , Carrollian particles and algebras
Bergshoeff:2014jla ; Hartong:2015xda ; Henneaux:2021yzg ; Marsot:2022imf ;
Bergshoeff:2022eog ; Bagchi:2022eui ; Kasikci:2023tvs ; Casalbuoni:2023bbh ;
Cerdeira:2023ztm ; Zhang:2023jbi ; Kamenshchik:2023kxi , Carrollian gravity
Henneaux:1979vn ; Bergshoeff:2017btm ; Matulich:2019cdo ; Gomis:2020wxp ;
Grumiller:2020elf ; Concha:2021jnn ; Perez:2021abf ; deBoer:2021jej ;
Hansen:2021fxi ; Campoleoni:2022ebj ; Ekiz:2022wbi ; Figueroa-OFarrill:2022mcy
; Miskovic:2023zfz  and black holes Ecker:2023uwm , supersymmetry and
supergravity Ravera:2019ize ; Ali:2019jjp ; Ravera:2022buz ; Kasikci:2023zdn ,
Carrollian strings Schild:1976vq ; Isberg:1993av ; Bagchi:2015nca ;
Fursaev:2023lxq ; Fursaev:2023oep , and fractons Bidussi:2021nmp ; Figueroa-
OFarrill:2023vbj ; Figueroa-OFarrill:2023qty ; Perez:2023uwt . See
deBoer:2023fnj  and references therein for a recent account on Carrollian
physics.

Here, rather than obtaining Carrollian physics as usually done via a limiting
procedure, we adopt an intrinsic perspective and construct a Carrollian
physical theory compatible with symmetries and geometry. It is reasonable to
expect that the intrinsic approach reveals physical scenarios that are not
reachable through a limiting procedure from another theory. Indeed, we show in
this paper that this is the case for the simplest model, the Carrollian scalar
field. This has been vastly studied in the literature, see Ciambelli:2018ojf ;
Bagchi:2019xfx ; Gupta:2020dtl ; Chen:2021xkw ; Bekaert:2022oeh ;
Bagchi:2022eav ; Chen:2023pqf ; Banerjee:2023jpi ; Casalbuoni:2023bbh ;
Zhang:2023jbi ; deBoer:2023fnj . In particular, it has been recently discussed
in the papers of Rivera-Betancour and Vilatte (RBV) Rivera-Betancour:2022lkc ,
and Baiguera, Oling, Sybesma, and Søgaard (BOSS) Baiguera:2022lsw . In both
works, the (conformal) Carrollian scalar field has been obtained as the limit
of the relativistic conformal scalar field. It has also been shown that the
so-called electric and magnetic actions cannot coexhist, albeit for different
reasons. In RBV, these two actions appear at different powers in $c\to 0$, and
so must be considered separately. In BOSS, the magnetic action is required to
be local Carroll boost invariant, which is the reason why their timelike
(electric) and spacelike (magnetic) actions do not mix.

This work is based on the simple observation – with dramatic consequences –
that these electric and magnetic actions coexist intrinsically on a Carrollian
manifold, without invoking a limiting procedure. This has been already
contemplated using Carrollian diffeomorphisms in Gupta:2020dtl , and in two
dimensions in Bagchi:2022eav . Here, we will study so retaining the full
diffeomorphism invariance. Indeed, the magnetic action we consider is
invariant under diffeomorphisms, but not under internal frame Carroll boost as
considered by BOSS. We accept this in order to uncover non-trivial dynamics,
and postpone its consequences to future studies. In a companion paper,
Grumiller and the author Ciambelli:2023tzb  explored a seemingly disconnected
yet ultimately related problem, the study of geodesics in Carrollian gravity
(intrinsically). We show that a strikingly similar result holds there: the
temporal and spatial parts of a particle trajectory mix together and lead to
geodesic motion. Therefore, the presence of dynamics in Carrollian physics
seems to be a common feature.

Our result indicates that $c^{2}$ might be treated as an intrinsic Carrollian
coupling constant for the scalar field, while it ceases to be interpreted as
the speed of light for the background. We show that the equation of motion is
nothing but the Klein-Gordon equation, with only one difference: the speed of
light is replaced by the relative coupling constant between the two actions.
This means that a Carrollian scalar field undergoes non-trivial dynamics, but
the dispersion relation is determined by the way the electric and magnetic
parts are coupled together. In the simplest geometric setup, we construct the
Feynman’s Green function and show that there are wave solutions with non-
trivial dispersion relations, such that the Carrollian vacuum can be
understood as a non-trivial medium. We defer to further studies the situation
on a general background, and the implications for flat holography. The former
inevitably involves technicalities concerning Carrollian connections (that we
overview in Appendix A), while the latter, which is one of our main
motivations, would require us to revisit the extrapolating dictionary. We want
here to emphasize our main result alone: while motion for zero energy
particles has been observed in deBoer:2023fnj , and multi-particles motion in
Casalbuoni:2023bbh ; Zhang:2023jbi , this is the first time that non-trivial
motion is observed for a single free Carrollian scalar field, with arbitrary
energy.

##  II Geometric Setup

Consider a $d+1$-dimensional manifold ${\cal M}$. A Carrollian structure is
determined by a nowhere vanishing Carrollian vector $v=v^{\mu}\partial_{\mu}$
and a rank-$d$ (co-rank $1$) degenerate metric $h_{\mu\nu}$ satisfying
$v^{\mu}h_{\mu\nu}=0$. A ruled Carrollian structure has an extra ingredient,
the Ehresmann connection, which is a one-form $E=E_{\mu}\text{d}x^{\mu}$
satisfying $E_{\mu}v^{\mu}=1$. This object is defined modulo shifts in the
kernel of $v^{\mu}$

$\displaystyle
E^{\prime}_{\mu}=E_{\mu}-\lambda_{\mu}\quad\text{with}\quad\lambda_{\mu}v^{\mu}=0.$
(1)

Such a shift is called a Carroll boost. It has two possible origins.

The first one is as an internal frame-bundle transformation, as done in
Hartong:2015xda ; Hansen:2021fxi ; Baiguera:2022lsw . In this case, this
transformation changes the field but does not act on the coordinates. The
vector $v^{\mu}$ and the degenerate metric $h_{\mu\nu}$ are invariant under
this internal boost. We will call this transformation a local Carroll boost.

A different origin of the Carroll boost is as a diffeomorphism, a
covariantization of the usual coordinate boost ($x^{\mu}=(t,x^{i})$)

$\displaystyle t^{\prime}=t+\lambda_{i}x^{i}\qquad x^{\prime i}=x^{i},$ (2)

which reads

$\displaystyle x^{\prime\mu}\simeq x^{\mu}+x^{\nu}\lambda_{\nu}v^{\mu}.$ (3)

This is a diffeomorphism generated by
$\xi^{\mu}=-v^{\mu}\lambda_{\alpha}x^{\alpha}$, and thus

$\displaystyle E^{\prime}_{\mu}-E_{\mu}\simeq{\cal
L}_{\xi}E_{\mu}=-\lambda_{\mu}-2v^{\alpha}\lambda_{\gamma}x^{\gamma}F_{\alpha\mu}-x^{\alpha}\partial_{\mu}\lambda_{\alpha},$
(4)

where we introduced the Carrollian field strength 111Related to the usual
Carrollian vorticity by  $\displaystyle
F_{\alpha\mu}=\omega_{\alpha\mu}-\frac{1}{2}(E_{\alpha}a_{\mu}-E_{\mu}a_{\alpha}),$
(5) where $a_{\mu}=v^{\nu}D_{\nu}E_{\mu}$ is the Carrollian acceleration, see
Ciambelli:2018wre .

$\displaystyle
F_{\alpha\mu}=\frac{1}{2}(\partial_{\alpha}E_{\mu}-\partial_{\mu}E_{\alpha}).$
(6)

So we recover eq. (1) under this change of coordinates assuming
$2v^{\alpha}\lambda_{\gamma}x^{\gamma}F_{\alpha\mu}=-x^{\alpha}\partial_{\mu}\lambda_{\alpha}$.
We refer to this change of coordinates as a diff Carroll boost. The action on
all the fields is the same for both diff and local Carroll boosts if we
further require $v^{\mu}\partial_{\mu}\lambda_{\alpha}=0$ and ${\cal
L}_{\xi}h_{\mu\nu}=x^{\alpha}\lambda_{\alpha}{\cal L}_{v}h_{\mu\nu}=0$. While
these conditions are needed to reproduce (1), we will not impose it in the
following, as our construction will be fully diffeomorphism invariant.

The Ehresmann connection allows us to specify the (boost-invariant) volume
form ${\varepsilon}_{{\cal M}}=E\wedge{\varepsilon}_{{\cal C}}$, where ${\cal
C}$ is a cut of ${\cal M}$ transverse to $v^{\mu}$. The degenerate metric,
Ehresmann connection, and Carrollian vector field allow us to introduce the
projector

$\displaystyle h_{\mu}{}^{\nu}=\delta_{\mu}^{\nu}-E_{\mu}v^{\nu}\qquad
v^{\mu}h_{\mu}{}^{\nu}=0=h_{\mu}{}^{\nu}E_{\nu},$ (7)

which can be used to raise and lower indices in the space orthogonal to
$v^{\mu}$. We can use this to introduce the symmetric tensor $h^{\mu\nu}$,
defined via

$\displaystyle E_{\mu}h^{\mu\nu}=0\qquad
h_{\mu\rho}h^{\rho\nu}=h_{\mu}{}^{\nu}.$ (8)

Both the projector and the tensor $h^{\mu\nu}$ transform non-trivially under
(local and diff) Carroll boosts. Under the conditions spelled above, we have

$\displaystyle{\cal L}_{\xi}h_{\mu}{}^{\nu}$ $\displaystyle=$ $\displaystyle
h_{\mu}{}^{\nu}+\lambda_{\mu}v^{\nu}$ (9) $\displaystyle{\cal
L}_{\xi}h^{\mu\nu}$ $\displaystyle=$ $\displaystyle
h^{\mu\nu}+v^{\mu}\lambda^{\nu}+v^{\nu}\lambda^{\mu},$ (10)

where we defined $\lambda^{\mu}=h^{\mu\nu}\lambda_{\nu}$.

There is no analogue of the Levi-Civita connection for a ruled Carrollian
structure. We relegate a full account of Carrollian connections to Appendix A,
and we here require the degenerate metric $h_{\mu\nu}$ to be time-independent
$v(h_{\mu\nu})=v^{\alpha}\partial_{\alpha}h_{\mu\nu}=0$, for which torsion-
free and metricity can be imposed. This choice is motivated by three reasons.
First of all, we will later impose the simplest possible setup, as it will be
enough to unravel the leitmotif of this paper, that is, that the Carrollian
scalar field is dynamical. Secondly, we prove in Ciambelli:2023tzb  that
Carrollian geodesics also have non-trivial dynamics, and we test that on
simple geometric backgrounds, where this assumption holds. Eventually, this
assumption also persists in celestial holography applications, where the
Carrollian scalar field lies on null infinity ${\cal I}$, and Einstein
equations impose geometric constraints on ${\cal I}$. Therefore, we assume
that our Carrollian connection satisfies the torsion-free condition, together
with

$\displaystyle D_{\mu}h_{\nu\rho}=0\quad(D_{\mu}-\omega_{\mu})v^{\nu}=0$ (11)
$\displaystyle(D_{\mu}+\omega_{\mu}){\varepsilon}_{{\cal
M}}=0\quad(D_{\mu}+\omega_{\mu})E_{\nu}=0,$ (12)

where we introduced

$\displaystyle\omega_{\mu}=\kappa E_{\mu}+\pi_{\mu},\qquad\pi_{\mu}v^{\mu}=0.$
(13)

See Chandrasekaran:2021hxc ; Freidel:2022vjq ; Ciambelli:2023mir  for more
details 222Dictionary for the comparison with Ciambelli:2023mir :
$x^{\mu}\leftrightarrow x^{a}$, $v^{\mu}\leftrightarrow\ell^{a}$,
$h_{\mu\nu}\leftrightarrow q_{ab}$, and $E_{\mu}\leftrightarrow k_{a}$.. Note
that we are assuming that $v^{\mu}$ is a Killing vector, ${\cal
L}_{v}h_{\mu\nu}=0$.

Finally, our connection parallel transports the projector and $h^{\mu\nu}$,

$\displaystyle D_{\mu}h_{\nu}{}^{\rho}=0\qquad D_{\mu}h^{\nu\rho}=0.$ (14)

##  III Scalar Field

We wish to construct a free Carrollian scalar field action on a general
background, where we require full diffeomorphism invariance and only assume
from now on $v^{\alpha}\partial_{\alpha}h_{\mu\nu}=0$. We focus here on the
massless scalar field. Given the objects at our disposal, there are $2$
independent terms that can be written for a scalar field action containing $2$
derivatives of the field 333Notice that a priori, being the action a sum of
two terms, there could be two independent scalar fields appearing. It is
trivial to see, however, that one can add a Lagrange multiplier that makes
these two fields proportional.

$\displaystyle S[\phi]=-\int_{{\cal M}}{\varepsilon}_{{\cal
M}}\left(g_{1}v^{\mu}\partial_{\mu}\phi
v^{\nu}\partial_{\nu}\phi+g_{2}h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right),$
(15)

where $g_{1}$ and $g_{2}$ are independent coupling constants. These could be
functionals of the field, but in the rest we assume that they are not.
Assuming $g_{2}\neq 0$ and introducing $g_{0}=-\frac{g_{1}}{g_{2}}$, we get

$\displaystyle S[\phi]$ $\displaystyle=$ $\displaystyle g_{2}\int_{{\cal
M}}{\varepsilon}_{{\cal M}}\left(g_{0}v^{\mu}\partial_{\mu}\phi
v^{\nu}\partial_{\nu}\phi-h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right).\
$ (16)

The first term is the kinetic term in the timelike action derived by BOSS,
while the second term is nothing but their free (i.e., without constraints)
spacelike action. Notice that BOSS and RBV studied conformal Carroll scalar
fields, so they have extra terms in the action to ensure conformality. While
classically one can assume that $g_{0}$ has definite sign – which we will
assume positive –, this is a dimensionful coupling constant, and thus it can
run in the quantum theory. One should then be careful that it does so keeping
definite sign, in order for our discussion later to be well-defined.

The electric action

$\displaystyle S_{e}[\phi]=-g_{1}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}v^{\mu}\partial_{\mu}\phi v^{\nu}\partial_{\nu}\phi$ (17)

is manifestly local Carroll boost invariant and diffeomorphism invariant. The
magnetic action

$\displaystyle S_{m}[\phi]=-g_{2}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ (18)

is not local Carroll boost invariant, as proven by BOSS, but it is by
construction diffeomorphism invariant. Aware of the non-invariance of the
magnetic action under local Carroll boosts, we continue our analysis, as our
result will be non-trivial and is supported by the analysis in
Ciambelli:2023tzb . A full account of the symmetries has been done by BOSS,
and pertains the electric and magnetic actions separately. For this reason, we
do not report details here, and refer to Baiguera:2022lsw  for more
information.

We then remark that the actions studied by Gupta:2020dtl  and RBV are also
included in this analysis. To see this, one introduces coordinates
$x^{\mu}=(t,x^{i})$, and uses the Papapetrou-Randers parameterization

$\displaystyle v^{\mu}\partial_{\mu}=\frac{1}{\Omega}\partial_{t}\quad
h_{\mu\nu}=a_{ij}\delta^{i}_{\mu}\delta^{j}_{\nu}\quad
E_{\mu}\text{d}x^{\mu}=\Omega\text{d}t-b_{i}\text{d}x^{i}.$ (19)

From this, defining $a^{ij}$ as the inverse of the (non-degenerate) tensor
$a_{ij}$, and $b^{i}=a^{ij}b_{j}$, we derive

$\displaystyle
h^{\mu\nu}=\begin{pmatrix}\frac{b^{k}b_{k}}{\Omega^{2}}&\frac{b^{j}}{\Omega}\\\
\frac{b^{i}}{\Omega}&a^{ij}\end{pmatrix}.$ (20)

Notice that our assumption that $v^{\mu}$ is a Killing vector becomes in this
parameterization $\partial_{t}a_{ij}=0$, but this assumption is not required
in the comparison with RBV. It is then a straightforward exercise to see that
our action (15) reduces to the electric and magnetic actions of RBV. As shown
in deBoer:2021jej  and by RBV, these two actions appear at different
$c$-orders in the Carrollian limit. Indeed, consider the action for the free
relativistic scalar field theory on a non-degenerate flat background

$\displaystyle S[\phi]=\int
c\text{d}t\text{d}^{d}x\left(\frac{1}{2c^{2}}(\partial_{t}\phi)^{2}-\frac{1}{2}\delta^{ij}\partial_{i}\phi\partial_{j}\phi\right).$
(21)

The limit $c\to 0$ yields the electric and magnetic actions, at different
powers of $c$. Therefore, as already stated, these two actions cannot coexist
in the Carrollian limit. The main message of this paper is that they can,
nevertheless, arise together from an intrinsic Carrollian viewpoint.

With our time-independent condition on $h_{\mu\nu}$, the equations of motion
can be derived on an otherwise general background. Linearity of the action
allows us to focus on the two sectors separately. We start with the electric
sector. Using (11) and (12), we get ($\partial_{\mu}\phi=D_{\mu}\phi$)

$\displaystyle\delta S_{e}[\phi]$ $\displaystyle=$
$\displaystyle-2g_{1}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}v^{\mu}D_{\mu}(\delta\phi)v^{\nu}D_{\nu}\phi$ (22) $\displaystyle=$
$\displaystyle-2g_{1}\int_{{\cal C}}{\varepsilon}_{{\cal C}}\delta\phi
v^{\mu}D_{\mu}\phi+2g_{1}\int_{{\cal M}}\delta\phi
D_{\mu}({\varepsilon}_{{\cal M}}v^{\mu}v^{\nu}D_{\nu}\phi)$ $\displaystyle=$
$\displaystyle-2g_{1}\int_{{\cal C}}{\varepsilon}_{{\cal C}}\delta\phi
v^{\mu}D_{\mu}\phi$ $\displaystyle+2g_{1}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}(v^{\mu}v^{\nu}D_{\mu}D_{\nu}\phi+\kappa v^{\mu}D_{\mu}\phi),$

where we used $v^{\mu}\omega_{\mu}=\kappa$, and chose the cut ${\cal C}$
orthogonal to $v^{\mu}$. From $\delta S[\phi]=\Theta+\int_{{\cal
M}}{\varepsilon}_{{\cal M}}EOM\delta\phi$, we read

$\displaystyle\Theta_{e}=-2g_{1}\int_{{\cal C}}{\varepsilon}_{{\cal
C}}\delta\phi v^{\mu}D_{\mu}\phi$ (23) $\displaystyle
EOM_{e}=2g_{1}(v^{\mu}v^{\nu}D_{\mu}D_{\nu}\phi+\kappa v^{\mu}D_{\mu}\phi).$
(24)

Note that the equations of motion can be rewritten in another suggestive way

$\displaystyle EOM_{e}=2g_{1}v^{\mu}D_{\mu}(v^{\nu}D_{\nu}[\phi]).$ (25)

For the magnetic sector, we have

$\displaystyle\delta S_{m}[\phi]$ $\displaystyle=$
$\displaystyle-2g_{2}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}h^{\mu\nu}D_{\mu}(\delta\phi)D_{\nu}\phi$ $\displaystyle=$ $\displaystyle
2g_{2}\int_{{\cal M}}{\varepsilon}_{{\cal
M}}\delta\phi(h^{\mu\nu}D_{\mu}D_{\nu}\phi-h^{\mu\nu}\pi_{\mu}D_{\nu}\phi),$

where we used that the cut is orthogonal to $v^{\mu}$, and thus there are no
contributions to the symplectic potential 444Had we not assumed that, we would
have had a contribution to the symplectic potential given by
$\displaystyle-2g_{2}\int_{{\cal C}}{\varepsilon}_{\mu}\delta\phi
h^{\mu\nu}D_{\nu}\phi$ (27) where ${\varepsilon}_{\mu}$ is the induced volume
to the cut. Our choice of cut is such that
$v^{\mu}{\varepsilon}_{\mu}={\varepsilon}_{{\cal C}}$ and
$h^{\mu\nu}{\varepsilon}_{\mu}=0$.. Therefore we have found

$\displaystyle\Theta_{m}=0\quad
EOM_{m}=2g_{2}(h^{\mu\nu}D_{\mu}D_{\nu}\phi-h^{\mu\nu}\pi_{\mu}D_{\nu}\phi).$
(28)

Putting the two contributions together, and using
$g_{0}=-\frac{g_{1}}{g_{2}}$, we get the equations of motion of our Carrollian
scalar field

$\displaystyle(g_{0}v^{\mu}v^{\nu}-h^{\mu\nu})D_{\mu}D_{\nu}\phi+(g_{0}\kappa
v^{\mu}+\pi_{\nu}h^{\mu\nu})D_{\mu}\phi=0,$ (29)

on a background that has time-independent degenerate metric $h_{\mu\nu}$. This
equation clearly indicates that there is non-trivial motion in this theory.
The first parenthesis is reminiscent of the relativistic Klein-Gordon
operator, except that the speed of light is replaced by (the inverse of the
square root of) $g_{0}$. It is as if there is a fictitious non-degenerate
inverse metric

$\displaystyle g^{\mu\nu}=-g_{0}v^{\mu}v^{\nu}+h^{\mu\nu}.$ (30)

The second parenthesis in (29) is dictated by the non-trivial connection
$D_{\mu}$: although $h_{\mu\nu}$ is parallel transported by $D_{\mu}$,
$v^{\mu}$ is not, and these terms keep that into account. They are useful to
understand the propagation of Carrollian fields on a curved background.
Although we set them to zero in the Section, we plan to focus on them in
future works, for the derivation of the equations of motion on a completely
general background is currently under investigation.

We conclude the Section with a comment on the symplectic structure. With our
choice of transverse cut ${\cal C}$, the symplectic potential is (23). The
symplectic $2$-form $\Omega=\delta\Theta$ is thus

$\displaystyle\Omega=2g_{0}g_{2}\int_{{\cal C}}{\varepsilon}_{{\cal
C}}v^{\mu}D_{\mu}\delta\phi\wedge\delta\phi.$ (31)

This means that the symplectic data are $\phi$ and its "temporal" derivative
$v^{\mu}\partial_{\mu}\phi=\dot{\phi}$, and the equal-time kinematic Poisson
bracket is

$\displaystyle\\{\dot{\phi}(x_{1}),\phi(x_{2})\\}=\frac{1}{2g_{0}g_{2}}\delta^{(d)}(x_{1}-x_{2}),$
(32)

where the covariant $\delta$ function is defined via

$\displaystyle\int_{{\cal C}}{\varepsilon}_{{\cal
C}}\delta^{(d)}\left(x_{1}-x_{2}\right)f\left(x_{1}\right)=f\left(x_{2}\right).$
(33)

##  IV Simplest Setup

While the dynamics is clear already from (29), we want to make further
simplifying assumptions, in order to reach its principal part. We impose that
the background is endowed with a strong Carroll structure, as originally
defined in Duval:2014lpa ; Duval:2014uva ; Duval:2014uoa . This implies that
both $h_{\mu\nu}$ and $v^{\mu}$ are parallel transported, and thus it further
imposes

$\displaystyle\kappa=0\qquad\pi_{\mu}=0.$ (34)

We then choose the vector field and degenerate metric to be flat, that is,
using coordinates $x^{\mu}=(u,x^{i})$ 555The null time coordinate is typically
called $u$ on null infinity ${\cal I}$ and $v$ on a black hole horizon. With
future applications to celestial holography in mind, we use $u$ here.,

$\displaystyle v^{\mu}\partial_{\mu}=\partial_{u}\qquad
h_{\mu\nu}=\delta_{ij}\delta^{i}_{\mu}\delta^{j}_{\nu}.$ (35)

Note that on this background we can use the diff Carroll boost invariance to
fix the Ehresmann connection to $E=\text{d}u$. This implies

$\displaystyle h^{\mu\nu}=\delta^{ij}\delta_{i}^{\mu}\delta_{j}^{\nu},$ (36)

but, while (35) is, this statement is not (local and diff) Carroll boost
invariant. Under the latter $h^{\mu\nu}$ transforms as in (10) and thus
$\delta_{ij}$ is boost invariant but $\delta^{ij}$ is not, as part of the
degenerate tensor $h^{\mu\nu}$ 666A simple way to see this is to perform a
generic Carrollian diffeomorphism $u^{\prime}(u,x)$ and $x^{\prime}(x)$, as
defined in Ciambelli:2018xat ; Ciambelli:2018wre . Calling the diffeomorphism
generators $\Omega(u,x)$ and $b_{i}(u,x)$, one obtains (19) (with
$t\leftrightarrow u$), and in particular (20), which shows the transformation
rule of $\delta^{ij}$. Note that the finite form of (10) is  $\displaystyle
h^{\prime\mu\nu}=h^{\mu\nu}+v^{\mu}\lambda^{\nu}+v^{\nu}\lambda^{\mu}+\lambda_{\alpha}\lambda^{\alpha}v^{\mu}v^{\nu}.$
(37) .

Under all these assumptions, the Carroll scalar action becomes

$\displaystyle S=g_{2}\int_{{\cal
M}}\text{d}u\text{d}^{d}x\left(g_{0}\partial_{u}\phi\partial_{u}\phi-\delta^{ij}\partial_{i}\phi\partial_{j}\phi\right).$
(38)

Its equations of motion are trivial to derive, either by direct computation or
using (29) 777This action can be derived from the construction in
Gupta:2020dtl , taking simplifying assumptions. Their starting point is a
Carroll-diffeomorpshims invariant action, while our is a full diffeormorphisms
invariant one.

$\displaystyle
g_{0}\partial_{u}^{2}\phi-\delta^{ij}\partial_{i}\partial_{j}\phi=0.$ (39)

We finally reached the main result of this manuscript. As advertised, the
equation of motion is simply the Klein-Gordon equation, where the speed of
light is replaced by the relative coupling constant between the electric and
magnetic sectors $c\leftrightarrow\frac{1}{\sqrt{g_{0}}}$. This is directly
seen comparing (38) with (21). While in relativistic physics this coupling
constant is the speed of light of the background, here the latter is not
defined, but, remarkably, this does not prevent motion. Notice also that
$g_{0}$ could be negative or positive. The negative branch could be understood
as an effective Euclidean scalar, where instead of Wick-rotating time one
makes the speed of light imaginary. Here we are mostly interested in the
positive branch.

Calling $x^{\mu}=(u,x^{i})$ and $x^{\prime\mu}=(u^{\prime},x^{\prime i})$, the
Green’s function for this scalar field satisfies

$\displaystyle
i(g_{0}\partial_{u}^{2}-\delta^{ij}\partial_{i}\partial_{j})G(x-x^{\prime})=\delta^{(d+1)}(x-x^{\prime}).$
(40)

Its Fourier transform is given by

$\displaystyle
G(x-x^{\prime})=\int\frac{\text{d}^{d+1}k}{(2\pi)^{d+1}}e^{-ik_{\mu}(x^{\mu}-x^{\prime\mu})}\tilde{G}(k),$
(41)

where $k_{\mu}=(k_{u},k_{i})$ is the $d+1$ Carrollian momentum. Therefore, we
have 888Conventions
$\displaystyle\delta^{(d+1)}(x-x^{\prime})=\int\frac{\text{d}^{d+1}k}{(2\pi)^{d+1}}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})},$
(42) and we recall $\delta(-x)=\delta(x)$.

$\displaystyle(g_{0}k_{u}^{2}-\delta^{ij}k_{i}k_{j})\tilde{G}(k)=i.$ (43)

The poles structure depends on the coupling constant $g_{0}$. Calling
$k^{2}=\delta^{ij}k_{i}k_{j}$ and assuming $g_{0}$ positive from now on, we
indeed have
$\tilde{G}(k)=\frac{i}{(\sqrt{g_{0}}k_{u}-\sqrt{k^{2}})(\sqrt{g_{0}}k_{u}+\sqrt{k^{2}})}$,
and the poles are located at $k_{u}=\pm\sqrt{\frac{k^{2}}{g_{0}}}$. Choosing
the time-ordered contour, that is, the $i{\varepsilon}$-prescription
$k_{u}=\pm\sqrt{\frac{k^{2}}{g_{0}}}\mp i{\varepsilon}$, we get Feynman’s
Green function

$\displaystyle
G_{F}(x-x^{\prime})=\int\frac{\text{d}^{d+1}k}{(2\pi)^{d+1}}\frac{ie^{-k_{\mu}(x^{\mu}-x^{\prime\mu})}}{g_{0}k_{u}^{2}-\delta^{ij}k_{i}k_{j}+i{\varepsilon}}.$
(44)

This shows that the $u$-time-ordered two point function
$G_{F}(x-x^{\prime})=\langle 0|T(\phi(x)\phi(x^{\prime}))|0\rangle$ is also
identical to the relativistic case, modulo the crucial replacement
$c\leftrightarrow\frac{1}{\sqrt{g_{0}}}$. The dispersion relation
$k_{u}=\pm\sqrt{\frac{k_{i}k_{j}\delta^{ij}}{g_{0}}}$ depends on the coupling
constant. Therefore, wave packets are solutions of the Carrollian scalar field
equation of motion, but their frequency is dictated by the coupling constant,
defining the time order in this theory is thus subtle. Here we simply used the
insertion of $\theta(u-u^{\prime})$ to define Feynman’s time-ordered Green
function. A deeper study is required to understand the nature of this
Carrollian motion, but from what we have achieved it seems that Carrollian
waves move at different speeds, each determined by their coupling constant.
Therefore, even when the energy is finite, motion is completely free, rather
than completely frozen.

Notice that in the relativistic vacuum we tune our variables such that
relativistic waves have a dispersion relation which is given by $\omega=ck$.
Generically, a wave propagating in a medium has a non-trivial dispersion
relation. For instance, a string vibration has a modelled dispersion relation
given by

$\displaystyle\omega=\sqrt{\frac{T}{\mu}}k,$ (45)

where $\omega$ is the frequency, $k$ is the wavenumber, $T$ is the string
tension and $\mu$ is the linear density of the string.

Similarly, and even more suggestively, for a plane relativistic
electromagnetic wave propagating in a uniform isotropic linear medium with
electric permittivity $\epsilon$ and magnetic permeability $\mu$, the
dispersion relation is

$\displaystyle\omega=\frac{k}{\sqrt{\mu\epsilon}},$ (46)

To convert to our notation, we note that $k=\sqrt{k_{i}k_{j}\delta^{ij}}$ and
$\omega=k_{u}$. Then, comparing this equation with our dispersion relation
$k_{u}=\pm\frac{k}{\sqrt{g_{0}}}$, we observe that it is as if a Carrollian
wave propagates on a medium possessing a non-trivial electric permittivity and
magnetic permeability such that

$\displaystyle g_{0}\approx\mu\epsilon.$ (47)

The main difference with the relativistic case is that each Carrollian scalar
field propagates with a different $\mu$ and $\epsilon$. Then, each Carrollian
scalar field would have a different dispersion in the Carrollian vacuum, a
freer principle of relativity. While this is merely an analogy at this point,
it indicates a potentially interesting feature of the Carrollian vacuum.

##  V Outlook

We have shown that one can combine the electric and magnetic Carroll scalar
action intrinsically on a Carrollian manifold. While in the $c\to 0$ limit
these two actions cannot coexist, from the intrinsic viewpoint they naturally
combine and lead to non-trivial dynamics. We discussed how the action behaves
both under local Carroll boosts and diffeomorphisms. Both sectors are
invariant under diffeomorphisms, but the magnetic part is not invariant under
local Carroll boost, which might be a cause of concern, and deserve further
study, although the possibility of this symmetry being (spontaneously) broken
has been advocated in Armas:2023dcz . This paper is a first exploration, and
many refinements and generalizations can be pursued. Among them, one should
perform this analysis on a general background, without requiring $h_{\mu\nu}$
to be time independent. Another aspect to explore is the symplectic analysis
and canonical quantization. Indeed, although our results are classically
sounded, there could be issues in the quantum theory, especially concerning
the time ordering and the $i{\varepsilon}$ prescription, since as we have seen
the notion of frequency is coupling dependent. For instance, what are the
properties of a Carrollian harmonic oscillator? Another urgent task is to
compute how the coupling constant $g_{0}$ runs in the quantum theory. Other
comments are in order at this stage.

Motion in Carroll physics has been already advocated from the hydrodynamics
perspective Ciambelli:2018xat . There however, we lacked a microscopic
analysis, and one could have argued that it was a fictitious macroscopic
evolution of thermodynamic quantities. The analysis performed here suggests
that there is microscopic motion, and opens the door to explorations in
Carrollian kinetic theory.

A similar avenue concerns the study of Carrollian algebras. The latter are
typically introduced as contractions of the Poincaré algebra. But we have seen
in this paper that intrinsic Carrollian physics might have sectors that are
not reachable through a limiting procedure. Thus, it would be interesting to
explore whether an intrinsic algebraic analysis can be done, such that the
Hamiltonian does not commute with all generators, and thus motion is
conceivable.

In a companion paper Ciambelli:2023tzb , we show that a strikingly similar
analysis pertains Carrollian geodesics in Carroll gravity. While not reachable
through a ultra-relativistic limit, there are intrinsic Carrollian geodesics
showing that particles move on a Carrollian manifold. We thus believe that we
have uncovered physical motion in Carrollian physics. This generalizes the
already found motion in deBoer:2023fnj  for zero-energy particles and in
Casalbuoni:2023bbh ; Zhang:2023jbi  for multi-particle systems. Furthermore,
it also generalizes the intrinsic analysis done in Gupta:2020dtl  for Carroll
diffeomorphisms and in Bagchi:2022eav  in two spacetime dimensions.

Understanding the microscopic structure of Carrollian physics could have
important repercussions in celestial holography, black hole physics and
quantum gravity. In celestial holography, what we have found here is an
intrinsic result in the bona fide boundary theory dual to asymptotically flat
gravity. Can we access this dynamical field from the bulk, and if so how?
Simultaneously, this might shed light on the physics at the black hole
horizon. The membrane paradigm predicts that Einstein equations become
Carrollian evolution equations at the horizon. Understanding the thermodynamic
and microscopic properties in Carroll could thus help us understand the
kinetic origin of black hole thermodynamics, and its repercussions in quantum
gravity. Indeed, as already stated, this work opens the doors to the study of
thermodynamics of Carrollian physics from a miscroscopic viewpoint, and thus
Carrollian kinetic theory.

In conclusion, there are many outlooks and future directions that we can
pursue, related to all the Carrollian applications mentioned in the
Introduction. This paper is just the starting point, and its aim is to show
that Carrollian theories have still much to explore, especially from the
intrinsic point of view.

#### Acknowledgements

This project started after the II Carroll workshop in Mons (Sep. 2022) and
reached its completion after the III Carroll workshop in Thessaloniki (Oct.
2023). I thank the organizers and participants of both workshops for the
stimulating environment. In particular, I am grateful to Stefano Baiguera and
Gerben Oling for discussions at very early stages of this project. I have
benefited from discussions with Luis Lehner, Andrea Puhm, and especially
Sabrina Pasterski. I also wish to thank Ankit Aggarwal and Florian Ecker for
important feedback on the manuscript. Furthermore, I am indebted with Daniel
Grumiller for precious help and suggestions during the final stages of this
project. I also thank Sruthi Narayanan, Marios Petropoulos, José Senovilla,
Simone Speziale, and Céline Zwikel for discussions on Carrollian connections.
Thank you Laurent Freidel, our long discussions have been essential, I greatly
appreciate them. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.

##  Appendix A Carrollian Connections

The goal of this Appendix is to show that for a time-independent degenerate
metric $h_{\mu\nu}$ one can impose that the connection is both torsion-free
and metric-compatible. For the sake of completeness, we perform a full
analysis of possible Carrollian connections. The latter have been studied in
Duval:2014uva ; Duval:2014uoa ; Bekaert:2015xua ; Ciambelli:2018xat ;
Ciambelli:2018wre ; Morand:2018tke ; Figueroa-OFarrill:2018ilb ;
Ciambelli:2019lap ; Figueroa-OFarrill:2020gpr ; Hansen:2021fxi ;
Chandrasekaran:2021hxc ; Petkou:2022bmz ; Freidel:2022vjq ; Ciambelli:2023mir
; Campoleoni:2023fug .

### Warm up, Pseudo-Riemannian

We start with a pseudo-Riemannian manifold ${\cal M}$ as a warm up. The metric
is a non-degenerate bilinear tensor, and we employ the notation

$\displaystyle g:T{\cal M}\otimes T{\cal M}\to C^{\infty}({\cal M})\quad
g(X,Y)=g_{ab}X^{a}Y^{b},$ (48)

for all $X,Y\in T{\cal M}$ vector fields. The indices $a,b$ refer to an
arbitrary basis $e_{a}$ of $T{\cal M}$, such that

$\displaystyle X=X^{a}e_{a},\quad Y=Y^{a}e_{a},\quad g(e_{a},e_{b})=g_{ab}.$
(49)

Typically, one considers a coordinate basis, or a specific non-coordinate
basis where $g_{ab}=\eta_{ab}$, and thus a Lorentz bundle. We will consider an
arbitrary frame, without particular restrictions on the G-structure. The basis
has thus generic structure constants

$\displaystyle[e_{a},e_{b}]=C_{ab}{}^{c}e_{c},$ (50)

and for $C_{ab}{}^{c}=0$ we get the usual coordinates basis. We introduce an
affine connection $\nabla_{X}:T{\cal M}\to T{\cal M}$. Affinity means, for all
$f\in C^{\infty}({\cal M})$,

$\displaystyle\nabla_{fX}=f\nabla_{X}\qquad\nabla_{X}(fY)=f\nabla_{X}Y+X(f)Y,$
(51)

where $X(f)=X^{a}e_{a}(f)$.

The key tensors are the torsion and the covariant derivative of the metric.
The torsion is a skew symmetric bilinear tensor defined as

$\displaystyle T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y],$ (52)

where the commutator ensures $T(fX,Y)=fT(X,Y)$. The covariant derivative of
the metric is given by

$\displaystyle\nabla_{X}g(Y,Z)=X(g(Y,Z))-g(\nabla_{X}Y,Z)-g(Y,\nabla_{X}Z).$
(53)

A connection satisfying $T(X,Y)=0$ for all $X,Y$ is called torsion-free, while
the condition $\nabla_{X}g(Y,Z)=0$ for all $X,Y,Z$ is called metricity. The
Christoffel symbols are defined by the action of the connection on the basis

$\displaystyle\nabla_{e_{a}}e_{b}=\Gamma_{ab}^{c}e_{c}.$ (54)

The Levi-Civita result Levi-Civita:1917pgo  is that there exists a unique
affine connection satisfying torsion-less and metricity. As usual, one solves
these conditions (for the basis elements) and gets the Christoffel symbols

$\displaystyle\Gamma_{ab}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{cd}\Big{(}e_{a}(g_{bd})+e_{b}(g_{ad})-e_{d}(g_{ab}))$
(55)
$\displaystyle+C_{ab}{}^{e}g_{ed}+C_{db}{}^{e}g_{ea}-C_{ad}{}^{e}g_{eb}\Big{)}.$

The crucial observation is that the connection is entirely solved by the
metric, and no conditions are imposed of the latter from setting (52) and (53)
to zero.

### Carrollian

Consider now the ruled Carrollian structure discussed in Section II. It is
convenient to go into a tangent bundle basis $e_{a}\in T{\cal M}$ adapted to
$v^{\mu}$. So we choose

$\displaystyle e_{a}=e^{\mu}_{a}\partial_{\mu}=(e_{0},e_{A})=(v,e_{A}),$ (56)

where $a=(0,A)$ is a $d+1$-dimensional index, and by construction
$e_{0}=v=v^{\mu}\partial_{\mu}$. The dual basis $e^{a}\in T^{*}{\cal M}$ is
defined via

$\displaystyle e^{a}e_{b}=\delta^{a}_{b},$ (57)

and thus

$\displaystyle e^{a}=e_{\mu}^{a}\text{d}x^{\mu}=(E,e^{A}),$ (58)

where $E=E_{\mu}\text{d}x^{\mu}$ is the Ehresmann connection. The merit of
this basis is to diagonalize the condition $v^{\mu}h_{\mu\nu}=0$. Indeed,
defining the metric is this basis

$\displaystyle q(e_{a},e_{b})=q_{ab},$ (59)

and using that $q_{ab}=e_{a}^{\mu}h_{\mu\nu}e^{\nu}_{b}$, we have

$\displaystyle
q(e_{0},e_{b})=q_{0b}=e_{0}^{\mu}h_{\mu\nu}e_{b}^{\nu}=v^{\mu}h_{\mu\nu}e_{b}^{\nu}=0.$
(60)

Therefore, only $q_{AB}$ is non-vanishing.

We then introduce the affine connection in this basis

$\displaystyle D_{a}e_{b}=D_{e_{a}}e_{b}=\Gamma_{ab}^{c}e_{c}.$ (61)

So far, the structure constants $C_{ab}{}^{c}$ are arbitrary. Following
Ciambelli:2019lap , given the underlying fibre bundle structure, we assume

$\displaystyle[v,e_{A}]=\varphi_{A}v,\qquad[e_{A},e_{B}]=\omega_{AB}v,$ (62)

where $\varphi_{A}$ and $\omega_{AB}$ are known as the Carrollian acceleration
and vorticity. This means that the only non-vanishing structure constants are

$\displaystyle C_{0A}{}^{0}=\varphi_{A},\qquad C_{AB}{}^{0}=\omega_{AB}.$ (63)

One can easily generalize this to arbitrary $C_{ab}{}^{c}$, without changing
the main result of this Appendix.

We then compute the torsion ($K_{[ab]}=\frac{1}{2}(K_{ab}-K_{ba})$)

$\displaystyle T(v,v)$ $\displaystyle=$ $\displaystyle 0$ (64) $\displaystyle
T(v,e_{A})$ $\displaystyle=$
$\displaystyle(2\Gamma_{[0A]}^{0}-\varphi_{A})v+2\Gamma_{[0A]}^{B}e_{B}$ (65)
$\displaystyle T(e_{A},e_{B})$ $\displaystyle=$
$\displaystyle(2\Gamma_{[AB]}^{0}-\omega_{AB})v+2\Gamma_{[AB]}^{C}e_{C}.$ (66)

So far, the analysis is similar to the relativistic case, the main difference
enters when discussing metricity, since we are dealing with a degenerate
metric $q_{ab}$. Its covariant derivative is

$\displaystyle D_{0}q_{00}=0\quad D_{A}q_{00}=0$ (67) $\displaystyle
D_{0}q_{0A}=-\Gamma_{00}^{B}q_{BA}\quad D_{A}q_{0B}=-\Gamma_{A0}^{C}q_{CB}$
(68) $\displaystyle
D_{0}q_{AB}=v(q_{AB})-\Gamma_{0A}^{C}q_{CB}-\Gamma_{0B}^{C}q_{CA}$ (69)
$\displaystyle
D_{A}q_{BC}=e_{A}(q_{BC})-\Gamma_{AB}^{D}q_{DC}-\Gamma_{AC}^{D}q_{DB},$ (70)

where many simplifications occurred thanks to the choice of adapted basis, and
we recall $D_{0}=D_{e_{0}}$ and $e_{0}=v$.

Now suppose we insist in requiring torsion-free and metricity. The torsion-
free condition gives

$\displaystyle\Gamma_{[0A]}^{0}=\frac{1}{2}\varphi_{A}\quad\Gamma_{[0A]}^{B}=0\quad\Gamma_{[AB]}^{0}=\frac{1}{2}\omega_{AB}\quad\Gamma_{[AB]}^{C}=0.$

For metricity, eq. (68) gives

$\displaystyle\Gamma_{00}^{B}=0\quad\Gamma_{A0}^{C}=0,$ (71)

while eqs. (69) and (70) read

$\displaystyle v(q_{AB})=\Gamma_{0A}^{C}q_{CB}+\Gamma_{0B}^{C}q_{CA}$ (72)
$\displaystyle e_{A}(q_{BC})=\Gamma_{AB}^{D}q_{DC}+\Gamma_{AC}^{D}q_{DB}.$
(73)

We immediately see that the condition $\Gamma_{[0A]}^{B}=0$ together with
$\Gamma_{A0}^{C}=0$ implies

$\displaystyle v(q_{AB})=0.$ (74)

This is the main message of this Appendix. Torsion-free and metricity on a
generic Carrollian structure imply that the degenerate metric is time
independent. This is different from the pseudo-Riemannian case, where these
two conditions have no restrictions on the underlying geometry. There are two
options, either we are on a time-independent background, and then we can
select a torsion-free and metric compatible connection $D$, or we need to give
up one of these conditions. In the main body of the manuscript, we decided to
work on a time-independent background, for the three reasons explained below
(10).

We note that torsion-free or metricity can be separately imposed without
leading to impositions on the background. Torsion-free is imposed in
Ciambelli:2023mir , and then there is some intrinsic leftover non-metricity.
Other accounts, such as Figueroa-OFarrill:2020gpr , impose metricity, and then
there is a leftover intrinsic torsion. Depending on the system under scrutiny,
a choice could be more helpful than the other. For instance, since the seminal
work of Ashtekar Ashtekar:1981hw , radiation in an asymptotically flat
spacetime is conveniently included in the connection, and thus the torsion-
free condition is preferred. Conversely, it is clearly advantageous to keep
torsion but impose metricity when discussing Carrollian fermions. The
important result to retain is that there is no unique choice for a Carrollian
connection, and that imposing too stringent conditions leads to constraints on
the geometric background.

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# Conductivity of infinite-layer NdNiO2 as a probe of spectator bands

Harrison LaBollita<EMAIL_ADDRESS>Department of Physics, Arizona State
University, Tempe, AZ 85287, USA  Alexander Hampel  Center for Computational
Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
Jonathan Karp  Department of Applied Physics and Applied Math, Columbia
University, New York, NY, 10027, USA  Antia S. Botana  Department of Physics,
Arizona State University, Tempe, AZ 85287, USA  Andrew J. Millis  Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA  Department of Physics, Columbia University, New York, NY,
10027, USA

###### Abstract

Using a density-functional theory plus dynamical mean-field theory
methodology, we compute the many-body electronic structure and optical
conductivity of NdNiO2 under the influence of large scattering rates on the
Nd($5d$) bands and including dynamical interactions on the Nd($5d$) orbitals
with shifts of the Nd-Ni $d$-level energy difference. We find a robust
conducting pathway in the out-of-plane direction arising from strong
hybridization between the Ni-$d_{z^{2}}$ and Nd($5d$) orbitals. This pathway
can be “short-circuited” if this hybridization is suppressed through large
electronic scattering rates but is not reduced to zero even by very large
beyond-DFT shifts of the Nd-Ni $d$-level energy splitting. The computed in-
plane conductivity for NdNiO2 predicts the material to be a “good metal” in
contrast to experiments indicating the material is a “bad metal” or “weak
insulator”. Our results motivate future experiments measuring the $c$-axis
resistivity as a proxy for the spectator bands and suggests the essential
difference between the infinite-layer nickelates and the cuprates is
dimensionality of their electronic structures.

##  I Introduction

The discovery of superconductivity in infinite-layer Li  _et al._ (2019);
Osada  _et al._ (2020a); Zeng  _et al._ (2021); Osada  _et al._ (2021) and
quintuple-layer Pan  _et al._ (2022) nickelates establishes the “$d^{9}$”
layered nickel compounds with the generic chemical formula $R_{n+1}$NinO2n+2
($n\geq 2$; $R$ = La, Pr, Nd) as a novel family of materials that can provide
new insights into superconductivity. In particular, the structural and
chemical similarities to the high-Tc cuprates Bednorz and Müller (1986)
suggest that the materials may provide an important perspective on
electronically mediated high transition temperature superconductivity.

Despite the structural and electronic similarities between the cuprates and
nickelates, there are many important differences in their physical properties
Botana and Norman (2020); Lechermann (2020a, b, 2021); Karp  _et al._ (2020a);
Chen  _et al._ (2022a); Liu  _et al._ (2020); Choi  _et al._ (2020);
Bandyopadhyay  _et al._ (2020); Kang and Kotliar (2021); Kapeghian and Botana
(2020); Krishna  _et al._ (2020); Been  _et al._ (2021); Chen  _et al._
(2022b). The parent-phase of the cuprates is an antiferromagnetic insulator
with a $\sim$ 1.5 eV charge gap and a room temperature resistivity $\gtrsim
100$ m$\Omega$cm, while the stoichiometric infinite-layer nickelates are at
most only weakly insulating with $\rho\sim 1$ m$\Omega$cm near room
temperature Li  _et al._ (2020); Osada  _et al._ (2020b); Lee  _et al._ (2022)
and no reported evidence of long-range antiferromagnetic (AFM) order Fowlie
_et al._ (2022); Hayward  _et al._ (1999).

Basic quantum chemical (formal valence) arguments as well as density
functional theory (DFT) electronic structure calculations indicate that in
both the high-Tc cuprates and the layered nickelates the transition metal ion
(Cu or Ni) is in or near a $d^{9}$ valence state with the hole in the
$d$-shell residing in the $d_{x^{2}-y^{2}}$ orbital so that an important
feature of the DFT-level electronic structure of both the cuprate and
nickelate materials is a quasi-two-dimensional transition metal derived
$d_{x^{2}-y^{2}}$ band crossing the Fermi level Keimer  _et al._ (2015);
Pickett (1989). In the cuprate materials the $d_{x^{2}-y^{2}}$-derived band is
the only relevant near Fermi surface band, whereas the electronic structure of
the nickelates includes additional bands Nd($5d$) orbitals hybridized with
other Ni $d$-orbitals (see Fig. 1) Lee and Pickett (2004); Botana and Norman
(2020). These additional bands, sometimes referred to as “spectator” or “self-
doping” bands, are a crucial difference between the cuprate and nickelate
materials, and their role in the low-energy physics of the infinite-layer
nickelate is a subject of great current interest Lechermann (2020b); Botana
and Norman (2020); Karp  _et al._ (2020b, a); Goodge  _et al._ (2021);
Petocchi  _et al._ (2020); Li and Louie (2022). At minimum the spectator bands
affect the low energy physics by changing the relation between the carrier
density in the Ni $d_{x^{2}-y^{2}}$ band and the chemical composition, so that
the stoichiometric NdNiO2 compound has a fractionally filled $d_{x^{2}-y^{2}}$
band and is not Mott insulating. However, it is possible that the “spectator”
bands play a more important role in the physics, for example by allowing other
$d$-orbital character near the Fermi level, opening the possibility of “Hund’s
physics” Kang and Kotliar (2021).

In this paper, we present the results of computational experiments designed to
shed light on the physics of the spectator bands and on one of the observables
that may enable experimental determination of their role. We show that the
different components of the conductivity tensor are sensitive reporters of the
presence and physics of spectator bands, and therefore present the
conductivities following from each many-body electronic structure calculations
in comparison to the experimental resistivity data Li  _et al._ (2020). Our
calculations study two possibilities. First, we consider what may be termed
the “standard model” of the nickelate materials in which all of the
interesting physics is carried by the Ni($3d$) orbitals (specifically, the
$x^{2}-y^{2}$ orbital), which are treated as correlated (within DMFT) and give
rise to physics rather similar to that of the cuprates, while the other bands
are treated on a non-interacting (DFT) level. In these “standard model”
computations we add in addition a phenomenological scattering rate to the
“spectator band” states; tuning this rate to a large value effectively removes
the contributions of the spectator bands to the transport enabling a
determination of transport signatures of the spectator bands. In a second set
of computational experiments we include interactions both on the Ni and on the
Nd $d$ states and manipulate the many-body electronic structure via an
adjustment of double counting potentials using a charge self-consistent
combination of density-functional theory and dynamical mean-field theory
(DFT+DMFT) in order to determine whether the presence of the spectator bands
is a robust consequence of the level on which the beyond-DFT correlations are
treated.

The rest of this paper is organized as follows. In Section II we present the
theoretical and computational methods. In Section III we present the “standard
model” electronic structure and optical conductivity of NdNiO2 assuming many-
body correlations only on the Ni orbitals but controlling the contribution of
the Nd orbitals to transport via a phenomenological scattering rate (Sec.
III.1) and then we consider a more general interacting model with correlations
included also on the Nd($5d$) orbitals within DMFT (Sec. III.2). Section IV is
a summary and conclusion, indicating also directions for future research.

Figure 1: (a) Crystal structure of infinite-layer NdNiO2 with Nd (green), Ni
(cyan), and O (red) atoms. (b) Brillouin zone with the path along high-
symmetry directions shown in green for plots of the DFT band structure
$\varepsilon_{\nu}({\bf k})$ and spectral function $A({\bf k},\omega)$. (c)
Band structure of NdNiO2 within DFT with orbital character shown for
Nd-$d_{z^{2}}$, Nd-$d_{xy}$, Ni-$d_{z^{2}}$, and Ni-$d_{x^{2}-y^{2}}$
depicted.

##  II Methodology

We use different charge self-consistent density-functional theory plus
dynamical mean-field theory (DFT+DMFT) frameworks to compute the many-body
electronic structure and optical conductivity of NdNiO2. The computations use
the experimental lattice parameters of the tetragonal P4/mmm symmetry
stoichiometric NdNiO2 compound Li  _et al._ (2019) (see Fig. 1a).

For DFT we used the all-electron, full potential augmented plane-wave plus
local orbital (APW+lo) basis set method as implemented in wien2k Blaha  _et
al._ (2020) with the Perdew-Burke-Ernzerhof (PBE) version Perdew  _et al._
(1996) of the generalized gradient approximation (GGA) for the exchange-
correlation functional. A dense $k$-mesh of 40$\times$40$\times$40 is used for
integration in the Brillouin zone for the self-consistent calculations. We
used $R_{\mathrm{MT}}K_{\mathrm{max}}=7$ and muffin-tin radii of 2.5, 1.93,
and 1.71 a.u. for Nd, Ni, and O, respectively. The Nd($4f$) states are treated
as core states.

For the DMFT calculations, we construct either a single impurity problem for
the full 5-orbital Ni($3d$) shell or two impurity problems: one for the
5-orbital Ni($3d$) shell and one for the 2-orbitals Nd($5d$) orbitals
{Nd-$d_{z^{2}}$, Nd-$d_{xy}$} which participate in the fermiology of this
material. Both scenarios are treated within the single-site DMFT
approximation. The atomic-like orbitals are created via the projection method
Aichhorn  _et al._ (2016, 2009) with a large energy window of size $-10$ to
$10$ eV around the Fermi level.

We determine the interaction parameters by calculating the static Coulomb
interaction $U(\omega=0)$ within the constrained random phase approximation
(cRPA) Aryasetiawan  _et al._ (2004) as implemented in VASP Kresse and Hafner
(1993); Kresse and Furthmüller (1996); Kresse and Joubert (1999). The Coulomb
matrix elements are evaluated from maximally localized Wannier functions
(MLWF) Miyake  _et al._ (2009) using Wannier90 Mostofi  _et al._ (2014). To
obtain similarly localized orbitals as used in DMFT we construct Wannier
functions in a large energy window for all Ni($3d$), Nd($5d$), and O($2p$)
orbitals. To evaluate the constrained polarization function we use the
projection scheme via the constructed MLWFs Kaltak (2015). The cRPA
calculation is performed on a $9\times 9\times 9$ ${\bf k}$-mesh (plus finite
size corrections), with $\sim$360 empty bands, and using a plane wave cut off
of 333 eV when evaluating the polarization function. The resulting Coulomb
tensor is then symmetrized in the Ni($3d$) and Nd($5d$) sub-block to obtain
the interaction parameters. The interactions on the Ni impurity are governed
by the rotationally-invariant Slater Hamiltonian parameterized by the Hubbard
$U=F^{0}=7.1$ eV and Hund’s coupling
$J_{\mathrm{H}}=\frac{1}{14}(F^{2}+F^{4})=1$ eV. For the Nd impurity, we apply
an appropriate two orbital Hubbard-Kanamori Hamiltonian with $U=4.2$ eV and
$J_{\mathrm{H}}=0.44$ eV. The fully-localized limit (FLL) formula is used for
the double counting correction, which has the following form:
$\Sigma_{\mathrm{DC}}=\frac{1}{2}U^{\prime}N(N-1)$ with $N$ being the total
occupation of the Ni or Nd site. The term proportional to $J$ is not written.
Throughout this work, we lave used $U^{\prime}=U$ unless otherwise indicated.

Figure 2: Summary of momentum resolved spectral data, $A({\bf k},\omega)$,
along high-symmetry lines in the Brillouin zone for a DFT+DMFT calculation
with the 5 orbital Ni($3d$) shells treated as correlated with constant self-
energies applied to the Nd($5d$) states. (a) $\Sigma_{\mathrm{Nd}}(\omega)=0$,
(b) $\Sigma_{\mathrm{Nd}}(\omega)=-i$ eV, and (c)
$\Sigma_{\mathrm{Nd}}(\omega)=-10i$ eV. (d) Spectral function $A({\bf
k},\omega)$ for CaCuO2 in the metallic state.

We employ a continuous-time quantum Monte Carlo (QMC) algorithm based on the
hybridization expansion method as implemented in TRIQS/cthyb Parcollet  _et
al._ (2015); Seth  _et al._ (2016) to solve the Ni and Nd impurity problems.
To reduce high-frequency noise in the QMC data, we represent the Green’s
function in a basis of Legendre polynomials and sample the Legendre
coefficients directly within the TRIQS/cthyb solver Boehnke  _et al._ (2011).
All calculations are performed at a system temperature of 290 K ($\beta=40$
eV-1) in the paramagnetic state. Maximum entropy methods are used to
analytically continue the QMC data and the diagonal components of the self-
energy from Matsubara space to real-frequency space Kraberger  _et al._
(2017).

From the electronic structure obtained from the various DFT+DMFT calculations,
we compute the frequency dependent optical conductivity within the Kubo
formalism, as implemented in the TRIQS/DFTtools software package Aichhorn  _et
al._ (2016). The locality of the DMFT self-energy means that vertex
corrections may be neglected. The frequency dependent optical conductivity is
given by

$\displaystyle\sigma^{\alpha\beta}(\Omega)=N_{\mathrm{sp}}\pi e^{2}\hbar$
$\displaystyle\int
d\omega\,\Gamma_{\alpha\beta}(\omega+\Omega/2,\omega-\Omega/2)$
$\displaystyle\times~{}~{}\frac{f(\omega-\Omega/2)-f(\omega+\Omega/2)}{\Omega}$

where

$\Gamma_{\alpha\beta}(\omega,\omega^{\prime})=\frac{1}{V}\sum_{{\bf
k}}\mathrm{Tr}\Big{[}v^{\alpha}({\bf k})A({\bf k},\omega)v^{\beta}({\bf
k})A({\bf k},\omega^{\prime})\Big{]}$ (2)

The spectral function $A$ and velocity operator $v$ are tensors in the space
of band indices and the velocity operator in direction $\alpha\in\\{x,y,z\\}$
is

$v^{\alpha}_{\nu\nu^{\prime}}({\bf k})=-i\langle\psi_{\nu}({\bf
k})|\nabla^{\alpha}|\psi_{\nu^{\prime}}({\bf k})\rangle/m_{e}$ (3)

the matrix elements of $v$ are computed within the WIEN2k optics code
Ambrosch-Draxl and Sofo (2006) on a dense $60\times 60\times 60$ ${\bf
k}$-mesh. For numerical stability, we use a broadening of 10 meV for the
calculation of all optical conductivity data.

##  III Results

###  III.1 Basic electronic structure and scattering effects

Fig 1c shows the non-interacting (DFT) band structure of NdNiO2 calculated
within DFT for NdNiO2 along the high-symmetry path in the Brillouin zone shown
in Fig 1b. The orbital characters of the bands are highlighted. The basic
features of the low-energy physics of NdNiO2, as described in previous works,
are revealed Botana and Norman (2020); Karp  _et al._ (2020a); Lee and Pickett
(2004). A quasi-two-dimensional Ni-$d_{x^{2}-y^{2}}$ derived band crosses
Fermi level and is analogous to the Cu-$d_{x^{2}-y^{2}}$ band found in DFT
calculations of the cuprates. In addition, the DFT calculation reveals an
electron pocket of mainly Nd-$d_{z^{2}}$ character centered at the $\Gamma$
point and a second pocket, of mixed Ni-$d_{z^{2}}/Nd_{xy}$ character centered
at the A point Botana and Norman (2020); LaBollita and Botana (2021). These
two bands accept carriers from the Ni-$d_{x^{2}-y^{2}}$ band; the consequences
of this “self-doping” effect are still up for debate Lechermann (2020b);
Botana and Norman (2020); Karp  _et al._ (2020b, a); Goodge  _et al._ (2021);
Petocchi  _et al._ (2020).

Figure 3: Calculated optical conductivity from DFT+DMFT (Ni($3d$) correlated)
(a) in-plane and (b) out-of-plane with different scattering rates applied to
the Nd($5d$) orbitals. Additionally, optical conductivity for CaCuO2 in the
out-of-plane is given for comparison in (b). The experimental DC conductivity
is denoted by the shaded red region.  Figure 4: Summary of momentum resolved
spectral data, $A({\bf k},\omega)$, along high-symmetry lines in the Brillouin
zone. (a) Conventional DFT+DMFT calculation with the 5 orbital Ni($3d$) shells
treated as correlated. (b) DFT+DMFT calculation where the 5 orbital Ni($3d$)
shell and the Nd-$\\{d_{z^{2}},d_{xy}\\}$ orbitals treated as correlated. (c)
Same as (b) with an adjusted double counting term
$U_{\mathrm{DC}}=U^{\prime}$. (d-f) Corresponding interacting Fermi surfaces
for (a-c), respectively, in the $k_{z}=0$ plane where the A-centered pocket
does not cross the chemical potential.

Using a DFT+DMFT framework where the five Ni($3d$) orbitals are treated as
correlated and all others are treated as uncorrelated, we perform many-body
electronic structure calculations for NdNiO2. We then introduce an additional
electronic scattering onto the Nd($5d$) states by adding a constant self-
energy of the form $\Sigma_{\mathrm{Nd}}(\omega)=-i\eta$, where $\eta$ is a
real, positive constant.

Figure 2 presents the results as a false-color plot of the trace of the band-
basis many-body spectral function for three choices of phenomenological
scattering rate. Fig. 2a presents the standard model results with no
broadening of the Nd-derived bands. The band structure is similar to the DFT
band structure, except that the Ni-$d_{x^{2}-y^{2}}$ band is substantially
narrowed. Panel Fig. 2d presents the results of an analogous DFT+DMFT
calculation for CaCuO2 which is isostructural and chemically analogous to
NdNiO2. Within the single-site DMFT approximation the accepted interaction
parameters ($U=7$ eV, $J_{\mathrm{H}}=1$ eV) leave CaCuO2 in its metallic
state and antiferromagentism or cluster DMFT methods are needed to reproduce
the observed insulating behavior Karp  _et al._ (2020a, 2022). Importantly,
there is no hybridization between the Cu($3d$)-Ca($3d$) orbitals exhibited by
no dispersing band present between the M and A points (see Fig. 2d). For
$\Sigma_{\mathrm{Nd}}=-1i$ eV (Fig. 2b), the Nd-$d_{z^{2}}$ band becomes so
strongly broadened that the $\Gamma$-centered spectator pocket is no longer
visible while the A spectator pocket, although broadened, remains visible, in
part because of the admixture of Ni $d_{z^{2}}$ states. Figure 2c shows that
for $\Sigma_{\mathrm{Nd}}=-10i$ eV all of the Nd($5d$) states are so broadened
that only the single Ni-$d_{x^{2}-y^{2}}$ band is visible at the chemical
potential.

Figure 3 summarizes the optical conductivity in the low-frequency regime
computed from the different spectral functions shown in Fig. 2. For the in-
plane conductivity (Fig. 3a), we find that additional scattering effects on
the Nd($5d$) states decreases the Drude peak with a maximum decrease by about
50% for $\Sigma_{\mathrm{Nd}}=-10i$ eV showing that the spectator bands
contribute about half of the in-plane conductivity in the standard model. The
out-of-plane conductivity shows a similar systematic trend with a much
stronger depletion dropping to essentially zero at larger scattering rates
showing that the spectator bands completely control the out of plane
conductivity.

This conclusion is reinforced by the CaCuO2 out of plane conductivity also
shown in Fig. 3b, which is nearly zero for CaCuO2 and matches the calculation
for NdNiO2 when the hybridizing $k_{z}$ band is destroyed via a large
scattering rate. The in-plane conductivity for CaCuO2 (not shown) is
significantly larger than NdNiO2, which is a consequence of the different
Fermi velocities at the chemical potential from the $d_{x^{2}-y^{2}}$-derived
bands in each material (compare Fig. 2a to Fig. 2d). This highlights an
important electronic difference between the cuprate and infinite-layer
nickelate in terms of the strong $c$-axis coupling exhibited by the nickelate.

Figure 5: (a) Local spectral functions for the Ni (only $e_{g}$ orbitals
shown) and Nd impurity problems for three different computational experiments
performed. (b) Matsubara self-energies for the Ni-$e_{g}$ orbitals:
$d_{x^{2}-y^{2}}$ (left) and $d_{z^{2}}$ (right). Figure 6: Calculated optical
conductivity for different DFT+DMFT electronic structure theories (a) in-plane
and (b) out-of-plane. The experimental DC conductivity is denoted by the
shaded red region.

###  III.2 Dynamical interactions applied to the Nd($5d$) orbitals

This subsection presents the results of a second computational experiment,
where we add electronic correlations within DMFT to the Nd($5d$) orbitals
($d_{z^{2}}$, $d_{xy}$), with interaction parameters obtained from cRPA, to
understand how additional beyond-DFT interactions might influence the
electronic structure of NdNiO2 and how these effects might be revealed in the
optical conductivity. Figure 4 summarizes the ${\bf k}$-resolved spectral data
for the second set of computational experiments. Figure 4a is repeated from
Fig. 2a for reference. The corresponding Fermi surface in the $k_{z}=0$ plane
is shown in Fig. 4d.

Figure 4b shows the spectral function, $A({\bf k},\omega)$ from a DFT+DMFT
calculation with the 5 Ni($3d$) and 2 Nd($5d$) orbitals treated as correlated
(referred to as Ni($3d$)+Nd($5d$)). Interestingly, the $\Gamma$-pocket band
arising from the Nd-$d_{z^{2}}$ has been shifted up with respect to the
overall spectrum. The strongly $k_{z}$ dispersing hybridized Nd band centered
at the $A$-point remains essentially unchanged between the two calculations
(Fig. 4a vs. Fig. 4b). The interacting Fermi surface reveals the significant
reduction in the size of the Nd-$d_{z^{2}}$ electron pocket (see Fig. 4e). The
area of the Ni-$d_{x^{2}-y^{2}}$ sheet remains essentially constant.

In DFT+DMFT calculations the double counting term plays a crucial role
especially for transition-metal oxides Park  _et al._ (2014); Wang  _et al._
(2012). Operationally, this term controls the relative energy shift of the
correlated states with respect to the uncorrelated states with the goal of
canceling out the correlation contributions encoded in the DFT exchange-
correlation functional. In Fig. 4c, we alter the double counting potential to
attempt to displace the Nd($5d$) states away from the chemical potential by
setting $U^{\prime}=U_{\mathrm{DC}}=10$ eV $\sim 2U_{\mathrm{Nd}}$. This
changes the double counting potential $\Sigma_{\mathrm{DC}}$ from $-0.78$ eV
for $U^{\prime}=U_{\mathrm{Nd}}$ to $-3.1$ eV for
$U^{\prime}=U_{\mathrm{DC}}$. This treatment results in a complete removal of
the Nd-$d_{z^{2}}$ electron pocket. However, the hybridizing $k_{z}$ band
remains unchanged. We note that this is in contrast to the mechanism of a
constant scattering rate shown in Fig. 2c. The Fermi surface sheets generated
from the Ni-$d_{x^{2}-y^{2}}$ bands remain constant across all calculations
(see Fig. 4(d-f)).

Figure 5 further reveals the low-energy spectrum for these three different
calculations. Across the three calculations, the quasiparticle spectral weight
of the Ni-$d_{x^{2}-y^{2}}$ states remains dominant around the chemical
potential. The Nd-$d_{z^{2}}$ states gradually decrease and eventually become
fully depleted at $\omega=0$ when the double counting potential is adjusted.
Spectral weight from the hybridized Ni-$d_{z^{2}}$ states also decreases at
the chemical potential in connection to the Nd-$d_{z^{2}}$ removal. The
electronic correlations captured quantitatively from the quasiparticle
renormalization factor
$Z=(1-\partial\mathrm{Im}\Sigma/\partial\omega_{n}\big{|}_{\omega_{n}\rightarrow
0})$ increase for the Ni-$d_{x^{2}-y^{2}}$ as the Nd($5d$) states are pushed
further away from the chemical potential (see Fig. 5b). This reveals the
proximity of the Ni-$d_{x^{2}-y^{2}}$ to a Mott-like state that may be masked
by the presence of the hybridized Nd($5d$)-Ni-$d_{z^{2}}$ states Gu  _et al._
(2020).

We now turn to the resulting optical conductivity shown in Fig. 6. The Drude
peak for the conductivity in the plane ($\sigma_{xx}$) shows a systematic
decrease across the three different DFT+DMFT calculations for NdNiO2 (see Fig.
6a). There are two contributions to this decrease. First, the the in-plane
conductivity contributions from the Nd($5d$) states are systematically
removed. Second, the electronic correlations of the Ni-$d_{x^{2}-y^{2}}$
increase resulting in a smaller (larger) quasiparticle renormalization $Z$
(mass enhancement $m^{\star}/m$). In the low-frequency regime, the optical
conductivity reads Petocchi  _et al._ (2020):

$\mathrm{Re}\,\sigma(\omega)=\frac{\sigma_{\mathrm{DC}}}{\pi}\frac{\tau}{1-(\omega\tau)^{2}}+\sigma_{\mathrm{inc}}(\omega),$
(4)

where $\sigma_{\mathrm{DC}}=(Zn)e^{2}$ and $n$ is the carrier density. Thus,
lowering $Z$ decreases the Drude peak $\sigma_{\mathrm{DC}}$. Interestingly,
the out-of-plane conductivity remains essentially unchanged across the three
different calculations as shown in Fig. 6b. In all three of these
calculations, there is no mechanism that alters this hybridizing $k_{z}$ band
which offers the only route for conduction out-of-plane. Therefore,
$\sigma_{zz}$ remains mostly unchanged.

##  IV Summary and Discussion

One motivation for the current interest in NdNiO2 and related materials is the
perspective that these compounds provide on the relationship between
superconductivity and electron correlation effects. The family of nickelate
materials shares with the family of cuprate materials a low energy electronic
structure prominently featuring a two dimensional approximately half-filled
band derived from the transition metal $d_{x^{2}-y^{2}}$ orbital and subject
to strong interactions. An important difference between the two material
families is the presence, in the Ni-compounds, of “spectator bands” derived
from Nd $d$-orbitals that are present near the Fermi level and provide both a
three dimensional dispersion (in the case of the infinite-layer nickelate) or
strong interlayer coupling (in the case of the 3 and 5 layer compounds) and a
“self doping” effect in which the occupancy of the $x^{2}-y^{2}$ derived band
becomes non-integer even at stoichiometric compositions and holes are
introduced into the $d_{3z^{2}-r^{2}}$ orbital. There are two spectator bands:
one centered at the $\Gamma$ point derived from Nd $d_{3z^{2}-r^{2}}$ and
apparently weakly coupled to the Ni states, and one centered at the A point
derived from Nd $d_{xy}$ states and coupled to Ni $d_{3z^{2}-r^{2}}$ states.

Understanding the role of the “spectator bands” in the physics of the
materials is an important open question. This paper explores the spectator
band issue via a set of computational experiments that treat the correlations
on the Nd site on the same level of theory as the correlations on the Ni site,
consider various modifications of the standard theory that change the
contributions of the spectator bands to the low energy physics, and present
the frequency dependent conductivity (especially the interband conductivity)
as an important experimental diagnostic of the effects of the spectator bands
because the Nd orbitals provide a robust conducting pathway along out-of-plane
direction (in the infinite layer material NdNiO2) or low-lying interband
transitions (in related materials such as $R_{n+1}$NinO2n+2 ($n\neq\infty$)
where a $R$O2 slab cuts the $c$-axis dispersion LaBollita and Botana (2021);
LaBollita  _et al._ (2022)).

We find, consistent with previous results Kang and Kotliar (2021); Petocchi
_et al._ (2020), that the standard correlation theory which treats the Nd
orbitals as weakly correlated predicts that NdNiO2 is a good metal with rather
isotropic conductance, in contrast to experiment which shows that the in-plane
conductance is large, characteristic of a “bad metal” (the out of plane
conductance is not known). Modifying the model by adding a large ad-hoc
scattering rate to the Nd orbitals completely suppresses the interplane
conductivity, but reduces the in-plane conductivity by only a factor of two or
so without changing the theoretically predicted good metal behavior.

We then extended the theory to treat correlations on the Nd sites on the same
DFT+DMFT level as the correlations on the Ni sites. The interactions seem to
deplete the $\Gamma$ pocket, leaving the minimum energy of this band very
close to the Fermi level, while not significantly changing the A pocket. We
further adjusted the relative energies of the Ni and Nd $d$-states by changing
the “double counting correction” in the calculation. With modest adjustment
the $\Gamma$ pocket can be entirely removed Si  _et al._ (2020) but the A
pocket is robust even to large changes, so that at this level of theory the
strong $c$-axis coupling is not altered by electronic correlation effects.
Indeed, previous studies have shown that in the vicinity of the A pocket there
is significant hybridization also with interstitial states that do not have a
clear atomic origin Gu  _et al._ (2020); Chen  _et al._ (2022b); Lechermann
(2020b). These interstitial states would not be subject to local correlation
effects, perhaps accounting for the resilience of the A pocket. Furthermore,
our adjustment of the spectator band states acts as a “governor” on the
correlations of the strongly correlated Ni-$d_{x^{2}-y^{2}}$ states, masking
via the self-doping effect a potential nearby Mott-like state.

In summary, the computational experiments performed here show that both the
“standard model” DFT+DMFT approach (correlations only on the Ni site) and any
reasonable deformation of it lead to a calculated in-plane DC conductivity
that is incompatible with available experiments, because the spectator bands
cannot be eliminated from the low energy theory. The dc and optical interplane
conductivity, as well as angle-resolved photoemission experiment measurements
especially of the A-pocket, are important tests of the theory. On the
theoretical side, our work sets the stage for a systematic examination of
beyond DMFT correlation effects ($d$-wave superconductivity, magnetism) on the
Nd bands.

###### Acknowledgements.

H.L and A.S.B acknowledge the support from NSF Grant No. DMR 2045826. The
Flatiron Institute is a division of the Simons Foundation.

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