aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USA

# Black Hole Horizons from the Double Copy

Samarth Chawla a and Cynthia Keeler<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We describe a procedure for locating black hole horizons in Kerr-Schild
spacetimes in the double copy paradigm. Using only single- and zeroth-copy
data on flat spacetime, our procedure predicts the existence of trapped
surfaces in the double-copy gravitational solution. We show explicitly how
this procedure locates the horizon of the Schwarzschild black hole and the
general Myers-Perry black hole.

##  1 Introduction

The classical double copy, in its most straightforward formulation
Monteiro:2014cda , maps classical solutions of the Abelian sector of Yang-
Mills theory and classical solutions of a biadjoint scalar theory to classical
solutions of general relativity. The gravitational solution is expressed as
two copies of the gauge theory solution, hence the moniker “double copy”.
Conversely, the gauge solution is often called the “single copy” of the
gravity solution, while the scalar is termed the “zeroth copy”. The
underpinnings of this double copy procedure lie in the color-kinematics
duality between gauge and gravity amplitudes (see Travaglini:2022uwo ;
Bern:2019prr ; Adamo:2022dcm  for recent reviews). Since the original
formulation of the double copy for Kerr-Schild spacetimes Monteiro:2014cda ,
several other examples of classical double copy relations have been found
Luna:2015paa ; Luna:2017dtq ; Ridgway:2015fdl ; Adamo:2017nia ; Bahjat-
Abbas:2017htu ; CarrilloGonzalez:2018ejf ; Ilderton:2018lsf ; Gurses:2018ckx ;
Carrillo-Gonzalez:2017iyj ; Lee:2018gxc ; Lescano:2020nve ; Lescano:2021ooe ;
Bah:2019sda ; Goldberger:2019xef ; Kim:2019jwm ; Luna:2020adi ; Easson:2020esh
; Berman:2018hwd ; Alkac:2021seh ; Mkrtchyan:2022ulc ; Adamo:2022rob ;
Alfonsi:2020lub ; Adamo:2020qru ; Gonzo:2021drq ; Luna:2018dpt ;
Keeler:2020rcv ; Easson:2021asd ; Easson:2022zoh ; Godazgar:2020zbv ;
White:2020sfn ; Chacon:2021wbr ; Chacon:2021lox ; Han:2022mze ;
Alawadhi:2020jrv ; Godazgar:2021iae ; Armstrong-Williams:2023ssz . For recent
reviews of the classical double copy, see Kosower:2022yvp  and chapter 8 of
Bern:2019prr .

The simplest non-trivial example of the classical double copy is a map between
the Schwarzschild metric, which can be interpreted as the gravitational field
of a static point mass, and the gauge field of a static point charge. While it
is unsurprising that these static and spherically symmetric solutions map to
each other, the distinctive event horizon of the Schwarzschild metric seems to
have no analogue in its single copy, the Coulomb solution. One of the broader
aims of the entire double copy program is to describe quantities in gravity
and gauge theory using a common set of building blocks Bern:2019prr . Thus a
natural question to ask is: how does the Schwarzschild event horizon emerge
out of its double copy construction from the Coulomb potential?

The teleological definition of a black hole (see for instance Hawking:1973uf )
is the region of spacetime outside the causal past of future null infinity.
The boundary of such a region, if it exists, is necessarily a null
hypersurface and is termed the event horizon. While this definition
effectively formalizes the notion of a black hole as a region of no escape, it
is a highly non-local definition that relies on knowledge of the global
structure of the metric all the way to asymptotic infinity. This non-locality
presents a potential obstacle in finding event horizons from the double copy,
which is a local map.111While the classical double copy in the formulations we
have relied on is indeed local in position space, the double copy of
amplitudes is local in momentum space. This raises the question of why the
classical version is a local product rather than a convolution, a question
addressed recently in Monteiro:2021ztt ; Luna:2022dxo . The statement of the
Kerr-Schild double copy (6), for instance, maps the gauge field $A_{\mu}$ at
each point to the metric $g_{\mu\nu}$ at the same point.

Fortunately, several local definitions of horizons have been proposed in the
literature. See Gourgoulhon:2008pu ; Ashtekar:2004cn ; Booth:2005qc ;
Senovilla:2011fk  for reviews of various local horizon definitions. All local
horizons found therein have one feature in common: the horizon hypersurface
must possess a foliation in terms of marginally trapped surfaces (we review
the definition of trapped surfaces in section 2.2). Thus, to remain agnostic
towards the various local horizon definitions, our general discussion focuses
primarily on deducing the existence of trapped surfaces in the double copy
spacetime using only single- and zeroth-copy data.

In the following section, we review the Kerr-Schild formulation of the
classical double copy and briefly survey a few local horizon definitions.
Section 3 builds up to a derivation of our main result: a prescription for
finding local horizons using only single- and zeroth-copy calculations on an
auxiliary flat spacetime. We then illustrate the use of the general
prescription by constructing the event horizons of the well-studied
Schwarzschild and Myers-Perry solutions in section 4. Finally, in section 5 we
discuss our result and directions for future work.

##  2 Review

###  2.1 Classical Double Copy

We review in this section the classical double copy for Kerr-Schild
spacetimes, first studied in Monteiro:2014cda . A Kerr-Schild spacetime is one
whose metric can be written as

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi k_{\mu}k_{\nu},$ (1)

where $\eta_{\mu\nu}$ is a metric with zero curvature, and $k_{\mu}$ is null
and geodesic. The inverse metric is 222We note here that the index on
$\eta_{\mu\nu}$ is not raised using the metric, that is $\eta^{\mu\nu}\neq
g^{\mu\rho}g^{\nu\lambda}\eta_{\rho\lambda}$. Instead, it is defined to
satisfy $\eta^{\mu\rho}\eta_{\rho\nu}=\delta^{\mu}_{\nu}$.

$g^{\mu\nu}=\eta^{\mu\nu}-\Phi k^{\mu}k^{\nu}.$ (2)

The index on $k_{\mu}$ can be raised with either $g_{\mu\nu}$ or
$\eta_{\mu\nu}$, since

$g^{\mu\nu}k_{\mu}=\eta^{\mu\nu}k_{\mu}-\Phi\left(k^{\mu}k_{\mu}\right)k^{\nu}=\eta^{\mu\nu}k_{\mu}.$
(3)

Thus, we can unambiguously write

$k^{\mu}k_{\mu}=0.$ (4)

Similarly, $k_{\mu}$ is geodesic with respect to both the flat metric
$\eta_{\mu\nu}$ or the curved metric $g_{\mu\nu}$,

$k^{\mu}\nabla_{\mu}k_{\nu}=k^{\mu}\mathring{\nabla}_{\mu}k_{\nu}=0,$ (5)

where $\mathring{\nabla}_{\mu}$ denotes the Levi-Civita connection
corresponding to the metric $\eta_{\mu\nu}$.

The single copy field is the Abelian potential $A_{\mu}=\Phi k_{\mu}$, and
$\Phi$ is the zeroth copy scalar field. They satisfy Maxwell’s equations and
the massless Klein-Gordon equation respectively on the flat background
$\eta_{\mu\nu}$ whenever the metric $g_{\mu\nu}$ satisfies the vacuum Einstein
equations.333 Much of the double copy literature uses a flat background owing
to its origins in the study of amplitudes in flat spacetime. However, the
classical double copy can still be performed in curved spacetimes, and the
curved equations of motion can be used; for relevant literature, see
Alawadhi:2020jrv ; Gurses:2018ckx ; Bahjat-Abbas:2017htu ; Carrillo-
Gonzalez:2017iyj ; Prabhu:2020avf ; Adamo:2017nia ; Alkac:2021bav ;
Han:2022mze ; Han:2022ubu .

The classical double copy construction of Kerr-Schild spacetimes thus maps
certain classical solutions of the Maxwell and Klein-Gordon equations to
classical solutions of the vacuum Einstein equations. The metric is
constructed “on top of” the flat metric $\eta_{\mu\nu}$ using a combination of
$A_{\mu}$ and $\Phi$,

$g_{\mu\nu}=\eta_{\mu\nu}+\frac{1}{\Phi}A_{\mu}A_{\nu}.$ (6)

There are other formulations of the double copy, most notably the Weyl double
copy introduced in Luna:2018dpt . However, we limit ourselves in scope to
Kerr-Schild spacetimes and the Kerr-Schild double copy.

As pointed out in the introduction, the teleological definition of a black
hole horizon requires knowledge of the global structure of spacetime, while
the Kerr-Schild double copy is a local map. To be able to find horizons using
the double copy we thus need a local definition of black hole horizons, which
we briefly review in the next section.

###  2.2 Local definitions of black hole horizons

Various local definitions of black hole horizons have been advanced in the
literature. In this section we review relevant material from
Gourgoulhon:2008pu . All local horizon definitions therein rely on the notion
of a trapped surface. Intuitively, a co-dimension $2$ spacelike surface is a
trapped surface when all light rays leaving the surface are condemned by the
ambient geometry to always fall “inward”. The notion of always falling
“inward” is made precise by looking at how the co-dimension $2$ volume element
evolves along null congruences leaving the surface.

Any co-dimension $2$ spacelike surface $\mathcal{S}$ has, at each point, two
distinct future-directed null vectors (call them $n^{\mu}$ and $l^{\mu}$) that
are normal to it. Using the conventional normalization $n_{\mu}l^{\mu}=-1$, we
can write the induced metric on $\mathcal{S}$ as

$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}l_{\nu}+l_{\mu}n_{\nu}.$ (7)

The co-dimension $2$ volume element is then $\sqrt{\det h}$. The expansion
$\theta^{(V)}$ of $\mathcal{S}$ along a normal vector $V^{\mu}$ is defined as

$\theta^{(V)}=\mathcal{L}_{V}\ln\sqrt{\det h}=h^{\mu\nu}\nabla_{\mu}V_{\nu},$
(8)

where $\mathcal{L}_{V}$ denotes the Lie derivative along $V^{\mu}$.

The surface $\mathcal{S}$ is a future-trapped surface whenever both
$\theta^{(n)}$ and $\theta^{(l)}$ are negative. Physically, this means that
the cross sectional volume of a bundle of null trajectories leaving
$\mathcal{S}$ always decreases, even along the nominally outgoing congruence
$l^{\mu}$. The limiting case, where $\theta^{(l)}$ vanishes, is a marginally
trapped surface. In flat spacetime, the expansions along $n^{\mu}$ and
$l^{\mu}$ have opposite sign. The infalling congruence is the one with
negative expansion, and the outgoing congruence has positive expansion. We
will assume throughout that $l^{\mu}$ is outgoing.

A marginally trapped surface is thus one for which the cross section of the
outgoing congruence $l^{\mu}$ is instantaneously constant. Say we have a
hypersurface $\mathcal{H}$ foliated by marginally trapped surfaces. Such a
surface is in general a trapping horizon. If additionally the outermost
condition $\mathcal{L}_{n}\theta^{(l)}<0$ is satisfied, $\mathcal{H}$ is a
future outer trapping horizon (FOTH) Hayward:1993wb . The outermost condition
ensures that the expansion $\theta^{(l)}$ turns negative inside $\mathcal{H}$,
which means that $\mathcal{H}$ encloses a trapped region of spacetime. If the
outermost condition is not satisfied, $\mathcal{H}$ could still be a local
horizon in terms of the other definitions outlined in Gourgoulhon:2008pu . For
simplicity, in this paper we pick examples where the notion of FOTH suffices.

##  3 Finding horizons in a Kerr-Schild spacetime

In this section we describe, for a Kerr-Schild spacetime, how to search for
the existence of horizons using quantities calculated on a flat spacetime. We
do so by foliating a given hypersurface into co-dimension $2$ surfaces
$\mathcal{S}_{\lambda}$ using the Kerr-Schild vector $k_{\mu}$ (1) and
computing their null expansions, again using only flat spacetime quantities.
This auxiliary flat spacetime is precisely the one that appears in (1) and
serves as the background spacetime for the single and zeroth copy theories.

###  3.1 The geometry of hypersurface foliations

In flat spacetime, let us pick an arbitrary hypersurface $\mathcal{H}$ with
normal one-form $\tilde{H}_{\mu}$. We will foliate $\mathcal{H}$ into
spacelike surfaces $\mathcal{S}_{\lambda}$ parametrized by $\lambda$, with
each leaf of the foliation of co-dimension $2$ with respect to the background
spacetime. At every point on a co-dimension $2$ spacelike surface there are
two independent future-directed null normals. We pick the foliation of
$\mathcal{H}$ such that $k_{\mu}$ can always be chosen as one of the two null
normals. We also assume that $k^{\mu}$ is infalling, so that in the convention
established in section 2.2, we set $n^{\mu}=k^{\mu}.$ See figure 1 for an
illustration of the foliation picked out by $k^{\mu}$.

$k^{\mu}$
---
$\mathscr{I}^{+}$
---
$\mathscr{I}^{-}$
---
$\mathcal{H}_{(\xi)}$
---
$i^{+}$
---
$i^{-}$
---
$i^{0}$
---
$\mathcal{S}_{1}$
---
$\mathcal{S}_{2}$
---
(a) Penrose diagram for an asymptotically flat spacetime, with a one-parameter
family of hypersurfaces $\mathcal{H}_{(\xi)}$. Trajectories in green are
integral curves of $k^{\mu}$. $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are
leaves of the foliation normal to $k^{\mu}$ on $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$.

$k^{\mu}$
---
$k^{\mu}$
---
$\mathcal{S}_{2}$
---
$\mathcal{S}_{1}$
---
(b) Diagram of $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ with one angular
direction restored. Schematically shows the evolution from $\mathcal{S}_{1}$
to $\mathcal{S}_{2}$ along $k^{\mu}$.

Figure 1: Foliation of hypersurfaces using $k_{\mu}$.

The normal subspace of each $\mathcal{S}_{\lambda}$ is by definition spanned
by $k_{\mu}$ and $\tilde{H}_{\mu}$, assuming the two to be linearly
independent. Thus we can find the second null normal $l^{(0)}_{\mu}$ uniquely
by looking for vectors in the span of $k_{\mu}$ and $\tilde{H}_{\mu}$ which
satisfy

$l^{(0)}_{\mu}l^{(0)}_{\nu}\eta^{\mu\nu}=0,\qquad
l^{(0)}_{\mu}k_{\nu}\eta^{\mu\nu}=-1.$ (9)

The flat metric $\eta^{\mu\nu}$ is used to contract indices here since we are
constructing the null normal $l^{(0)}_{\mu}$ in flat spacetime. Note that
since the Kerr-Schild vector $k_{\mu}$ satisfies
$g^{\mu\nu}k_{\mu}=\eta^{\mu\nu}k_{\mu}$, in all that follows we raise and
lower indices on $k$ with impunity. The two conditions (9) are enough to
specify $l^{(0)}_{\mu}$ uniquely:

$l^{(0)}_{\mu}=\frac{\tilde{H}_{\rho}\tilde{H}_{\sigma}\eta^{\rho\sigma}}{2(k^{\rho}\tilde{H}_{\rho})^{2}}k_{\mu}-\frac{\tilde{H}_{\mu}}{k^{\rho}\tilde{H}_{\rho}}.$
(10)

The case where $k^{\rho}\tilde{H}_{\rho}=0$ has the integral curves of
$k^{\mu}$ lying on the $\mathcal{H}$, and there is no foliation of the
hypersurface with spacelike surfaces normal to $k_{\mu}$. As long as
$k^{\rho}H_{\rho}\neq 0$ then, we define for simplicity of notation

$H_{\mu}=-\frac{\tilde{H}_{\mu}}{k^{\rho}\tilde{H}_{\rho}},$ (11)

and use $H_{\mu}$ as the hypersurface normal. Rewriting $l^{(0)}_{\mu}$ in
terms of $H_{\mu}$,

$l^{(0)}_{\mu}=\frac{H_{\rho}H_{\sigma}\eta^{\rho\sigma}}{2}k_{\mu}+H_{\mu}.$
(12)

The intrinsic geometry of each $\mathcal{S}_{\lambda}$, inherited from
$\eta_{\mu\nu}$, is given by the induced metric and its inverse,

$\begin{split}\mathring{h}_{\mu\nu}&=\eta_{\mu\nu}+k_{\mu}l^{(0)}_{\nu}+l^{(0)}_{\mu}k_{\nu},\\\
\mathring{h}^{\mu\nu}&=\eta^{\mu\nu}+k^{\mu}l^{(0)}_{\rho}\eta^{\nu\rho}+l^{(0)}_{\rho}\eta^{\rho\mu}k^{\nu},\end{split}$
(13)

where we use a circular accent to denote quantities defined on flat spacetime.

We now need to map $\mathcal{H}$ and its chosen foliation to the Kerr-Schild
spacetime with metric $g_{\mu\nu}$ (1). Since $\mathcal{H}$ and its foliation
$\mathcal{S}_{\lambda}$ are described in terms of the differential forms
$H_{\mu}$ and $k_{\mu}$, they are in fact independent of the metric. Thus we
can use the same hypersurface $\mathcal{H}$ and its foliation
$\mathcal{S}_{\lambda}$ with metric $g_{\mu\nu}$. However, with a different
metric the null normals of $\mathcal{S}_{\lambda}$ may no longer be the same.

Since $k_{\mu}$ is null with respect to both $\eta_{\mu\nu}$ and $g_{\mu\nu}$,
it is still one of the null normals. As in the previous case, the second null
normal $l_{\mu}$ can be found in the span of $k_{\mu}$ and $H_{\mu}$ by
imposing

$l_{\mu}l_{\nu}g^{\mu\nu}=0,\qquad l_{\mu}k_{\nu}g^{\mu\nu}=-1.$ (14)

Using the relation (2) between $g^{\mu\nu}$ and $\eta^{\mu\nu}$, these
conditions specify $l_{\mu}$ uniquely as

$l_{\mu}=l^{(0)}_{\mu}-\frac{\Phi}{2}k_{\mu}.$ (15)

The intrinsic geometry of $\mathcal{S}_{\lambda}$ inherited from the metric
$g_{\mu\nu}$ is given by the induced metric and its inverse444Interestingly,
owing to the definition of $l_{\mu}$ in terms of $l^{(0)}_{\mu}$ (15) and the
form of the metric (1), it turns out that $h_{\mu\nu}=\mathring{h}_{\mu\nu}$.

$\begin{split}h_{\mu\nu}&=g_{\mu\nu}+k_{\mu}l_{\nu}+l_{\mu}k_{\nu}\\\
h^{\mu\nu}&=g^{\mu\nu}+k^{\mu}l_{\rho}g^{\nu\rho}+l_{\rho}g^{\rho\mu}k^{\nu}.\end{split}$
(16)

###  3.2 The dynamics of hypersurface foliations

Having described the choice of foliation, we now examine how each leaf
$\mathcal{S}_{\lambda}$ evolves along its null normals. We specifically look
at the expansion, defined as in (8). Using the induced metric from (13) we can
write the expansions as

$\mathring{\theta}^{(k)}=\mathring{h}^{\mu\nu}\mathring{\nabla}_{\mu}k_{\nu},\qquad\mathring{\theta}^{(l^{(0)})}=\mathring{h}^{\mu\nu}\mathring{\nabla}_{\mu}l^{(0)}_{\nu}.$
(17)

In flat spacetime there are no trapped surfaces, and thus the two expansions
are always of opposite sign. Since we picked the convention that $k^{\mu}$ is
ingoing and $l^{\mu}$ is outgoing, $\mathring{\theta}^{(k)}\leq 0$ and
$\mathring{\theta}^{(l^{0})}\geq 0$. In other words, $k_{\mu}$ describes a
converging null congruence and $l^{(0)}_{\mu}$ describes a diverging null
congruence.

In curved spacetime with metric $g_{\mu\nu}$, we use the induced metric (16)
to calculate the expansions as

$\theta^{(k)}=h^{\mu\nu}\nabla_{\mu}k_{\nu},\qquad\theta^{(l)}=h^{\mu\nu}\nabla_{\mu}l_{\nu}.$
(18)

Here, the two may have the same sign. We are interested in spacetimes where
$\theta^{(k)}\leq 0$ everywhere and $\theta^{(l)}>0$ at asymptotic infinity.
Regions where $\theta^{(l)}<0$ are trapped, and the boundary of such regions,
where $\theta^{(l)}=0$, are trapping horizons.

###  3.3 The double copy relation

An explicit computation of the expansions using the metric (1) (outlined in
appendix A) shows that for Kerr-Schild spacetimes the expansions on the
$\eta_{\mu\nu}$ and the $g_{\mu\nu}$ background are related via

$\theta^{(k)}=\mathring{\theta}^{(k)},\qquad\theta^{(l)}=\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)}.$
(19)

Our assertion that $\theta^{(k)}<0$ everywhere is automatically satisfied. We
also see that there is a trapped region if the term
$\frac{\Phi}{2}\mathring{\theta}^{(k)}$ becomes larger in magnitude than
$\mathring{\theta}^{(l^{(0)})}$.

These relations constitute a “double copy” map in the sense that they relate
the $\eta_{\mu\nu}$ and $g_{\mu\nu}$ dynamics of the foliation of
$\mathcal{H}_{(\xi)}$ picked out by the Kerr-Schild vector $k^{\mu}$. Note
that this choice of foliation is the same as the one normal to the single copy
field $A_{\mu}=\Phi k_{\mu}$. In the spirit of the double copy, these allow us
to predict the existence of a trapped region using quantities that can be
defined using just the single and zeroth copy fields in flat space.
Explicitly, the term $\frac{\Phi}{2}\mathring{\theta}^{k}$ can be written as
$1/2$ times the expansion along $A_{\mu}$,

$\begin{split}\frac{1}{2}\mathring{\theta}^{A}&=\frac{1}{2}\mathring{h}^{\mu\nu}\partial_{\mu}A_{\nu}\\\
&=\frac{1}{2}\mathring{h}^{\mu\nu}\partial_{\mu}\left(\Phi k_{\nu}\right)\\\
&=\frac{\Phi}{2}\mathring{h}^{\mu\nu}\partial_{\mu}k_{\nu},\end{split}$ (20)

since $\mathring{h}^{\mu\nu}k_{\nu}=0$ by construction. In other words, since
$A_{\mu}$ and $k_{\mu}$ are parallel, evolving a co-dimension 2 volume element
along either results in the same trajectory, albeit at differing rates. The
fractional rate of change is thus enhanced by a factor of $\Phi$ for evolution
along $A_{\mu}$ when compared to that along $k_{\mu}$.

To summarize, our findings here constitute a diagnostic test for the presence
of a trapping horizon using flat space computations, provided the existence of
a classical double copy (in the Kerr-Schild sense) for fields $A_{\mu}$ and
$\Phi$. The test proceeds as follows: first pick a hypersurface $\mathcal{H}$
to test. Foliate it by spacelike slices normal to
$k_{\mu}=\frac{1}{\Phi}A_{\mu}$, and find the complementary null normal
$l^{(0)}$ as in (12). Compute the expansions $\mathring{\theta}^{(k)}$ and
$\mathring{\theta}^{(l^{(0)})}$. If the combination
$\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)}$ vanishes
all along $\mathcal{H}$, it is a trapping horizon.

##  4 Examples

We now demonstrate how the approach outlined in the previous section applies
to black holes whose horizon structure as well as double copy description are
already known. A large class of stationary black hole spacetimes whose metric
can be put in the Kerr-Schild form (1) is the Myers-Perry class. The double
copy construction of the Myers-Perry metric in general dimensions is given in
Monteiro:2014cda . We first demonstrate the double copy description of the
Schwarzschild horizon, which we hope illustrates our approach more
transparently than the somewhat messier expressions which arise in our
analysis of the more general Myers-Perry metric.

###  4.1 Schwarzschild black hole

We write the Schwarzschild metric in Kerr-Schild coordinates as

$ds^{2}=-d\tau^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}+\frac{2M}{r}\left(-d\tau-dr\right)^{2}.$ (21)

The first four terms in (21) constitute the flat metric $\eta_{\mu\nu}$ in
spherical polar coordinates. The zeroth copy $\Phi$ and the Kerr-Schild vector
$k_{\mu}$ are

$\Phi=\frac{2M}{r},\qquad k_{\mu}dx^{\mu}=-d\tau-dr,$ (22)

from which we can also construct the single copy gauge field

$A_{\mu}=\Phi k_{\mu}=-\frac{2M}{r}\left(d\tau+dr\right).$ (23)

This is the gauge field of a static point charge, albeit in an unusual gauge.

The negative signs in $k_{\mu}$ ensure that its integral curves are future-
directed and ingoing. Note that while the spatial coordinates are precisely
the same as the usual spherical polar coordinates, the time coordinate $\tau$
here relates to the usual Schwarzschild $t$ coordinate via

$d\tau=dt+\frac{2M}{r}dr.$ (24)

The Kerr-Schild coordinate patch thus covers the same region as the infalling
Eddington-Finkelstein coordinate patch, which is obtained by one further
coordinate transformation from $\tau$ to $u$ given by $du=d\tau+dr$.

The event horizon $\mathcal{H}$ for the Schwarzschild black hole in Kerr-
Schild coordinates is at $r=2M$. We thus look at the family of hypersurfaces
$\mathcal{H}_{(r)}$ at constant $r$. The normals to these surfaces are along
the one-form $dr$. Normalizing the one-form as described in (11), we find that

$H_{\mu}dx^{\mu}=dr.$ (25)

On the auxiliary flat spacetime, the hypersurfaces $\mathcal{H}_{(r)}$ are
foliated by co-dimension 2 spacelike surfaces $\mathcal{S}_{\lambda}$ normal
to $k_{\mu}$ and to $l^{(0)}_{\mu}$, which in Kerr-Schild coordinates is

$\begin{split}l^{(0)}_{\mu}dx^{\mu}&=\frac{H_{\rho}H_{\sigma}\eta^{\rho\sigma}}{2}k_{\mu}dx^{\mu}+H_{\mu}dx^{\mu}\\\
&=\frac{\eta^{rr}}{2}\left(-d\tau-dr\right)+dr\\\
&=\frac{1}{2}\left(-d\tau+dr\right).\end{split}$ (26)

The null normal on Schwarzschild, from the general form (15), is

$\begin{split}l_{\mu}dx^{\mu}&=l^{(0)}_{\mu}dx^{\mu}-\frac{\Phi}{2}k_{\mu}dx^{\mu}\\\
&=\frac{1}{2}\left(-d\tau+dr\right)-\frac{M}{r}\left(-d\tau-dr\right)\\\
&=-\frac{1}{2}\left(1-\frac{M}{r}\right)d\tau+\frac{1}{2}\left(1+\frac{M}{r}\right)dr.\end{split}$
(27)

See figure 2 for an illustration of the hypersurfaces $\mathcal{H}_{(r)}$ and
their foliation on the Schwarzschild background.

$k^{\mu}$
---
$\mathscr{I}^{+}$
---
$\mathscr{I}^{-}$
---
$\mathcal{H}_{(r)}$
---
$i^{-}$
---
$i^{0}$
---
$i^{+}$
---
$i^{+}$
---
$r=0$
---

$r=2M$
---

$\mathcal{S}_{1}$
---
$\mathcal{S}_{2}$
---

$r=2M$
---

$r=2M$
---

(a) Penrose diagram for the infalling coordinate patch of the Schwarzschild
solution. Blue curves represent the family of test hypersurfaces
$\mathcal{H}_{(r)}$ at constant $r$, green lines are the integral curves of
$k^{\mu}$. $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are leaves of the foliation
of $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ normal to $k^{\mu}$.

$k^{\mu}$
---
$k^{\mu}$
---
$\mathcal{S}_{2}$
---
$\mathcal{S}_{1}$
---
(b) Two of the constant $r$ hypersurfaces with one angular direction restored,
showing the evolution from $\mathcal{S}_{1}$ to $\mathcal{S}_{2}$ along
$k^{\mu}$.

Figure 2: Foliation of constant $r$ surfaces in Schwarzschild using $k^{\mu}$.

In Schwarzschild spacetime, the vectors $l^{\mu}$ and $l^{(0)\mu}$ are
parallel to geodesics with respect to $g_{\mu\nu}$ and $\eta_{\mu\nu}$
respectively. In fact, $l^{\mu}$ lies along the generators of the horizon at
$r=2M$.

Thus, in this case our prescription (15) for relating $l_{\mu}$ and
$l_{\mu}^{(0)}$ maps a family of null geodesics in flat spacetime to null
geodesics on Schwarzschild. This map, however, is distinct from the one
proposed in Gonzo:2021drq , which does not map freely-falling trajectories.
Instead, it maps on-shell trajectories of a test charge in the single copy
gauge field to geodesics of a test particle in the double copy metric.

Writing out the induced metric and computing expansions, we find on the flat
spacetime

$\mathring{\theta}^{(k)}=-\frac{2}{r},\qquad\mathring{\theta}^{(l^{(0)})}=\frac{1}{r},$
(28)

and thus from our general result (19)

$\theta^{(k)}=-\frac{2}{r},\qquad\theta^{(l)}=\frac{1}{r}+\frac{M}{r}\left(-\frac{2}{r}\right)=\frac{r-2M}{r^{2}}.$
(29)

We see that, as expected on physical grounds, $\theta^{(k)}$ is always
negative while $\theta^{(l)}$ is positive outside the horizon at $r=2M$ and
negative inside. We can also explicitly check the outermost condition
$\mathcal{L}_{k}\theta^{(l)}<0$ on the marginal surface $r=2M$,

$\begin{split}\mathcal{L}_{k}\theta^{(l)}\big{|}_{r=2M}&=k^{\mu}\partial_{\mu}\theta^{(l)}\big{|}_{r=2M}\\\
&=\left(\partial_{\tau}-\partial_{r}\right)\theta^{(l)}\big{|}_{r=2M}\\\
&=-\frac{1}{4M^{3}}.\end{split}$ (30)

Thus, the surface $r=2M$ is a future-outer trapping horizon Hayward:1993wb .
From the perspective of the double copy construction we have described, the
change in sign of $\theta^{(l)}$ is a consequence of the interplay between
$\mathring{\theta}^{(l^{(0)})},\mathring{\theta}^{(k)}$ and $\Phi$ as in (19).
As the horizon $\Phi=1$ is approached from the outside, the term
$\frac{\Phi}{2}\mathring{\theta}^{(k)}$ starts to dominate over
$\mathring{\theta}^{(l^{(0)})}$.

As noted in section 3.3, the second term in our expression for $\theta^{(l)}$
(29) can be equivalently written as $1/2$ the expansion of the single copy
field $A_{\mu}$. The expansion of the outgoing null vector $l^{\mu}$ with
respect to $g_{\mu\nu}$ is thus related to the expansion of $l^{(0)\mu}$ with
respect to $\eta_{\mu\nu}$ via

$\theta^{(l)}=\mathring{\theta}^{(l^{(0)})}+\frac{1}{2}\mathring{\theta}^{(A)}.$
(31)

The second term has a clear single copy interpretation, while the first term
depends upon the arbitrary choice of hypersurfaces $\mathcal{H}_{(r)}$ on the
flat background $\eta_{\mu\nu}$.

We mentioned earlier in this section that $l^{\mu}$ is parallel to a family of
geodesics. Explicitly, the vector $\hat{l}^{\mu}$ defined as

$\hat{l}^{\mu}=-\left(\frac{1}{2}-\frac{M}{r}\right)^{-1}l^{\mu},$ (32)

generates past-directed, radially-infalling null geodesics. The Schwarzschild
metric can, in fact, be written in a set of Kerr-Schild coordinates adapted to
$\hat{l}_{\mu}$ instead of $k_{\mu}$, which cover the outgoing patch instead
(including the past horizon). This second set of Kerr-Schild coordinates map
$g_{\mu\nu}$ to a distinct (albeit gauge-equivalent) single copy
$\hat{A}_{\mu}=\Phi\hat{l}_{\mu}$ living on a _different_ flat background
$\hat{\eta}_{\mu\nu}$. The same family of hypersurfaces $\mathcal{H}_{(r)}$
can now be used to locate the past horizon of the eternal Schwarzschild black
hole at $r=2M$. The interested reader might find the discussion of the various
Kerr-Schild coordinate patches in Boyer:1966qh  useful. We omit delving into
this issue at present, and instead move on to the more general Myers-Perry
class of black holes.

###  4.2 Myers-Perry black holes

The Myers-Perry spacetimes in arbitrary dimension $D\geq 4$ generalize the
$D=4$ Kerr black hole to arbitrary dimension. Its double copy description
first appeared in Monteiro:2014cda , which we re-state here. Just as for the
Kerr black hole, Myers-Perry spacetimes have in general two horizons, an outer
event horizon and an inner Cauchy horizon.

For $D\geq 4$, there are $(D-2)/2$ independent rotation planes for even $D$,
and $(D-1)/2$ for odd $D$ (see for instance Myers:2011yc  or appendix E of
Frolov:2017kze ). Call this number of independent rotations $m$. The black
hole can be parametrized by $m$ spin parameters $a_{i}$ and a mass $M$. In
even dimensions we use coordinates $\left(\tau,x_{1},y_{1},\dots
x_{m},y_{m}\right)$ where the pair $\left(x_{i},y_{i}\right)$ spans the
rotation plane corresponding to $a_{i}$. In odd dimensions an additional
spacelike coordinate $z$ is added. The relation between these Kerr-Schild
coordinates and the usual Myers-Perry coordinates
$\left(t,r,\phi_{i},\mu_{i}\right)$ can be found in, for instance, appendix
E.3 of Frolov:2017kze .

The radial variable $r$ solves the equation

$\left(1-\varepsilon\right)\frac{z^{2}}{r^{2}}+\sum_{i=1}^{m}\frac{x_{i}^{2}+y_{i}^{2}}{r^{2}+a_{i}^{2}}=1,$
(33)

where $\varepsilon=0$ for even $D$ and $\varepsilon=1$ for odd $D$. Surfaces
of constant $r$ within a constant time-slice are thus ellipsoids, which become
spheres in the non-rotating limit. We also define the functions $U$ and $V$,

$V=\frac{1}{r^{1+\varepsilon}}\prod_{i=1}^{m}\left(r^{2}+a_{i}^{2}\right),\qquad
U=V\left(1-\sum_{i=1}^{m}\frac{a_{i}^{2}\left(x_{i}^{2}+y_{i}^{2}\right)}{r^{2}+a_{i}^{2}}\right).$
(34)

With these definitions, the metric in Kerr-Schild coordinates is

$ds^{2}=-d\tau^{2}+\left(1-\varepsilon\right)dz^{2}+\sum_{i=1}^{m}\left(dx_{i}^{2}+dy_{i}^{2}\right)\\\
+\frac{2M}{U}\left(-d\tau-\left(1-\varepsilon\right)\frac{z\,dz}{r}-\sum_{i=1}^{m}\frac{r\left(x_{i}\,dx_{i}+y_{i}\,dy_{i}\right)}{r^{2}+a_{i}^{2}}-\sum_{i=1}^{m}\frac{a_{i}\left(x_{i}\,dy_{i}-y_{i}\,dx_{i}\right)}{r^{2}+a_{i}^{2}}\right)^{2}.$
(35)

The first line constitutes the flat metric $\eta_{\mu\nu}$. The zeroth copy
$\Phi$ and the Kerr-Schild vector $k_{\mu}$ are

$\Phi=\frac{2M}{U},\qquad
k_{\mu}dx^{\mu}=-d\tau-\left(1-\varepsilon\right)\frac{z\,dz}{r}-\sum_{i=1}^{m}\frac{r\left(x_{i}\,dx_{i}+y_{i}\,dy_{i}\right)}{r^{2}+a_{i}^{2}}-\sum_{i=1}^{m}\frac{a_{i}\left(x_{i}\,dy_{i}-y_{i}\,dx_{i}\right)}{r^{2}+a_{i}^{2}}.$
(36)

The single copy field can be constructed using $A_{\mu}=\Phi k_{\mu}$.

Much like the choice of coordinates in Schwarzschild, the radial variable $r$
here is defined such that the horizons are always a surface of constant $r$.
Specifically, the horizons solve the equation $V-2M=0$. We thus look at the
family of hypersurfaces $\mathcal{H}_{(r)}$ at constant $r$. The normals to
this family are along the one-form $dr$, which in Kerr-Schild coordinates is

$dr=\frac{V}{U}\left(\sum_{i=1}^{m}\frac{r\left(x_{i}\,dx_{i}+y_{i}\,dy_{i}\right)}{r^{2}+a_{i}^{2}}+\left(1-\varepsilon\right)\frac{z\,dz}{r}\right).$
(37)

Normalizing using (11), we find $H_{\mu}dx^{\mu}=dr$.

The foliation of $\mathcal{H}_{(r)}$ in flat spacetime is given by the null
normals $k_{\mu}$ and $l^{(0)}_{\mu}$,

$\begin{split}l^{(0)}_{\mu}dx^{\mu}&=\frac{H_{\rho}H_{\sigma}\eta^{\rho\sigma}}{2}k_{\mu}dx^{\mu}+H_{\mu}dx^{\mu}\\\
&=\frac{V}{2U}k_{\mu}dx^{\mu}+dr.\\\ \end{split}$ (38)

Similar to the Schwarzschild case, we can find the corresponding null normal
on the Myers-Perry background using (15),

$l_{\mu}dx^{\mu}=\frac{V-2M}{2U}k_{\mu}dx^{\mu}+dr.$ (39)

We calculate the flat space expansions,

$\begin{split}\mathring{\theta}^{(k)}&=\frac{\varepsilon}{r}-\frac{2}{r}\sum_{i=1}^{m}\left[\frac{r^{2}}{r^{2}+a_{i}^{2}}+\frac{V}{U}\frac{a_{i}^{2}r^{2}\left(x_{i}^{2}+y_{i}^{2}\right)}{\left(r^{2}+a_{i}^{2}\right)^{3}}\right],\\\
\mathring{\theta}^{(l^{(0)})}&=-\frac{V}{2U}\mathring{\theta}^{(k)},\end{split}$
(40)

from which we obtain using (19),

$\begin{split}\theta^{(k)}&=\frac{\varepsilon}{r}-\frac{2}{r}\sum_{i=1}^{m}\left[\frac{r^{2}}{r^{2}+a_{i}^{2}}+\frac{V}{U}\frac{a_{i}^{2}r^{2}\left(x_{i}^{2}+y_{i}^{2}\right)}{\left(r^{2}+a_{i}^{2}\right)^{3}}\right],\\\
\theta^{(l)}&=-\frac{V-2M}{2U}\theta^{(k)}.\end{split}$ (41)

Since $\mathring{\theta}^{(k)}<0$, we see that $\theta^{(l)}$ vanishes
whenever $V=2M$.

Just as for Schwarzschild, the term $\frac{M}{U}\theta^{(k)}$ can be written
as $\frac{1}{2}\mathring{\theta}^{(A)}$, using the single-copy gauge field
$A_{\mu}=\frac{2M}{U}k_{\mu}$. The past horizon can also be similarly found,
using Kerr-Schild coordinates adapted to $l_{\mu}$ instead of $k_{\mu}$. See
section 3.3 of Myers:1986un  for a discussion of the Kerr-Schild coordinate
patch adapted to $l_{\mu}$.

##  5 Discussion

We have demonstrated a procedure for finding hoirzons within the Kerr-Schild
classical double copy as first proposed in Monteiro:2014cda . The procedure is
as follows: pick a hypersurface $\mathcal{H}$ to test, then foliate the
surface using $k_{\mu}$. Find the complementary null normal $l^{(0)}_{\mu}$ on
the flat $\eta_{\mu\nu}$ background, and compute its expansion
$\mathring{\theta}^{(l^{(0)})}$. If the expansion $\mathring{\theta}^{(A)}$ of
the single copy field $A_{\mu}=\Phi k_{\mu}$ equals
$-2\mathring{\theta}^{(l^{(0)})}$ everywhere along the test hypersurface, then
the double copy will have a trapping horizon there.

Importantly, our procedure relies on information available in the single copy
alone. The null normal $l^{(0)}_{\mu}$ is an object on the flat
$\eta_{\mu\nu}$ background. We do use the zeroth copy, in that
$k_{\mu}=A_{\mu}/\Phi$, but no double-copy specific information is required.
We should point out that our perspective here assumes the single copy
$A_{\mu}$ and the zeroth copy $\Phi$ live on the flat background
$\eta_{\mu\nu}$.

Additionally, what we have actually found are marginally trapped surfaces.
When we choose both $k_{\mu}$ and $l_{\mu}$ to be future directed, the future
outer marginally trapped surface which occurs where $\theta^{(l)}=0$ will
indicate an event horizon, but only if we additionally assume e.g. the null
energy condition. As in the famous singularity theorems Penrose:1964wq ,
future-directed light rays following a null congruence with zero expansion
will inevitably collapse to a singularity due to the null Raychaudhuri
equation. Allowing matter which disobeys the null energy condition could let
the expansion turn positive again, letting the light escape.

Our horizon condition
$\mathring{\theta}^{(l^{(0)})}=-\mathring{\theta}^{(A)}/2$ assumed the charge
for the single copy gauge field was $M$. If we wish to allow for the
replacement $M\rightarrow q$, then the condition should be written as
$\mathring{\theta}^{(l^{(0)})}=-M\mathring{\theta}^{(A)}/2q$. Importantly,
this refined condition indicates that changing the sign of $q$ in the single
copy does not change the sign of the mass in the double copy; any choice of
point charge in the $U(1)$ sector of the gauge theory results in a
Schwarzschild black hole with a horizon.

Throughout this paper, our approach has been to consider the codimension-1
surface $\mathcal{H}$ as being at the same coordinate location in both the
flat spacetime of the single copy and the curved spacetime of the double copy.
The paths tangent to $k^{\mu}$ are the same in both spacetimes, while the
$l^{\mu}$ and $l^{(0)\mu}$ paths are chosen by the same leaves of $k^{\mu}$ in
the hypersurface $\mathcal{H}$. Even though $l^{\mu}$ for the cases we
examined ends up being tangent to a set of geodesics, we have not followed
previous work on the double copy of geodesics Gonzo:2021drq . In Gonzo:2021drq
, the authors instead replace the conserved charges associated with the
curved-space geodesics with conserved charges for particles moving in the
single-copy gauge field.

Our work does extend to non-vacuum Kerr-Schild spacetimes; as an example, it
is straightforward to show that metrics of the general form
$ds^{2}=g_{tt}(r)dt^{2}-1/g_{tt}(r)\,dr^{2}+r^{2}d\Omega^{2}$, as studied in
Easson:2020esh , have a Kerr-Schild form and thus a simple single-copy.
Following our procedure here, the trapping horizons unsurprisingly appear
where $\theta^{(l)}=g_{tt}/r=0$. Just as in the current work, the data needed
to find this result is entirely in the single copy and its flat background.

For the vacuum Schwarzschild, Kerr, and Myers-Perry black holes we
investigated, the $k^{\mu}$ in the Kerr-Schild coordinates is also a (doubled)
principal null vector of the Weyl tensor. These spacetimes are Petrov type D,
so their Weyl tensor should have two doubled principal null vectors. When we
have correctly chosen the $\mathcal{H}$ surface to be along the horizon, then
$l^{\mu}$ actually turns out to be exactly this second doubled principal null
vector. So far we have investigated horizons only for single Kerr-Schild
spacetimes, but the classical double copy extends to multi-Kerr-Schild
Chawla:2022ogv ; Luna:2015paa  as well as Weyl formalisms Luna:2018dpt ;
Godazgar:2020zbv . We defer further investigations of the relationship between
local horizon definitions and these more general classical double copy
formalisms to future work.

###### Acknowledgements.

The authors thank Kwinten Fransen, Damien Easson, Tucker Manton, and Raphaela
Wutte for helpful discussions. The authors are supported by the U.S.
Department of Energy under grant number DE-SC0019470 and by the Heising-Simons
Foundation “Observational Signatures of Quantum Gravity” collaboration grant
2021-2818.

##  Appendix A Calculation of curved spacetime expansions from flat spacetime
expansions

In this appendix we give a detailed derivation of our main result relating the
expansions on the $\eta_{\mu\nu}$ and the $g_{\mu\nu}$ spacetimes (19) for a
generic Kerr-Schild metric (1)

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi k_{\mu}k_{\nu}.$ (42)

We will use coordinate-basis methods throughout. Computing expansions is also
possible in an orthonormal basis using the Newman-Penrose formalism, where the
expansion appears as one of the various spin coefficients. Our main result
(19) is corroborated by just such a calculation of “shifted spin coefficients”
performed by BenTov:2017kyf  in a different context.

First, for notational simplicity, we define the vectors $l^{(0)\mu}$ and
$l^{\mu}$, which live respectively on the $\eta_{\mu\nu}$ and $g_{\mu\nu}$
backgrounds,

$\displaystyle l^{(0)\mu}$ $\displaystyle\coloneqq\eta^{\mu\nu}l^{(0)}_{\nu}$
(43) $\displaystyle l^{\mu}$ $\displaystyle\coloneqq g^{\mu\nu}l_{\mu}.$ (44)

Thus, including (15), we have the relations

$\displaystyle l^{\mu}$ $\displaystyle=l^{(0)\mu}+\frac{\Phi}{2}k^{\mu}$ (45)
$\displaystyle l_{\mu}$ $\displaystyle=l^{(0)}_{\mu}-\frac{\Phi}{2}k_{\mu}.$
(46)

In terms of $l^{\mu}$ and $l^{(0)\mu}$, we can write the induced (inverse)
metrics $h^{\mu\nu}$ (13) and $\mathring{h}^{\mu\nu}$ (16) more succinctly as

$\displaystyle h^{\mu\nu}$
$\displaystyle=g^{\mu\nu}+k^{\mu}l^{\nu}+l^{\mu}k^{\nu}$ (47)
$\displaystyle\mathring{h}^{\mu\nu}$
$\displaystyle=\eta^{\mu\nu}+k^{\mu}l^{(0)\nu}+l^{(0)\mu}k^{\nu}.$ (48)

Using the explicit inverse Kerr-Schild metric (2) and the relation (45),

$\begin{split}h^{\mu\nu}&=\eta^{\mu\nu}-\Phi
k^{\mu}k^{\nu}+k^{\mu}\left(l^{(0)\nu}+\frac{\Phi}{2}k^{\nu}\right)+\left(l^{(0)\mu}+\frac{\Phi}{2}k^{\mu}\right)k^{\nu}\\\
&=\eta^{\mu\nu}+k^{\mu}l^{(0)\nu}+l^{(0)\mu}k^{\nu}\\\
&=\mathring{h}^{\mu\nu}.\end{split}$ (49)

We have thus found that the metric on $\mathcal{S}_{\lambda}$ induced from the
flat and the curved background are equal. We henceforth drop the circular
accent on $\mathring{h}^{\mu\nu}$.

To relate the flat- and curved-spacetime expansions defined in (17) and (18),
we will also need to relate the derivatives $\mathring{\nabla}_{\mu}$ and
$\nabla_{\mu}$. It is a standard result in differential geometry that the
difference between the action of any two covariant derivatives on a one-form
is a linear map (see e.g. chapter 3 of Wald:1984rg),

$\nabla_{\mu}V_{\rho}-\mathring{\nabla}_{\mu}V_{\rho}=-\Gamma_{\mu\rho}^{\sigma}V_{\sigma}.$
(50)

We can thus write

$\displaystyle\begin{split}\theta^{(k)}&=h^{\mu\nu}\nabla_{\mu}k_{\nu}\\\
&=h^{\mu\nu}\mathring{\nabla}_{\mu}k_{\mu}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}k_{\sigma}\\\
&=\mathring{\theta}^{(k)}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}k_{\sigma},\end{split}$
(51) and
$\displaystyle\begin{split}\theta^{(l)}&=h^{\mu\nu}\nabla_{\mu}l_{\nu}\\\
&=h^{\mu\nu}\mathring{\nabla}_{\mu}l_{\nu}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}l_{\sigma}\\\
&=h^{\mu\nu}\mathring{\nabla}_{\mu}\left(l_{\nu}^{(0)}-\frac{\Phi}{2}k_{\nu}\right)-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}l_{\sigma}\\\
&=\mathring{\theta}^{(l^{(0)})}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}l_{\sigma}-h^{\mu\nu}\mathring{\nabla}_{\mu}\left(\frac{\Phi}{2}k_{\sigma}\right)\\\
&=\mathring{\theta}^{(l^{(0)})}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}l_{\sigma}-\frac{\Phi}{2}\mathring{\theta}^{(k)}.\end{split}$
(52)

We demonstrated that the last equality holds in section 3.3. Clearly, we will
need the contractions $h^{\mu\nu}\Gamma_{\sigma\mu\nu}k^{\sigma}$ and
$h^{\mu\nu}\Gamma_{\sigma\mu\nu}l^{\sigma}$.

First, from the metric compatibility of the two derivative operators with
$g_{\mu\nu}$ and $\eta_{\mu\nu}$ respectively, we can write

$\Gamma_{\sigma\mu\nu}=g_{\sigma\lambda}\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}\left(\mathring{\nabla}_{\nu}\left[\Phi
k_{\mu}k_{\sigma}\right]+\mathring{\nabla}_{\mu}\left[\Phi
k_{\nu}k_{\sigma}\right]-\mathring{\nabla}_{\sigma}\left[\Phi
k_{\mu}k_{\nu}\right]\right).$ (53)

Let us simplify $h^{\mu\nu}\Gamma_{\sigma\mu\nu}$,

$h^{\mu\nu}\Gamma_{\sigma\mu\nu}=\eta^{\mu\nu}\Gamma_{\sigma\mu\nu}+2k^{\mu}l^{(0)\nu}\Gamma_{\sigma\mu\nu},$
(54)

where we used the index symmetry of $\Gamma_{\sigma\mu\nu}$. Using the metric
compatibility of $\mathring{\nabla}$ with $\eta_{\mu\nu}$ and the nullness of
$k_{\mu}$, the first term in (54) simplifies as

$\frac{1}{2}\eta^{\mu\nu}\left(\mathring{\nabla}_{\nu}\left[\Phi
k_{\mu}k_{\sigma}\right]+\mathring{\nabla}_{\mu}\left[\Phi
k_{\nu}k_{\sigma}\right]-\mathring{\nabla}_{\sigma}\left[\Phi
k_{\mu}k_{\nu}\right]\right)=\mathring{\nabla}_{\mu}\left[\Phi
k^{\mu}k_{\sigma}\right].$ (55)

To simplify the second term of (54), we compute

$\begin{split}k^{\mu}\Gamma_{\sigma\mu\nu}&=\frac{1}{2}k^{\mu}\mathring{\nabla}_{\mu}\left[\Phi
k_{\nu}k_{\sigma}\right]+\frac{1}{2}k^{\mu}\mathring{\nabla}_{\nu}\left[\Phi
k_{\mu}k_{\sigma}\right]-\frac{1}{2}k^{\mu}\mathring{\nabla}_{\sigma}\left[\Phi
k_{\mu}k_{\nu}\right]\\\ &=\frac{1}{2}\Phi
k_{\nu}k^{\mu}\mathring{\nabla}_{\mu}k_{\sigma}+\frac{1}{2}\Phi
k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}k_{\nu}+\frac{1}{2}k_{\nu}k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}\Phi+\frac{1}{2}k^{\mu}k_{\mu}\mathring{\nabla}_{\nu}\left[\Phi
k_{\sigma}\right]\\\ &\phantom{=\ }\ +\frac{1}{2}\Phi
k_{\sigma}k^{\mu}\mathring{\nabla}_{\nu}k_{\mu}-\frac{1}{2}k^{\mu}k_{\mu}\mathring{\nabla}_{\sigma}\left[\Phi
k_{\nu}\right]-\frac{1}{2}\Phi
k_{\nu}k^{\mu}\mathring{\nabla}_{\sigma}k_{\mu}\\\
&=\frac{1}{2}k_{\nu}k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}\Phi.\end{split}$
(56)

The last line here follows from the previous by observing that the nullness
and geodeticity of $k_{\mu}$ forces all but the third term to vanish. The
second term of (54) is therefore

$2k^{\mu}l^{(0)\nu}\Gamma_{\sigma\mu\nu}=l^{(0)\nu}k_{\nu}k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}\Phi=-k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}\Phi,$
(57)

where we used the normalization condition $l^{(0)\mu}k_{\mu}=-1$. We have thus
simplified (54) to

$\begin{split}h^{\mu\nu}\Gamma_{\sigma\mu\nu}&=\mathring{\nabla}_{\mu}\left[\Phi
k^{\mu}k_{\sigma}\right]-k_{\sigma}k^{\mu}\mathring{\nabla}_{\mu}\Phi\\\
&=\Phi\mathring{\nabla}_{\mu}\left[k^{\mu}k_{\sigma}\right]\\\ &=\Phi
k_{\sigma}\mathring{\nabla}_{\mu}k^{\mu},\end{split}$ (58)

where to obtain the final equality we used the geodeticity of $k_{\sigma}$. In
just a moment we will need also the equality
$\mathring{\theta}^{(k)}=\mathring{\nabla}_{\mu}k^{\mu}$, which we now show.
The difference between the two terms can be simplified as,

$\begin{split}\mathring{\theta}^{(k)}-\mathring{\nabla}_{\mu}k^{\mu}&=\left(h^{\mu\nu}-\eta^{\mu\nu}\right)\mathring{\nabla}_{\mu}k_{\nu}\\\
&=\left(k^{\mu}l^{(0)\nu}+l^{(0)\mu}k^{\nu}\right)\mathring{\nabla}_{\mu}k_{\nu}\\\
&=l^{(0)\nu}k^{\mu}\mathring{\nabla}_{\mu}k_{\nu}+\frac{1}{2}l^{(0)\mu}\mathring{\nabla}_{\mu}\left(k^{\nu}k_{\nu}\right)\\\
&=0,\end{split}$ (59)

where the first term vanishes due to geodeticity and the second due to the
nullness of $k^{\mu}$. We can finally compute the two contractions we needed,

$\displaystyle h^{\mu\nu}\Gamma_{\sigma\mu\nu}k^{\sigma}$
$\displaystyle=\Phi\left(k^{\sigma}k_{\sigma}\right)\mathring{\nabla}_{\mu}k^{\mu}=0,$
(60) $\displaystyle h^{\mu\nu}\Gamma_{\sigma\mu\nu}l^{\sigma}$
$\displaystyle=\Phi\left(l^{\sigma}k_{\sigma}\right)\mathring{\nabla}_{\mu}k^{\mu}=-\Phi\mathring{\theta}^{(k)}.$
(61)

Substituting (60) into (51) and (61) into (52),

$\displaystyle\theta^{(k)}$
$\displaystyle=\mathring{\theta}^{(k)}-h^{\mu\nu}\Gamma_{\sigma\mu\nu}k^{\sigma}=\mathring{\theta}^{(k)}.$
(62)
$\displaystyle\begin{split}\theta^{(l)}&=\mathring{\theta}^{(l^{(0)})}-h^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}l_{\sigma}-\frac{\Phi}{2}\mathring{\theta}^{(k)}=\mathring{\theta}^{(l^{(0)})}+\Phi\mathring{\theta}^{(k)}-\frac{\Phi}{2}\mathring{\theta}^{(k)}\\\
&=\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)}.\end{split}$
(63)

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# Tailoring $T_{c}$ by symmetry principles: The concept of Superconducting
Fitness

Aline Ramires  Institute for Theoretical Studies, ETH Zurich, CH-8092, Zurich,
Switzerland.  Daniel F. Agterberg  Department of Physics, University of
Wisconsin, Milwaukee, WI 53201, USA  Manfred Sigrist  Institute for
Theoretical Physics, ETH Zurich, CH-8093, Zurich, Switzerland

###### Abstract

We propose a generalization of the concept of _superconducting fitness_ ,
which allows us to make statements analogous to _Anderson’s theorems_
concerning the stability of different superconducting states. This concept can
be applied to complex materials with several orbital, layer, sublattice or
valley degrees of freedom. The superconducting fitness parameters
$F_{A}({\mathbb{k}})$ and $F_{C}({\mathbb{k}})$ give a direct measure of the
robustness of the weak coupling instability and of the presence of detrimental
terms in the Hamiltonian, respectively. These two parameters can be employed
as a guide to engineer normal state Hamiltonians in order to favour or
suppress superconducting order parameters with different symmetries and
topological properties. To illustrate the applicability and power of this
concept we study three cases: the non-centrosymmetric heavy fermion CePt3Si,
the hole doped iron pnictide KFe2As2 and the doped topological insulator
CuxBi2Se3.

Simple superconductors are usually well described by a single band of doubly
degenerate electrons with only the spin degree of freedom (DOF). Their
phenomenology can be addressed by strong symmetry-based arguments, such as
Anderson’s Theorems, which predict the vulnerability of different
superconducting states through the presence of key symmetry breaking fields 1;
2 . The most interesting superconductors available today are in fact complex
materials, which have extra DOFs, such as orbital 3; 4; 5; 6; 7; 8; 9; 10; 11;
12; 13; 14; 15; 16, layer 17; 18, sublattice or valley 19; 20; 21; 22; 23; 24.
Many of them host unconventional superconducting states with phenomenology
which seems to go beyond the intuition developed for simple systems. In
particular, unexpected response under key symmetry breaking fields, such as
unusual upper critical field anisotropy has been observed 3; 4; 24. Effective
models which do not carefully take into account the underlying symmetries and
properties of the extra electronic DOF usually fail to describe their
behaviour. This requires us to start with a microscopic description in the
basis in which the all symmetry properties are explicit (here we refer to this
basis as the local orbital basis, but it can be associated with other
microscopic DOF). On the other hand, superconductivity is well understood as a
weak-coupling Fermi surface instability, which can be arbitrarily complex and
is described in the band basis. The robustness of the instability is
ultimately guaranteed by the presence of the desired states to be paired at
the Fermi energy with opposite momenta, which can be quantified by the
_superconducting fitness_3; 5. This concept captures the complexity of the
electronic states and encodes all the symmetry aspects in a concise form,
allowing one to make statements analogous to Anderson’s Theorems for complex
superconducting materials.

This paper generalizes the concept of superconducting fitness previously
introduced by some of the authors3; 5, now within a non-perturbative scheme,
extending the range of applicability. The original concept of superconducting
fitness (characterized by the function $F_{C}({\mathbb{k}})$ defined below),
assumes there is a weak-coupling instability and provides a way to check how
different kinds of perturbations would destabilize a given superconducting
order parameter. From this point, valid questions are: How to guarantee the
presence of a weak coupling instability to start with? Can we assure that
there is _some_ component of the order parameter in the orbital basis which
will lead to an intra-band order parameter in the band basis? The answer to
these questions is yes, and here we introduce a second superconducting fitness
function, $F_{A}({\mathbb{k}})$, which allows us to assess the presence of a
weak coupling instability. The generalization of the concept of
superconducting fitness also allows us to better understand the role of each
term in the normal state Hamiltonian, and how it supports or suppresses
different kinds of superconducting order parameters. With this concept in
hand, we have a very portable and inexpensive tool to, in principle, engineer
the normal electronic state in order to support the emergence of specific
superconducting order parameters, which can be exotic or topological. In this
paper we apply this generalized concept to the non-centrosymmetric material
CePt3Si, the hole doped iron-pnictide KxFe2As2, and the doped topological
insulator CuxBi2Se3.

We focus here on a generic two-orbital model, a minimal model for the
discussion of multi-band superconductivity. The Hamiltonian can be written in
terms of an ‘orbital $\otimes$ spin’ basis,
$\Psi_{OS}^{\dagger}=(c_{1\uparrow}^{\dagger},c_{1\downarrow}^{\dagger},c_{2\uparrow}^{\dagger},c_{2\downarrow}^{\dagger})$,
and parametrized as:

$\displaystyle
H_{0}({\mathbb{k}})=\sum_{a,b}h_{ab}({\mathbb{k}})\tau_{a}\otimes\sigma_{b},$
(1)

where the Pauli matrices $\tau_{a}$ and $\sigma_{b}$ correspond to the extra
(orbital) and spin DOF, respectively. $h_{ab}({\mathbb{k}})$ are real
functions of momenta ${\mathbb{k}}$ due to the hermiticity of the Hamiltonian.
In presence of parity ($P:{\mathbb{k}}\rightarrow-{\mathbb{k}}$, for orbitals
of same parity) and time-reversal symmetry (TRS) defined as
$\Theta=K\tau_{0}\otimes(i\sigma_{2})$, the only pairs allowed for $(a,b)$ are
$(0,0),(1,0),(3,0),(2,1),(2,2),(2,3)$. For orbitals with opposite parity
($P:\tau_{3}\otimes\sigma_{0}$ and ${\mathbb{k}}\rightarrow-{\mathbb{k}}$),
the allowed pairs in presence of TRS are
$(0,0),(2,0),(3,0),(1,1),(1,2),(1,3)$. For the case of a sublattice DOF, such
as in graphene, parity also takes one sublattice into another
($P:\tau_{1}\otimes\sigma_{0}$ and ${\mathbb{k}}\rightarrow-{\mathbb{k}}$),
and as a consequence the allowed pairs are
$(0,0),(1,0),(2,0),(3,1),(3,2),(3,3)$. In all cases, the set of matrices
$\tau_{a}\otimes\sigma_{b}$ is completely anti-commuting, an important
property to simplify the equations below.

In order to introduce a more concise notation, we substitute the
$\tau_{a}\otimes\sigma_{b}$ matrices by $T_{i}$-operators labelled by
$i=1,...,15$, which can be identified with the generators of SU(4), traceless
hermitian matrices, plus $T_{0}=\tau_{0}\otimes\sigma_{0}$. The generators
follow the relation:
$T_{i}T_{j}=\delta_{ij}I_{0}+\sum_{c}(if_{ijk}+d_{ijk})T_{k},$ where
$f_{ijk}=Tr[[T_{i},T_{j}]T_{k}]$ and $d_{ijk}=Tr[\\{T_{i},T_{j}\\}T_{k}]$ are
completely anti-symmetric and symmetric structure constants, respectively. We
distinguish the special subset of 5 completely anti-commuting generators by a
different letter $O_{i}$, such that $T_{1...5}$ are identified with
$O_{1...5}$. These follow $\\{O_{i},O_{j}\\}=2\delta_{ij}I_{0}$. The
Hamiltonian can then be written as:

$\displaystyle H_{0}({\mathbb{k}})$ $\displaystyle=$ $\displaystyle
c_{0}({\mathbb{k}})T_{0}+\textbf{c}({\mathbb{k}})\cdot\textbf{O},$ (2)

where $c_{0}({\mathbb{k}})$ is an even function of ${\mathbb{k}}$ ,
$\textbf{O}=(O_{1},...,O_{5})$ and $\textbf{c}({\mathbb{k}})$ is a
5-dimensional vector. In presence of parity and TRS the problem is doubly
degenerate, so we can parametrize the Green’s functions (GF) in terms of two
poles in a convenient way:

$\displaystyle
G_{0}(k)=G_{a}(k)\frac{(I_{0}+\hat{\textbf{c}}({\mathbb{k}})\cdot\textbf{O})}{2}+G_{b}(k)\frac{(I_{0}-\hat{\textbf{c}}({\mathbb{k}})\cdot\textbf{O})}{2},$

where $G_{a,b}(k)=G_{a,b}({\mathbb{k}},i\omega_{n})$ and
$\hat{\textbf{c}}({\mathbb{k}})=\textbf{c}({\mathbb{k}})/|\textbf{c}({\mathbb{k}})|$.
From $(i\omega_{n}T_{0}-H_{0})G_{0}(k)=T_{0}$ we find
$G_{a,b}(k)=(i\omega_{n}-\epsilon_{a,b}({\mathbb{k}}))^{-1}$, where
$\epsilon_{a,b}({\mathbb{k}})=c_{0}({\mathbb{k}})\pm|\textbf{c}({\mathbb{k}})|$.

We can also write the order parameter in terms of the generators of SU(4) as a
gap matrix
$\Delta({\mathbb{k}})=\mathbb{d}({\mathbb{k}})\cdot\textbf{T}(i\Sigma_{2})$,
where $\mathbb{d}({\mathbb{k}})$ is a 16-dimensional vector,
$\textbf{T}=(T_{0},T_{1},...,T_{15})$ and
$i\Sigma_{2}=\tau_{0}\otimes(i\sigma_{2})$. Given the fact that the ‘two
orbital $\otimes$ spin’ problem can also be thought in terms of a $j=3/2$
problem 25; 26; 27, we can analogously classify the order parameter as
singlet, triplet, quintuplet, and septuplet. The singlet is described by the
identity matrix $T_{0}$, while the states in the quintuplet by the set of 5
(non-symplectic) generators introduced above in the context of orbitals with
same parity, satisfying $\Sigma_{2}O_{i}\Sigma_{2}=O_{i}^{T}$. The states in
the triplet and septuplet are described by the remaining 10 (symplectic)
generators, following $\Sigma_{2}T_{i}\Sigma_{2}=-T_{i}^{T}$.

We can then insert the parametrized GF and gap matrix in the linearized gap
equation

$\displaystyle
1=-Tv\sum_{{\mathbb{k}},n}Tr[\hat{\Delta}^{\dagger}({\mathbb{k}})G_{0}(k)\hat{\Delta}({\mathbb{k}})G^{T}_{0}(-k)],$
(4)

where $v$ is the magnitude of the attractive interaction in the symmetry
channel of the respective $\Delta({\mathbb{k}})=d\hat{\Delta}({\mathbb{k}})$,
where $\hat{\Delta}({\mathbb{k}})$ is the normalized gap matrix satisfying
$\langle|\hat{\Delta}({\mathbb{k}})|^{2}\rangle_{FS}=1$. After some
manipulation we find the suggestive form:

$\displaystyle 1$ $\displaystyle=$
$\displaystyle-\frac{Tv}{8}\sum_{{\mathbb{k}},n}\Big{[}(G_{a}\bar{G}_{a}+G_{b}\bar{G}_{b})Tr[|F_{A}({\mathbb{k}})|^{2}]$
$\displaystyle+$
$\displaystyle(G_{a}\bar{G}_{b}+G_{b}\bar{G}_{a})Tr[|F_{C}({\mathbb{k}})|^{2}]\Big{]},$

where $G_{a,b}=G_{a,b}(k)$ and $\bar{G}_{a,b}=G_{a,b}(-k)$ and here we
introduce the superconducting fitness parameters:

$\displaystyle
F_{A}({\mathbb{k}})(i\sigma_{2})=\tilde{H}_{0}({\mathbb{k}})\hat{\Delta}({\mathbb{k}})+\hat{\Delta}({\mathbb{k}})\tilde{H}_{0}^{*}(-{\mathbb{k}}),$
(6) $\displaystyle
F_{C}({\mathbb{k}})(i\sigma_{2})=\tilde{H}_{0}({\mathbb{k}})\hat{\Delta}({\mathbb{k}})-\hat{\Delta}({\mathbb{k}})\tilde{H}_{0}^{*}(-{\mathbb{k}}),$

where
$\tilde{H}_{0}({\mathbb{k}})=(H_{0}({\mathbb{k}})-c_{0}({\mathbb{k}})T_{0})/|\textbf{c}({\mathbb{k}})|$.
From the definition above it becomes clear that
$Tr[|F_{A}({\mathbb{k}})|^{2}]$ quantifies the presence of _intra-band
pairing_ , therefore guaranteeing the robustness of the SC instability in the
weak-coupling limit; while $Tr[|F_{C}({\mathbb{k}})|^{2}]$ quantifies the
presence of _inter-band pairing_ , usually associated with detrimental effects
to the respective superconducting state 3.

The first line in Eq. Tailoring $T_{c}$ by symmetry principles: The concept of
Superconducting Fitness can be treated as the usual BCS equation, leading to
the familiar form, now carrying the superconducting fitness parameter
$F_{A}({\mathbb{k}})$:

$\displaystyle-2v\alpha
ln\left(\frac{4e^{\gamma}}{\pi}\frac{\omega_{C}}{2k_{B}T}\right),$ (7)

where

$\displaystyle 16\alpha=\left[N_{a}(0)\langle
Tr[|F_{A}({\mathbb{k}})|^{2}]\rangle_{FS_{a}}+(a\rightarrow b)\right],$ (8)

$\omega_{C}$ is a characteristic cutoff energy, $N_{a,b}(0)$ are the density
of states at the Fermi energy and $\langle...\rangle_{FS_{a,b}}$ denotes the
average over the Fermi surface for bands $a$ and $b$, respectively.

The second line, after the sum over the Matsubara frequencies, can be written
as:

$\displaystyle-\frac{v}{8}\sum_{{\mathbb{k}}}\frac{\text{tanh}\left(\frac{\beta\epsilon_{a}({\mathbb{k}})}{2}\right)}{(\epsilon_{a}({\mathbb{k}})+\epsilon_{b}({\mathbb{k}}))}Tr[|F_{C}({\mathbb{k}})|^{2}]+(a\rightarrow
b).$ (9)

At this point, for the single band scenario, one usually turns the integral
over momenta into an integral over energy, introducing a DOS and integrating
over a narrow range over the FS between $\pm\omega_{C}$. Note, though, that
this integral is different because its denominator is written as a sum of two
dispersions, and the integral is dominated by the region where
$\epsilon_{a}+\epsilon_{b}=0$, but for well separated bands, this condition is
normally satisfied far away from the FS. We introduce the quantity
$q({\mathbb{k}})=\epsilon_{a}({\mathbb{k}})-\epsilon_{b}({\mathbb{k}})$ and
assume well separated bands, such that $q({\mathbb{k}})>>\omega_{C}$ in the
range of energies close to the FS. Within this approximation, we find that the
second line in Eq. Tailoring $T_{c}$ by symmetry principles: The concept of
Superconducting Fitness is equal to $2v\delta$, where

$\displaystyle\\!\\!\\!\\!\\!\\!\\!16\delta=\frac{\omega_{C}^{2}}{2}\left[N_{a}(0)\left\langle\frac{Tr[|F_{C}({\mathbb{k}})|^{2}]}{q({\mathbb{k}}_{Fa})^{2}}\right\rangle_{\\!\\!FS_{a}}\\!\\!\\!\\!\\!+(a\rightarrow
b)\right].$ (10)

Now the full gap equation can be concisely written as:

$\displaystyle 1$ $\displaystyle=$ $\displaystyle-2v\alpha
ln\left(\frac{4e^{\gamma}}{\pi}\frac{\omega_{C}}{2k_{B}T}\right)+2v\delta,$
(11)

and the critical temperature is explicitly obtained as:

$\displaystyle k_{B}T_{C}$ $\displaystyle=$
$\displaystyle\frac{4e^{\gamma}}{\pi}\frac{\omega_{C}}{2}e^{-1/2|v|\alpha}e^{-\delta/\alpha},$
(12)

which makes clear that the larger the $\alpha$, the larger the critical
temperature, and if $\delta$ is finite $T_{C}$ is suppressed.

Note that this closed form for $T_{C}$ was now obtained non-perturbativelly,
within the usual assumptions in the weak coupling limit of
$T_{C}<<\omega_{C}$, under the requirement of
$q({\mathbb{k}}_{F})>>\omega_{C}$, which is realistic for many materials of
interest.

Within this non-perturbative formalism one can also find a closed form for
$T_{C}$ for the single band scenario. Starting from
$H_{0}=\epsilon({\mathbb{k}})\sigma_{0}+\textbf{s}({\mathbb{k}})\cdot\bm{\sigma}$,
where $\textbf{s}({\mathbb{k}})$ is a three-dimensional vector associated with
an external symmetry breaking field, we find $\langle
Tr[|F_{A}({\mathbb{k}})|^{2}]+Tr[|F_{C}({\mathbb{k}})|^{2}]\rangle_{FS}=8$ for
any superconducting state. Performing a calculation in the same spirit as in
28; 29, the gap equation leads to:

$\displaystyle 1$ $\displaystyle=$
$\displaystyle-2vN(0)ln\left(\frac{4e^{\gamma}}{\pi}\frac{\omega_{C}}{2k_{B}T_{C}}\right)$
$\displaystyle-$ $\displaystyle\frac{v}{8}N(0)\langle
2f(\rho_{\mathbb{k}})Tr[|F_{C}({\mathbb{k}})|^{2}]\rangle_{FS},$

where $\rho_{\mathbb{k}}=|\mathbb{s}_{\mathbb{k}}|/\pi T$ and
$f(x)=2\text{Re}\sum_{n}[(2n+1+ix)^{-1}-(2n+1)^{-1}]$. The maximum critical
temperature is achieved when $|\mathbb{s}_{\mathbb{k}}|\rightarrow 0$, such
that:

$\displaystyle ln\left(\frac{T_{C}}{T_{C}^{Max}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{16}\langle
2f(\rho_{\mathbb{k}})Tr[|F_{C}({\mathbb{k}})|^{2}]\rangle_{FS}.$ (14)

Expanding in $\rho_{\mathbb{k}}$ we find:

$\displaystyle T_{C}$ $\displaystyle\approx$ $\displaystyle
T_{C}^{Max}\left(1-\frac{7\zeta(3)}{64\pi^{2}}\frac{\langle\rho_{\mathbb{k}}^{2}Tr[|F_{C}({\mathbb{k}})|^{2}]\rangle_{FS}}{(T_{C}^{Max})^{2}}\right),$
(15)

which is the same as Eq. 27 from our previous work 3, up to a change in the
normalization of $F_{C}({\mathbb{k}})$.

Now we apply the analysis of the two superconducting fitness parameters to
several superconducting materials in order to show the power of this concept
and what kind of robust statements one can make concerning the stability of
different superconducting states, or how to better engineer the normal state
in order to support the most interesting or exotic order parameter.

_CePt 3Si_: is a heavy fermion superconductor without inversion symmetry 30.
It has a $T_{C}\approx 0.75K$ and an upper critical field $H_{c2}\approx 5T$,
much larger than the Pauli limit, naively suggesting it is a triplet
superconductor 30. Simply modelling this system as a single band
superconductor29; 31, the absence of inversion symmetry leads to the presence
of a Rashba-type spin-orbit coupling (SOC) of the form
$H_{SOC}=\lambda\textbf{g}_{\mathbb{k}}\cdot\bm{\sigma}$, where $\lambda$ is
the strength of the SOC coupling and
$\textbf{g}_{\mathbb{k}}=(-k_{y},k_{x},0)$. The normalized triplet order
parameters in accordance with the $C_{4v}$ point group and the respective
averages of $Tr[|F_{C}({\mathbb{k}})|^{2}]$ and
$Tr[|F_{A}({\mathbb{k}})|^{2}]$ are summarized in Table I in the supplemental
material (SM). Note that all the spin triplet states with $A_{1}$ symmetry are
strongly suppressed as soon as SOC is turned on, given the large value of
$Tr[|F_{C}({\mathbb{k}})|^{2}]$ and the small (or zero 15) value of
$Tr[|F_{A}({\mathbb{k}})|^{2}]$. Interestingly enough, the last state within
this family has a non-zero $Tr[|F_{A}({\mathbb{k}})|^{2}]$, which always
guarantees the presence of a weak coupling instability in presence of SOC.
These aspects can be related to Fig. 1 in Tanaka et al.31, where one can see
all $A_{1}$ states are strongly suppressed in presence of SOC, with the last
state displaying a small tail for large $\lambda$. The state with $A_{2}$
symmetry is completely robust in presence of SOC since
$Tr[|F_{C}({\mathbb{k}})|^{2}]=0$. For the states with $B_{1}$, $B_{2}$ and
$E$ symmetries, $Tr[|F_{A}({\mathbb{k}})|^{2}]=Tr[|F_{C}({\mathbb{k}})|^{2}]$,
which tells us that the effect of SOC is two-fold: it is partially detrimental
(by a finite $Tr[|F_{C}({\mathbb{k}})|^{2}]$), but also supports the presence
of the weak coupling instability (by a finite
$Tr[|F_{A}({\mathbb{k}})|^{2}]$), therefore these states are suppressed with
increasing $\lambda$, but much more slowly than those in the $A_{1}$ family,
as can be also observed in Tanaka et al.31. In conclusion, it is easily
inferred from the superconducting fitness parameters that all the order
parameters are suppressed in presence of SOC, with the exception of $A_{2}$
channel.

_KFe 2As2_: is a strongly hole doped iron-based superconductor with
$T_{C}\approx 3K$ 32. Recently it was proposed that s-wave superconductivity
in the spin triplet $A_{2g}$ channel is realized once the Hund’s coupling is
larger than inter-orbital Hubbard repulsion in presence of SOC 33. A minimal
model with 2 orbitals in the basis
$\Psi_{\mathbb{k}}^{\dagger}=(d_{yz\uparrow}^{\dagger},d_{yz\downarrow}^{\dagger},d_{xz\uparrow}^{\dagger},d_{xz\downarrow}^{\dagger})$,
can be written as
$H_{0}=a({\mathbb{k}})\tau_{0}\otimes\sigma_{0}+b({\mathbb{k}})\tau_{3}\otimes\sigma_{0}+c({\mathbb{k}})\tau_{1}\otimes\sigma_{0}$,
where $a({\mathbb{k}})={\mathbb{k}}^{2}/2m-\mu$ and
$b({\mathbb{k}})=bk_{x}k_{y}$ characterize intra-orbital hopping, while
$c({\mathbb{k}})=c(k_{x}^{2}-k_{y}^{2})$ characterizes inter-orbital hopping.
For this model, even though the interaction in the $A_{2g}$ channel can be
attractive, the pairing is purely inter-band (with order parameter
$\Delta_{A_{2g}}({\mathbb{k}})=d_{A_{2g}}({\mathbb{k}})\tau_{2}\otimes\sigma_{1}$),
therefore the superconducting instability is not robust since the pairing
susceptibility is not logarithmically divergent 33. From the perspective of
the superconducting fitness, this is captured by the fact that
$F_{A}({\mathbb{k}})=0$ (the fitness parameters for all superconducting states
are presented in Table II in the SM). Once SOC is turned on,
$H_{SOC}=\lambda\tau_{2}\otimes\sigma_{3}$, the $A_{1g}$ and $A_{2g}$ channels
mix, and the authors in 33 argue that a weak-coupling instability develops
because now the component with $A_{1g}$ symmetry (with order parameter
$\Delta_{A_{1g}}({\mathbb{k}})=d_{A_{1g}}({\mathbb{k}})\tau_{0}\otimes\sigma_{2}$)
has intra-band character. From the superconducting fitness it actually becomes
evident that in fact pairing in the $A_{2g}$ channel itself develops a weak-
coupling instability in presence of SOC since $F_{A}({\mathbb{k}})\sim
8\lambda\tau_{0}\otimes\sigma_{2}$ is finite, even before considering the
admixture with a $A_{1g}$ component. As a consequence, the prefactor in the
term carrying the Cooper logarithm is proportional to $\lambda^{2}$, in
accordance with previous results 33. So the ultimate reason why there is a
weak-coupling instability in this scenario is SOC leading to a mixture of
orbitals such that intra-band pairing in the $A_{2g}$ channel is now possible,
not by the admixture of a $A_{1g}$ component to the order parameter. This is
corroborated by the fact that this conclusion is unaffected by the presence of
a strong repulsive U in the $A_{1g}$ channel 33. We would like to note here
that SOC has a different role than in the previous example: now SOC is
important for the stabilization of a weak coupling instability.

Figure 1: Fermi surfaces from the effective model of KFe2As2. The insets
display the form factors with $B_{1g}$ (left bottom) and $B_{2g}$ (top right)
symmetries.

The detailed analysis of the superconducting fitness parameters allows us to
infer that SOC is generally detrimental to superconductors with d-wave
character, while it supports states with s-wave symmetry. We can also discuss
the competition between the two order parameters with d-wave symmetry. The SC
instability in the $B_{1g}\sim k_{x}^{2}-k_{y}^{2}$ and $B_{2g}\sim
k_{x}k_{y}$ channels is guaranteed by the presence of the $b({\mathbb{k}})$
and $c({\mathbb{k}})$ term, respectively. Note that the product of the form
factors determining $Tr[|F_{A}({\mathbb{k}})|^{2}]$ is the same for both
channels. Ultimately, assuming the same pairing interaction in both channels,
the magnitude of the constants $b$ and $c$ in the Hamiltonian is what
determines which channel is favoured: in this case enhancing inter-orbital
hopping favours the $B_{2g}$ channel. The parameter
$Tr[|F_{C}({\mathbb{k}})|^{2}]$ is also useful in determining the presence of
detrimental effects: since it is proportional to $1/q({\mathbb{k}}_{F})$, it
is minimized when the Fermi surfaces are far apart in the Brillouin zone. As a
consequence, the position of the nodes would tend to be in the direction where
the two Fermi surfaces are the closest, in this case favouring the $B_{1g}$
channel, as schematically shown in Fig. 1. These arguments suggest interesting
directions for engineering of superconducting states based in the application
of strain.

_Cu xBi2Se3_: is a candidate topological superconductor 34. Zhang et al. 35
proposed a two-orbital effective model based on two effective $p_{z}$-like
orbitals which are a superposition of $p_{z}$ orbitals from Bi and Se. They
label the orbitals as $P_{1z+}$ and $P_{2z-}$, with the $\pm$ sign indicating
the parity of the effective orbital. In the basis
$\Phi^{\dagger}=(c_{1\uparrow}^{\dagger},c_{1\downarrow}^{\dagger},c_{2\uparrow}^{\dagger},c_{2\downarrow}^{\dagger})$,
the Hamiltonian can be parametrized as in Eq. 1, with non-zero
$h_{ab}({\mathbb{k}})$ only for $(a,b)=(0,0),(2,0),(3,0),(1,1),(1,2),(1,3)$ in
presence of TRS and parity. The explicit general form and character of these
terms are summarized in Table III in the SM. The order parameter can also be
parametrized as
$\Delta=\sum_{a,b}d_{ab}({\mathbb{k}})\tau_{a}\otimes\sigma_{b}$, and for the
$D_{3d}$ point group symmetry there are four irreducible representations,
displayed in Table IV in the SM. For simplicity we focus on
${\mathbb{k}}$-independent order parameters. One can now evaluate the
superconducting fitness parameters for each term in the Hamiltonian
separately. Table V in the SM summarizes which fitness parameters are zero or
finite for each of the possible order parameters. Since all the terms in the
Hamiltonian, when evaluated at the Fermi surface, have approximately the same
magnitude (see Table III), with the exception of the $(1,3)$ term
corresponding to trigonal warping, we can assign a score to each term as
follows: a $-1$ score is given to each element in the Hamiltonian which
contributes to a non-zero $F_{C}({\mathbb{k}})$, while a $+1$ score is
assigned to each element in the Hamiltonian contributing to a finite
$F_{A}({\mathbb{k}})$. These results are summarized in Table IV, and allows us
to make the following observations: i) The dominant term in the Hamiltonian
$(3,0)$ supports even and suppresses odd order parameters; ii) The terms
$(1,1)$ and $(1,2)$, associated with SOC, support only the order parameters
with $A_{1u}$ and $A_{2u}$ symmetry amongst the odd order parameters and the
first order parameter with $A_{1g}$ symmetry; iii) The $(2,0)$ term supports
order parameters in the $A_{1g}$ (first), $A_{1u}$ and $E_{u}$, while it
supresses order parameters in the $A_{1g}$ (second) and $A_{2u}$
representations; iv) The trigonal warping term $(1,3)$ favours order
parameters in the $A_{1g}$ (first), $A_{2u}$ and $E_{u}$ channels, while it
suppresses order parameters in the $A_{1g}$ (second) and $A_{1u}$ channels. Fu
and Berg 34 proposed a criterium for the realization of topological SC which
states: a time-reversal invariant, centro-symmetric superconductor is
topological if it has odd-parity pairing with a full superconducting gap and
if the Fermi surface encloses an odd number of time-reversal invariant momenta
in the Brillouin zone. From the perspective of the superconducting fitness we
can understand why the authors find that the most stable state with odd parity
is in the $A_{1u}$ symmetry channel, since this is the only odd order
parameter with only one sizeable detrimental term. We can now go further and
propose how this material should be engineered such that an odd
superconducting state is favoured. One direction is to make the $(3,0)$ term
smaller. This term is associated with the difference in intra-orbital hopping
for the two effective orbitals, and is dominated by in-plane hopping (see
parameters in the SM and Ref. (36)). The study of the dependence of the
magnitude of this parameter under different kinds of strain from DFT
calculations would be insightful. Other terms can also be engineered in order
to favour a specific odd parity order parameter, such as enhancement of the
SOC, associated with $(1,1)$ and $(2,1)$ terms for the development of an order
parameter with $A_{1u}$ or $A_{2u}$ symmetry. Interestingly enough, nematic
superconductivity 12; 37, emerging from the two-dimensional $E_{u}$
representation is favoured by the enhancement of trigonal warping.

In conclusion, we have introduced the concept of superconducting fitness and
the parameters $F_{A}({\mathbb{k}})$ and $F_{C}({\mathbb{k}})$ which quantify
the robustness of the superconducting instability and the presence of
detrimental effects, respectively. These two parameters provide a very handy
and useful computational tool to evaluate the effects of each symmetry allowed
term in the Hamiltonian and how it supports or suppresses different
superconducting order parameters. Ultimately, this scheme can be used as a
first guide to engineer complex materials in order to support the desired
exotic superconducting states.

We thank Philip Brydon, Mark Fischer, Dirk Manske, Mathias Scheurer, Jörg
Schmalian, Carsten Timm, and Jose Luis Lado Villanueva for useful discussions.
This work was partially supported by Dr. Max Rössler, the Walter Haefner
Foundation and the ETH Zurich Foundation (A.R.) and by the Swiss National
Science Foundation (M.S.). D.F.A. acknowledges the hospitality of the Pauli
Center of the ETH Zurich.

Basis | Order Parameter ($\textbf{d}({\mathbb{k}})$) | $Tr[|F_{C}({\mathbb{k}})|^{2}]$ | $Tr[|F_{A}({\mathbb{k}})|^{2}]$ | Suppression of $T_{c}$ | Relation to SOC ($\textbf{g}_{\mathbb{k}}$)
---|---|---|---|---|---
$A_{1}$ | $\sqrt{\frac{3}{2}}\frac{1}{k_{F}}(k_{x},k_{y},0)$ | $8$ | 0 | Strong | $\bot$
$\frac{1}{k_{F}}(k_{x},k_{y},k_{z})$ | $8$ | 0 | Strong | $\bot$
$\sqrt{\frac{7}{2}}\frac{1}{k_{F}^{3}}(k_{x}^{3},k_{y}^{3},0)$ | $7.6$ | $0.4$ | Strong + Tail | other
$A_{2}$ | $\sqrt{\frac{3}{2}}\frac{1}{k_{F}}(-k_{y},k_{x},0)$ | 0 | $8$ | No | $\parallel$
$B_{1}$ | $\sqrt{\frac{3}{2}}\frac{1}{k_{F}}(k_{x},-k_{y},0)$ | $4$ | $4$ | Slight | other
$B_{2}$ | $\sqrt{\frac{3}{2}}\frac{1}{k_{F}}(k_{y},k_{x},0)$ | $4$ | $4$ | Slight | other
$E$ | $\frac{\sqrt{3}}{k_{F}}(k_{z},0,0)$ | $4$ | $4$ | Slight | other
$\frac{\sqrt{3}}{k_{F}}(0,k_{z},0)$ | $4$ | $4$ | Slight | other
Table 1: Table summarizing the effects of SOC for the example of CePt3Si. The table enumerates the allowed triplet superconducting states with $C_{4v}$ point group symmetry parametrized as $\hat{\Delta}({\mathbb{k}})=\textbf{d}({\mathbb{k}})\cdot\bm{\sigma}(i\sigma_{2})$, the respective superconducting fitness parameters, the suppression $T_{C}$ and the relation between the $\textbf{d}_{\mathbb{k}}$ and $\textbf{g}_{\mathbb{k}}$ vectors in each case. Basis | Spin | Orbital | Parity | Matrix Form |  $Tr[|F_{C}({\mathbb{k}})|^{2}]$* |  $Tr[|F_{A}({\mathbb{k}})|^{2}]$*
---|---|---|---|---|---|---
$A_{1g}$ | Singlet | Intra | Even (s) | $\tau_{0}\otimes(i\sigma_{2})$ | $0$ | $|d_{A_{1g}}({\mathbb{k}})|^{2}(|b({\mathbb{k}})|^{2}+|c({\mathbb{k}})|^{2}+\lambda^{2})$
$B_{1g}$ | Singlet | Intra | Even (d) | $\tau_{3}\otimes(i\sigma_{2})$ | $|d_{B_{1g}}({\mathbb{k}})|^{2}(\lambda^{2}+|c({\bf k})|^{2})$ | $|d_{B_{1g}}({\mathbb{k}})|^{2}|b({\mathbb{k}})|^{2}$
$B_{2g}$ | Singlet | Inter | Even (d) | $\tau_{1}\otimes(i\sigma_{2})$ | $|d_{B_{2g}}({\mathbb{k}})|^{2}(\lambda^{2}+|b({\bf k})|^{2})$ | $|d_{B_{2g}}({\mathbb{k}})|^{2}|c({\mathbb{k}})|^{2}$
$A_{2g}$ | Triplet | Inter | Even (s) | $\tau_{2}\otimes\sigma_{1}$ |  $|d_{A_{2g}}({\mathbb{k}})|^{2}(|c({\mathbb{k}})|^{2}+|b({\bf k})|^{2}$) | $|d_{A_{2g}}({\mathbb{k}})|^{2}\lambda^{2}$
Table 2: Table summarizing the superconducting fitness parameters of different superconducting states for the example of KFe2As2. * Values should be normalized by $(|b({\mathbb{k}})|^{2}+|c({\mathbb{k}})|^{2}+\lambda^{2})/64$, where $b({\mathbb{k}})=bk_{x}k_{y}$ and $c({\mathbb{k}})=c(k_{x}^{2}-k_{y}^{2})$, characterizing intra- and inter-orbital hopping, repectively. The momentum dependency of the order parameters are the following: $d_{A_{1g}}\sim\text{cte}$, $d_{A_{2g}}\sim\text{cte}$, $d_{B_{1g}}\sim(k_{x}^{2}-k_{y}^{2})$ and $d_{B_{2g}}\sim k_{x}k_{y}$. In order to reproduce the Fermi surface in Fig. 1 in the main text, the following parameters were used: $m=1$, $b=0.63$, $c=0.35$, $\mu=0.25$ 33. $(a,b)$ | General form | Magnitude (eV) | Character
---|---|---|---
$(0,0)$ | $a_{0}+a_{p}(k_{x}^{2}+k_{y}^{2})+a_{z}k_{z}^{2}$ | $\approx 0.3$ | Intra-orbital hopping
$(2,0)$ | $b_{1}k_{z}+b_{3}k_{z}^{3}+b_{T}k_{y}(3k_{x}^{2}-k_{y}^{2})$ | $\approx 0.2$ | Inter-orbital hopping
$(3,0)$ | $c_{0}+c_{p}(k_{x}^{2}+k_{y}^{2})+c_{z}k_{z}^{2}$ | $\approx 0.45$ | Intra-orbital hopping
$(1,1)$ | $d_{1}k_{x}+d_{2}k_{x}(k_{x}^{2}+k_{y}^{2})$ | $\approx 0.3$ | SOC
$(1,2)$ | $-d_{1}k_{y}-d_{2}k_{y}(k_{x}^{2}+k_{y}^{2})$ | $\approx 0.3$ | SOC
$(1,3)$ | $e_{T}k_{x}(k_{x}^{2}-3k_{y}^{2})$ | $\approx 0.05$ | Trigonal warping
Table 3: Symmetry allowed terms for the Hamiltonian $H_{0}({\mathbb{k}})=\sum_{a,b}h_{ab}({\mathbb{k}})\tau_{a}\otimes\sigma_{b},$ of CuxBi2Se3 with their momentum dependence up to $O({\mathbb{k}}^{2})$, the respective order of magnitude at the Fermi surface and the character of the term. For the magnitudes above we used the maximum value of the parameters cited in Liu et al.36 at the Fermi surface with $k_{F}\approx 0.1\AA$ (this can be obtained from the ARPES experiment 38 or by the value of the electron density upon doping39 $\sim 10^{20}cm^{-3}$). Note that all but the last trigonal warping term are of the same order of magnitude. Irrep | Spin | Orbital | Parity | Matrix Form | $\langle|F_{C}({\mathbb{k}})|^{2}\rangle_{FS}$ | $\langle|F_{A}({\mathbb{k}})|^{2}\rangle_{FS}$
---|---|---|---|---|---|---
$A_{1g}$ | Singlet | Intra | Even | $\tau_{0}\otimes(i\sigma_{2})$ | 0 | +4
$\tau_{3}\otimes(i\sigma_{2})$ | -3 | +1
$A_{1u}$ | Triplet | Inter | Odd | $(i\tau_{2})\otimes\sigma_{1}$ | -1 | +3
$A_{2u}$ | Singlet | Inter | Odd | $\tau_{1}\otimes(i\sigma_{2})$ | -2 | +2
$E_{u}$ | Triplet | Inter | Odd | $(i\tau_{2})\otimes\sigma_{0}$ | -2 | +2
$(i\tau_{2})\otimes\sigma_{3}$ | -2 | +2
Table 4: Table summarizing the score for superconducting fitness parameters of
different superconducting states for the example of CuxBi2Se3. The previous to
last column gives a $-1$ score to each element in the Hamiltonian which
contributes to a non-zero $F_{C}({\mathbb{k}})$, while the last column gives a
$+1$ score to each element in the Hamiltonian contributing to a finite
$F_{A}({\mathbb{k}})$ (see table below). The table does not count the
contribution from the trigonal warping term (1,3) since this is one order of
magnitude smaller than the other terms.

$A_{1g},(c,d)=(0,2)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $\neq 0$ | $=0$
$(3,0)$ | $\neq 0$ | $=0$
$(1,1)$ | $\neq 0$ | $=0$
$(1,2)$ | $\neq 0$ | $=0$
$(1,3)$ | $\neq 0$ | $=0$

$A_{1g},(c,d)=(3,2)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $=0$ | $\neq 0$
$(3,0)$ | $\neq 0$ | $=0$
$(1,1)$ | $=0$ | $\neq 0$
$(1,2)$ | $=0$ | $\neq 0$
$(1,3)$ | $=0$ | $\neq 0$

$A_{1u},(c,d)=(2,1)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $\neq 0$ | $=0$
$(3,0)$ | $=0$ | $\neq 0$
$(1,1)$ | $\neq 0$ | $=0$
$(1,2)$ | $\neq 0$ | $=0$
$(1,3)$ | $=0$ | $\neq 0$

$A_{2u},(c,d)=(1,2)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $=0$ | $\neq 0$
$(3,0)$ | $=0$ | $\neq 0$
$(1,1)$ | $\neq 0$ | $=0$
$(1,2)$ | $\neq 0$ | $=0$
$(1,3)$ | $\neq 0$ | $=0$

$E_{u},(c,d)=(2,0)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $\neq 0$ | $=0$
$(3,0)$ | $=0$ | $\neq 0$
$(1,1)$ | $\neq 0$ | $=0$
$(1,2)$ | $=0$ | $\neq 0$
$(1,3)$ | $\neq 0$ | $=0$

$E_{u},(c,d)=(2,3)$
---
$(a,b)$ | $F_{A}({\mathbb{k}})$ | $F_{C}({\mathbb{k}})$
$(2,0)$ | $\neq 0$ | $=0$
$(3,0)$ | $=0$ | $\neq 0$
$(1,1)$ | $=0$ | $\neq 0$
$(1,2)$ | $\neq 0$ | $=0$
$(1,3)$ | $\neq 0$ | $=0$

Table 5: Table summarizing when the superconducting fitness parameters are
zero or finite for all the symmetry allowed terms in the Hamiltonian of
CuxBi2Se3, for all possible superconducting order parameters. These results
were used to generate the scores in Table IV above. In blue we highlight the
terms $(a,b)$ in the Hamiltonian which guarantee the weak coupling instability
without introducing any detrimental effect, while in red we highlight the
terms which only lead to a suppression of the critical temperature for the
respective order parameter.

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