# Singularity resolution in gravitational collapse

Karim Mosani<EMAIL_ADDRESS>BITS Pilani K.K. Birla Goa Campus,
Sancoale, Goa 403726, India  Dipanjan Dey<EMAIL_ADDRESS>International Center for Cosmology, Charusat University, Anand 388421,
Gujarat, India  Kaushik Bhattacharya<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Kanpur, Kanpur 208016, India  Pankaj
S. Joshi<EMAIL_ADDRESS>International Center for Cosmology,
Charusat University, Anand 388421, Gujarat, India

###### Abstract

We investigate the unhindered gravitational collapse of a homogeneous scalar
field with nonzero potential, a two-dimensional analog of the Mexican hat-
shaped Higgs field potential. We prove that the density dependence on the
scale factor cannot be expressed as an algebraic function in such a scenario.
For a certain transcendental expression of the density of such field as a
function of scale factor, we then show that the collapse evolves to a
singularity at an infinite comoving time, which is equivalent to saying that
the singularity is avoided altogether. An ultra high density region (UHDR) of
the order of Planck length can, however, be reached in a finite comoving time.
The absence of the formation of trapped surfaces makes this UHDR globally
visible.

key words: Gravitational Collapse, Naked Singularity, Scalar field

##  I Introduction

Hawking and Penrose depicted the formation of singularities in the
gravitational collapse and cosmology in what is known as the singularity
theorems Hawking_73 ; Wald_84 . However, it is widely believed that no past
inextendible nonspacelike geodesic can exist between the singularity and any
point on the spacetime manifold. In other words, no nonspacelike geodesics
could have a positive tangent at the singularity. This statement is known as
the strong cosmic censorship hypothesis Penrose_69 ; Joshi_07 . One motivation
tempting us towards the strong cosmic censorship hypothesis is our desire
always to have a globally hyperbolic spacetime metric Joshi_07 .

Global hyperbolicity is the most potent form of the causality condition
Wald_84 . Even if the spacetime is strongly causal, which ensures no formation
of closed nonspacelike curves, such spacetime metric should retain the
property of no violation of causality if there is a small perturbation in the
spacetime metric. Thus, there should be stability in the spacetime metric in
that all “nearby” spacetime metrics should also have a similar property as far
as the causality condition is concerned. However, different topologies can be
defined on the set of all Lorentz metrics, each giving a different meaning to
the word “nearby”. This drawback can be overcome by considering the spacetime
to be globally hyperbolic, an alternative, and the strongest description of
the causality condition. Globally hyperbolic spacetime harbors a unique
topology that is homeomorphic to $S\times\mathbb{R}$, where $S$ is a Cauchy
surface admitted by the spacetime manifold Geroch_70  (any two Cauchy surfaces
are homeomorphic to each other). A global hyperbolic spacetime is stably
causal Hawking_73 . One can show that if the strong cosmic censorship is not
valid, then the spacetime metric is not globally hyperbolic Joshi_82 .
Nevertheless, one can also show that a sufficiently inhomogeneous spherically
symmetric dust cloud collapse can end up in a singularity such that null
geodesics can escape from the singularity without getting trapped by the
trapped surfaces Joshi_93 ; Mosani_20(2) . Such singularities are also stable
under small perturbation in a certain subset of the entire initial data
leading to the collapse Deshingkar_98 ; Mena_2000 ; Mosani_20 . These may act
as a counterexample to the strong cosmic censorship hypothesis. Apart from the
scenario mentioned above, examples of gravitational collapse of various matter
fields have been shown to give rise to a naked singularity Christodoulou_91 ;
Magli_97 ; Magli_98 ; Harada_99 ; Harada_02 ; Goswami_04 ; Goswami_04b ;
Giambo_03 ; Giambo_06 . The intriguing question is whether cosmic censorship
is respected in the gravitational collapse of fundamental matter fields
deduced from a suitable Lagrangian.

In the case of a massless scalar field, it was shown by Christodoulou that the
cardinality of the linearly independent elements in the set of initial data
giving rise to a naked singularity as an end state is strictly less than that
of the entire collection of initial data Christodoulou_94 ; Christodoulou_99 .
In other words, in the case of scalar field collapse with zero $\phi^{2}$ term
in the scalar field Lagrangian, the set of initial data giving rise to naked
singularity has positive codimension in the entire set of initial data (the
whole set includes those initial data giving rise to a black hole, and those
giving rise to a naked singularity). This outcome concludes that a massless
scalar field collapses to a naked singularity that is non-generic (by the
generic outcome of the gravitational collapse, we mean that the set of the
initial data giving rise to the outcome has a nonzero measure in the entire
initial data set Joshi_07 ). Genericity aspects of the naked singularity
formed due to massive scalar field collapse are yet to be studied.

In Goswami_04b , a massive homogeneous scalar field having a certain potential
$V(\phi)\propto e^{-\phi}$, is shown to form a naked singularity at the end of
its collapse. Here, we are interested in a more realistic potential of the
scalar field, whose Lagrangian has a $\mathbb{Z}_{2}$ symmetry. Symmetries in
scalar field theories are interesting as we know the Higgs field potential
Higgs_64  also has symmetries. In particle physics, the Higgs field plays a
pivotal role in determining the properties of the matter content of the known
universe. In our case, we work with a much-simplified model consisting of only
one real scalar field whose potential has a particular symmetry. Such
potential is motivated by the theory of phase transitions by Landau Melo_17 .
At temperatures above a certain cut-off $T_{c}$, the scalar field has zero
average value in its lowest energy state, i.e., the vacuum. At temperatures
below $T_{c}$, such field has two nonzero average values in vacuum (the value
of these two states are the same in magnitude but differ in polarity). As
there appears multiple vacua in the theory below a certain temperature, the
system has to choose any one of the vacua, thereby breaking the symmetry of
the system. All the perturbations of the field now have to be done with
respect to a particular vacuum, and consequently, the overall symmetry is
broken by the vacuum. This mode of symmetry breaking is known as spontaneous
symmetry breaking. In particle physics, a considerable amount of vacuum energy
in the universe is contributed by the Higgs field due to its nonzero average
lowest energy state Rugh_02 . In this paper, we will discuss the dynamics of
gravitational collapse in the presence of a scalar field which shows
spontaneous symmetry breaking.

In this paper, we show that for a certain homogeneous density dynamics
(gravitational collapse) of our toy model scalar field with a potential which
has $\mathbb{Z}_{2}$ symmetry, the cloud collapses eternally, thereby the
singularity formation is avoided. However, a very UHDR of the size of the
order of Planck length is obtained in a finite comoving time. Trapped surfaces
do not form, causing the exposure of this strong gravity region to the outside
observer. The paper is arranged as follows: In section II, we derive basic
dynamical equations of the massive scalar field using the Einsteins field
equations, which will be used in further investigations. Section III shows
that for the scalar field mentioned above, the density can not be an algebraic
function of the scale factor but a transcendental function. For a particular
transcendental function as a density configuration, satisfying the regularity
requirements, we discuss the visibility aspects of the UHDR, which may be
governed by quantum gravity, and see if $\mathbb{Z}_{2}$ symmetry in the
scalar field potential is obeyed by the causal property of the UHDR. We end
the paper by concluding remarks. We use the geometrized units $c=8\pi G=1$.

##  II Einstein’s field equations and collapse dynamics

Consider a homogeneous gravitational collapse of the perfect fluid scalar
field $\phi=\phi(t)$ having the potential $V(\phi)$. The components of the
stress-energy tensor are given by

$T^{\mu}_{\nu}=\textrm{diag}\left(\rho,p,p,p\right).$ (1)

The spacetime geometry is governed by the Friedmann–Lemaître–Robertson–Walker
(FLRW) metric

$ds^{2}=-dt^{2}+a^{2}dr^{2}+R^{2}d\Omega^{2}\,,$ (2)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. Here $a=a(t)$ is the
scale factor such that $a(0)=1$ and $a(t_{s})=0$, where $t_{s}$ is the time of
formation of the singularity. $R=R(t,r)$ is the physical radius of the
collapsing cloud and can be written as

$R(t,r)=ra(t).$ (3)

The Lagrangian of the scalar field is given by

$\mathcal{L}_{\phi}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi),$
(4)

The energy-momentum tensor is then

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{\phi}\right)}{\delta
g^{\mu\nu}}.$ (5)

The density and the isotropic pressure are subsequently expressed in terms of
the time derivative of the scalar field and its potential as

$\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\frac{3\dot{a}^{2}}{a^{2}},$ (6)

and

$p=\frac{1}{2}\dot{\phi}^{2}-V(\phi)=-\frac{2\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}.$
(7)

The overhead dot denotes the time derivative of the functions. The Klien-
Gordan equation

$\ddot{\phi}+\frac{3\dot{a}}{a}\dot{\phi}+V_{,\phi}=0,$ (8)

can be obtained from the Einstein’s field Eq.(6) and Eq.(7) (or from Eq.(6)
along with the Bianchi identity) if $\dot{\phi}$ does not vanish identically.
Hence the Klein-Gordan equation should not be seen as an independent equation
constraining the choice of free functions. From Eq.(6) and Eq.(7), and from
using the chain rule $\dot{\phi}=\phi_{,a}\dot{a}$, we get

$\rho+p=\phi_{,a}^{2}\dot{a}^{2}.$ (9)

Eq.(6) can be rewritten to obtain the dynamics of the collapse as

$\dot{a}=-\sqrt{\frac{\rho(a)}{3}}a,$ (10)

differentiating which, we obtain

$\ddot{a}=\frac{1}{3}a\left(\frac{a\rho_{,a}}{2}+\rho\right).$ (11)

Using Eq.(10) in Eq.(9), we get

$\rho\left(1-\frac{\phi_{,a}^{2}a^{2}}{3}\right)+p=0.$ (12)

From Eq.(6) and Eq.(10), we get

$p=\rho-2V.$ (13)

Using Eq.(12) and Eq.(13), we get

$\rho=\frac{V(\phi)}{1-\frac{\phi_{,a}^{2}a^{2}}{6}}.$ (14)

Using Eq.(7), Eq.(10), and Eq.(11) in Eq.(9) and rearranging, we obtain,

$\frac{\rho_{,a}}{\rho}=-\phi_{,a}^{2}a.$ (15)

Differentiating Eq.(14) with respect to $a$ and substituting in Eq.(15), we
obtain a second order non-linear differential equation

$\frac{V_{,\phi}\phi_{,a}}{V}+\frac{a}{3}\frac{\phi_{,a}\left(a\phi_{,aa}+\phi_{,a}\right)}{1-\frac{\phi_{,a}^{2}a^{2}}{6}}+a\phi_{,a}^{2}=0.$
(16)

Now, in case $\phi_{,a}\neq 0$, we can get the reduced form of the above
differential equation as

$\frac{V_{,\phi}}{V}+\frac{a}{3}\frac{\left(a\phi_{,aa}+\phi_{,a}\right)}{\left(1-\frac{\phi_{,a}^{2}a^{2}}{6}\right)}+a\phi_{,a}=0.$
(17)

For a given potential $V=V(\phi)$, one can solve the differential Eq.(17) by
choosing two suitable initial conditions to get $\phi=\phi(a)$. In the next
section, we derive the exact solution of the collapsing toy model scalar field
with potential having $\mathbb{Z}_{2}$ symmetry.

Figure 1: The scalar field $\phi$ and the potential $V(\phi)$ are represented
by the horizontal and vertical axis respectively. $q>0$ in $(a)$ and $q<0$ in
$(b)$. Here, $\gamma=\sqrt{\frac{-q}{4\lambda}}$ and
$\delta=\sqrt{\frac{-q}{2\lambda}}$.

##  III Exact solution: Eternally collapsing cloud

The toy model scalar field which we consider, is described by the potential
(Fig.(1))

$V(\phi)=\frac{1}{2}q\phi^{2}+\lambda\phi^{4},$ (18)

At high temperature $T$, above a certain cut-off $T_{c}$, $q>0$. The potential
in such case is a parabola (see Fig.(1.a)), with the average of its lowest
energy state at

$\phi_{v}=0.$ (19)

This is the vacuum. As the temperature cools down below $T_{c}$, $q<0$, and
the potential is no more a parabola (see Fig.(1.b)). Now there are two states
of vacuum, i.e. the lowest energy states, represented by

$\phi_{v}=\pm\sqrt{-\frac{q}{4\lambda}}.$ (20)

Both the vacuum states are allowed to exist with equal probability. One can
see that at $T>T_{c}$, the lowest energy state is symmetric with respect to
$\phi\longleftrightarrow-\phi$ reflection. However, at $T<T_{c}$, choosing any
one of the ground states as the lowest energy state, we see that the symmetry
$\phi\longleftrightarrow-\phi$ is not respected any more.

We can get a differential equation by using Eq.(18), and its derivative with
respect to $\phi$, and substituting in the differential Eq.(17). Then, we can,
in principle, get the solution, an expression of $\phi$ in terms of $a$, which
corresponds to the potential in Eq.(18). However, the problem with this
approach is that the differential Eq.(17) is non-linear and is not easy to
solve analytically. We, therefore, don’t proceed with this approach. Instead,
we carefully choose the density configuration as a function of the scale
factor such that the underlying potential turns out to be of the form in
Eq.(18). This can be done due to one degree of freedom available to us.

However, choosing a particular density configuration so as to ensure that the
scalar field follows the “Higgs like” potential is a hit and trial method. It
is obviously not practical to try each and every possible kind of density
configuration and see if the corresponding underlying potential of the scalar
field is the one that we desire. The following theorem, however, helps us to
choose a suitable density profile that can lead to a desirable potential.

Theorem: In a collapsing homogeneous scalar field, if $\rho(a)$ is an
algebraic function, then either $V(\phi)$ is a transcendental function, or

$\lim_{a\to 0}V(\phi(a))=\frac{1}{\phi^{\alpha}};\hskip 28.45274pt\alpha>0.$
(21)

(If (21) is true, then $V(\phi)$ can be algebraic or transcendental.)

Proof: If we consider that the density configuration of the collapsing cloud
can be expressed as an algebraic function in $a$, then we can say that the
density close to the time of formation of the singularity is of the order as
follows:

$\lim_{a\to 0}\rho\sim O\left(\frac{1}{a^{n}}\right).$ (22)

From Eq.(10), we then obtain

$\lim_{a\to 0}\dot{a}\sim O\left(a^{1-\frac{n}{2}}\right),$ (23)

differentiating which, we get

$\lim_{a\to 0}\ddot{a}\sim O\left(a^{1-n}\right).$ (24)

Substituting Eq.(23) and Eq.(24) in Eq.(7), the behavior of the dynamics of
pressure close to the time of formation of the singularity is obtained as

$\lim_{a\to 0}p\sim O\left(\frac{1}{a^{n}}\right).$ (25)

The above equation along with Eq.(9) gives

$\lim_{a\to 0}\dot{\phi}^{2}\leq O\left(\frac{1}{a^{n}}\right).$ (26)

Now, since we assume the density to have an algebraic expression in terms of
$a$, we imply using Eq.(25) and Eq.(9) that $\dot{\phi}^{2}$ has an algebraic
expression in terms of $a$. We therefore have from Eq.(26) that

$\lim_{a\to 0}\dot{\phi}^{2}\sim O\left(\frac{1}{a^{m}}\right);\hskip
28.45274ptm\leq n.$ (27)

Using the chain rule $\dot{\phi}^{2}=\phi_{a}^{2}\dot{a}^{2}$ and Eq.(23), we
can then conclude

$\lim_{a\to 0}\phi_{,a}\sim O\left(\frac{1}{a^{1+\frac{m-n}{2}}}\right).$ (28)

Using (28) and (22) in Eq.(14), we obtain the behavior of the potential close
to the singularity as

$\lim_{a\to 0}V\sim O\left(\frac{1}{a^{n}}\right).$ (29)

Case I: If $m=n$, then,

$\lim_{a\to 0}\phi_{,a}\sim O\left(\frac{1}{a}\right).$ (30)

Integrating the expression (30), we obtain

$\lim_{a\to 0}\phi\sim O\left(\log a\right).$ (31)

Hence, for (29) to hold true, $V$ should be a transcendental function of
$\phi$.

Case II: If $m<n$, then integrating the expression (28), we obtain

$\lim_{a\to 0}\phi\sim O\left(a^{\frac{n-m}{2}}\right).$ (32)

Hence, for (29) to hold true, (21) should be satisfied where

$\alpha=\frac{2n}{n-m}.$ (33)

$\hskip 239.00298pt\Box$

We now utilize the result of this theorem for our special case where the
potential is of the form (18).

Corollary: The density configuration of the collapsing homogeneous scalar
field having the potential (18) is not an algebraic function of $a$.

Proof: For such potential, substituting either (31) or (32) in (18) does not
satisfy $(\ref{Vorder})$ which holds true for algebraic $\rho(a)$. The
corollary is valid for both $q>0$ and $q<0$ in Eq.(18).

Using the above corollary, let us set the density configuration of the
homogeneous scalar field as the following transcendental function in $a$:

$\rho(a)=64\lambda(k-\log a)^{2},\hskip 28.45274ptk>0.$ (34)

For such density profile, using Eqs.(10, 11, 7) in Eq.(9), we get

$\phi_{,a}^{2}=\frac{2}{a^{2}\left(k-\log a\right)}.$ (35)

Taking the square root gives us

$\phi_{,a}=\pm\frac{\sqrt{2}}{a\sqrt{k-\log a}}.$ (36)

Integrating, the above equation, we get

$\phi(a)=\mp 2\sqrt{2}\sqrt{k-\log a}.$ (37)

Using Eq.(34) and Eq.(35) in Eq.(14), we then obtain

$V(\phi)=-\frac{8}{3}\lambda\phi^{2}+\lambda\phi^{4},$ (38)

which resembles the potential depicted in Eq.(18) with

$q=\frac{16}{3}\lambda.$ (39)

If we choose $\phi_{,a}>0$, then $\phi<0$, and vice versa. We can see that as
the cloud evolves from $a=1$ to $a=0$ (singularity), the scalar field evolves
from $-2\sqrt{2}\sqrt{k}$ to $-\infty$ if we choose $\phi_{,a}>0$. It evolves
from $2\sqrt{2}\sqrt{k}$ to $\infty$ if we choose $\phi_{,a}<0$.

As a special case example, for $k=k_{v}$ given by

$k_{v}=\frac{1}{6},$

the collapse initiates from the vacuum state

$\left(\phi_{v},V_{v}\right)=\left(\pm\frac{2}{\sqrt{3}},-\frac{23\lambda}{9}\right),$
(40)

Eq.(20). It ends at the singularity, for which the scalar field and the
potential blows up as

$\lim_{a\to 0}\left(\phi,V\right)=\left(\pm\infty,\infty\right).$ (41)

We, therefore, have a class of scalar field collapse with the potential
Eq.(38), which starts from some finite nonzero value of $\phi$ and blows up in
the end, maintaining the polarity of $\phi$ throughout the collapse. It is
worth mentioning that Eq.(37) and Eq.(38) satisfies the differential Eq.(17).

To understand the time evolution of the cloud, we solve differential Eq.(10)
by substituting Eq.(34) and setting the constant of integration such that
$a(0)=1$. We then obtain

$a(t)=\exp\left(k\left(1-\exp\left(\frac{8\sqrt{\lambda}t}{\sqrt{3}}\right)\right)\right).$
(42)

We can see that $a\to 0$ at $t\to\infty$. Hence, the singularity is formed at
the infinite comoving time. As mentioned before, the UHDR is, however, reached
in a finite comoving time. We now discuss the property of this region in the
next section.

##  IV Nature of the ultra high density region

To check the nature of this strong gravity region as far as its visibility is
concerned, we have to investigate the formation of trapped surfaces around the
singularity. Trapped surfaces do not form at time $t$ if

$\frac{\rho R^{2}}{3}<1.$ (43)

This ensures the positivity of the expansion scalar of the outgoing null
geodesic congruence, which in our case is given by

$\theta_{l}=\frac{2}{R}\left(1-\sqrt{\frac{\rho}{3}}R\right).$ (44)

We can hence conclude from Eq.(43) that for a given $r_{c}$ (where $r_{c}$ is
the largest comoving radius of the collapsing cloud) when the redefined time
$a$ becomes zero, $\theta_{l}>0$ if

$r_{c}^{2}<\lim_{a\to 0}\frac{3}{a^{2}\rho}.$ (45)

We see that for the density configuration mentioned in Eq.(34), the inequality
(43) which is the condition to avoid the formation of trapped surface, is
satisfied at $a\to 0$. This is because

$\lim_{a\to 0}\frac{\rho a^{2}r^{2}}{3}=0.$ (46)

Hence, there is always a causal connection from any point in the collapsing
cloud to an external observer.

As mentioned before, the singularity is formed at the infinite comoving time.
One could argue that the knowledge of the causal structure of the singularity
is relevant physically (or astrophysically) only if it is formed in a finite
comoving time. However, since we have shown that trapped surfaces never form
in such eternal collapse, the collapsing region is always visible to the
external observer, in principle. The density monotonically increases, and
beyond a specific threshold density, the quantum effects dominate. These
quantum effects will be hidden behind the event horizon if the collapse ends
in a black hole. However, in the scenario discussed here, these effects will
be visible since there are no trapped surfaces to trap the outcoming light
from the collapsing region.

At different epochs as we go in the future, along different outgoing radial
null geodesics that emanate from the center, we can calculate the quantity
$R_{ij}K^{i}K^{j}$. It can be shown that as we go forward in time (in other
words, as $a$ decreases) $R_{ij}K^{i}K^{j}$ increases, blowing up at $a=0$.

Figure 2: $R_{ij}K^{i}K^{j}$ along the outgoing radial null geodesic increases
unboundedly as the scale factor decreases.

To see this, consider the geodesic equation of the radial null geodesic
escaping from the center written as follows:

$\dot{K}^{t}+\frac{{K^{r}}^{\prime}}{a}+\frac{\dot{a}}{a}K^{t}=0.$ (47)

The outgoing radial null geodesic with the tangent having components

$K^{t}=\frac{1}{a},\hskip 28.45274ptK^{r}=\frac{1}{a^{2}},\hskip
28.45274ptK^{\theta}=K^{\phi}=0,$ (48)

is one of the kind described by Eq.(47). For this particular geodesic,

$R_{ij}K^{i}K^{j}=-\frac{3\dot{a}}{a^{3}}+\frac{2\dot{a}^{2}}{a^{4}}+\frac{\ddot{a}}{a^{3}}.$
(49)

In terms of the scale factor, we can rewrite the above equation using Eq.(10,
11, 34) as

$R_{ij}K^{i}K^{j}=\frac{\left(k-\log
a\right)}{3a^{2}}\left(64\lambda\left(-1+3k-3\log
a\right)+24\sqrt{3\lambda}\right).$ (50)

One can see that $R_{ij}K^{i}K^{j}$ increases progressively as $a$ decreases,
and blows up at $a=0$. So whereas the actual singularity is never approached
or reached, $R_{ij}K^{i}K^{j}$ becomes larger and larger with time, as seen in
Fig.(2). What it means is that the projection of the Ricci scalar in the null
frame with four-velocity $K^{i}$ increases monotonically as one progress
forward in time.

##  V Conclusions

The density of a collapsing scalar field with potential, which is a two-
dimensional analog of the “Mexican hat” shaped Higgs field potential, is not
an algebraic function of the scale factor. We hence choose a suitable
transcendental function for the density configuration and show that the
singularity is formed at an infinite comoving time. However, in a finite but
large time, the density of the cloud goes beyond a certain cut-off, above
which the laws of quantum gravity governs. The absence of trapped surfaces
allows one to observe the UHDR where new physics takes place.

The collapsing scalar field obeys the strong energy condition only if

$\rho+3p=-\frac{6\ddot{a}}{a}>0.$

Using Eq.(42), and some rearrangement leads to

$\frac{1}{k}>\exp{\left(\frac{8t\sqrt{\lambda}}{3}\right)}$

for the strong energy condition to hold. Since the collapsing system reaches
the UHDR in a finite comoving time (beyond which the laws of physics are not
governed by general relativity), the above inequality may or may not hold
continually, depending on the values of $k$ and $\lambda$. However, the weak
energy condition is always obeyed by the collapsing cloud as seen from Eq.(9.

We find that the causal structure of UHDR respects the $\mathbb{Z}_{2}$
(reflection: $\phi\longleftrightarrow-\phi$) symmetry.

Unlike massless scalar field collapse, which is proven to have a non-generic
naked singularity Christodoulou_94 ; Christodoulou_99 , we do not know whether
the UHDR formed due to the scalar field with a nonzero $\phi^{2}$ term gives a
generic outcome. The visible nature of the UHDR is nevertheless stable under
small perturbation in the parameters $\lambda$ and $k$, as seen from Eq.(34)
and Eq.(46).

It is worth mentioning that we have followed the philosophy of Misner
Misner_1969 , which say that even though we are well aware of the possibility
of the failure of general relativity as one approaches the UHDR, one should
take into consideration the predictions of general relativity in this regime
since it may give us some indication about what one should expect from a more
general theory of gravity which works in this regime.

##  VI Acknowledgement

K.M. would like to acknowledge the support of the Council of Scientific and
Industrial Research (CSIR, India, Ref: 09/919(0031)/2017-EMR-1) for funding
the work.

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# Hall mass and transverse Noether spin currents in noncollinear
antiferromagnets

Luke Wernert  Department of Physics, Colorado State University, Fort Collins,
CO 80523, USA  Bastián Pradenas  William H. Miller III Department of Physics
and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA  Oleg
Tchernyshyov  William H. Miller III Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218, USA  Hua Chen  Department of Physics,
Colorado State University, Fort Collins, CO 80523, USA  School of Advanced
Materials Discovery, Colorado State University, Fort Collins, CO 80523, USA

###### Abstract

Noncollinear antiferromagnets (AFMs) in the family of Mn${}_{3}X$ ($X$=Ir, Sn,
Ge, Pt, etc.) have recently attracted attention in the emerging field of
antiferromagnetic spintronics because of their various interesting transport,
magnetic, and optical properties. Due to the noncollinear magnetic order, the
localized electron spins on different magnetic sublattices are not conserved
even when spin-orbit coupling is neglected, making it difficult to understand
the transport of spin angular momentum. Here we study the conserved Noether
current associated with spin-rotation symmetry of the local spins in
noncollinear AFMs. We found that a Hall component of the d.c. spin current can
be created by a longitudinal driving force associated with a propagating spin
wave, and is proportional to a response coefficient that we denote as the Hall
(inverse) mass. Such a Hall spin current can be generated by spin pumping in a
ferromagnet (FM)-noncollinear AFM bilayer structure as we demonstrated
numerically. Finally we showed that the Hall mass is an isotropic quantity,
similar to the isotropic spin Hall conductivity, and should generally exist in
noncollinear AFMs and their polycrystals. Our results shed light on the
potential of noncollinear AFMs in manipulating the polarization and flow of
spin currents in general spintronic devices.

Antiferromagnets (AFMs) with noncollinear magnetic order have recently become
a topic of interest in spintronics. In spite of the usual challenge associated
with vanishing net magnetization pertinent to all AFMs, which has been
significantly mitigated in recent years due to the rapid development of
antiferromagnetic spintronics [1, 2, 3, 4, 5, 6], the complex magnetic
ordering of noncollinear AFMs leads to exotic transport phenomena that open up
opportunities otherwise unavailable in common collinear ones. A prominent
example is the anomalous Hall effect (AHE) in the noncollinear AFM family
Mn${}_{3}X$ (where $X$ = Ir, Sn, Ge, etc.)[7, 8, 9, 10, 11, 12, 13, 4] as well
as other transport and optical properties with the same symmetry requirements
as the AHE [10, 11, 12, 13, 14, 15], such as the anomalous Nernst effect [14]
and the magneto-optical Kerr effect [16, 17]. The low symmetry of the magnetic
structure also allows the existence of the magnetic spin-Hall (MSHE) and
inverse spin-Hall (MISHE) effects [18, 19, 20, 21], anisotropic
magnetorestriction and piezomagnetic effects [22, 23], and nontrivial spin-
transfer torques [24, 25], etc. Interesting features of the electronic
structure in Mn${}_{3}X$ such as Weyl nodes near the Fermi surface and their
associated transport signatures have also been extensively studied
theoretically and experimentally [26, 27, 9, 28]. In addition to Mn${}_{3}X$,
transport phenomena in other noncollinear AFMs such as antiperovskite
Mn${}_{3}AB$ with $A=$ Ga, Ni, Cu and $B=$ C, N [29, 30, 31], orthoferrites
[32], vector spin Seebeck effect [33], and many effective Ising magnets [34,
35, 36, 37, 38] have also garnered significant interest recently.

In a magnet, spin current can be carried by itinerant electrons or magnons.
Both mechanisms have been well characterized for ferromagnets (FMs) and
collinear AFMs with isotropic (exchange) interactions [39]. The magnetic order
parameter in the form of uniform or staggered magnetization lowers the SO(3)
symmetry of global spin rotations to rotations about the magnetization
direction. As a result, only the longitudinal component of spin current
carried by quasiparticle excitations is usually considered, with spin
superfluids in easy-plane magnets an exception, where it is carried by the
magnetic ground state understood as magnon condensates. In the presence of
finite spin-orbit coupling, an approximate diffusive picture of spin transport
can usually be established, which has historically played a powerful role in
discovering and understanding many remarkable phenomena in FM-based
spintronics [40, 41, 42, 43] and more recently in collinear AFM spintronics as
well [44, 45, 6]. In contrast, in noncollinear AFMs, the magnetic order
parameter breaks spin rotation symmetry completely. It is not clear whether a
conserved spin current can even be defined for AFMs with noncollinear magnetic
order. In spite of this conceptual difficulty, recent experimental and
theoretical discoveries in noncollinear AFMs, including MSHE/MISHE [18, 19],
and tunneling magnetoresistance [46, 47] have been interpreted using a spin
current language. These heuristic arguments suggest that some form of a
conserved spin current might nonetheless exist in noncollinear AFMs.

In this Letter, we study the spin current carried by the magnetization fields
in a general noncollinear AFM. A noteworthy aspect of the spin current in a
noncollinear AFM is the relationship between the spatial gradient of the
magnetic order parameter and the direction of the flow of the spin current. In
FMs and collinear AFMs, the spin current flows along the spatial gradient. We
find that this is generally not the case in noncollinear AFMs. A spatial
gradient of the magnetic order in the $x$ spatial direction may induce a spin
current flowing along the $y$ direction.

We first derive a conserved current for global spin rotations by applying
Noether’s theorem to a continuum theory [48, 49, 50, 51] of a Heisenberg
antiferromagnet with 3 sublattices in 2 spatial dimensions with a hexagonal
lattice symmetry exemplified by the Mn${}_{3}X$ family [52, 53, 54, 55, 56,
57, 58, 59, 60]. The conserved Noether charge is the total spin. Most
interestingly, the Noether spin current can have a component transverse to the
gradient of the order parameter, i.e., a Hall spin current. The theory is
verified by simulating the spin dynamics in a lattice model consisting of an
interface between an FM and a noncollinear kagome AFM, where the dynamical
d.c. spin current on the AFM side driven by spin pumping on the FM side has a
component flowing parallel to the interface. Such Hall spin currents originate
from off-diagonal components of the inverse effective mass tensor in the
continuum Lagrangian, which survives under averaging the Lagrangian over SO(3)
rotations. Such off-diagonal components of the inverse effective mass tensor,
named as the Hall (inverse) mass in this Letter, therefore exist in
polycrystalline, or effectively isotropic, noncollinear AFM systems in
general.

_Noether spin current of noncollinear AFMs._ We first use a prototypical two-
dimensional kagome lattice model [51, 52] to study some generic properties of
Noether spin currents in noncollinear AFMs. The model has classical spins of
length $S$ with antiferromagnetic nearest-neighbor exchange coupling $J$ on a
kagome lattice. We consider two representative noncollinear antiferromagnetic
ground states in this model, direct [Fig. 1 (a) inset] and inverse triangular
order [Fig. 1 (b) inset], relevant to that in Mn${}_{3}X$.

The low-energy behavior of the model is captured by a continuum Lagrangian
obtained through gradient expansion [51, 52, 58, 59, 55, 60, 61]. The
magnetization fields of the three sublattices are coplanar and point at angles
of $120^{\circ}$ to each other, thus forming a rigid body. The orientation of
this object can be encoded in terms of a spin frame—three mutually orthogonal
unit vectors $\mathbf{n}_{\alpha}$, where $\alpha=x,y,z$, rigidly attached to
the magnetic order parameter [60, 61]. The low-energy Lagrangian is

$\displaystyle\mathcal{L}=\frac{\rho}{4}\partial_{t}\mathbf{n}_{\alpha}\cdot\partial_{t}\mathbf{n}_{\alpha}-\frac{1}{4}\Gamma_{ab}^{\alpha\beta}\partial_{a}\mathbf{n}_{\alpha}\cdot\partial_{b}\mathbf{n}_{\beta}$
(1)

Here the inertia density $\rho=\frac{1}{2JA_{c}}$ is related to the
paramagnetic susceptibility. $A_{c}=2\sqrt{3}a_{0}^{2}$ is the unit cell area
and $a_{0}$ is the nearest-neighbor distance. Latin indices $a$ and $b=x,y$
label spatial directions and Greek indices denote spin components. Summation
over doubly repeated Greek or Latin indices is implied. The spin-frame vectors
can be chosen as certain superpositions of sublattice magnetizations in such a
way that $\mathbf{n}_{x}$ and $\mathbf{n}_{y}$ transform in terms of each
other under point-group symmetries of the lattice in the same way as spatial
gradients $\partial_{x}$ and $\partial_{y}$ (hence the labels) [60]. The third
spin-frame vector $\mathbf{n}_{z}=\mathbf{n}_{x}\times\mathbf{n}_{y}$ then has
the meaning of the vector spin chirality. The fourth-rank tensor $\Gamma$ is
generally symmetric under the simultaneous exchange of the Latin and Greek
indices [61]. For the kagome models considered above we found,

$\Gamma_{ab}^{\alpha\beta}=\pm\frac{\sqrt{3}}{4}JS^{2}(\delta_{a\alpha}\delta_{b\beta}+\delta_{a\beta}\delta_{b\alpha}),$
(2)

where $+(-)$ corresponds to the direct (inverse) triangular order, $\alpha$
and $\beta$ take on values $x$ or $y$; it vanishes if either $\alpha=z$ or
$\beta=z$. Alternatively, the spin-frame vectors $(\mathbf{n}_{\alpha})_{m}$
can be understood as column vectors of a rotation matrix $R_{m\alpha}$ [61],
which leads to an equivalent form of Eq. (1) [51, 52]:

$\mathcal{L}=-\frac{\rho}{4}\operatorname{Tr}{[(R^{-1}\partial_{t}R)^{2}]}+\frac{1}{4}\operatorname{Tr}[\Gamma_{ab}(R^{-1}\partial_{a}R)(R^{-1}\partial_{b}R)]$
(3)

where the trace is over spin indices. The latter will be omitted below for
brevity when possible.

The equation of motion for the spin frame is obtained by minimizing the action
with respect to the fields $\mathbf{n}_{\alpha}$ while maintaining the
orthonormality constraints
$\mathbf{n}_{\alpha}\cdot\mathbf{n}_{\beta}=\delta_{\alpha\beta}$ [60]. They
read

$\frac{\rho}{2}\mathbf{n}_{\alpha}\times\partial_{t}^{2}\mathbf{n}_{\alpha}-\frac{1}{2}\Gamma_{ab}^{\alpha\beta}\,\mathbf{n}_{\alpha}\times\partial_{a}\partial_{b}\mathbf{n}_{\beta}=0.$
(4)

Lagrangian (1) is invariant under global spin rotations,
$\delta\mathbf{n}_{\alpha}=\bm{\theta}\times\mathbf{n}_{\alpha}$ for an
infinitesimal rotation angle $\bm{\theta}$. By Noether’s theorem, the spin
current flowing along spatial direction $a$ is

$\bm{\mathcal{J}}_{a}=\mathbf{n}_{\alpha}\times\frac{\partial\mathcal{L}}{\partial(\partial_{a}\mathbf{n}_{\alpha})}=-\frac{1}{2}\Gamma_{ab}^{\alpha\beta}\,\mathbf{n}_{\alpha}\times\partial_{b}\mathbf{n}_{\beta}.$
(5)

The spin density is

$\displaystyle\bm{\mathcal{J}}_{0}$ $\displaystyle=$
$\displaystyle\mathbf{n}_{\alpha}\times\frac{\partial\mathcal{L}}{\partial(\partial_{t}\mathbf{n}_{\alpha})}=\frac{\rho}{2}\mathbf{n}_{\alpha}\times\partial_{t}\mathbf{n}_{\alpha}=\rho\bm{\Omega}$
$\displaystyle=$ $\displaystyle\frac{1}{A_{c}}\sum_{i=1}^{3}\mathbf{S}_{i},$

where $\bm{\Omega}$ is the rotation frequency of the spin frame. The last
equality comes from the equation of motion for the canting field [61]. The
continuity equation for the spin current,

$\partial_{t}\bm{\mathcal{J}}_{0}+\partial_{a}\bm{\mathcal{J}}_{a}=0,$ (7)

follows directly from the equation of motion (4).

The Noether spin current Eq. (5) generally becomes nonzero whenever the spin
configuration is nonuniform, even if it is static, similar to the case of spin
supercurrents in collinear FMs and AFMs [62, 63, 64, 65, 66, 67, 68, 58]. By
way of example, consider a static spin configuration with the spin frame
twisting about $\mathbf{n}_{x}$ as one moves along the spatial $x$ direction,
$\partial_{x}\mathbf{n}_{\alpha}=\partial_{x}\phi\,\mathbf{n}_{x}\times\mathbf{n}_{\alpha}$,
where $\phi(x)$ is a twist angle. The only nonzero spin current components
are:

$\bm{\mathcal{J}}_{y}=\pm\frac{\sqrt{3}}{8}JS^{2}\,\partial_{x}\phi\,\mathbf{n}_{y},$
(8)

whose spatial direction is orthogonal to the gradient of $\phi$, and are
therefore reminiscent of a Hall current. This will be our focus below.

_Hall mass and transverse Noether spin currents.—_ In this section we
illustrate the richness of the Hall spin currents mentioned above by
considering dynamical noncollinear spins which are more relevant to typical
spintronics experiments. We consider d.c. spin currents due to spin waves, for
which
$\mathbf{n}_{\alpha}(\mathbf{r},t)=\mathbf{n}^{0}_{\alpha}+\bm{\theta}(\mathbf{r},t)\times\mathbf{n}^{0}_{\alpha}$,
$\mathbf{n}^{0}_{\alpha}$ being the ground state spin frame vectors and
$\bm{\theta}(\mathbf{r},t)={\rm
Re}[\bm{\theta}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}]$. Taking a time
average on both sides of Eq. (5) and subtracting any static contributions in
equilibrium, we obtain

$\displaystyle\langle\mathcal{J}_{a}^{\alpha}\rangle$ $\displaystyle=$
$\displaystyle\Gamma_{ab}^{\gamma\beta}\left[-\frac{1}{2}\left\langle\mathbf{n}_{\gamma}\times\partial_{b}\mathbf{n}_{\beta}\right\rangle_{\alpha}\right]$
$\displaystyle\equiv$
$\displaystyle\mathrm{Tr}\left(\Gamma_{ab}\langle\mathcal{P}^{\alpha}_{b}\rangle\right)$

which resembles a linear response. Indeed, we show in [61] that
$\mathcal{P}_{b}^{\alpha}$ can be understood as an SO(3) gauge potential [69]
due to spatial translation along $b$ projected onto the $\alpha$-th spin
angular momentum component, and can therefore be viewed as a spin current
driving force [57]. Moreover, $\langle\mathcal{P}\rangle$ can be compactly
expressed in terms of time-averaged densities of energy, linear momentum, and
spin for spin waves, which are separately conserved and can hence be
controlled externally, as presented below.

From Eq. (Hall mass and transverse Noether spin currents in noncollinear
antiferromagnets), the time averaged spin density is

$\displaystyle\langle\mathcal{J}_{0}^{\gamma}\rangle=-\frac{\rho\omega}{4}\mathrm{Im}(\bm{\theta}\times\bm{\theta}^{*})_{\gamma}$
(10)

Note that due to the last equality of Eq. (Hall mass and transverse Noether
spin currents in noncollinear antiferromagnets), the time-averaged canting of
the noncollinear spins is equal to the
$\langle\mathcal{J}_{0}^{\gamma}\rangle$ above and represents the other part
of the angular momentum that is carried by the spin waves, which will be
discussed in a future work. Eq. (10) suggests that linearly polarized AFM spin
waves, for which $\bm{\theta}$ are all real and
$\bm{\theta}\times\bm{\theta}^{*}=0$, cannot carry spin. However, the kagome
AFM models considered here all have degenerate spin wave modes that can be
circularly polarized [60]. More general cases will be discussed in the next
section. The time-averaged linear momentum ($\mathcal{T}_{b0}$) and energy
($\mathcal{T}_{00}$) densities are [61]

$\displaystyle\langle\mathcal{T}_{b0}\rangle=\frac{\rho\omega}{2}|\bm{\theta}|^{2}k_{b},\;\langle\mathcal{T}_{00}\rangle=\frac{\rho\omega^{2}}{2}|\bm{\theta}|^{2}$
(11)

where $|\bm{\theta}|^{2}=\bm{\theta}\cdot\bm{\theta}^{*}$. As a result,

$\displaystyle\langle\mathcal{P}^{\alpha}_{b}\rangle_{\beta\gamma}=\frac{1}{\rho}\frac{\langle\mathcal{J}_{0}^{\gamma}\rangle\langle\mathcal{T}_{b0}\rangle}{\langle\mathcal{T}_{00}\rangle}\delta_{\alpha\beta}\equiv
P_{b}^{\gamma}\delta_{\alpha\beta},$ (12)

We therefore have
$\langle\mathcal{J}_{a}^{\alpha}\rangle=\Gamma_{ab}^{\alpha\gamma}P_{b}^{\gamma}$
which even better illustrates the meaning of $\Gamma$ as a response tensor
than Eq. (Hall mass and transverse Noether spin currents in noncollinear
antiferromagnets). It is then remarkable that, due to the off-diagonal
components of $\Gamma$, neither the spin nor the spatial directions of the
spin current have to be aligned with that of the driving force, or
equivalently with that of the spin wave’s spin and linear momentum, suggesting
the existence of Hall spin currents.

To demonstrate the Hall spin currents explicitly, we consider a bilayer system
consisting of an FM layer interfaced with a noncollinear AFM layer in 2D, both
on a kagome lattice (Fig. 1 insets). The ground state configurations are
calculated using an LLG solver to let the system relax [61]. We then solve the
linearized LLG equation for harmonic excitations created by an external a.c.
magnetic field applied on a few leftmost layers on the FM side. In the FM
layer, such excitations correspond to FM spin waves propagating towards the
interface, carrying a spin angular momentum whose direction is determined by
that of the static magnetization, set to $\hat{\mathbf{x}}$. This is clearly
shown by the d.c. spin currents on the FM side in Fig. 1. However, the spin
currents on the AFM side have several components not naively expected from
that on the FM side, and the transversely flowing $\mathcal{J}_{y}^{y}$
component has opposite signs for the direct and inverse triangular
noncollinear states.

(a)

(b)

(c)

(d)

Figure 1: (a) and (b), Dynamical d.c. Noether spin currents in (a) FM-Mn3Ir
(direct triangular order) and (b) FM-Mn3Sn (inverse triangular order)
interfaces outside of the interface region (gray rectangles). (c) and (d),
Spin current driving force $P$ for (c) FM-Mn3Ir and (d) FM-Mn3Sn interfaces
outside of the interface region. All numerical LLG calculations were performed
with the following parameters: gyromagnetic ratio $\gamma=1$, damping
$\alpha=0.05$, $\omega=1$, and linear polarization along $\hat{\mathbf{y}}$
for the external a.c. field. The interface region has a width of approximately
10 unit cells.

To understand this nontrivial behavior, we plot in Figs. 1 (c) and (d) the
corresponding driving force $P$ in both systems. One can see that different
from $\langle\mathcal{J}\rangle$, both sides of the interface have the same
dominant component of $P$, $P_{x}^{x}$. ($P_{x}^{y}$ and $P_{x}^{z}$ arise due
to uncontrolled scattering at the interface region.) Moreover, the nonzero
components of $\langle\mathcal{J}\rangle$ for the two noncollinear AFMs
considered here are straightforwardly obtained from $P_{x}^{x}$ and
$P_{x}^{y}$ together with the nontrivial components of $\Gamma$ in Eq. (2).
($P_{x}^{z}$ does not contribute since $\Gamma$ has no $z$-spin indices.) For
example, the $\langle\mathcal{J}_{y}^{y}\rangle$ component is due to
$\Gamma_{yx}^{yx}=\pm\frac{\sqrt{3}JS^{2}}{4}$, whose different signs for the
two types of order leads to the opposite $\langle\mathcal{J}_{y}^{y}\rangle$
in Figs. 1 (a) and (b). Conversely, the $\langle\mathcal{J}^{x}_{y}\rangle$
component has the same sign in both systems due to ${P}^{y}_{x}$ and
$\Gamma_{xy}^{yx}=\Gamma_{yx}^{yx}$ both changing sign.

The above observation suggests that it is meaningful to associate certain
components in $\Gamma$ with an analogue of the Hall effect for the Noether
spin current. Since $\Gamma$ has the units of inverse mass per area or volume
(times $\hbar^{2}$), we name the part of $\Gamma$ responsible for the Hall
spin currents the Hall (inverse) mass. However, it is not as simple as deeming
the off-diagonal components such as $\Gamma_{xy}^{xy}$ the Hall mass since the
individual components change their values under rotations of the coordinate
system, while both the charge Hall effect and the spin Hall effect have
certain invariance under coordinate transformations. More specifically, the
charge Hall effect is a pseudovector $\sigma_{\rm
H}^{a}=\frac{1}{2}\epsilon_{abc}\sigma_{bc}$ [70], with $\sigma_{bc}$ the
conductivity tensor, whose length is invariant under O(3) transformations; the
spin Hall effect has an isotropic part
$\frac{1}{6}\epsilon_{abc}\sigma_{abc}^{\rm s}$, with $\sigma_{abc}^{\rm s}$
the spin conductivity tensor, which is a scalar and invariant under O(3)
transformations as well. In the next section we will propose an appropriate
definition of the Hall mass based on a general field theory of noncollinear
AFMs.

_General definition of Hall mass in noncollinear AFMs.—_ We start by
considering an arbitrary single-crystalline noncollinear AFM in three
dimensions with spin $S$ and Heisenberg exchange coupling between any two
spins that depends on their spatial separation. The $\Gamma$ tensor of such an
AFM can be obtained through gradient expansion as [61]

$\displaystyle\Gamma_{ab}^{\alpha\beta}=-\frac{S^{2}}{V_{c}}\sum_{j,pq}J_{0p,jq}(\mathbf{r}_{0p,jq})_{a}(\mathbf{r}_{0p,jq})_{b}{m}_{p}^{\alpha}{m}_{q}^{\beta}$
(13)

where $V_{c}$ is the volume of the unit cell, $i,j$ label unit cells, $p,q$
label sublattices, $\hat{\mathbf{m}}_{p}$ is a unit vector along the spin
direction on sublattice $p$, and $\mathbf{r}_{ip,jq}$ is the position vector
of site $jq$ relative to site $ip$. Such a $\Gamma$ is symmetric under
separate permutations of its spatial and spin indices and is also independent
of unit cell choices [61].

The $\Gamma$ as defined in Eq. (13) generally has off-diagonal components
whose values will also change under rotations of the coordinate system.
Moreover, the low-energy spin wave modes that depend on $\Gamma$ are in
general non-degenerate and are linearly polarized, different from that in
ferromagnets, making them unable to carry spin according to Eq. (10). (Spin
can also be carried by large-angle precession, or supercurrents, of the
noncollinear spins, which we do not discuss in this work.) Both of these
hurdles can be overcome if we consider a polycrystal of the given noncollinear
AFM. When the grain sizes of the polycrystal are smaller than the typical
wavelengths of the spin wave modes, its low-energy Lagrangian should have the
same form as Eq. (3), but with $\rho$ replaced by an effective isotropic
paramagnetic susceptibility $\bar{\rho}$ and $\Gamma$ by an angular-averaged
$\bar{\Gamma}$ that depends only on two parameters [71, 61]:

$\displaystyle\bar{\Gamma}_{ab}^{\alpha\beta}=g_{\rm
H}(\delta_{a\alpha}\delta_{b\beta}+\delta_{a\beta}\delta_{b\alpha})+g_{0}\delta_{ab}\delta_{\alpha\beta}$
(14)

We define the $g_{0}$ and $g_{\rm H}$ in Eq. (14) as the longitudinal and the
Hall mass, respectively, since they are now invariant under rotations. (Note
that $\delta_{ab}\delta_{\alpha\beta}$, $\delta_{a\alpha}\delta_{b\beta}$, and
$\delta_{a\beta}\delta_{b\alpha}$ are the only three linearly independent
isotropic rank-4 tensors in three dimensions.) In terms of the $\Gamma$ of the
corresponding single crystal, one can get $g_{\rm
H}=\frac{1}{10}\left(\Gamma^{ab}_{ab}-\frac{1}{3}\Gamma^{aa}_{bb}\right)$,
$g_{0}=\frac{2}{15}\left(\Gamma^{aa}_{bb}-\frac{1}{2}\Gamma^{ab}_{ab}\right)$.

Using the isotropic $\bar{\Gamma}$ and the polycrystal Lagrangian, one can
easily get the spin wave dispersions in terms of $g_{0}$, $g_{\rm H}$ and
$\bar{\rho}$:

$\displaystyle\omega_{i}$ $\displaystyle=$ $\displaystyle c_{i}k,~{}i={\rm
I,II,III}$ (15) $\displaystyle c_{\rm I}$ $\displaystyle=$
$\displaystyle\sqrt{g_{0}/\bar{\rho}}$ $\displaystyle c_{\rm II,III}$
$\displaystyle=$ $\displaystyle\sqrt{(g_{0}+g_{\rm H})/\bar{\rho}}$

where the mode I has its polarization parallel to $\mathbf{k}$ while II, III
have transverse polarizations, similar to phonons in isotropic elastic media
[61]. The two transverse modes can thus be circularly polarized and carry the
spin that is parallel to the propagation direction. For such spin waves, the
Hall mass $g_{\rm H}$ gives rise to Hall spin currents with their spin
polarization parallel to the directions of spatial flow, i.e., “axial” spin
currents. Moreover, Eq. (15) suggests another way of determining the Hall mass
from spectroscopic measurements on noncollinear AFM polycrystals.

_Discussion.—_ A key prediction from our spin pumping calculation Fig. 1 is
the transverse spin current $\langle\mathcal{J}_{y}^{y}\rangle$. Such an axial
spin current, however, cannot be directly detected by ordinary inverse spin
Hall effect, for which the directions of the spin current flow and of the spin
polarization must be orthogonal to each other. A possible workaround is to use
a low-symmetry crystal that has a spin Hall conductivity component
$\sigma^{\rm s}_{yya}$, so that a charge current along $a$ can be induced by
$\langle\mathcal{J}_{y}^{y}\rangle$ [72, 73, 24, 74, 18]. Alternatively, if
the noncollinear AFM under consideration has spin wave modes that can carry
$y$-spin, one can pump $y$-spin from the FM side to create
$\langle\mathcal{J}_{y}^{x}\rangle$ which is then detectable by ISHE, since
$\Gamma_{yx}^{yx}=\Gamma_{yx}^{xy}$. Such a phenomenon is, however, more
analogous to the spin swapping effect [75, 76] despite the different
microscopic origins.

Separately, in real 3D noncollinear AFMs the $\Gamma$ tensor can be quite
different from those of our 2D kagome models. On the one hand, one can use Eq.
(13) to calculate $\Gamma$ for a given system once the spin Hamiltonian is
determined from, e.g., inelastic neutron diffraction and linear spin-wave
theory. On the other hand, nonzero components of $\Gamma$ can be identified by
symmetry analysis once the symmetry of the magnetic ground state is known. For
example, we have used Eq. (13) to calculate $\Gamma$ for cubic and hexagonal
Mn${}_{3}X$ by keeping the nearest-neighbor exchange coupling only [61]. We
found that even when the individual kagome planes in these materials have the
same orientations as that in our 2D models, the 3D materials have different
$\Gamma$ components.

That the Hall mass $g_{\rm H}$ generally exists in isotropic noncollinear AFMs
is remarkable. It suggests that all noncollinear AFMs can be classified by the
signs of their Hall mass, which is also reflected by whether the two
degenerate transverse-polarization spin wave modes are below or above the
longitudinal-polarization mode as indicated by Eq. (15). We therefore propose
inelastic magnetic neutron or x-ray measurements of powder samples of
noncollinear AFMs which can reveal the existence and signs of their Hall mass.

Our work has ignored effects of conduction electrons for metallic noncollinear
AFMs. When spin-orbit coupling is negligible, spin rotation symmetry is
preserved for the whole system including conduction electrons and local
moments. Therefore the Noether spin current is carried by the conduction
electrons and local magnetic moments together, and our discussion on
anisotropic spin current response induced by spin pumping, including the Hall
mass, should qualitatively hold.

###### Acknowledgements.

The authors are grateful to Collin Broholm and Satoru Nakatsuji for useful
discussions. L.W. and H.C. acknowledge support by NSF CAREER grant
DMR-1945023. B.P. and O.T. acknowledge support by the U.S. Department of
Energy under Award No. DE-SC0019331 and by the U. S. National Science
Foundation under Grants No. PHY-1748958 and PHY-2309135.

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# The Dataset Nutrition Label (2nd Gen): Leveraging Context to Mitigate Harms
in Artificial Intelligence

Kasia S. Chmielinski
Berkman Klein Center
Harvard University, Cambridge MA
<EMAIL_ADDRESS>
Sarah Newman
metaLAB
Harvard University, Cambridge MA
<EMAIL_ADDRESS>
Matt Taylor
Data Nutrition Project
<EMAIL_ADDRESS>
Josh Joseph
MIT Quest for Intelligence
MIT, Cambridge MA
<EMAIL_ADDRESS>
Kemi Thomas
Data Nutrition Project
<EMAIL_ADDRESS>
Jessica Yurkofsky
Massachusetts College of Liberal Arts
<EMAIL_ADDRESS>
Yue Chelsea Qiu
Data Nutrition Project
<EMAIL_ADDRESS>

###### Abstract

As the production of and reliance on datasets to produce automated decision-
making systems (ADS) increases, so does the need for processes for evaluating
and interrogating the underlying data. After launching the Dataset Nutrition
Label in 2018, the Data Nutrition Project has made significant updates to the
design and purpose of the Label, and is launching an updated Label in late
2020, which is previewed in this paper. The new Label includes context-
specific Use Cases & Alerts presented through an updated design and user
interface targeted towards the data scientist profile. This paper discusses
the harm and bias from underlying training data that the Label is intended to
mitigate, the current state of the work including new datasets being labeled,
new and existing challenges, and further directions of the work, as well as
Figures previewing the new label.

##  1 Introduction

Automated decision-making systems (ADS) are increasingly used to make rapid
and broad decisions about people [6, 15, 16], creating the possibility of
negative unintended consequences that often affect the very communities that
are already marginalized, underserved, or underrepresented [7, 3]. The assumed
neutrality of technical systems, combined with “black-box” opacity in some
cases, can prevent interrogation into possible biases of the system, leading
to unchecked and potentially harmful decision-making [10]. Building on work
that started in late 2017, The Data Nutrition Project has spent the past year
redesigning and improving the concept, design, depth, and utility of the
Dataset Nutrition Label. This short paper offers an overview of the current
state of the work, including: a) the motivation and need for this work, b)
updates to the label, including specific methodology, design, and current
datasets under analysis, and c) related work, areas of further development,
and new challenges that have arisen. This paper also includes some figures
from the second generation of the Label, which is scheduled for release in
late 2020.

##  2 Motivation

A growing awareness of the possible potential harms or biases caused by ADS,
driven in part through high-profile, high-visibility incidents, has motivated
conversation about our responsibility as technologists to understand potential
sites of bias and use these insights to build better systems.

An often-overlooked site of harm is the data that is used to train the model
itself. Problematic, incomplete, or otherwise biased datasets used to train
ADS will cause the model to replicate the very issues found in the training
dataset. With vast amounts of raw data becoming increasingly available
globally [8], understanding or ’sense-making’ of the data becomes critical as
data practitioners share and reuse ‘found data’ with insufficient
documentation: either none at all, incomplete, or provided within the context
of a domain’s epistemic norms [18], which can be unfamiliar to data
practitioners [17, 23]. Additionally, there is no global agreement on
standards when it comes to dataset quality investigation, and this is
exacerbated by the ‘move fast’ culture endemic in the tech industry that often
prioritizes rapid deployment above all else [19].

##  3 Current State of the Work

With an aim of mitigating harms caused by automated decision-making systems,
The Dataset Nutrition Label tool (hereafter referred to as ‘the Label’)
enhances context, contents, and legibility of datasets. Drawing from the
analogy of the Nutrition Facts Label on food, the Label highlights the
‘ingredients’ of a dataset to help shed light on how (or whether) the dataset
is healthy for a particular algorithmic use case.

The first generation of the Label (2018) introduced a diagnostic framework
that provided a distilled overview of a dataset before ADS development,
presented in a flexible ‘modular’ fashion, such that the Label could be
adjusted for relevance across domains and data types [11]. The first
prototype, built on ProPublica’s Dollars for Docs dataset [26], highlighted a
number of qualitative and quantitative modules as examples of Label content.

Since the launch of the Label, The Data Nutrition Project team solicited
feedback from data practitioners including dataset owners, data scientists,
data analysts, product managers, among others. Some of the main points of
feedback included: a) A single static dataset Label cannot be relevant to
everyone using that dataset, due to innumerable use cases of ADS built on that
data; b) Many data scientists want to do the ‘right thing’ when it comes to
addressing issues in data, but often don’t have the time or expertise, or an
available tool, to identify the relevant information; and c) Data
practitioners evaluating datasets often have a particular use case in mind for
the dataset, and are thus looking for specific issues relating to their
intended use rather than browsing general information. These insights are also
reflected more broadly in the literature as a shift towards the
‘contextualization’ of artificial intelligence [17] and the need for
frameworks that are ‘personalized’ [25]. Reflecting this awareness, the second
generation of the Label presents the dataset in a novel graphical user
interface (GUI) that promotes relevant, targeted information based on a
selected Use Case.

###  3.1 The Second Generation Dataset Nutrition Label

The second generation of the Label is presented in a novel, interactive, web-
based GUI with three distinct panes of information: the Overview, Use Cases &
Alerts, and Dataset Info panes. This redesign was as a result of interviews
and feedback from data scientists and other practitioners who clearly stated
the need for insights centered around intended use: the dataset selection
process generally begins with a business or research question, and the
evaluation of the dataset happens within the context of that question. The
redesign of the Label thus aims to balance general information alongside known
issues and relevant information for a particular Use Case, and includes:

  * •

The default Overview pane, which presents an at-a-glance set of modules about
the dataset (figure 1, which shows the prototype Label for the COVID Tracking
Project dataset)

  * •

The Use Cases & Alerts view, which allows a user to select the intended Use
Case for the ADS or model being trained on the data (figure 2). Once a Use
Case has been selected, the user is prompted to choose a Prediction, which is
the mathematical method or strategy employed in order to address that Use
Case. This then triggers the display of flagged information (Alerts and FYIs)
that are specifically relevant to the selected Use Case and Prediction. The
three-point color scale on Alerts indicates whether a mitigation strategy is
known. The FYI content is coded Green as an indication that there is no
mitigation necessary; this is just information that could be useful to the
practitioner.

  * •

The third and final view of the Label is the Dataset Info pane, which is a
qualitative overview of the dataset broken into several categories:
_Description, Composition, Provenance, Collection, Management_ (figure 3). For
this view, we have pulled many questions directly from the thorough and
thoughtful work of the Datasheets for Datasets project [9], with some
modifications based on additional frameworks from AI Global, data.world,
DrivenData, and feedback from colleagues at the Department of Education,
Memorial Sloan Kettering, and SafeGraph. The purpose of this view is to
provide more traditional qualitative documentation for data practitioners to
consult. Based on the responses to the questions, much of the content also
appears as Alerts or FYIs and highlighted on the Use Case / Alerts pane
(Section 2, above).

###  3.2 Prototypes and Dataset Collaborators

The Data Nutrition Project is currently working with several collaborators on
the second generation of the Dataset Nutrition Label. In particular, we are
working on three Labels: 1) The COVID Tracking Project US state level data
(The Atlantic) [2]; 2) Evictions and housing complaint data for New York City
(JustFix.nyc) [14]; 3) Melanoma image datasets ISIC 2018 [12] and ISIC 2020
[13] (Imaging Informatics Group at Memorial Sloan Kettering). All of these
Labels will be made public on the Data Nutrition Project website at
www.datanutrition.org

##  4 Related Work, Challenges, Further Directions

###  4.1 Related Work

In the last few years, the emerging field of ethical artificial intelligence
has shifted towards a principles-based framework to provide a path towards
defining, building and deploying ’trustworthy AI’. In ethical AI documents,
the principle of transparency is especially prevalent; other principles
include justice, fairness, non-maleficence, responsibility, and privacy, among
others [28, 22]. Under the banner of increased transparency, there are a
number of initiatives and an ever-growing list of tools deployed towards the
mitigation of bias in ADS. Broadly speaking, these tools aim to provide
transparency at the system, model, and data levels [29].

At the system level, related work includes guidance around managing complex
build systems. Examples include internal audit frameworks [27], which aim to
provide scaffolding for risk management across siloed departments or roles.
Several interventions have been researched and deployed at the model level,
both within industry [20, 1] and at the government level [4]. There are also
automated tools built to programmatically assess or perturb algorithms, such
as TRIPOD [21]. Finally, there are several related projects in the dataset
transparency space, including the manually-generated Datasheets for Datasets
work which our project leverages [9], data statements [30], and several
automated tools [31, 23]. The Dataset Nutrition Label provides transparency at
the dataset level in a semi-automated fashion, with a strong novel emphasis on
an interaction model that prioritizes relevant known issues based on a
practitioner’s chosen Use Case. We focus on usability in order to make our
Label actionable and immediately useful within a common data scientist’s
workflow.

###  4.2 Challenges

While the second generation of the Label addresses some of the particular
limitations of the original Label (see Section II), it also introduces some
additional challenges. These challenges are enumerated below.

  * •

_Labels on changing datasets_. It is nontrivial to create a Label that remains
relevant for a continually changing dataset. We address this in two ways: 1)
Our Labels include a ‘Date’ field so that their applicability can be
contextualized to the time at which they were produced; and 2) much of the
Label content is agnostic to data changes, so long as the Use Cases and the
structure of the dataset itself has not changed (i.e., adding new columns).

  * •

_Proprietary Datasets_. For ease of build, the DNP team leveraged open
datasets to produce the second set of Label prototypes. However, this process
requires that a third party (DNP) be able to access the data in order to build
the Label. To mitigate this in the future, we are considering automated
frameworks for Label generation.

  * •

_Qualitative vs. Quantitative_. Based on feedback about the importance of
context, the second generation of the Label explores the generation of
qualitative information about the dataset rather than the quantitative modules
explored in the first generation. For future versions of the Label, we will
likely focus on reintegrating quantitative measures as additional modules.

###  4.3 Further Directions

The new Labels (second generation) and expanded methodology paper are
scheduled for release in December 2020. Planned work in 2021 includes:
building additional Labels, creating a front-end form-like tool and ingestion
engine for more streamlined production of the Labels, and building a Label
comparison user interface that enables the comparison of datasets for a
particular use case. The team will also be seeking feedback on the second
generation of the Label from stakeholders. As was the case with feedback on
the 2018 Label Prototype, this feedback will serve in continuing to improve
the Label for various dataset domains and stakeholders.

##  5 Conclusion

ADS are trained to be opinionated, decision-making systems; as such, they will
always exhibit some form of bias. Each decision made during the course of ADS
development - from the original business use case to the dataset selection and
model training process - further biases the algorithm towards an intended
outcome. The goal, therefore, is to understand and manage that bias since
complete elimination of bias is not realistic (nor is it desired), and instead
to mitigate and minimize harms that may arise from the decisions made by the
algorithm or system [5, 24]. Through the Dataset Nutrition Label, Data
Nutrition Project team is committed to continuing to advance this important
area of ADS development through well designed, carefully vetted, and context-
aware Labels that help mitigate harm while also improving public understanding
of the risks and opportunities of working with ADS, and with particular focus
on their underlying data.

## Figures

Figure 1: The Overview pane for the COVID Tracking Project dataset Figure 2:
The Use Cases / Alerts pane for the COVID Tracking Project dataset Figure 3:
The Dataset Info pane for the COVID Tracking Project dataset

## Broader Impact

As stated in the main body of the paper, the anticipated broader impact of
this work includes the following:

  * •

_Who may benefit from this research._ The intended recipients of this work are
manifold: from the data scientists and practitioners building models on
datasets, students and researchers interested in interrogating, understanding,
and building more ethical ADS, and, most importantly, vulnerable,
marginalized, or otherwise mis-represented populations that are often harmed
by lack of (or incomplete) representation in underlying training data.

  * •

_Who may be put at disadvantage from this research._ This research is intended
to mitigate some of the harms caused by infelicities or other issues in
underlying data. However, as indicated in the New Challenges addressed in the
main paper, there may still be concerns about identifying every possible use
case and thus identifying harms caused by Use Cases that are not addressed on
the Label. As many of the possible harms that we are identifying and intending
to mitigate are emergent, particularly in the case of new and frequently
changing data, as is the case with Covid-19 data, it is possible that only
future versions of the work will address issues that we have not yet
identified, or that will not immediately emerge. Since the Label is in
English, those who do not have command of the English language may also not be
able to benefit from this work.

  * •

_What are the consequences of failure of the system?_ As outlined in the main
body of the paper, if the Label were used incorrectly, or if it does not
identify possible Harms for the Use Cases specified, or if the Label were
intentionally misused to “ethics-wash” a dataset with known issues (this is
not possible with the current version of the Label but might be possible in
future versions), this would be considered a failure of the system.

  * •

_Whether the task/method leverages biases in the data._ The DNP Method
specifically intends to mitigate harms caused by biases in data. So this work
it only leverages biases in the central goal of the work, which is to identify
and help mitigate the problems with bias in datasets.

## Acknowledgments and Disclosure of Funding

Funding in direct support of this work: from the Miami Foundation, a
subsidiary of the Knight Foundation, administered through grant support on the
Ethics & Governance of Artificial Intelligence, from July 2019-December 2020.
Additional revenues related to this work: The first four authors listed
(Chmielinski, Newman, Taylor, Joseph) were previously Fellows in the Assembly
Fellowship co-hosted by the Berkman Klein Center at Harvard University and the
MIT Media Lab. Chmielinkski was previously a Mozilla Open Leader, and
currently a Fellow with the Consumer Reports Digital Lab, though none of these
additional awards directly supported this work.

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