###### Abstract

The DBI and special galileon theories exhibit a conformal symmetry at
unphysical values of the spacetime dimension. We find the Lagrangian form of
this symmetry. The special conformal transformations are non-linearly realized
on the fields, even though conformal symmetry is unbroken. Commuting the
conformal transformations with the extended shift symmetries, we find new
symmetries, which when taken together with the conformal and shift symmetries
close into a larger algebra. For DBI this larger algebra is the conformal
algebra of the higher dimensional bulk in the brane embedding view of DBI. For
the special galileon it is a real form of the special linear algebra. We also
find the Weyl transformations corresponding to the conformal symmetries, as
well as the necessary improvement terms to make the theories Weyl invariant,
to second order in the coupling in the DBI case and to lowest order in the
coupling in the special galileon case.

On the Conformal Symmetry of
Exceptional Scalar Theories

Kara<EMAIL_ADDRESS>Kurt
<EMAIL_ADDRESS>Ondřej
<EMAIL_ADDRESS>
aCERCA, Department of Physics,

Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106

bInstitute of Physics of the Czech Academy of Sciences, CEICO,

Na Slovance 2, 182 21 Prague 8, Czech Republic

###### Contents

  1. 1 Introduction
  2. 2 DBI
    1. 2.1 Conformal symmetry
    2. 2.2 Including shift symmetries
    3. 2.3 Brane point of view
  3. 3 Special Galileon
    1. 3.1 Conformal symmetry
    2. 3.2 Including shift symmetries
  4. 4 Weyl symmetry
    1. 4.1 DBI
    2. 4.2 Special Galileon
  5. 5 Conclusions

##  1 Introduction

Soft theories are effective field theories whose amplitudes go to zero as one
of the external momenta are scaled to zero, scaling with some power of this
momentum as we approach zero. Among soft theories consisting of a single
scalar $\phi$, there are two exceptional theories that scale with the highest
possible power, a power higher than expected given the number of derivatives
per field in the Lagrangian [1, 2]. These are the Dirac-Born-Infeld (DBI)
theory, whose action is given in (2.1), and the special galileon [1, 3], whose
action is given in (3.1). Both theories depend on a single dimensionful
coupling constant $\alpha$, and are otherwise completely fixed.

These two theories, among others [4], have a privileged place among scalar
effective field theories; they naturally appear in the Cachazo-Yuan-He (CHY)
representation [5, 6, 7] and act as nodes in a web of theories related by the
double copy and other procedures [8, 9] (see [10] for a nice visual
representation of this web). Understanding their properties and what makes
these theories unique is a step towards being able to uncover how fundamental
these tools are and their ability to relate gauge and gravitational theories.

DBI and the special galileon exist for any spacetime dimension $d$, however in
[11] it was noted that if we consider these theories at unphysical values of
$d$, $d=0$ for DBI and $d=-2$ for the special galileon, then the coupling
$\alpha$ becomes dimensionless and the theory is scale invariant. In fact,
each term in the Lagrangian is separately scale invariant, as classical scale
invariance is automatic and trivial once there are no dimensionful couplings.
In [11] it is shown that these theories also have full conformal invariance in
addition to scale invariance, and that the conformal invariance is non-trivial
and fixes the structure of the Lagrangian to be that of DBI/special galileon.
For example, it was argued that the stress tensor on flat space could be
improved to be traceless, and that the amplitudes satisfy conformal Ward
identities, only if the structure of the theory was that of DBI/special
galileon.

However, conformal symmetry as we usually understand it is linearly realized
on the fields, and a linear symmetry can never relate terms with different
powers of the field, as would be required to fix the full non-linear structure
of these theories. We therefore expect that this conformal symmetry is
realized in a novel non-linear way. On the other hand, conformal symmetry
should still be unbroken, so that the usual Ward identities on the amplitudes
are satisfied. Here we resolve this puzzle and find the field transformations
that leave the Lagrangians invariant and fix their non-linear structure,
without spontaneously breaking conformal symmetry. The transformations include
non-linearities in the special conformal generators (shown in (2.6) for DBI
and (3.3) for the special galileon), but preserve the $\phi=0$ vacuum and
satisfy the same conformal algebra as their linear counterparts.

Another defining feature of these theories is that they possess non-trivial
shift symmetries, where the field shifts by powers of the spacetime coordinate
$x^{\mu}$. The shift symmetries for DBI have up to one power of $x^{\mu}$
(shown in (2.11)), and the shift symmetries for the special galileon have up
to two powers of $x^{\mu}$ (shown in (3.4)). These shift symmetries also fix
the full non-linear structure of the theory, and are responsible for the
enhanced soft limits.

These shift symmetries do not close with the conformal symmetries under
commutation, and instead produce new symmetries. In the DBI case, we find a
new symmetry (2.12) which has 2 powers of $x^{\mu}$, and in the special
galileon case we find several new symmetries (3.5) which have up to 4 powers
of $x^{\mu}$. These new symmetries combined with the conformal symmetries and
shift symmetries then close to form a larger algebra. In the DBI case, this
larger algebra is the conformal algebra of one dimension higher. In Section
2.3, we explain this from the brane embedding point of view; the DBI theory
can be thought of as the worldvolume theory of a $d$ dimensional brane
embedded in a $d+1$ dimensional bulk, and from this perspective, the
symmetries all descend from conformal symmetries of the bulk. In the special
galileon case, the full symmetry algebra is a real form of the special linear
algebra.

Finally, we consider coupling these theories to a metric and look for a local
Weyl symmetry that descends to the new non-linear global conformal symmetries
when the metric is frozen to flat space. This Weyl symmetry requires new non-
linear terms in the scalar field transformation in order to reproduce the non-
linear terms in the global special conformal transformations. We find the
improvement terms necessary to add to the theories so that they are Weyl
invariant, to lowest order in the coupling for the special galileon, and to
second order in the coupling for DBI.

#### Conventions:

The spacetime dimension is $d$. We use the mostly plus metric. The curvature
conventions are those of [12].

##  2 DBI

The DBI action for a single scalar $\phi$ in $d$ spacetime dimensions is

${S}=\int d^{d}x\ -{1\over\alpha}\sqrt{1+\alpha(\partial\phi)^{2}}\,.$ (2.1)

It depends on a single coupling constant, $\alpha$.

In any $d$, this action is manifestly invariant under the standard
translations and Lorentz transformations,

$\displaystyle P^{\mu}\,\phi=-\partial^{\mu}\phi,\ \ \ \
J^{\mu\nu}\,\phi=(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})\phi\,,$ (2.2)

which satisfy the commutation relations of the Poincare algebra,

$\displaystyle\left[J^{\mu\nu},J^{\sigma\rho}\right]$ $\displaystyle=$
$\displaystyle\eta^{\mu\sigma}J^{\nu\rho}-\eta^{\nu\sigma}J^{\mu\rho}+\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}\,,$
$\displaystyle\left[J^{\mu\nu},P^{\rho}\right]$ $\displaystyle=$
$\displaystyle\eta^{\rho\mu}P^{\nu}-\eta^{\rho\nu}P^{\mu}\,,$
$\displaystyle\left[P^{\mu},P^{\nu}\right]$ $\displaystyle=$ $\displaystyle
0\,.$ (2.3)

###  2.1 Conformal symmetry

The DBI coupling constant in (2.1) has mass dimension $[\alpha]=-d$. As was
noted in [11], if we consider the unphysical value $d=0$, the coupling becomes
dimensionless, and the theory becomes scale invariant under the standard
dilation symmetry $D\phi=-\left(x^{\mu}\partial_{\mu}+\Delta\right)\phi$,
where $\Delta$ is the scaling dimension of the field. Scale invariance
requires the field $\phi$ to have dimension $\Delta={d-2\over 2}$, so when
$d=0$ we have $\Delta=-1$, and the action (2.1) is invariant under

$D\phi=-x^{\mu}\partial_{\mu}\phi+\phi\,.$ (2.4)

This scale symmetry, like the Poincare transformations (2.2), is linear in the
fields, and each term in the expansion of the square root in the action (2.1)
in powers of $\alpha$ is separately scale invariant when $d=0$111In checking
invariance for this and other symmetries below in $d=0$, we proceed as in
dimensional regularization, manipulating everything in general $d$, setting
$d=0$ only at the end and never using any dimensionally dependent identities..

Conformal symmetry also includes special conformal transformations, and the
standard linear action of these on a field of weight $\Delta$ is

$K^{\mu}\,\phi=\left(-2x^{\mu}x^{\nu}\partial_{\nu}+x^{2}\partial^{\mu}-2x^{\mu}\Delta\right)\phi.$
(2.5)

In [11] it was argued that conformal symmetry should fix the square root
structure in the DBI action when $d=0$; a generic scalar effective theory in
$d=0$ with one derivative per field would be scale invariant, but not
conformally invariant, and only the particular square root structure of DBI
would have full conformal invariance. However this cannot be achieved with the
standard linear special conformal transformations (2.5), since they cannot
relate terms with differing powers of $\phi$ in order to fix the square root
structure.

The non-linear special conformal transformation which does accomplish this is

$K^{\mu}\,\phi=\left(-2x^{\mu}x^{\nu}\partial_{\nu}+x^{2}\partial^{\mu}+2x^{\mu}\right)\phi+\alpha\phi^{2}\partial^{\mu}\phi.$
(2.6)

This transformation contains a new non-linear piece which depends on $\alpha$,
and when $\alpha=0$ it reduces to the standard linear special conformal
transformation (2.5) with $\Delta=-1$. The non-linear transformation (2.6) is
a symmetry of the DBI action when $d=0$, and it fixes the square root
structure of the action. Note that despite the non-linearities, the
transformation (2.6) preserves the vacuum $\phi=0$, so we can still say that
conformal symmetry is unbroken222A special conformal transformation similar to
(2.6) occurs in the worldvolume theory of a flat brane probing an AdS bulk,
but in this case conformal symmetry is broken [13].. Usually non-linear
transformations are associated with broken symmetries, but in these cases
there is always a leading term in the transformation that is independent of
the fields so that the vacuum is not preserved. Here we have no such term, the
transformation starts at linear order in $\phi$.

We can now compute the commutators of the scale symmetry (2.4) and non-linear
special conformal symmetries (2.6) with the Poincare symmetries (2.2) and we
get

$\displaystyle\left[P^{\mu},D\right]$ $\displaystyle=$ $\displaystyle
P^{\mu}\,,$ $\displaystyle\left[K^{\mu},D\right]$ $\displaystyle=$
$\displaystyle-K^{\mu}\,,$ $\displaystyle\left[K^{\mu},P^{\nu}\right]$
$\displaystyle=$ $\displaystyle-2(\eta^{\mu\nu}D-J^{\mu\nu})\,,$
$\displaystyle\left[J^{\mu\nu},K^{\rho}\right]$ $\displaystyle=$
$\displaystyle\eta^{\rho\mu}K^{\nu}-\eta^{\rho\nu}K^{\mu},$
$\displaystyle\left[K^{\mu},K^{\nu}\right]$ $\displaystyle=$
$\displaystyle\left[J^{\mu\nu},D\right]=0\,,$ (2.7)

along with the Poincare commutators (2.3). These are the standard commutation
relations of the conformal algebra. They hold independently of $\alpha$, so
despite the non-linear term in (2.6), we have the usual conformal algebra. In
computing these we have not fixed the spacetime dimension $d$, though we did
need to fix $\Delta=-1$ in writing the generators (2.4), (2.6). As is well
known, we can define $J^{-1,0}=D,\ J^{-1,\mu}={1\over 2}(P^{\mu}-K^{\mu}),\
J^{0,\mu}={1\over 2}(P^{\mu}+K^{\mu}),$ and assemble the conformal generators
into a $d+2$ dimensional anti-symmetric matrix $J^{AB}$ with
$A,B=-1,0,1,\cdots,d$

$J^{AB}\equiv\left(\begin{array}[]{c|c|c}0&D&{1\over 2}(P^{\nu}-K^{\nu})\\\
\hline\cr-D&0&{1\over 2}(P^{\nu}+K^{\nu})\\\ \hline\cr-{1\over
2}(P^{\mu}-K^{\mu})&-{1\over
2}(P^{\mu}+K^{\mu})&J^{\mu\nu}\end{array}\right).$ (2.8)

Then the commutation relations (2.7), (2.3) become

$\left[J^{AB},J^{CD}\right]=g^{AC}J^{BD}-g^{BC}J^{AD}+g^{BD}J^{AC}-g^{AD}J^{BC}\,,$
(2.9)

where

$g^{AB}\equiv\left(\begin{array}[]{c|c|c}-1&&\\\ \hline\cr&1&\\\
\hline\cr&&\eta^{\mu\nu}\end{array}\right)\,,$ (2.10)

showing that the conformal algebra is $so(2,d)$.

Since the field $\phi$ has conformal dimension $\Delta=-1$ in dimension $d=0$,
it might be thought that it satisfies the standard scalar unitarity bound
$\Delta\geq{d-2\over 2}$ [14, 15]. This unitarity bound comes from demanding
positivity of the second level descendent state $P^{2}|\psi\rangle$, given
that the primary $|\psi\rangle$ has positive norm. But there is also a bound
$\Delta\geq 0$ coming from positivity of the first level descendant
$P^{\mu}|\psi\rangle$. When $d\geq 2$ this bound is subsumed by the standard
bound, but in our case where $d=0$, $\Delta=-1$ it is stronger and it is
violated. This is a moot point, however, since the non-linear action (2.6) of
the conformal algebra is not the standard one that leads to these bounds, and
so the bounds may not apply.

###  2.2 Including shift symmetries

The DBI action (2.1) is also invariant in any $d$ under the extended shift
symmetries,

$\displaystyle C\,\phi=1,\ \ \
B^{\mu}\,\phi=x^{\mu}+\alpha\,\phi\partial^{\mu}\phi,$ (2.11)

where $x^{\mu}$ is the spacetime coordinate. These are spontaneously broken,
since they do not preserve the vacuum $\phi=0$. Any theory with one derivative
per field has the simple shift symmetry $C$, but the extended shifts
$B^{\mu}$, which include terms linear in the spacetime coordinate $x^{\mu}$,
are $\alpha$ dependent and fix the square root structure of the action in any
$d$.

We can now ask how these shift symmetries interplay with the conformal
symmetries (2.2), (2.4), (2.6) by computing the commutators. In fact, taken
together, the conformal symmetries and shift symmetries (2.11) do not quite
close under the commutators. By computing $[K^{\mu},B^{\nu}]$, we find a new
scalar symmetry,

$N\phi=-x^{2}+\alpha\left(\phi^{2}-2x^{\mu}\phi\partial_{\mu}\phi\right).$
(2.12)

This new transformation is also a symmetry of DBI when $d=0$, and it starts
with a quadratic shift in the spacetime coordinate. (For example, we can
quickly see that the kinetic term $\sim(\partial\phi)^{2}$ is invariant under
the lowest order shift $N^{(0)}\phi\sim x^{2}$:
$N^{(0)}\left((\partial\phi)^{2})\right)\sim\partial_{\mu}\phi\partial^{\mu}(N^{(0)}\phi)\sim\partial_{\mu}\phi
x^{\mu}$, which integrates by parts to $\sim d\phi$, which vanishes when
$d=0$.)

After including $N$, the algebra closes. The complete set of commutators, in
addition to the conformal algebra (2.7), (2.3), is

$\displaystyle\left[J^{\mu\nu},B^{\rho}\right]=\eta^{\rho\mu}B^{\nu}-\eta^{\rho\nu}B^{\mu}\,,\
\ \ \ \left[J^{\mu\nu},C\right]=\left[J^{\mu\nu},N\right]=0\,,$ (2.13)
$\displaystyle\left[N,D\right]=-N\,,\ \ \ \left[B^{\nu},D\right]=0\,,\ \ \
\left[C,D\right]=C\,,$ (2.14)
$\displaystyle\left[P^{\mu},N\right]=-2B^{\mu}\,,\ \ \
\left[P^{\mu},B^{\nu}\right]=\eta^{\mu\nu}C\,,\ \ \
\left[P^{\mu},C\right]=0\,,$ (2.15)
$\displaystyle\left[K^{\mu},C\right]=-2B^{\mu}\,,\ \ \
\left[K^{\mu},B^{\nu}\right]=\eta^{\mu\nu}N\,,\ \ \
\left[K^{\mu},N\right]=0\,,$ (2.16)
$\displaystyle\left[B^{\mu},B^{\nu}\right]=\alpha J^{\mu\nu},\ \ \
\left[C,B^{\mu}\right]=-\alpha P^{\mu}\,,\ \ \ \left[N,B^{\mu}\right]=-\alpha
K^{\mu}\,,\ \ \ \left[C,N\right]=2\alpha D\,.$ (2.17)

As before, we do not need to fix $d=0$ in computing these, though we have used
$\Delta=-1$ in defining the generators.

The relations (2.13) indicate that the shift symmetries $N,B^{\mu},C$
transform under Lorentz transformations simply as their tensor indices
indicate, with spins $0,1,0$ respectively. The relations (2.14) indicate that
they are eigenvectors under the action of commutation with $D$ to the right,
with eigenvalues $-1,0,1$ respectively. The relations (2.15) and (2.16)
indicate that $P^{\mu}$ and $K^{\mu}$ are like raising and lowering operators
respectively between the different $D$ eigenstates. These relations (2.13),
(2.14), (2.15), (2.16) are all the commutators between elements of the
conformal algebra and shift symmetries, and are independent of $\alpha$. They
can be summarized in this figure:

(2.18)

The DBI symmetries thus form a finite dimensional Verma module under the
conformal algebra, where the new symmetry $N$ is like a conformal primary and
$B^{\mu}$, $C$ are descendants. This short finite dimensional module, which
occurs when a scalar conformal primary has dimension $-1$ as is the case
here333This is a type I shortening condition in the language of [16], which
occur for scalar primaries when they have weight $0,-1,-2,\cdots$. The trivial
case with weight 0 would describe a simple shift symmetric field with
conformal symmetry. The next case with weight $-2$ will occur in Section 3.2
for the special galileon. , is nothing but the fundamental vector
representation of the $so(2,d)$ conformal algebra. To see this, define
$S^{-1}=-{1\over 2}\left(C-N\right),\ S^{0}=-{1\over 2}\left(C+N\right),\
S^{\mu}=B^{\mu},$ and assemble the DBI symmetries into a $d+2$ dimensional
vector $S^{A}$,

$S^{A}=\left(\begin{array}[]{c}-{1\over 2}\left(C-N\right)\\\
\hline\cr-{1\over 2}\left(C+N\right)\\\ \hline\cr
B^{\mu}\end{array}\right)\,.$ (2.19)

Then the commutators (2.13), (2.14), (2.15), (2.16) become

$\left[J^{AB},S^{C}\right]=g^{AC}S^{B}-g^{CB}S^{A},\ \ $ (2.20)

with the conformal generators packaged into $J^{AB}$ as in (2.8) with $g^{AB}$
as in (2.10). This is the statement that $S^{A}$ is a vector under the
conformal algebra $so(2,d)$. The relations (2.17), which give the commutators
between the shift symmetries, are proportional to $\alpha$, and can now be
summarized as

$\left[S^{A},S^{B}\right]=\alpha J^{AB}\,.$ (2.21)

To identify the full algebra, we assemble $J^{AB}$ and $S^{A}$ into an $d+3$
dimensional anti-symmetric matrix $J^{{\cal A}{\cal B}}$ with ${\cal A},{\cal
B}=-1,0,1,\cdots,d,d+1$,

$J^{{\cal A}{\cal
B}}=\left(\begin{array}[]{cc}J^{AB}&-{1\over\sqrt{\alpha}}S^{B}\\\
{1\over\sqrt{\alpha}}S^{A}&0\end{array}\right)\,,\ \ \alpha>0\,.$ (2.22)

The commutators (2.21), (2.20), (2.9) can now all be summarized as

$\left[J^{{\cal A}{\cal B}},J^{{\cal C}{\cal D}}\right]=g^{{\cal A}{\cal
C}}J^{{\cal B}{\cal D}}-g^{{\cal B}{\cal C}}J^{{\cal A}{\cal D}}+g^{{\cal
B}{\cal D}}J^{{\cal A}{\cal C}}-g^{{\cal A}{\cal D}}J^{{\cal B}{\cal C}},$
(2.23)

where

$g^{{\cal A}{\cal B}}=\left(\begin{array}[]{cc}g^{AB}&0\\\
0&1\end{array}\right),\ \ \ \alpha>0\,.$ (2.24)

These are the commutation relations of $so(2,d+1)$. In writing (2.22), (2.24),
we have assumed that $\alpha>0$, which is the “right sign” from the standard
UV completion point of view [17]. If instead we have the “wrong sign”
$\alpha<0$ [18], then the $d+3$ metric (2.24) will have a minus sign in the
lower right corner, and the full algebra will be $so(3,d)$.

The smallest symmetry algebra of DBI that includes the conformal symmetry of
[11] and the shift symmetries is thus isomorphic to the conformal algebra of a
space of dimension $d+1$. The breaking pattern is to the $d$ dimensional
conformal algebra, since the shifts (2.11), (2.12) are the only ones that do
not preserve the vacuum $\phi=0$,

$so(2,d+1)\rightarrow so(2,d)\,,\ \ \ \alpha>0.$ (2.25)

In fact, this is isomorphic to the symmetry breaking pattern of the AdS DBI
theory, the theory of a $d+1$ dimensional AdS brane in a $d+2$ dimensional AdS
bulk [19, 20] (or the $k=1$ non-abelian AdS scalar shift theory, in the
language of [21]).

###  2.3 Brane point of view

The DBI theory (2.1) can be thought of as a gauge fixed worldvolume theory of
a $d$ dimensional flat brane probing a $d+1$ dimensional flat bulk [22]. The
fact that the full symmetry algebra including the conformal and shift
symmetries is the conformal algebra of one dimension higher suggests that they
have their origin as conformal symmetries of the bulk. Here we will see that
this is indeed the case. (In this subsection we’ll assume $\alpha>0$ and
suppress the coupling, for clarity.)

As detailed in [23, 19], we consider the theory of scalars $X^{A}(x)$,
$A=1,\cdots,d+1$, describing the embedding of a $d$ dimensional worldvolume
into a $d+1$ dimensional bulk. The fixed bulk metric is $G_{AB}(X)$, and the
induced metric on the brane is

$\bar{g}_{\mu\nu}(x)={\partial X^{A}\over\partial x^{\mu}}{\partial
X^{B}\over\partial x^{\nu}}G_{AB}(X(x)).$ (2.26)

The action is

$S=-\int d^{d}x\ \sqrt{-\bar{g}},$ (2.27)

and is manifestly invariant under worldvolume diffeomorphisms $\xi^{\mu}(x)$,
under which the $X^{A}$ transform as scalars
$\delta_{\xi}X^{A}=\xi^{\mu}\partial_{\mu}X^{A}$.

Given a transformation by a bulk diffeomorphism $K^{A}(X)$, $\delta
X^{A}=K^{A}(X)$, the induced metric is invariant if $K^{A}$ is a Killing
vector of the bulk metric,

${\cal L}_{K}G_{AB}\equiv
K^{C}\partial_{C}G_{AB}+\partial_{A}K^{C}G_{CB}+\partial_{B}K^{C}G_{AC}=0,$
(2.28)

and this is a global symmetry of the action in any $d$.

Suppose instead that $K^{A}$ is a conformal Killing vector,

${\cal L}_{K}G_{AB}=\Phi G_{AB},$ (2.29)

where $\Phi={2\over d+1}\nabla_{A}K^{A}$. Then under such a diffeomorphism we
have

$\delta\bar{g}_{\mu\nu}=\Phi\bar{g}_{\mu\nu},$ (2.30)

and the change in the action (2.27) is then

$\delta S=-\int d^{d}x\ \delta\left(\sqrt{-\bar{g}}\right)=-{d\over 2}\int
d^{d}x\ \sqrt{-\bar{g}}\Phi\,.$ (2.31)

This vanishes when $d=0$, which means that the shift by a bulk conformal
Killing vector is a symmetry of (2.27) when $d=0$.

To recover the DBI theory (2.1), we consider a flat bulk metric
$G_{AB}(X)=\eta_{AB}$, and we fix the worldvolume diffeomorphisms by going to
the unitary gauge $X^{\mu}=x^{\mu}$, $X^{d+1}=\phi$. The induced metric (2.26)
becomes $\bar{g}_{\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}\phi\partial_{\nu}\phi$,
and the action (2.27) becomes the DBI action (2.1). The transformation
resulting from a bulk diffeomorphism $K^{A}$ must be compensated by a
worldvolume diffeomorphism with $\xi^{\mu}=-K^{\mu}$ in order to preserve the
unitary gauge condition, so that the resulting global transformation acting on
$\phi$ is

$\delta_{K}\phi=-K^{\mu}(x,\phi)\partial_{\mu}\phi+K^{d+1}(x,\phi).$ (2.32)

When the $K^{A}$ are isometries of $\eta_{AB}$, we get global symmetries valid
in any $d$; the isometries along the brane directions $\mu$ give the Poincare
symmetries (2.2), the translation in the $d+1$ direction gives the shift $C$
in (2.11), and the Lorentz boosts mixing the $d+1$ and $\mu$ directions give
the extended shifts $B^{\mu}$ in (2.11).

We can now ask about the conformal Killing vectors of $\eta_{AB}$ that are not
Killing vectors, which will give symmetries only when $d=0$. The bulk dilation
vector field is

$K^{A}(X)=X^{A},$ (2.33)

which plugging into (2.32) leads to

$\delta\phi=-x^{\mu}\partial_{\mu}\phi+\phi,$ (2.34)

which is precisely the dilation in (2.4), including the correct scaling weight
$\Delta=-1$. The bulk special conformal vector fields are

$K^{A}_{(B)}(X)=2X_{B}X^{A}-X^{2}\delta^{A}_{B}.$ (2.35)

Looking at the $B=\mu$ components, plugging into (2.32) leads to

$\delta_{(\mu)}\phi=\left(-2x^{\mu}x^{\nu}\partial_{\nu}+x^{2}\partial^{\mu}+2x^{\mu}\right)\phi+\phi^{2}\partial^{\mu}\phi,$
(2.36)

which, once the coupling is restored, is precisely the special conformal
transformation in (2.6), with the correct scaling weight $\Delta=-1$, and
including the new non-linear term. Looking at the $B=d+1$ component, plugging
into (2.32) leads to

$\delta_{(d+1)}\phi=-x^{2}+\left(\phi^{2}-2x^{\mu}\phi\partial_{\mu}\phi\right)\,,$
(2.37)

which, once the coupling is restored, is precisely the new transformation in
(2.12).

This construction makes it clear that new symmetries for $d=0$ will also be
present in any brane theory where the bulk has conformal Killing vectors which
are not Killing vectors. This includes the AdS and dS DBI theories discussed
in [19, 20, 24, 25, 21], and more general cases such as the cosmological setup
in [26].

##  3 Special Galileon

We now turn to the special galileon. The action can be written as

$\displaystyle S_{\rm sgal}$ $\displaystyle=-\frac{1}{2}\int
d^{d}x\sum_{n=1}^{\left\lfloor\frac{d+1}{2}\right\rfloor}\frac{\alpha^{n-1}}{(2n-1)!}\left(\partial\phi\right)^{2}\mathcal{L}^{\rm
TD}_{2n-2}$ (3.1) $\displaystyle=\int
d^{d}x-\frac{1}{2}(\partial\phi)^{2}-\frac{\alpha}{12}(\partial\phi)^{2}\Big{[}(\square\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}\Big{]}+\cdots\,,$

where the total derivative combinations are $\mathcal{L}^{\rm
TD}_{n}\equiv\sum_{p}(-1)^{p}\eta^{\mu_{1}p({\nu_{1}})}\cdots\eta^{\mu_{n}p({\nu_{n}})}\partial_{\mu_{1}}\partial_{\nu_{1}}\phi\cdots\partial_{\mu_{n}}\partial_{\nu_{n}}\phi$
with the sum running over all permutations of the $\nu$ indices with
$(-1)^{p}$ the sign of the permutation. As with DBI, the theory depends on a
single coupling constant, $\alpha$, and in any $d$ it is manifestly invariant
under the standard translations and Lorentz transformations (2.2).

###  3.1 Conformal symmetry

The galileon coupling constant in (3.1) has dimension $[\alpha]=-(d+2)$. As
was noted in [11], if we consider the unphysical value $d=-2$, then the
coupling becomes dimensionless, and the theory becomes scale invariant. In
this case the theory will have the standard dilation symmetry
$D\phi=-\left(x^{\mu}\partial_{\mu}+\Delta\right)\phi$, where $\Delta$ is the
conformal dimension of the field. The field has dimension $\Delta={d-2\over
2}$, so when $d=-2$ we have $\Delta=-2$, and each term in the action (3.1) is
separately invariant under

$D\phi=-x^{\mu}\partial_{\mu}\phi+2\phi\,.$ (3.2)

As with DBI, if conformal symmetry is to fix the theory as claimed in [11],
the special conformal transformations should include non-linear parts. The
form of the special conformal generators which does this is

$K^{\mu}\,\phi=\left(-2x^{\mu}x^{\nu}\partial_{\nu}+x^{2}\partial^{\mu}+4x^{\mu}\right)\phi-\alpha(\partial\phi)^{2}\partial^{\mu}\phi\,.$
(3.3)

When $\alpha=0$, it reduces to the standard linear special conformal
transformation (2.5) with $\Delta=-2$. The non-linear transformation (3.3) is
a symmetry of the galileon action (3.1) when $d=-2$, and it fixes the
coefficients of the various galileon terms [27] relative to each other into
the special galileon combination. As with DBI, the transformation (3.3)
preserves the vacuum $\phi=0$, so we can still say that conformal symmetry is
unbroken, even though it is non-linearly realized. The transformations (3.2),
(3.3) along with the Poincare symmetries (2.2) satisfy the conformal algebra
(2.7), (2.3) in any $d$, despited the non-linear $\alpha$ dependent terms in
(3.3).

###  3.2 Including shift symmetries

The generators of the galileon shift symmetries are [3]

$\displaystyle
C\phi=1~{},~{}~{}{B^{\mu}}\phi=x^{\mu}~{},~{}~{}~{}{S^{\mu\nu}}\phi=x^{\mu}x^{\nu}-\frac{1}{d}x^{2}\eta^{\mu\nu}-\alpha\left[\partial^{\mu}\phi\,\partial^{\nu}\phi-\frac{1}{d}(\partial\phi)^{2}\eta^{\mu\nu}\right].$
(3.4)

The shifts $C$, $B^{\mu}$ are zeroth and first other in the spacetime
coordinate $x^{\mu}$, and they are symmetries of a generic galileon [27]. The
symmetric traceless tensor shifts $S_{\mu\nu}$ are second order in $x^{\mu}$,
and they are symmetries of the special galileon that fix the coefficients of
the galileon terms relative to each other.

The shift symmetries (3.4) do not close with the conformal symmetries of
Section 3.1 under the commutator. We find 3 new transformations, two scalars
and a vector and involving up to four powers of $x^{\mu}$, appearing on the
right hand side of commutators before everything closes,

$\displaystyle
A^{\mu}\phi=x^{2}x^{\mu}+\alpha\left[4\phi\partial^{\mu}\phi-2\partial^{\mu}\phi
x^{\nu}\partial_{\nu}\phi-x^{\mu}(\partial\phi)^{2}\right]\,,$ $\displaystyle
T\phi=x^{2}-\alpha(\partial\phi)^{2}\,,$ $\displaystyle
N\phi=x^{4}-2\alpha\left[x^{2}(\partial\phi)^{2}+2x^{\mu}x^{\nu}\partial_{\mu}\phi\partial_{\nu}\phi-8x^{\mu}\phi\partial_{\mu}\phi+8\phi^{2}\right]+\alpha^{2}(\partial\phi)^{4}\,.$
(3.5)

The new transformations $N,A^{\mu}$ are indeed new symmetries of the special
galileon action (3.1) when $d=-2$. (For example, we can quickly see that the
kinetic term $\sim(\partial\phi)^{2}$ is invariant under the lowest order
shift $N^{(0)}\phi\sim x^{4}$:
$N^{(0)}\left((\partial\phi)^{2})\right)\sim\partial_{\mu}\phi\partial^{\mu}(N^{(0)}\phi)\sim\partial_{\mu}\phi\,x^{2}x^{\mu}$,
which integrates by parts to $\sim(d+2)x^{2}\phi$, which vanishes when
$d=-2$.) The transformation $T$ is not a symmetry, however, as we’ll see it
always appears in the commutators with a factor of $d+2$ which vanishes when
$d=-2$.

The commutators of the original shift transformations (3.4) and new
transformations (3.5) with the conformal generators are as follows. They
transform in the expected way as tensors under Lorentz transformations,

$\displaystyle\left[J^{\mu\nu},S^{\lambda\sigma}\right]=\eta^{\mu\lambda}S^{\nu\sigma}-\eta^{\nu\lambda}S^{\mu\sigma}+\eta^{\mu\sigma}S^{\lambda\nu}-\eta^{\nu\sigma}S^{\lambda\mu}\,,$
$\displaystyle\left[J^{\mu\nu},B^{\rho}\right]=\eta^{\rho\mu}B^{\nu}-\eta^{\rho\nu}B^{\mu}\,,$
$\displaystyle\left[J^{\mu\nu},A^{\rho}\right]=\eta^{\rho\mu}A^{\nu}-\eta^{\rho\nu}A^{\mu}\,,$
$\displaystyle\left[J^{\mu\nu},C\right]=\left[J^{\mu\nu},T\right]=\left[J^{\mu\nu},N\right]=0\,.$
(3.6)

They are eigenstates of the dilation operator,

$\displaystyle\left[C,D\right]=2C\,,$
$\displaystyle\left[B^{\mu},D\right]=B^{\mu}\,,$
$\displaystyle\left[S^{\mu\nu},D\right]=\left[T,D\right]=0\,,$
$\displaystyle\left[A^{\mu},D\right]=-A^{\mu}\,,$
$\displaystyle\left[N,D\right]=-2N\,,$ (3.7)

and they transform into each other under $K,P$,

$\displaystyle\left[P^{\mu},N\right]=4A^{\mu}\,,$
$\displaystyle\left[P^{\mu},A^{\nu}\right]=2S^{\mu\nu}+{d+2\over
d}\eta^{\mu\nu}T\,,$ $\displaystyle\left[P^{\mu},T\right]=2B^{\mu}\,,$
$\displaystyle\left[P^{\mu},S^{\nu\lambda}\right]=\eta^{\mu\nu}B^{\lambda}+\eta^{\mu\lambda}B^{\nu}-\frac{2}{d}B^{\mu}\eta^{\nu\lambda}~{},$
$\displaystyle\left[P^{\mu},B^{\nu}\right]=\eta^{\mu\nu}C\,,$
$\displaystyle\left[P^{\mu},C\right]=0\,,$ (3.8)
$\displaystyle\left[K^{\mu},C\right]=-4B^{\mu}\,,$
$\displaystyle\left[K^{\mu},B^{\nu}\right]=-2S^{\mu\nu}-{d+2\over
d}\eta^{\mu\nu}T\,,$
$\displaystyle\left[K^{\mu},S^{\nu\lambda}\right]=-\eta^{\mu\nu}A^{\lambda}-\eta^{\mu\lambda}A^{\nu}+\frac{2}{d}A^{\mu}\eta^{\nu\lambda}~{},$
$\displaystyle\left[K^{\mu},T\right]=-2A^{\mu}\,,$
$\displaystyle\left[K^{\mu},A^{\nu}\right]=-\eta^{\mu\nu}N\,,$
$\displaystyle\left[K^{\mu},N\right]=0\,.$ (3.9)

These relations can be summarized in this figure,

(3.10)

The shift transformations thus form a finite dimensional Verma module of the
conformal algebra where $N$ is a conformal primary of weight $-2$. This Verma
module describes a symmetric traceless tensor representation of the conformal
algebra $so(2,d)$. Group the shift generators into a $d+2$ dimensional
symmetric $g$-traceless matrix, $S^{AB}=S^{BA},\ S^{AB}g_{AB}=0$, with $g$ as
in (2.10),

$S^{AB}=\left(\begin{array}[]{c|c|c}\frac{1}{4}(C+N+2T)&{1\over
4}(C-N)&-{1\over 2}\left(A^{\mu}+B^{\mu}\right)\\\ \hline\cr{1\over
4}(C-N)&\frac{1}{4}(C+N-2T)&{1\over 2}\left(A^{\mu}-B^{\mu}\right)\\\
\hline\cr-{1\over 2}\left(A^{\mu}+B^{\mu}\right)&{1\over
2}\left(A^{\mu}-B^{\mu}\right)&S^{\mu\nu}+{1\over
D}\eta^{\mu\nu}T\end{array}\right).$ (3.11)

Grouping the conformal generators as in (2.8), the commutators (3.6), (3.7),
(3.8), (3.9) become simply

$\left[J^{AB},S^{CD}\right]=g^{AC}S^{BD}-g^{BC}S^{AD}+g^{AD}S^{CB}-g^{BD}S^{CA}\,,$
(3.12)

which is the statement that $S^{AB}$ transforms as a symmetric traceless
tensor representation under the conformal algebra.

The commutators of the shift generators with each other give back conformal
generators, and are all proportional to $\alpha$,

$\displaystyle\left[S^{\mu\nu},S^{\lambda\sigma}\right]$
$\displaystyle=-\alpha\left(\eta^{\mu\lambda}J^{\nu\sigma}+\eta^{\nu\lambda}J^{\mu\sigma}+\eta^{\nu\sigma}J^{\mu\lambda}+\eta^{\mu\sigma}J^{\nu\lambda}\right),$
$\displaystyle\left[B^{\mu},S^{\nu\lambda}\right]$
$\displaystyle=\alpha\Big{(}\eta^{\mu\nu}P^{\lambda}+\eta^{\mu\lambda}P^{\nu}-\frac{2}{d}P^{\mu}\eta^{\nu\lambda}\Big{)},$
$\displaystyle\left[A^{\mu},S^{\nu\lambda}\right]$
$\displaystyle=-\alpha\Big{(}\eta^{\mu\nu}K^{\lambda}+\eta^{\mu\lambda}K^{\nu}-\frac{2}{d}K^{\mu}\eta^{\nu\lambda}\Big{)},$
$\displaystyle\left[S^{\mu\nu},C\right]$
$\displaystyle=\left[S^{\mu\nu},T\right]=\left[S^{\mu\nu},N\right]=0\,,$
$\displaystyle\left[A^{\mu},A^{\mu}\right]$
$\displaystyle=\left[B^{\mu},B^{\nu}\right]=0\,,$
$\displaystyle\left[A^{\mu},B^{\mu}\right]$
$\displaystyle=-2\alpha\left(\eta^{\mu\nu}D-J^{\mu\nu}\right)\,,$
$\displaystyle\left[A^{\mu},T\right]$ $\displaystyle=-2\alpha K^{\mu},\ \ \
\left[A^{\mu},C\right]=4\alpha P^{\mu},\ \ \left[A^{\mu},N\right]=0,$
$\displaystyle\left[B^{\mu},T\right]$ $\displaystyle=2\alpha P^{\mu},\ \ \
\left[B^{\mu},N\right]=-4\alpha K^{\mu},\ \ \left[B^{\mu},C\right]=0,$
$\displaystyle\left[N,C\right]$ $\displaystyle=16\alpha D,\ \
\left[N,T\right]=\left[C,T\right]=0.$ (3.13)

In terms of (3.11) and (2.8), the commutators (3.13) become simply

$\left[S^{AB},S^{CD}\right]=-\alpha\left(g^{AC}J^{BD}+g^{BC}J^{AD}+g^{BD}J^{AC}+g^{AD}J^{BC}\right)\,.$
(3.14)

We can now identify the full algebra. Put together the anti-symmetric
conformal generators (2.8) and the symmetric traceless shift generators (3.11)
into a traceless matrix as follows

$M^{AB}\equiv-{1\over 2}\left(J^{AB}+{1\over\sqrt{\alpha}}S^{AB}\right)\,,\ \
\alpha>0.$ (3.15)

Then the commutators (3.14), (3.12), (2.9) can be summarized as

$\left[M^{AB},M^{CD}\right]=g^{CB}M^{AD}-g^{AD}M^{CB}\,,$ (3.16)

which are the commutation relations of $sl(d+2,{\mathbb{R}})$. The smallest
symmetry algebra of the special galileon that includes the conformal symmetry
of [11] and all the extended shift symmetries is thus isomorphic to
$sl(d+2,{\mathbb{R}})$. In writing (3.15) we have assumed $\alpha>0$, the
“right sign” from the point of view of positivity bounds [28, 29]. If
$\alpha<0$, then the algebra would instead be the maximally split real form.

The breaking pattern is to the $d$ dimensional conformal algebra, since the
shifts $S^{AB}$ are the only ones that do not preserve the vacuum $\phi=0$,

$sl(d+2,{\mathbb{R}})\rightarrow so(2,d)\,,\ \ \ \alpha>0.$ (3.17)

In fact, this is the same symmetry breaking pattern as the AdS special
galileon theory in $d+1$ dimensions (the $k=2$ non-abelian AdS scalar shift
theory of [21]).

##  4 Weyl symmetry

A conformal field theory can generally be coupled to a background metric
$g_{\mu\nu}$ in such a way that the conformal symmetry descends from a local
Weyl symmetry [30, 31] (there are some exceptions in non-unitary theories, see
e.g. [32, 33, 34]). The usual form of the Weyl symmetry for a field $\phi$ of
weight $\Delta$ is

$\delta g_{\mu\nu}=2\sigma g_{\mu\nu},\ \ \ \delta\phi=-\Delta\sigma\phi,$
(4.1)

where $\sigma(x)$ is the local scalar gauge parameter. This symmetry is
present in addition to ordinary diffeomorphism (diff) symmetry with local
parameter $\xi^{\mu}(x)$,

$\delta g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu},\ \ \
\delta\phi=\xi^{\mu}\partial_{\mu}\phi.$ (4.2)

The global conformal transformations emerge when we go to flat space,
$g_{\mu\nu}=\eta_{\mu\nu}$, and restrict to those diff+Weyl symmetries which
preserve the flat metric $\eta_{\mu\nu}$. Such symmetries are those for which
$\xi^{\mu}$ is a conformal Killing vector,
$\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\nu}={2\over
d}\partial^{\lambda}\xi_{\lambda}\eta_{\mu\nu}$, and $\sigma=-{1\over
d}\partial^{\mu}\xi_{\mu}$. The conformal Killing vector corresponding to
dilations is $\xi^{\mu}=-x^{\mu}$ with $\sigma=1$. Plugging this into the
scalar transformations in (4.1), (4.2) gives the standard dilation
transformation $\delta\phi=-\left(x^{\mu}\partial_{\mu}+\Delta\right)\phi.$
The conformal Killing vector corresponding to special conformal
transformations is $\xi^{\mu}=-2b\cdot x\,x^{\mu}+x^{2}b^{\mu}$ with
$\sigma=2b\cdot x$, where $b^{\mu}$ is a constant vector parametrizing the
transformations. Plugging this into the scalar transformations in (4.1), (4.2)
gives the standard linear special conformal transformation
$\delta\phi=b_{\mu}\left(-2x^{\mu}x^{\nu}\partial_{\nu}+x^{2}\partial^{\mu}-2x^{\mu}\Delta\right)\phi$.

###  4.1 DBI

To reproduce the flat space non-linear terms in (2.6) after coupling to a
general background metric, we need new non-linear terms proportional to
$\alpha$ in the Weyl transformation. We will assume that the Weyl
transformation for the metric, and the diffeomorphisms, retain their standard
forms and do not depend on $\alpha$. The Weyl transformation for the scalar
will be modified. Each power of $\alpha$ comes with 2 derivatives and 2 powers
of $\phi$, so the most general possible modification at order $\alpha$ that
reduces to (2.4) for $\sigma=1$ and (2.6) for $\sigma=2b\cdot x$ is

$\delta\phi=\sigma\phi+\alpha\left({1\over
2}\phi^{2}\partial_{\mu}\phi\partial^{\mu}\sigma+a_{1}\phi^{3}R\sigma+a_{2}\phi^{3}\square\sigma\right)+{\cal
O}\left(\alpha^{2}\right)\,,$ (4.3)

with constants $a_{1},a_{2}$.

We now want to couple (2.1) to the metric such that this new Weyl
transformation is a symmetry when $d=0$. The most general coupling to the
metric, to order $\alpha$, is

$\displaystyle S=\int d^{d}x\
\sqrt{-g}\bigg{[}{1\over\alpha}-{1\over\alpha}\sqrt{1+\alpha(\partial\phi)^{2}}+b_{0}R\phi^{2}$
(4.4)
$\displaystyle+\alpha\left(b_{1}R^{\mu\nu}\phi^{2}\partial_{\mu}\phi\partial_{\nu}\phi+b_{2}R\phi^{2}(\partial\phi)^{2}+b_{3}R\phi^{3}\square\phi+b_{4}W_{\mu\nu\lambda\rho}^{2}\phi^{4}+b_{5}R_{\mu\nu}^{2}\phi^{4}+b_{6}R^{2}\phi^{4}\right)+{\cal
O}\left(\alpha^{2}\right)\bigg{]}\,,$

where $W_{\mu\nu\lambda\rho}$ is the Weyl tensor and $b_{0},\ldots,b_{6}$ are
constants. Requiring Weyl invariance up to order $\alpha$ when $d=0$ then
fixes $a_{1}=0$, $b_{0}=-1/4$, $b_{1}=1/4$, $b_{2}=0$, $b_{3}=-a_{2}/2$,
$b_{5}=1/32$, $b_{6}=a_{2}/4$, while leaving $a_{2},b_{4}$ free.

Thus the Weyl symmetry and invariant action to order $\alpha$ is

$\displaystyle S=\int d^{d}x\
\sqrt{-g}\bigg{[}{1\over\alpha}-{1\over\alpha}\sqrt{1+\alpha(\partial\phi)^{2}}-{1\over
4}R\phi^{2}$ $\displaystyle+\alpha\left({1\over
4}R^{\mu\nu}\phi^{2}\partial_{\mu}\phi\partial_{\nu}\phi+{1\over
32}R_{\mu\nu}^{2}\phi^{4}-{a\over 2}R\phi^{3}\square\phi+{a\over
4}R^{2}\phi^{4}+bW_{\mu\nu\lambda\rho}^{2}\phi^{4}\right)+{\cal
O}\left(\alpha^{2}\right)\bigg{]}\,,$ $\displaystyle\delta g_{\mu\nu}=2\sigma
g_{\mu\nu},\ \ \ \ \delta\phi=\sigma\phi+\alpha\left({1\over
2}\phi^{2}\partial_{\mu}\phi\partial^{\mu}\sigma+a\phi^{3}\square\sigma\right)+{\cal
O}\left(\alpha^{2}\right)\,.$ (4.5)

where $a$, $b$ are the two unfixed free parameters. The parameter $a$ can be
further removed by the field redefinition

$\phi\rightarrow\phi+\tfrac{1}{2}a\alpha R\phi^{3}\,,$ (4.6)

as in [30]. This is the only allowed field redefinition at order $\alpha$ that
does not transform the scalar, and hence does not modify the DBI form of the
Lagrangian, when restricted to flat space. The parameter $b$ remains arbitrary
since it comes in front of the Weyl-squared term, which is separately Weyl
invariant to lowest order. This term would presumably be completed at higher
order into some invariant of the $\alpha$-deformed Weyl symmetry, or else
forced to vanish if no such invariant exists. Note that the coupling to the
metric generically breaks all the shift symmetries.

This new Weyl symmetry satisfies the same algebra as the standard Weyl
symmetry, namely that the Weyl transformations are abelian,

$\left[\delta_{\sigma_{1}},\delta_{\sigma_{2}}\right]=0,$ (4.7)

up to order $\alpha$. In fact this requirement alone is enough to fix the form
of the transformation: demanding (4.7) alone for (4.3) when $d=0$ fixes
$a_{1}=0$, and then taking into account the possible field re-definition (4.6)
removes $a_{2}$.

In [11], an action linear in the curvatures and including all powers of $\phi$
was provided which makes the stress tensor on flat space traceless. The terms
linear in the curvature in the action (4.5) match the terms linear in the
curvature from [11], up to order $\alpha$ (and when $a=0$). However the action
in [11] cannot itself be Weyl invariant in the manner we are requiring,
because, as we can see from (4.5), terms quadratic in the curvature are
required.

Pushing to next order in $\alpha$ we find that the action can be made Weyl
invariant when $d=0$ to quadratic order in $\alpha$, with the addition of the
following terms to the action,

$\displaystyle S\bigg{|}_{\alpha^{2}}=$ $\displaystyle\ \int d^{d}x\
\sqrt{-g}\,\bigg{\\{}-\frac{\alpha^{2}}{32}R\phi^{2}(\partial\phi)^{4}-\frac{\alpha^{2}}{8}R^{\mu\nu}\phi^{2}(\partial\phi)^{2}\nabla_{\mu}\phi\nabla_{\nu}\phi$
(4.8)
$\displaystyle+\frac{\alpha^{2}}{64}\phi^{4}\bigg{[}\left(R_{\mu\nu}^{2}+\tfrac{1}{2}R^{2}\right)(\partial\phi)^{2}-2RR^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-6R^{\mu\nu}R_{\nu\sigma}\nabla_{\mu}\phi\nabla^{\sigma}\phi\bigg{]}$
$\displaystyle+\frac{\alpha^{2}}{48}W_{\mu\nu\rho\sigma}\bigg{[}2\phi^{3}\nabla^{\mu}\phi\nabla^{\rho}\phi\nabla^{\nu}\nabla^{\sigma}\phi-3R^{\mu\rho}\phi^{4}\nabla^{\nu}\phi\nabla^{\sigma}\phi-\tfrac{1}{6}\phi^{6}\nabla^{\sigma}\nabla^{\nu}R^{\mu\rho}-\phi^{5}\nabla^{\nu}R^{\mu\rho}\nabla^{\sigma}\phi\bigg{]}$
$\displaystyle-\frac{\alpha^{2}}{192}\phi^{6}\bigg{[}W_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma}+R^{\mu\nu}R_{\nu\tau}R^{\tau}_{\mu}-\tfrac{1}{4}R^{3}\bigg{]}\bigg{\\}}\,,$

along with the additional non-linear terms in the Weyl transformation,

$\displaystyle\delta\phi\bigg{|}_{\alpha^{2}}=\frac{\alpha^{2}}{4}S^{\mu\nu}\phi^{4}\nabla_{\mu}\phi\nabla_{\nu}\sigma.$
(4.9)

Here $S_{\mu\nu}$ is the Schouten tensor, defined in general $d$ as

$\displaystyle
S_{\mu\nu}=\frac{1}{(d-2)}\left(R_{\mu\nu}-\frac{1}{2(d-1)}Rg_{\mu\nu}\right).$
(4.10)

This action and Weyl transformation will also have ambiguities due to field
re-definitions that vanish in flat space (coming from order $\alpha^{2}$ terms
in (4.6)), as well as ambiguities from possible Weyl invariants. Here we have
not attempted to parameterize all these ambiguities but have simply made a
choice for which the action and Weyl transformation are manageable, in order
to illustrate that there is no obstruction to extending the Weyl symmetry to
order $\alpha^{2}$. The first line of (4.8) matches those found in the action
linear in the curvatures constructed in [11] by demanding the theory have a
vanishing stress tensor in flat space. In the third line of (4.8) there are
additional terms linear in the curvature that do not appear in [11], but these
are proportional to the Weyl tensor and so they do not contribute to the flat
space stress tensor. Thus what we find is consistent with [11] up to order
$\alpha^{2}$. It would be interesting if the action in [11] could be extended
to an action which is fully Weyl invariant to all orders.

###  4.2 Special Galileon

A similar procedure can be applied in the case of the special galileon in
$d=-2$, and once again we find that the theory can be improved to be Weyl
invariant, at least to $\mathcal{O}(\alpha)$. The non-linear conformal
transformations (3.2), (3.3) get uplifted to

$\displaystyle\delta\phi=2\sigma\phi-\frac{\alpha}{2}(\partial\phi)^{2}\nabla_{\mu}\sigma\nabla^{\mu}\phi\,,$
(4.11)

in a general background metric. A priori there are many more terms that could
enter this transformation, however requiring that the Weyl transformations
commute as in (4.7) reduces this number, and employing field redefinitions
that vanish on flat space analogous to (4.6) removes the rest. We find the
following action is Weyl invariant under (4.11) in $d=-2$, up to order
$\alpha$,

$\displaystyle S=$ $\displaystyle\int
d^{d}x\sqrt{-g}\,\bigg{\\{}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{6}R\phi^{2}-\frac{\alpha}{12}(\partial\phi)^{2}\Big{[}(\square\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}\Big{]}$
(4.12)
$\displaystyle-\frac{\alpha}{12}R^{\mu\nu}\phi^{2}\nabla_{\mu}\nabla^{\tau}\phi\nabla_{\nu}\nabla_{\tau}\phi+\frac{\alpha}{72}R\phi^{2}\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi$
$\displaystyle+\frac{\alpha}{12}R^{\mu\nu}\phi^{2}\Box\phi\nabla_{\mu}\nabla_{\nu}\phi-\frac{\alpha}{72}R\phi^{2}(\Box\phi)^{2}$
$\displaystyle-\frac{\alpha}{35}W^{\mu\nu\rho\sigma}\phi\bigg{[}\frac{13}{6}\nabla_{\rho}\phi\nabla_{\mu}\phi\nabla_{\sigma}\nabla_{\nu}\phi-\phi\nabla_{\rho}\nabla_{\mu}\phi\nabla_{\sigma}\nabla_{\nu}\phi+W_{\tau\rho\nu\sigma}\phi\nabla_{\mu}\phi\nabla^{\tau}\phi\bigg{]}$
$\displaystyle-\frac{\alpha}{48}\bigg{[}9R^{\mu\nu}R_{\mu\tau}\phi^{2}\nabla^{\tau}\phi\nabla_{\nu}\phi-\frac{7}{2}R_{\mu\nu}^{2}(\partial\phi)^{2}\phi^{2}-\frac{1}{6}R^{2}(\partial\phi)^{2}\phi^{2}-3RR^{\mu\nu}\phi^{2}\nabla_{\mu}\phi\nabla_{\nu}\phi\bigg{]}$
$\displaystyle-\frac{\alpha}{210}\bigg{[}15R^{\nu\rho}\nabla^{\sigma}W_{\mu\nu\rho\sigma}\phi^{3}\nabla^{\mu}\phi-14W_{\mu\nu\rho\sigma}R^{\mu\rho}\phi^{2}\nabla^{\sigma}\phi\nabla^{\nu}\phi-\frac{1}{2}R^{\mu\rho}\nabla^{\nu}\nabla^{\sigma}W_{\mu\nu\rho\sigma}\phi^{4}\bigg{]}$
$\displaystyle+\frac{\alpha}{48}\bigg{[}\frac{1}{7}W_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma}-\frac{1}{2}R_{\mu\nu}R^{\mu\rho}R_{\rho}^{\nu}+\frac{1}{4}R_{\mu\nu}^{2}R+\frac{1}{36}R^{3}\bigg{]}\phi^{4}\bigg{\\}}+{\cal
O}\left(\alpha^{2}\right)\,.$

The terms in the second line are the same as those in [11] found by demanding
tracelessness of the stress tensor in flat space444Our expressions match those
in [11] for $\alpha=-1$.. In the third line there are additional terms linear
in the curvature that do not appear in [11], but these can be removed by field
re-definitions that vanish in flat space with $\sigma=2b\cdot x$, at the
expense of making the Weyl transformation (4.11) more complicated. Although
admittedly not very illuminating, we present these results both to demonstrate
there is no immediate obstruction to Weyl invariance, and in the hope that
these actions may be found to arise from a more fundamental understanding of
the Weyl invariance of these theories to all orders in $\alpha$.

##  5 Conclusions

We have found the field transformations behind the conformal symmetries for
DBI in dimension $d=0$ and the special galileon in dimension $d=-2$ that were
found in [11]. Using these, we have studied how these symmetries interplay
with the extended shift symmetries, finding new symmetries that close them
into a larger algebra. In the DBI case, this larger algebra can be seen as the
conformal algebra of the $d+1$ dimensional target space in the embedded brane
way of understanding DBI. In the special galileon case it is a real form of
the special linear algebra (it would be interesting if this could be
understood from a geometric picture of the special galileon [35, 29]). We have
also found the corresponding Weyl invariances, to second order in the coupling
in the DBI case and to lowest order in the coupling in the special galileon
case.

A natural question is whether theories with higher order shift symmetries [36,
37], $\delta\phi\sim x^{k}$ for $k>2$, could emerge and have conformal
symmetry. Continuing the pattern we see here for DBI ($k=1$) and special
galileon $(k=2)$, we would expect such a theory to be conformal in dimension
$d=-2(k-1)$, where the scalar has weight $\Delta=-k$. We would expect the
shift symmetries and new symmetries found by commuting with the conformal
generators to form a rank $k$ symmetric-traceless representation of the
$so(2,d)$ conformal algebra, where the conformal primary has weight $-k$.
However, as shown in [21], the commutators of these generators among
themselves cannot close back to the conformal algebra when $k>2$, except in
the trivial case where such commutators are all abelian, which would
presumably correspond to a free theory or one whose structure is not fixed in
a non-trivial way by the symmetry (this is consistent with the results of [38,
39] forbidding a non-trivial algebra for $k>0$ theories in flat space). This
conclusion does not apply if multiple fields of higher spin are allowed, for
which non-trivial algebras are possible [40, 21, 41].

Given that these theories have a formal conformal invariance in unphysical
$d$, it would be interesting to explore whether the tools of the modern
conformal bootstrap [42, 43] could be extended to these unphysical dimensions
and brought to bear in order to non-perturbatively explore these theories. An
immediate obstacle is that the conformal transformations we have are non-
linear and so the fields do not transform as conformal primaries in the usual
way, so the form of correlators, crossing symmetry, etc. are all presumably
modified. Alternatively, perhaps there are composite operators that transform
in the usual linear way as conformal primaries, given that the basic field
$\phi$ transforms non-linearly. If such operators exist, their correlators
could be a target for the bootstrap. But there is no guarantee that they
should exist, especially given expectations that these soft scalar theories
are not true local field theories with local operators all the way to the UV
[44, 45].

#### Acknowledgments:

We would like to thank James Bonifacio, Austin Joyce and David Stefanyszyn for
helpful conversations. KH acknowledges support from DOE grant DE-SC0019143, KH
and KF acknowledge support from Simons Foundation Award Number 658908. OH was
supported by the GAČR Grant EXPRO 20-25775X.

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# Current induced torques between ferromagnets and compensated
antiferromagnets: symmetry and phase coherence effects

Karthik Prakya1, Adrian Popescu2,3, and Paul M. Haney2 1\. The MITRE
Corporation, Bedford, MA 01730
2\. Center for Nanoscale Science and Technology, National Institute of
Standards and Technology, Gaithersburg, MD 20899
3\. Maryland NanoCenter, University of Maryland, College Park, MD 20742, USA

###### Abstract

It is shown that the current-induced torques between a ferromagnetic layer and
an antiferromagnetic layer with a compensated interface vanish when the
ferromagnet is aligned with an axis of spin-rotation symmetry of the
antiferromagnet. For properly chosen geometries this implies that the current
induced torque can stabilize the out-of-plane (or hard axis) orientation of
the ferromagnetic layer. This current-induced torque relies on phase coherent
transport, and we calculate the robustness of this torque to phase breaking
scattering. From this it is shown that the torque is not linearly dependent on
applied current, but has an absolute maximum.

###### pacs:

85.35.-p, 72.25.-b,

##  I Introduction

Current-induced torques result from the interaction between conduction
electron spins and the magnetization of a sample when current flows through
it. This torque is generally present when the magnetization is spatially
nonuniform, and has been extensively studied in the context of magnetic domain
walls and spin valve structures. Since its theoretical predictionslonc ;
berger , extensive studies have led to a theoretical framework of current-
induced torque in ferromagnets that describes experimental results with
quantitative success.stiles:jmmm  It has been proposed that current-induced
torques also exist in antiferromagnetic systems.nunez ; haney:jmmm  Previous
theoretical studies considered systems composed entirely of antiferromagnetic
layers nunez ; duine:prb ; haney:prb  as well as experimental wei  and
theoreticalhaney:prl ; loktev ; loktev2  systems with both ferromagnetic and
antiferromagnetic layers. Theoretical work has also focused on antiferromagnet
textures.brataas1 ; brataas2 ; duine:2011 ; niu  Experiments have demonstrated
current-induced torque in materials with other types of complex magnetic
ordering, such as skyrmion lattices. Recent theoryshick  and
experimentjungwirth  have shown that antiferromagnets exhibit anisotropic
magnetoresistance, demonstrating a coupling between magnetic order and charge
transport these materials.

Antiferromagnets exhibit an array of magnetic ordering, such as spin density
waves that are commensurate or incommensurate with the lattice, and
configurations with multiple spin density waves. As shown in Ref. haney:prl, ,
the symmetry properties of the antiferromagnetic layer can lead to torques in
multilayers with qualitatively different properties than conventional spin
valves. In particular, a collinear compensated antiferromagnetic layer
interface (with each spin in the $\pm{\hat{z}}$ direction, which we call a 1Q
spin structure) leads to a torque which vanishes when the ferromagnet is
perpendicular to the $\hat{z}$ direction. This torque can stabilize the hard-
axis orientation of the ferromagnet in systems where the antiferromagnet is
pinned. Here we treat similar systems (see Fig. 1a), and compute the current-
induced torque on the ferromagnetic layer. (Previous works have investigated
the current-induced torque on the antiferromagnetic layer in such
systems.loktev ; loktev2 )

In this work we consider a system where the antiferromagnetic layer has a 3Q
spin structure (see Fig. 1b). This is qualitatively different than the
previously studied 1Q antiferromagnet because the 3Q structure has only a
single axis of spin rotational symmetry (3-fold in this case), whereas for the
1Q antiferromagnet all directions perpendicular to the $\hat{z}$ direction are
axes of 2-fold spin rotational symmetry. We show that an important consequence
of the reduced symmetry of the 3Q antiferromagnet is that the current-induced
torque stabilizes the out-of-plane magnetic orientation only when the
magnetization is initialized nearby this orientation (in contrast, the 1Q
antiferromagnet drives any initial orientation out-of-plane). In this work we
additionally determine the effects of phase breaking scattering: The current-
induced torque relies on phase coherence, and quantifying the robustness with
respect to scattering is important to gauge the feasibility of observing these
effects in real systems.

Our results are easily generalized to multilayers composed of a free
ferromagnet layer, and a fixed magnetic layer whose spin configuration has an
axis of $n$-fold rotational symmetry. The key property of the torque is that:
if the ferromagnetic layer is aligned with an axis of spin-rotational symmetry
of the fixed layer, then the current-induced torque (in fact, all torques)
must vanish. This is seen by recognizing that, by assumption, the system is
invariant with respect to spin rotations about the ferromagnet orientation by
some angle $\phi_{n}$, and any nonzero torque (which must be perpendicular to
the ferromagnet orientation) does not respect this symmetry. For conventional
spin valves, this statement implies the well known fact that the current-
induced torque vanish when the ferromagnet layers are aligned or anti-aligned.
Identifying the points where the current-induced torque vanishes is important
because the torque may drive the magnetization to these fixed points. For
properly designed antiferromagnet-ferromagnet multilayers this property of the
torque can stabilize the out-of-plane magnetic orientation.haney:prl  This is
because this orientation, being a maximum of the magnetic free energy,
represents a fixed point for the conventional micromagnetic torques. In the
absence of current-induced torques, this out-of-plane configuration is an
unstable fixed point; however if the current-induced torque drives the
ferromagnet to this orientation and exceeds the damping torque, it can
stabilize this configuration, as shown by micromagnetic simulations in Ref.
haney:prl, .

##  II Method

To calculate the current-induced torques, we use the nonequilibrium Green’s
function technique within a tight binding representation. This is a well
established approach to calculating the transport properties of magnetic thin
films. We highlight the most important details here. The system is taken to
consist of two semi-infinite electrodes, with a scattering region placed
between them. There is a difference $V_{\rm app}$ in the electrochemical
potential of the two electrodes. The central quantity is the density matrix
$\rho$:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{-\infty}^{E_{\rm F}-V_{\rm
app/2}}\left[G^{r}\left(E\right)-G^{a}\left(E\right)\right]dE+$ (1)
$\displaystyle~{}~{}~{}\int_{E_{\rm F}-V_{\rm app}/2}^{E_{\rm F}+V_{\rm
app/2}}G^{r}\left(E\right)\Gamma_{L}\left(E\right)G^{a}\left(E\right)dE.$

where
$G^{r,a}\left(E\right)=\left(E-H_{C}-\Sigma^{r,a}_{L}\left(E\right)-\Sigma^{r,a}_{R}\left(E\right)\right)^{-1}$.
$H_{C}$ is the scattering region Hamiltonian, and $\Sigma^{r}_{L}$ is the self
energy which describes the electronic coupling between the scattering region
and the semi-infinite left lead; it is given by
$\Sigma^{r}_{L}=V_{C,L}^{\dagger}g_{0,L}^{r}\left(E\right)V_{C,L}$, where
$V_{C,L}$ is the coupling matrix element between the left lead and central
region, and $g_{0,L}$ is the surface Green’s function of the isolated semi-
infinite left lead. The same form of self energy holds for the right lead.

As noted in previous works,duine:prb  phase coherence plays a central role in
a number of the antiferromagnetic systems studied so far. To explore the
robustness of the torques in this system, we include an additional self energy
$\Sigma_{S}$ in Green’s function which describes elastic, phase breaking
scattering. Its form is:

$\displaystyle\Sigma_{S}\left(E\right)=iD\left(G^{r}\left(E\right)-G^{a}\left(E\right)\right)$
(2)

where $D$ parameterizes the scattering. A discussion of the parameter $D$ in
terms of real material properties and temperature is given in Sec. (III).

We assume the spin-orbit coupling is negligible, so that the current-induced
torque on the ferromagnet layer is simply given by the transverse component of
incoming spin current flux. For our geometry, the net spin current has real
space velocity in the $\hat{y}$ direction. The spin current operator
$\vec{J}\left(y\right)$ is then:

$\displaystyle\hat{\vec{J}}\left(y\right)=\sum_{\begin{subarray}{c}j\in
R\left(y\right)\\\ k\in L\left(y\right)\\\
s,s^{\prime}\end{subarray}}i\left[c^{\dagger}_{j,s}\vec{\sigma}_{s,s^{\prime}}c_{k,s^{\prime}}t_{j,k}-{\rm
h.c.}\right],$ (3)

where $R\left(y\right)$ are the set of sites with coordinate $y^{\prime}$
greater than $y$, and $L\left(y\right)$ are the set of sites with coordinate
$y^{\prime}$ less than $y$. $\vec{\sigma}$ sigma is the vector of Pauli
matrices, and we take the hopping $t_{j,k}$ between all sites $j$ and $k$ to
be spin independent. We present results in terms of torque per current (units
of ${\mu_{B}}/e$), which represents the spin torque efficiency. The absolute
value of this quantity determines the critical current needed to drive
magnetic dynamics.

As discussed in Refs. haney2007,  and nikolic, , it is sometimes necessary to
compute the entire energy integral (both terms in Eq. 1) in order to find the
current-induced torques. This is particularly the case when the torques in
question are present in equilibrium (which is itself dependent on the
symmetries of the system, as discussed in Ref. haney2007, ). We checked
explicitly that the current-induced torque in question for these systems are
dominated by the nonequilibrium contribution to the density matrix (the second
term of Eq. 1), and present only this contribution in the results (the
remaining “energy integral” contribution is several orders of magnitude
smaller in all the cases we checked). We take the Fermi energy to be
$E_{F}=3.75~{}t$, and use a dense $k$-point mesh to converge the transport
integrals, up to $1000^{2}$ k-points for a unit cell having 4 atoms per layer.

Figure 1: (a) Overall system geometry (b) The crystal and spin structure for
the 3Q state. The spins at the corners of the interior box all point inward.
(c) The spin on the [111] interface of the lattice from (b). The small black,
medium red, and large gray dots represent atoms in different layers (i.e with
different y-values). The spin of dots without an arrow is completely in the
$\hat{y}$ direction, while other spins are partially canted in the ${\hat{y}}$
direction. (d) Spherical coordinate system used to describe the torques on the
ferromagnet. The blue (dark) spins in the $x-z$ plane represent the 3-fold
symmetric spins of the antiferromagnetic layer, and the skinnier red arrow
represents the orientation of the ferromagnet layer.

A schematic of the overall system is shown in Fig. 1a. It consists of semi-
infinite ferromagnetic and antiferromagnetic layers, separated by a
nonmagnetic spacer which is 3 atomic layers thick. The layers are fcc, with
interfaces in the [111] direction. We use two different spin structures for
the antiferromagnet. One is a 3Q spin structure, depicted in Fig. 1b. The spin
structure in the $(111)$ planes is shown in Fig. 1c, which shows the 3-fold
symmetry of the spin in the $x-z$ plane. Each spin also has a component along
the $y$ axis (into or out of the page). Atoms with no arrow in the figure have
a spin fully aligned in the $+\hat{y}$ direction, while other atoms’ spins are
partially canted in the $-\hat{y}$ direction, so that the net bulk spin
vanishes. Common antiferromagnetic materials such as FeMn are predicted to
have a 3Q ground state,schulthess ; footnote2 ; stocks  consistent with
measurementskawarazaki ; kennedy , although there is not complete consensus
between all the experimental data. To further explore the consequences of the
antiferromagnet symmetry, we also consider a system where the $y$ component of
the spins are set to 0. This artificial system retains the 3-fold symmetry in
the $x-z$ plane, but is also symmetric under $s_{y}\leftrightarrow-s_{y}$. We
refer to this as the “no-canting” antiferromagnet. We emphasize that our
primary results generalize to any antiferromagnet for which there is an axis
of $n$-fold spin rotational symmetry, as explained in the introduction.

We present the angular variation of the torque on the ferromagnet layer in
terms of spherical coordinates, as shown in Fig. 1d. The $\hat{y}$ direction
is the hard axis of the F, which is taken to coincide with the axis of 3-fold
symmetry of the antiferromagnet. As explained in the introduction, this
alignment of hard axis and the antiferromagnet axis of spin rotational
symmetry is crucial for the out-of-plane orientation to be stabilized by the
current-induced torque. The $\hat{z}$ direction is along one of the spins of
the antiferromagnetic layer. We utilize similar schematics as Fig. 1d in the
next section to show the relative orientation of the ferromagnet layer with
the spins of the antiferromagnet.

##  III Results

The current-induced torque on the ferromagnetic layer for a no-canting
antiferromagnetic system is shown in Fig. 2. Unlike the current-induced torque
in a conventional spin valve, whose magnitude has a simple $\sin(\theta)$
dependence, we find a more complex angular dependence for the torque. We first
fix $\phi=0^{\circ}$ and vary the ferromagnet orientation from $\theta=0$ to
$360^{\circ}$. These orientations are in the easy plane. The torques conform
to the 3-fold symmetry, varying approximately as $\sin\left(3\theta\right)$,
as shown in Fig. 2a. For fixed $\phi=90^{\circ}$, sweeping the polar angle
$\theta$ takes the magnetization out of the easy plane, through the hard axis
direction. The torques in this case are shown in Fig. 2c. The torques vary as
$\sin\left(2\theta\right)$, again as required by symmetry. For fixed
$\phi=45^{\circ}$, varying $\theta$ takes the ferromagnet on an “off-axis”
orbit, and the torque exhibits more complex angular dependence.

Figure 3 shows similar results for the 3Q spin structure for the same set of
magnetic orientations. The reduction in symmetry due to the inequivalence of
$s_{y}$ and $-s_{y}$ leads to more complex behavior of the torque. For
$\phi=0^{\circ}$, we note the invariance of the torque under
$\theta\rightarrow\theta+120^{\circ}$. Key data points are shown in Fig. 3c by
the black arrows. As argued in the introduction, when the ferromagnet layer is
aligned to the axis of 3-fold symmetry, the current-induced torque vanishes.

Figure 2: The angular dependence of the current-induced torque (CIT) on the
ferromagnet for the system with no antiferromagnetic canting in the
$y$-direction (the “no canting” system). The black dashed line is the torque
in the $\hat{\phi}$ direction, and the gray line with markers is the torque in
the $\hat{\theta}$ direction. (a) shows the torque as when the ferromagnet is
coplanar with the antiferromagnet spins, which shows a $\sin(3\theta)$
dependence. (b) shows an intermediate angle, and (c) shows the torque as the
ferromagnet orientation is normal to the plane of the antiferromagnet spins.
In this case, the torque varies as $\sin\left(2\theta\right)$. The diagrams to
the right of the plots show the direction of antiferromagnet spins in the x-z
plane, and with a circle representing the angles of the ferromagnet layer in
the plot. Figure 3: The angular dependence of the torque on the ferromagnet
for the system with no 3Q spin ordering of the antiferromagnet. The black
dashed line is the torque in the $\hat{\phi}$ direction, and the gray line
with markers is the torque in the $\hat{\theta}$ direction. (a) shows that the
torque again varies as $\sin(3\theta)$ when the ferromagnet layer is confined
to the $x-z$ plane (easy plane). (b) shows complex angular dependence for the
ferromagnet layer oriented along an axis of low symmetry. (c) shows that the
torque vanishes when the ferromagnet is aligned to the axis of 3-fold symmetry
of the antiferromagnet (arrows indicate these points).

To gain a fuller view of the current-induced torque near the out-of-plane
fixed point, we show the torque in the vicinity of these points in Fig. 4. For
the no-canting system, the $+\hat{y}$ and $-\hat{y}$ fixed point are
equivalent. For electrons flowing from the antiferromagnet to the ferromagnet,
these are stable fixed points. For the 3Q antiferromagnet, on the other hand,
the $+\hat{y}$ and $-\hat{y}$ fixed points are inequivalent. In this case, we
find the $+\hat{y}$ is a stable attractor, while the $-\hat{y}$ is an elliptic
fixed point. The nature of the fixed point (stable, unstable, elliptic, etc.)
is parameter dependent, making it difficult to make general statements about
the prevalence of different fixed points.

For antiferromagnetic systems it is also important to distinguish between
stable fixed points to which any initial magnetization vector is driven
(global attractors), and those fixed points for which only an initial
magnetization vector nearby is driven (local attractors). Inspection of Fig.
2a shows that, if the magnetization is in the $x-z$ plane, the torque driving
it to the out-of-plane direction is quite weak (in this case, the relevant
torque is in the $\hat{\phi}$ direction). On the other hand, if the
magnetization is near the $z-y$ plane (Fig. 2c), the torque driving it to the
out-of-plane orientation ($\Gamma_{\theta}$) is much stronger. Rather than
characterizing the flow of the current-induced torque field for any particular
system in detail (which is highly parameter dependent), we simply emphasize
that an experiment is more likely to observe these torques if the
magnetization is initially in the out-of-plane before the current is applied.
Application of a current can stabilize this configuration, so that subsequent
removal of the applied field does not result in the magnetization returning to
the easy plane.

Figure 4: A zoom-in view of the torques on the ferromagnet layer near the
fixed point of the current-induced torque. (a) shows the result for the “no
canting” system, where the $\pm y$ fixed points are equivalent. The red dot on
the sphere on the right represents the magnetic orientation shown in the left
panel. The dark blue arrows represent the orientation of the antiferromagnet
spins. (b) shows the result for the 3Q system. The torques indicate that the
$+{\hat{y}}$ orientation is a stable fixed point. (c) shows that the
$-{\hat{y}}$ orientation is an elliptic fixed point. (The three blue (dark)
arrows of (a) have no $\hat{y}$ component, while for (b) and (c), the three
similar blue (dark) arrows are canted, acquiring a small positive $\hat{y}$
component.)

In contrast to the current-induced torque in noncollinear ferromagnets, the
current-induced torque in many antiferromagnet systems rely on phase
coherence.duine:prb  This is because the eigenstates of the bulk
antiferromagnet are degenerate Kramer’s doublets with opposite spins. A
distribution of these eigenstates carries no net spin current. However, spin-
dependent reflection at the ferromagnet interface leads to a superposition of
these degenerate states, which results in a nonzero spin polarization of the
current in the antiferromagnet. The component of this spin current
perpendicular to the ferromagnet is responsible for the torque on the F, and
vanishes as the coherence between the states is destroyed. The requirement of
ballistic (or quasi-ballistic) transport imposes more stringent requirements
on the existence of current-induced torques in antiferromagnets than in
ferromagnets. Materials should be nearly single crystal, and scattering (from
e.g. phonons) should be minimized. In order to estimate the acceptable limits
of electron-phonon scattering, we add an elastic scattering channel to the
Green’s function self-energy as described in Sec. II. Fig. 5 shows how
increased scattering decreases current-induced torque near the out-of-plane
fixed point of the no-canting system. Here the scattering parameter $D$ is
normalized by the square of the hopping matrix element $t$.

Figure 5: (a) The magnitude of current-induced torque near the fixed point of
the “no-canting” system as a function of the elastic scattering parameter
${D/t^{2}}$. The parameters used in the curve fit are:
$\Gamma_{0}=0.0478~{}\left(\mu_{B}/e\right),~{}A=670~{}t^{-2}$. (b) The same
plot for the 3Q system. The fit applies only to (a).

To place the result of Fig. 5 in context, we write $D$ in terms of material
properties. For simplicity, we focus on just one phase breaking process:
elastic acoustic phonon scattering. Our aim is to explicitly show that the
current-induced torque, as a function of the applied current, has a maximum
absolute value. Depending on materials properties and temperature, other
scattering processes may be more important. In any event, for acoustic phonon
scattering, $D$ takes the form:lundstrom

$\displaystyle D=\frac{E_{a}^{2}k_{\rm B}T}{\rho v^{2}a^{3}}\equiv D_{0}T$ (4)

where $E_{a}$ is the elastic deformation potential, $\rho$ is the material
density, $v$ is the speed of sound, $a$ is the lattice spacing, and $T$ is the
temperature. The linear $T$ dependence reflects the increased thermal
population of phonons with increasing temperature. Other scattering process
(e.g. electron-electron scattering, inelastic phonon scattering) depend on $T$
differently; generally $D\propto T^{p}$ where $p$ varies from 0.5 to 3 (see
Ref. mohanty,  and references within).

Joule heating may increase the importance of thermal effects: for current
density $J$ flowing through a material with resistivity $\Omega$, thermal
conductivity $\kappa$, and length $L$ along the current direction (in this
case, the $\hat{y}$-direction), the spatially averaged temperature increases
by a factor on the order of $J^{2}L^{2}\Omega/\kappa$. To stabilize the out-
of-plane magnetic orientation requires a current density of $\alpha\gamma
M_{s}t_{\rm F}/2g$footnote1 , where $g$ is the current-induced torque per
current, $\alpha$ is the damping, $\gamma$ is the gyromagnetic ratio, $M_{s}$
is the saturation magnetization of the ferromagnet layer, and $t_{\rm F}$ is
the thickness of the ferromagnet layer. For the no-canting system, the
current-induced torque per current is $g=0.05~{}\mu_{\rm B}/e$. According to
this estimate and typical material parameters, this leads to a critical
current density on the order of $10^{12}~{}{\rm A/m^{2}}$. Taking
$\rho=10^{-7}~{}{\rm\Omega\cdot m},~{}\kappa=50~{}{\rm W/\left(m\cdot
K\right)},L=50~{}{\rm nm}$ leads to only a modest increase in temperature,
less than $10~{}{\rm K}$. The other parameters of Eq. 4 for metals are
typically $E_{a}=10~{}{\rm eV},~{}\rho=10^{4}~{}{\rm
kg/m^{3}},~{}v=5000~{}{\rm m/s},~{}a=0.35~{}{\rm nm}$. In total, we find a $D$
parameter on the order of $10^{-5}~{}{\rm eV}^{2}$ to $10^{-4}~{}{\rm
eV}^{2}$. In light of Fig. 5, this implies that elastic phonon scattering does
not immediately destroy the current-induced torque for the no-canting system.
On the other hand, the much weaker current-induced torque per current of the
3Q system ($g=4\times 10^{-4}~{}\mu_{\rm B}/e$) requires a 100-fold increase
in the current to stabilize the out-of-plane orientation, a current density
which exceeds the maximum these systems can accommodate.

We’ve observed that the current-induced torque decays as $1/D$ for the no-
canting system. This is not universal behavior. Indeed, the current-induced
torque in the 3Q system is nonmonotonic with scattering parameter
$D$.footnote3  Despite its non-universality, we find it instructive to assume
such a dependence in order to derive closed form expressions for the maximum
current-induced torque as a function of applied current density. Recalling
that $D$ is proportional to $T$, we find the absolute current-induced torque
$\Gamma_{\rm abs}$ (units of torque) varies with current as:

$\displaystyle\Gamma_{\rm
abs}\left(J\right)=\frac{\Gamma_{0}J}{1+AD_{0}\left(T_{0}+BJ^{2}\right)},$ (5)

where $\Gamma_{0}$ is the current-induced torque in the absence of scattering
(recall $\Gamma_{0}$ has units of torque per current), $T_{0}$ is the sample
temperature in the absence of current, $B=L^{2}\rho/\kappa$ describes the
system’s susceptibility to current-induced heating, and $D_{0}$ is defined in
Eq. 4.footnote2  The absolute current-induced torque has a maximum - for
current densities that are too large, the magnitude of the current-induced
torque decreases due to increased scattering from Joule heating. The maximum
absolute current-induced torque is given by:

$\displaystyle\Gamma_{\rm abs}^{\rm
max}=\frac{\Gamma_{0}}{3L}\sqrt{\frac{2\kappa}{\rho
D_{0}A\left(1+D_{0}T_{0}A\right)}},$ (6)

The parameters $\Gamma_{0}$ and $A$ are entirely system specific, and related
to the spin dependent transport properties of a system, and their robustness
with respect to scattering. If the above maximum torque exceeds the damping
torque $\alpha\gamma M_{s}t_{\rm F}/2g$, then the out-of-plane configuration
can be stabilized by the current-induced torque. Intuitively, it’s
advantageous to use a low $M_{s}$ material in order to reduce the critical
current, and a thin multilayer to reduce heating. For scattering processes
with different functional dependence on $T$, a similar line of reasoning
applies, although the specific form of the maximum absolute current-induced
torque will differ. It’s straightforward to show that a $T^{p}$ dependence of
$D$ results in a maximum current-induced torque expression similar to Eq. 6,
where the expression inside the square root is taken to the power $1/2p$.

##  IV Conclusion

This work demonstrates the role of symmetry and phase coherence effects in the
current-induced torque present between ferromagnet and antiferromagnetic
layers with a compensated interface. Basic symmetry arguments identify the
fixed points of the current-induced torque. We demonstrate that for an
antiferromagnetic layer with a 3Q spin structure, the current-induced torque
has a complex angular dependence, and the fixed points for the current-induced
torque are generally only local attractors. This is important because
experiments designed to drive the ferromagnet to these fixed points must
initialize the ferromagnet sufficiently nearby. We also show via explicit
calculations the primary role played by phase coherence for these torques, and
show an inverse relationship between the magnitude of the current-induced
torque and the phase breaking scattering parameter. In the antiferromagnetic
system with planar spins (the no-canted system), we find the current-induced
torque to be sufficiently robust to scattering to stabilize the out-of-plane
magnetic orientation, while for the 3Q ordered antiferromagnet, the current-
induced torque is too weak to stabilize this orientation. We expect that the
robustness of this torque to scattering should be system specific, determined
by which scattering processes are dominant and the system electronic
structure.

##  V Acknowledgements

A.P. acknowledges support under the Cooperative Research Agreement between the
University of Maryland and the National Institute of Standards and Technology
Center for Nanoscale Science and Technology, Award 70NANB10H193, through the
University of Maryland.

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