# Scalar-spinor fields interaction
in de Sitter ambient space formalism

Y. Ahmadi<EMAIL_ADDRESS>Department of Physics, Razi University,
Kermanshah, Iran  F. Jalilifard<EMAIL_ADDRESS>Department of
Physics, Razi University, Kermanshah, Iran  M.V. Takook<EMAIL_ADDRESS>Department of Physics, Razi University, Kermanshah, Iran

###### Abstract

In de Sitter ambient space formalism, the massless minimally coupled scalar
field can be constructed from a massless conformally coupled scalar field and
a constant five-vector $A^{\alpha}$. Also, a constant five-vector $B^{\alpha}$
appears in the interaction Lagrangian of massless minimally coupled scalar and
spinor fields in this formalism. These constant five-vector fields can be
fixed in the interaction case in the null curvature limit. Here we will
calculate the $\cal S$ matrix elements of scalar-spinor fields interaction in
the tree level approximation. Then the constant five-vectors $A^{\alpha}$ and
$B^{\alpha}$, will be fixed by comparing the $\cal S$ matrix elements in the
null curvature limits with the Minkowskian counterparts.

##  I Introduction

The interaction Lagrangian in standard model is usually constructed from gauge
theory, but the scalar-spinor field interaction is presented by the Yukawa
potential. In quantum field theory (QFT) the Yukawa potential cannot be
written from any group theoretical point of view. In the previous paper, the
interaction Lagrangian of scalar-spinor fields is defined by a new
transformation in de Sitter (dS) ambient space formalism, which was free of
any infrared divergence higgs . The Yukawa potential can be obtained from this
Lagrangian in the null curvature limit. In this model two constant five-vector
fields appear, one is from the new transformation and the other is from the
massless minimally coupled (mmc) scalar field two-point function higgs . In
this article we will fixed this two constant five-vector fields by calculating
the scattering $(\cal S)$ matrix elements of the scalar-spinor fields
interaction in the null curvature limit.

The experimental data confirm that the our universe can be described by the dS
metric in the large scale and also in the inflationary epoch Riess ; Perl .
Thus it is important to construct QFT in dS space-time. In last years, the
efforts have been made to do it in ambient space formalism ta97 ; 77 ; ta96 ;
taazba ; taro12 ; berotata ; derotata . The advantage of using ambient space
is due to the linearity of the group action on the ambient space formalism,
then the calculations are very similar to the Minkowski counterpart.

We know that, in the Minkowski space-time, the analyticity properties of the
two-point functions are the fundamental basis for calculation of the
probability amplitude or the Green functions of the interaction fields. The
analyticity properties of the two-point functions in dS ambient space
formalism have been proved by Bros et al. brgamo ; brmo ; brmo03 . Therefore,
this formalism is more appropriate for constructed the quantum interaction
fields in dS space-time. In the previous article, we studied the Compton
scattering in dS ambient space formalism and showed that the $\cal S$ matrix
elements in the null curvature limit reduce to the Minkowskian counterpart
electron-photon . The interaction fields are important in this article for two
reason: First, the effects of the classical gravitational field on the quantum
field are better seen in the QFT in the dS ambient space formalism. Secondly,
in the formulation of the QFT in dS ambient space formalism, two constant
five-vector fields appear, which can be fixed in the null curvature limit ta97
; 77 ; higgs .

The organization of this article is as follow. We present the notation and
terminology that has been used in this article in section II. In section III,
the interaction Lagrangian in the small curvature approximation has been
presented and then the $\cal S$ matrix elements for the spinor-scalar
interaction have been calculated in dS ambient space formalism. In section IV,
the null curvature limit of the $\cal S$ matrix element has been calculated
and then the constant five-vectors $A^{\alpha}$ and $B^{\alpha}$ have been
fixed in this limit. Finally, the conclusion has been presented in section V.

##  II notation and terminology

The dS space-time can be considered as a 4-dimensional hyperboloid embedded in
5-dimensional Minkowski space with the following relation:

$M_{H}=\\{x\in{\rm IR}^{5}|\;\;x\cdot
x=\eta_{\alpha\beta}x^{\alpha}x^{\beta}=-H^{-2}\\};\;\;\alpha,\beta=0,1,2,3,4.$

The dS metric is:

$ds^{2}=\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}|_{x^{2}=-H^{-2}}=g_{\mu\nu}^{dS}dX^{\mu}dX^{\nu};\;\;\mu=0,1,2,3,$

where $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1,-1)$, $H$ is Hubble parameter,
$X^{\mu}$ is dS intrinsic coordinate and $x^{\alpha}$ is the 5-dimensional dS
ambient space formalism ($x\cdot x=-H^{-2}$). The ambient space coordinate
$x^{\alpha}$ can be represented in terms of the intrinsic coordinate
$X^{\mu}$. For defining time evolution operator and scattering matrix, one can
choose the following static coordinate system:

$\left\\{\begin{array}[]{clcr}x^{0}&=\sqrt{H^{-2}-r^{2}}\sinh Ht_{s},\\\
x^{1}&=\sqrt{H^{-2}-r^{2}}\cosh Ht_{s},\\\ x^{2}&=r\cos\theta,\\\
x^{3}&=r\sin\theta\cos\phi,\\\ x^{4}&=r\sin\theta\sin\phi,\\\
\end{array}\right.$ (II.1)

where $-\infty<t_{s}<\infty\;,\;0\leq
r<H^{-1}\;,\;0\leq\theta\leq\pi\;,\;0\leq\phi<2\pi$. This coordinate system
does not covered the total of the dS hyperboloid. For investigating the null
curvature limit, the following coordinate system is useful:

$\left\\{\begin{array}[]{clcr}x^{0}&=H^{-1}\sinh(HX^{0}),\\\
\vec{x}&=H^{-1}\dfrac{\overrightarrow{X}}{\lVert\overrightarrow{X}\lVert}\cosh(HX^{0})\sinh(H\lVert\overrightarrow{X}\lVert),\\\
x^{4}&=H^{-1}\cosh(HX^{0})\cosh(H\lVert\overrightarrow{X}\lVert),\\\
\end{array}\right.$ (II.2)

where the
$\lVert\overrightarrow{X}\lVert=(X_{1}^{2}+X_{2}^{2}+X_{3}^{3})^{\frac{1}{2}}$
is the norm of three-vector $\overrightarrow{X}$.

The Fourier transformation cannot be generally defined in curved space-time,
but in the dS space-time, due to the maximally symmetric properties of dS
hyperboloid the Fourier-Helgason-type transformation or the Bros-Fourier
transformation can be defined hel62 ; hel94 ; brmo03 ; brmo . Then,
corresponding to any space-time variable $x^{\alpha}$ another transformed
variable, $\xi^{\alpha}=(\xi^{0},\vec{\xi},\xi^{4})$ can be defined in the
positive null-cone $C^{+}=\left\\{\xi\in{\rm
IR}^{5}|\;\;\xi\cdot\xi=0,\;\;\xi^{0}>0\right\\}$ brmo03 ; brmo . One can
parameterize the $\xi$ in terms of massive particle four-momentum in flat
Minkowski space-time as:

$\xi=\left(\xi^{0},\overrightarrow{\xi},\xi^{4}\right)=\left(\dfrac{k^{0}}{m}=\sqrt{\dfrac{\overrightarrow{k}^{2}}{m^{2}}+1},\dfrac{\overrightarrow{k}}{m},-1\right).$
(II.3)

In the dS ambient space formalism, dS-Dirac first-order field equation is
bagamota :

$\left(-i\not\\!x\not\\!\partial^{\top}+2i\pm\nu\right)\psi(x)=0,$ (II.4)

where $\not\\!x=\eta_{\alpha\beta}\gamma^{\alpha}x^{\beta}$ and
$\partial^{\top}_{\alpha}=\theta_{\alpha\beta}\partial^{\beta}=\partial_{\alpha}+Hx_{\alpha}x\cdot\partial$.
$\theta_{\alpha\beta}=\eta_{\alpha\beta}+H^{2}x_{\alpha}x_{\beta}$ is the
projection operator on dS hyperboloid. In this equation, $\nu$ is related to
dS mass parameter $m^{2}_{f,\nu}=H^{2}(2+\nu^{2}\pm i\nu)$ and in the null
curvature limit it is defined the Minkowskian mass parameter $m$ 77 :

$\lim_{H\rightarrow 0,\nu\rightarrow\infty}(H\nu)^{2}=m^{2}.$

The five matrices $\gamma^{\alpha}$, which satisfy the conditions
$\gamma^{\alpha}\gamma^{\beta}+\gamma^{\beta}\gamma^{\alpha}=2\eta^{\alpha\beta}$
and $\gamma^{\alpha\dagger}=\gamma^{0}\gamma^{\alpha}\gamma^{0}$, can be
chosen as ta97 ; ta96 ; bagamota :

$\gamma^{0}=\left(\begin{array}[]{clcr}I&\;\;0\\\ 0&-I\\\
\end{array}\right),\;\;\;\gamma^{4}=\left(\begin{array}[]{clcr}0&I\\\ -I&0\\\
\end{array}\right),$ $\gamma^{1}=\left(\begin{array}[]{clcr}0&i\sigma^{1}\\\
i\sigma^{1}&0\\\
\end{array}\right),\;\;\gamma^{2}=\left(\begin{array}[]{clcr}0&-i\sigma^{2}\\\
-i\sigma^{2}&0\\\
\end{array}\right),\;\;\gamma^{3}=\left(\begin{array}[]{clcr}0&i\sigma^{3}\\\
i\sigma^{3}&0\\\ \end{array}\right),$ (II.5)

where $\sigma^{i}$ $(i=1,2,3)$ are the Pauli matrices. These matrices are
different from Minkowski gamma matrices $\gamma^{\prime\mu}$. The relation
between the $\gamma$ matrices in Minkowski and dS ambient space formalism are
bagamota :

$\gamma^{\prime\mu}=\gamma^{\mu}\gamma^{4}.$ (II.6)

The charged spinor field operator, which satisfies the equation (II.4), is 77
:

$\psi(x)=\int_{S^{3}}d\mu(\xi)\sum_{n}\left[a(\tilde{\xi},n)(Hx.\xi)^{-2-i\nu}{\cal
U}(x,\xi,n)+b^{{\dagger}}(\xi,n)(Hx.\xi)^{-1+i\nu}{\cal V}(x,\xi,n)\right],$
(II.7)

where $n=-\frac{1}{2},\frac{1}{2}$, $\mathrm{d}\mu(\xi)$ is the
$SO(4)$-invariant normalized volume element and
$\tilde{\xi}^{\alpha}=(\xi^{0},-\vec{\xi},\xi^{4})$. The explicit form of
$\cal{U}$ and $\cal{V}$ has been defined in ta97 ; bagamota . The adjoint
spinor $\overline{\psi}(x)$ in ambient space formalism is defined as
$\overline{\psi}(x)=\psi^{\dagger}(x)\gamma^{0}\gamma^{4}$ bagamota . The
$a(\tilde{\xi},n),\;a^{\dagger}(\tilde{\xi},n)\;\text{and}\;b(\tilde{\xi},n),\;b^{\dagger}(\tilde{\xi},n)$
are annihilation and creation operators 77 :

$a^{\dagger}(\xi,n)\left|\Omega\right>\equiv\left|1_{\xi,n}^{a}\right\rangle,\;\;b^{\dagger}(\xi,n)\left|\Omega\right>\equiv\left|1_{\xi,n}^{b}\right\rangle\,.$
(II.8)

The vacuum state $|\Omega>$, with norm
$\left<\Omega\right.\left|\Omega\right>=1$, is invariant under the action of
the UIR of the dS group. This vacuum state can be identified with the Bunch-
Davies or Hawking-Ellis vacuum state. The anti-commutation relations for
creation and annihilation operators are bagamota ; 77 :

$\left[a(\tilde{\xi}^{\prime},n^{\prime}),a^{\dagger}(\xi,n)\right]_{+}={\cal
N}_{p}(\xi,n)\delta_{s^{3}}(\xi-\xi^{\prime})\delta_{nn^{\prime}},$
$\left[b(\tilde{\xi}^{\prime},n^{\prime}),b^{\dagger}(\xi,n)\right]_{+}={\cal
N}_{q}(\xi,n)\delta_{s^{3}}(\xi-\xi^{\prime})\delta_{nn^{\prime}},$ (II.9)

where ${\cal N}_{p}(\xi,r)\equiv{\cal N}_{p}$ and ${\cal
N}_{q}(\xi,r)\equiv{\cal N}_{q}$ are the normalization constants.

In dS ambient space formalism, the field equation for massless conformally
coupled (mcc) scalar field is brmo ; ta97 ; 77 :

$\left(Q_{0}-2\right)\phi_{mcc}(x)=0,$ (II.10)

that
$Q_{0}=-H^{-2}\partial^{\top}\cdot\partial^{\top}\equiv-H^{-2}\square_{H}\;.$
The mcc scalar field operator, which satisfies the field equation (II.10), is
brmo ; 77 :

$\phi_{mcc}(x)=\sqrt{c_{0}}\int_{S^{3}}d\mu({\xi})\left[d({\tilde{\xi}},1)(Hx\cdot\xi)^{-2}+d^{\dagger}(\xi,0)(Hx\cdot\xi)^{-1}\right],$
(II.11)

where $c_{0}$ is the normalization constant. The $d(\tilde{\xi},1)$ and
$d^{\dagger}(\xi,0)$ are annihilation and creation operators respectively:

$d^{\dagger}(\xi,0)\left|\Omega\right>\equiv\left|1_{\xi}^{d}\right\rangle,\;\;d(\tilde{\xi},1)\left|\Omega\right>=0.$

Between the massless minimally coupled (mmc) scalar field, and the mcc scalar
fields in the dS ambient space formalism, the following relation exist higgs ;
77 ; khrota ; gareta :

$\phi_{mmc}(x)=\left[A\cdot\partial^{\top}+2A\cdot
H^{2}x\right]\phi_{mcc}(x),$ (II.12)

where $A^{\alpha}$ is an arbitrary constant five-vector. The mmc scalar field
operator can be obtained by inserting (II.11) in (II.12) and doing some simple
calculations higgs ; 77 . The analytic field operator is defined in complex dS
space-time $\phi_{mmc}(x)=\lim_{y\rightarrow
0}\phi_{mmc}(z)=\lim_{y\rightarrow 0}\phi_{mmc}(x+iy).$

The analytic two-point function of mcc scalar field can be calculated in terms
of the generalized Legendre function brmo ; brgamo :

$W_{mcc}(z,z^{\prime})=\left<\Omega|\phi_{mcc}(z)\phi_{mcc}(z^{\prime})|\Omega\right>=$
$c_{0}\int
d\mu(\xi)(Hz\cdot\xi)^{-2}(Hz^{\prime}.\cdot\xi)^{-1}=\dfrac{-H^{2}}{8\pi^{2}}\dfrac{1}{1-{\cal
Z}(z,z^{\prime})}=\dfrac{1}{(2\pi)^{2}}\dfrac{1}{(z-z^{\prime})^{2}}\,,$
(II.13)

where ${\cal Z}(z,z^{\prime})=1+\dfrac{H^{2}}{2}(z-z^{\prime})^{2}$ is the
geodesic distance between two point $z$ and $z^{\prime}$ brmo ; chta .

One can define the mmc scalar analytic two-point function in terms of mcc
scalar analytic two-point function as 77 ; higgs :

$W_{mmc}(z,z^{\prime})=\left[A\cdot\partial^{\top}+2A\cdot
H^{2}z\right]\left[A\cdot\partial^{\prime\top}+2A\cdot
H^{2}z^{\prime}\right]W_{mcc}(z,z^{\prime}).$ (II.14)

For defining the real two-point function one can use the boundary value of the
analytic two-point function $W_{mmc}(z,z^{\prime})$ 77 ; khrota ,

${\cal W}_{mmc}(x,x^{\prime})=\lim_{y,y^{\prime}\rightarrow
0}W_{mmc}(z,z^{\prime})=\lim_{y,y^{\prime}\rightarrow
0}W_{mmc}(x+iy\;,\;x^{\prime}+iy^{\prime}).$

##  III scattering matrix

In ambient space formalism, the scalar field equation in dS space-time can be
written as:

$(Q_{0}+\sigma(\sigma+3))\phi(x)=0\,.$

The solutions of this equation are named dS plane waves
$(x\cdot\xi)^{\sigma}$. The five-vector $\xi^{\alpha}$ plays the role of
momentum parameter and it is lying in the null-cone in ambient space higgs .
The action of mcc scalar and spinor free fields in dS universe is 77 ;
bagamota ; higgs :

$S(\psi,\phi)=\int\mathrm{d}\mu(x){\cal
L}(\psi,\phi)=\int\mathrm{d}\mu(x)\left[H\overline{\psi}\gamma^{4}\left(-i\not\\!x\not\\!\partial^{\top}+2i\right)\psi+\phi\;Q_{0}\;\phi\right],$
(III.15)

where $\mathrm{d}\mu(x)$ is dS invariant volume element. The above action is
invariant under the global U(1) symmetry. By defining the covariant derivative
$D_{\alpha}\psi=(\partial_{\alpha}^{\top}+i{\cal
G}B^{\top}_{\alpha}\phi_{mmc})\psi$, where $B^{\alpha}B_{\alpha}=0$ and ${\cal
G}$ is the coupling constant that determines interaction intensity, the action
(III.15) will be invariant under the following transformations higgs :

$\left\\{\begin{array}[]{clcr}\phi\rightarrow\phi^{\prime}=\phi+(x.B)^{-3}\\\
\psi\rightarrow\psi^{\prime}=e^{\frac{i}{2}{\cal G}(x\cdot
B)^{-2}}\psi\end{array}\right..$

So by replacing this new derivative $D_{\alpha}$ with transverse derivative
$\partial_{\alpha}^{\top}$, the interaction Lagrangian is obtained as higgs :

${\cal L}_{int}={\cal
G}\;H\overline{\psi}\gamma^{4}\not\\!x\not\\!B^{\top}\phi_{mmc}\psi.$ (III.16)

The constant five-vector $B^{\alpha}$ in Lagrangian and constant five-vector
$A^{\alpha}$ in mmc scalar two-point function can be fixed in the interaction
case in the null curvature limit. The interaction Lagrangian plays an
important role for calculation of the probability amplitude or the $\cal S$
matrix elements.

The time evolution operator cannot be explicitly defined in a general curved
space time. In previous article we show that the time evolution operator
$U(t,t_{0}),$ such that $|\alpha,t\rangle=U(t,t_{0})|\alpha,t_{0}\rangle$, can
be defined in the dS static coordinate system (II.1), but it is very
complicated for calculation of the $\cal S$ matrix element bh :

$|\phi_{out}\;,+\infty>=U(\infty\;,-\infty)|\phi_{in}\;,-\infty>={\cal
S}|\phi_{in}\;,-\infty>.$ (III.17)

Then we presented a logical approximation when the interaction occurs in the
atomic dimension electron-photon , which the $\cal S$ matrix elements can be
calculated. In this approximation the direct effect of curvature can be
ignored but the indirect effect is presented. In this case the time evolution
operator can be expanded on the Minkowski space-time as electron-photon :

$U(t,t_{0})={U_{M}}(t,t_{0})+Hf(t,t_{0})+\cdots\;,$

the term $f(t,t_{0})$ is due to the direct effect of curvature that in atomic
dimension and in null curvature limit, it is disappeared. The mathematical
form of $U_{M}$ is exactly similar to the time evolution operator in
Minkowskian space-time. Therefore the scattering matrix may be presented with
the following approximate equation from the Minkowskian counterpart mandl shaw
; kaku ; zuber :

${\cal S}=\frac{(-1)^{\tau}}{{\tau}!}\sum_{{\tau}=0}^{\infty}{\cal
S}^{({\tau})}+\cdots\,,$

where

${\cal
S}^{({\tau})}=\int\mathrm{d}\mu(x_{1})\int\mathrm{d}\mu(x_{2})\cdots\int\mathrm{d}\mu(x_{\tau})\;\;\;T[{\cal
L}_{int}(\phi(x_{1})){\cal L}_{int}(\phi(x_{2}))\cdots{\cal
L}_{int}(\phi(x_{\tau}))].$ (III.18)

In this equation, ${\cal L}_{int}(\phi(x))$ is the interaction Lagrangian and
$T[{\cal L}_{int}(\phi(x_{1})){\cal L}_{int}(\phi(x_{2})){\cal
L}_{int}(\phi(x_{3}))...]$ is the time order product of fields.

By inserting interaction Lagrangian of spinor-scalar fields (III.16) in
(III.18), the ${\cal S}^{(2)}$ is obtained as:

${\cal S}^{(2)}={\cal
G}^{2}H^{2}\int\int\mathrm{d}\mu(x)\mathrm{d}\mu(x^{\prime})\;\;T\left[\left(\overline{\psi}\gamma^{4}\not\\!x\not\\!B^{\top}\phi_{mmc}\psi\right)_{x}\left(\overline{\psi}\gamma^{4}\not\\!x^{\prime}\not\\!B^{\top}\phi_{mmc}\psi\right)_{x^{\prime}}\right].$
(III.19)

By using Wick’s theorem, one can write the time order product of fields in
terms of the normal order product of fields. After some simple calculations,
the time order product of (III.19) turns into 8 normal order product. Since we
consider the interaction, which is depicted in figure 1, only one term is
remained :

$q$$p$$p+q$$p^{\prime}$$q^{\prime}$$x$$x^{\prime}$ Figure 1: spinor-spinor
interaction diagram with mmc scalar propagator

${\cal S}^{(2)}={\cal
G}^{2}H^{2}\int\int\mathrm{d}\mu(x)\mathrm{d}\mu(x^{\prime})\;\;N\left[(\overline{\psi}\gamma^{4}\not\\!x\not\\!B^{\top}\underleftrightarrow{\phi_{mmc}\psi)_{x}(\overline{\psi}\gamma^{4}\not\\!x^{\prime}\not\\!B^{\top}\phi}_{mmc}\psi)_{x^{\prime}}\right],$
(III.20)

where, the symbol $\leftrightarrow$ is the mmc scalar two-point function
(II.14) and $N[...]$ is normal order product,

$\underleftrightarrow{\phi_{mmc}(x)\;\;\phi}_{mmc}(x^{\prime})=\left<\Omega|\phi_{mmc}(x)\;\;\phi_{mmc}(x^{\prime})|\Omega\right>={\cal
W}_{mmc}(x,x^{\prime}).$

Since in (III.20) the spinor field, $\gamma$ matrices and other quantities are
matrix, one can write them in terms of their components:

${\cal S}^{(2)}={\cal
G}^{2}H^{2}\int\int\mathrm{d}\mu(x)\mathrm{d}\mu(x^{\prime})\;{\cal
W}_{mmc}(x,x^{\prime})\left(\gamma^{4}\not\\!x\not\\!B^{\top}(x)\right)_{gh}\left(\gamma^{4}\not\\!x^{\prime}\not\\!B^{\top}(x^{\prime})\right)_{kl}\;$
$\times
N\left[\overline{\psi}_{g}(x)\psi_{h}(x)\overline{\psi}_{k}(x^{\prime})\psi_{l}(x^{\prime})\right].$
(III.21)

By breaking the spinor field (II.7) into $+,-$ parts, the normal order product
in above relation (III.21) converted to 16 terms but just one of them is
described the figure 1,

$\psi(x)=\psi^{+}(x)+\psi^{-}(x),\;\;\;\;\;\overline{\psi}(x)=\overline{\psi}^{+}(x)+\overline{\psi}^{-}(x)$
(III.22) $\mbox{physical term in normal order
product}=\overline{\psi}^{-}_{g}(x)\psi^{+}_{h}(x)\psi^{-}_{l}(x^{\prime})\overline{\psi}^{+}_{k}(x^{\prime}).$

Thus the ${\cal S}^{(2)}$ is obtained as:

${\cal S}^{(2)}={\cal
G}^{2}H^{2}\int\int\mathrm{d}\mu(x)\mathrm{d}\mu(x^{\prime})\;{\cal
W}_{mmc}(x,x^{\prime})\;\left(\gamma^{4}\not\\!x\not\\!B^{\top}(x)\right)_{gh}\left(\gamma^{4}\not\\!x^{\prime}\not\\!B^{\top}(x^{\prime})\right)_{kl}$
$\times\\{\overline{\psi}^{-}_{g}(x)\psi^{+}_{h}(x)\psi^{-}_{l}(x^{\prime})\overline{\psi}^{+}_{k}(x^{\prime})\\}.$

In this interaction, incoming sate $|i>$ is two spinor fields with
$\xi_{p},\xi_{q}$ ”momentums” and $r,n$spin polarizations. Also outgoing state
$|f>$ is two spinor fields with $\xi^{\prime}_{p},\xi^{\prime}_{q}$
”momentums” and $r^{\prime},n^{\prime}$ spin polarizations,

$|i>=|1_{p}^{a^{\dagger}},1_{q}^{b^{\dagger}}>=a^{\dagger}_{r}(\xi_{p})b^{\dagger}_{n}(\xi_{q})|\Omega>$
$|f>=|1_{p^{\prime}}^{a^{\dagger}},1_{q^{\prime}}^{b^{\dagger}}>=a^{\dagger}_{r^{\prime}}({\xi}^{\prime}_{p})b^{\dagger}_{n^{\prime}}({\xi}^{\prime}_{q})|\Omega>\;\;\;\;\;\Longrightarrow\;\;\;\;\;<f|=<1_{p^{\prime}}^{a},1_{q^{\prime}}^{b}|=<\Omega|b_{n^{\prime}}({\xi}^{\prime}_{q})a_{r^{\prime}}({\xi}^{\prime}_{p}).$

Then the ${\cal S}^{(2)}_{fi}$ is:

${\cal S}^{(2)}_{fi}=<f|{\cal S}^{(2)}|i>={\cal N}_{p}{\cal
N}_{p^{\prime}}{\cal N}_{q}{\cal N}_{q^{\prime}}{\cal G}^{2}H^{2}$
$\times\int\int\mathrm{d}\mu(x)\;\mathrm{d}\mu(x^{\prime})\;{\cal
W}_{mmc}(x,x^{\prime})\;\left(\gamma^{4}\not\\!x\not\\!B^{\top}(x)\right)_{gh}\left(\gamma^{4}\not\\!x^{\prime}\not\\!B^{\top}(x^{\prime})\right)_{kl}$
$\times(Hx.{\xi}_{q})^{-1-i\nu}(Hx.\xi_{p})^{-2-i\nu}(Hx^{\prime}.\xi^{\prime}_{q})^{-1+i\nu}(Hx^{\prime}.\xi^{\prime}_{p})^{-2+i\nu}$
$\times\overline{\cal V}_{g}(x,\xi_{q},n)\;\;{\cal
U}_{h}(x,\xi_{p},r)\;\;\overline{{\cal
U}}_{k}(x^{\prime},\xi^{\prime}_{p},r^{\prime})\;\;{\cal
V}_{l}(x^{\prime},\xi^{\prime}_{q},n^{\prime}).$ (III.23)

This $\cal S$ matrix elements may be calculated by numerical methods, but in
this paper we would like to fix the constant five-vectors $A^{\alpha}$ and
$B^{\alpha}$, which will be discussed in the next section.

##  IV flat limit

The radius of dS hyperboloid is $H^{-1}$. The curvature of space-time
diminishes for large radius, or equivalently ${H\rightarrow 0}$, and dS space-
time matches with Minkowski space-time. In this limit one can obtain these
relations ta97 :

$\lim_{H\rightarrow
0}(x\cdot\xi)^{-2-i\nu}=e^{-ik.X},\;\;\;\;\;\;\;\;\;\;\lim_{H\rightarrow
0}H\not\\!x=\lim_{H\rightarrow
0}H\eta_{\alpha\beta}\gamma^{\alpha}x^{\beta}=-\gamma^{4},$
$\lim_{H\rightarrow 0}\theta_{\alpha\beta}=\lim_{H\rightarrow
0}\left(\eta_{\alpha\beta}+H^{2}x_{\alpha}x_{\beta}\right)=\eta_{\mu\nu},\;\;\;\;\;\;\left\\{\begin{array}[]{clcr}\mu=\nu=0,1,2,3\\\
\alpha=\beta=0,1,2,3,4\;.\\\ \end{array}\right.$

Since the five-vector $B^{\alpha}$ is constant, at null curvature limit
$B^{4}$ can be chosen as zero and then

$\lim_{H\rightarrow
0}\not\\!B^{\top}=-B_{\mu}\gamma^{{}^{\prime}\mu}\gamma^{4}\;;\;\;\;\;\;\mu=0,1,2,3.$

The Minkowski $\gamma^{\prime}$ matrices are related to ambient $\gamma$
matrices as (II.6). The analytic mmc scalar two-point function (II.14) in null
curvature limit, $H\rightarrow 0$, is:

$\lim_{H\rightarrow 0}W_{mmc}(z,z^{\prime})=\lim_{H\rightarrow
0}\left[A\cdot\partial^{\top}+2A\cdot
H^{2}z\right]\left[A\cdot\partial^{\prime\top}+2A\cdot
H^{2}z^{\prime}\right]W_{mcc}(z,z^{\prime})$
$=(A\cdot\partial)(A\cdot\partial^{\prime})\lim_{H\rightarrow
0}W_{mcc}(z,z^{\prime})$

and by using the (II.13), the real mmc scalar two-point function in null
curvature limit, $H\rightarrow 0$, obtains as:

$\lim_{H\longrightarrow 0}{\cal W}_{mmc}(x,x^{\prime})\equiv
w(X,X^{\prime})=\dfrac{1}{(2\pi)^{2}}(A.\partial)(A.\partial^{\prime})\dfrac{1}{(X-X^{\prime})^{2}}\;.$

By using this result for mmc scalar field, adjoint spinor definition in
ambient space formalism (II.7) and delta function definition in Minkowski
space-time ,
$\delta^{4}(p-p^{\prime})=\int\dfrac{\mathrm{d}^{4}X}{(2{\pi})^{4}}e^{-iX.(p-p^{\prime})}$,
the the null curvature limit of ${\cal S}^{(2)}_{fi}$ obtains as:

$\lim_{H\rightarrow 0}{\cal S}^{(2)}_{fi}=\dfrac{1}{(2\pi)^{2}}{\cal
N}_{p}{\cal N}_{p^{\prime}}{\cal N}_{q}{\cal N}_{q^{\prime}}{\cal
G}^{2}\left\\{\int\int\mathrm{d}^{4}X\;\mathrm{d}^{4}X^{\prime}e^{-iX.(p+q)}e^{iX^{\prime}.(p^{\prime}+q^{\prime})}\;\;\;(A.\partial)(A.\partial^{\prime})\dfrac{1}{(X-X^{\prime})^{2}}\right\\}$
$\times\left({\cal V}^{\dagger}(q,n)\gamma^{\prime
0}\right)_{g}\;B_{\mu}\left(\gamma^{\prime\mu}\gamma^{4}\right)_{gh}\;{\cal
U}_{h}(p,r)\;\;\;\left({\cal U}^{\dagger}(p^{\prime},r^{\prime})\gamma^{\prime
0}\right)_{k}B_{\nu}\left(\gamma^{\prime\nu}\gamma^{4}\right)_{kl}\;{\cal
V}_{l}(q^{\prime},n^{\prime}).$ (IV.24)

By notice that $\int
d^{4}X\partial_{\mu}\left(e^{-iX.(p+q)}\dfrac{1}{(X-X^{\prime})^{2}}\right)=0$,
one can prove the following relation:

$\int
d^{4}Xe^{-iX.(p+q)}\;\partial_{\mu}\left(\dfrac{1}{(X-X^{\prime})^{2}}\right)=-\int
d^{4}X\dfrac{1}{(X-X^{\prime})^{2}}\left(\partial_{\mu}e^{-iX.(p+q)}\right)$
$=-i(p+q)_{\mu}\int d^{4}X\dfrac{e^{-iX.(p+q)}}{(X-X^{\prime})^{2}},$

then we obtain:

$\int\int\mathrm{d}^{4}X\;\mathrm{d}^{4}X^{\prime}e^{-iX.(p+q)}e^{iX^{\prime}.(p^{\prime}+q^{\prime})}\;\;\;(A.\partial)(A.\partial^{\prime})\dfrac{1}{(X-X^{\prime})^{2}}=$
$A.(p+q)A.(p^{\prime}+q^{\prime})\int\int\mathrm{d}^{4}X\;\mathrm{d}^{4}X^{\prime}e^{-iX.(p+q)}e^{iX^{\prime}.(p^{\prime}+q^{\prime})}\;\;\;\dfrac{1}{(X-X^{\prime})^{2}}.$

After some straightforward calculations one can arrive the following relation:

$\int\int\mathrm{d}^{4}X\;\mathrm{d}^{4}X^{\prime}e^{-iX.(p+q)}e^{iX^{\prime}.(p^{\prime}+q^{\prime})}\;\;\;\dfrac{1}{(X-X^{\prime})^{2}}=(2\pi)^{8}\delta(p+q-p^{\prime}-q^{\prime})\dfrac{1}{(p+q)^{2}+i\epsilon}.$

By inserting these results in (IV.24) one can obtain the null curvature limit
of ${\cal S}^{(2)}$ matrix as follow:

$\lim_{H\rightarrow 0}{\cal S}^{(2)}_{fi}=(2\pi)^{6}{\cal N}_{p}{\cal
N}_{p^{\prime}}{\cal N}_{q}{\cal N}_{q^{\prime}}{\cal
G}^{2}\delta(p+q-p^{\prime}-q^{\prime})$ $\times\left({\cal
V}^{\dagger}(q,n)\gamma^{\prime
0}\right)_{g}\;B_{\mu}\left(\gamma^{\prime\mu}\gamma^{4}\right)_{gh}\;{\cal
U}_{h}(p,r)\;\;\;\dfrac{[A.(p+q)]^{2}}{(p+q)^{2}+i\epsilon}\;\;\;\left({\cal
U}^{\dagger}(p^{\prime},r^{\prime})\gamma^{\prime
0}\right)_{k}B_{\nu}\left(\gamma^{\prime\nu}\gamma^{4}\right)_{kl}\;{\cal
V}_{l}(q^{\prime},n^{\prime}).$

By comparing this equation with the Minkowski result mandl shaw , the constant
five-vectors $A_{\mu}$ and $B_{\mu}$ can be fixed as:

$A_{\mu}=a\;\dfrac{(p_{\mu}+q_{\mu})}{(p+q).(p+q)}\,,\;\;\;\;\;B_{\mu}=b\;\gamma^{\prime}_{\mu}\gamma^{4},$

where $a$ and $b$ can be written in therms of the normalization constants.

##  V Conclusion

A constant five-vector field $A^{\alpha}$ appears in dS mmc scalar field
quantization. Also another constant five-vector field $B^{\alpha}$ appears in
definition of the interaction Lagrangian of dS spinor-scalar fields
interaction. These constant five-vectors $A^{\alpha}$ and $B^{\alpha}$ can be
fixed in calculation of the ${\cal S}$ matrix elements in tree diagrams
approximation in the null curvature limit. The constant five-vector
$B^{\alpha}$ come from the gauge theory and depended on $\gamma$ matrices and
the constant five-vector $A^{\alpha}$ come from the propagator. It is
interesting to note that the constant five-vector $A^{\alpha}$ depended on the
laboratory energy scale i.e. the energy-momentum of the incoming fields, which
is a logical result. These calculations may be open the door to measuring the
gravitational effects on quantum fields which may be measurable in the
laboratory scale of energy. The other diagrams and interactions may be
investigated in the next works.

Acknowledgments: The authors wish to express their particular thanks to M.
Amiri, M. Rastiveis, R. Raziani and S. Tehrani-Nasab for discussions.

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# Imaging and Manipulation of Skyrmion Lattice Domains in Cu2OSeO3

S. L. Zhang  Clarendon Laboratory, Department of Physics, University of
Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom  A. Bauer  Physik
Department, Technische Universität München, James-Franck-Strasse 1, 85748
Garching, Germany  H. Berger  Crystal Growth Facility, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  C. Pfleiderer
Physik Department, Technische Universität München, James-Franck-Strasse 1,
85748 Garching, Germany  G. van der Laan  Magnetic Spectroscopy Group, Diamond
Light Source, Didcot, OX11 0DE, United Kingdom  T. Hesjedal  Clarendon
Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford,
OX1 3PU, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Nanoscale chiral skyrmions in noncentrosymmetric helimagnets are promising
binary state variables in high-density, low-energy nonvolatile memory.
Nevertheless, they normally appear in an ordered, single-domain lattice phase,
which makes it difficult to write information unless they are spatially broken
up into smaller units, each representing a bit. Thus, the formation and
manipulation of skyrmion lattice domains is a prerequisite for memory
applications. Here, using an imaging technique based on resonant magnetic
x-ray diffraction, we demonstrate the mapping and manipulation of skyrmion
lattice domains in Cu2OSeO3. The material is particularly interesting for
applications owing to its insulating nature, allowing for electric field-
driven domain manipulation.

Skyrmions are particle-like magnetization swirls in magnetic
materials,Mühlbauer  _et al._ (2009); Nagaosa and Tokura (2013); Sun  _et al._
(2013); Jiang  _et al._ (2015); Büttner  _et al._ (2015); Kezsmarki  _et al._
(2015); Boulle  _et al._ (2016) which are promising candidates for advanced
magnetic memory applications.Nagaosa and Tokura (2013) In these memory
schemes, individual skyrmions are used to encode the binary information ‘1’
and ‘0’, e.g., via their presenceRomming  _et al._ (2013); Zhang  _et al._
(2015); Woo  _et al._ (2016) or their internal structures.Zhang  _et al._
(2016a); Díaz and Troncoso (2016) Skyrmions are topological objects, which
makes them robust against superparamagnetism. Consequently, it should be
possible to reduce the bit size beyond the limits of conventional
ferromagnetic memory. Further, the energy required to manipulate skyrmions is
several orders of magnitude less than domain wall-based ferromagnetic
memory.Jonietz  _et al._ (2010); Schulz  _et al._ (2012); Mochizuki  _et al._
(2014)

Prominent skyrmion-carrying materials are noncentrosymmetric helimagnets, such
as MnSi,Mühlbauer  _et al._ (2009) Fe1-xCoxSi,Münzer  _et al._ (2010) FeGe,Yu
_et al._ (2012a) Cu2OSeO3,Seki  _et al._ (2012a); Langner  _et al._ (2014) and
$\beta$-type Co8Zn8Mn4.Tokunaga  _et al._ (2015) In such materials, the broken
crystalline inversion symmetry induces the Dzyaloshinskii-Moriya interaction
(DMI), leading to periodic, incommensurate, modulated spiral spin structures.
Assisted by thermal fluctuations at finite temperature and in an external
field, the topologically protected skyrmion lattice phase forms, consisting of
chiral skyrmions.Mühlbauer  _et al._ (2009) The advantages of chiral skyrmions
for memory devices are twofold. First, compared to magnetic bubbles with
similar topological spin configurations,Giess (1980) skyrmions are more robust
and can be manipulated with ease. The size of an individual skyrmion in these
materials is usually between 3-100 nm,Mühlbauer  _et al._ (2009); Yu  _et al._
(2010, 2011); Kanazawa  _et al._ (2011) i.e., smaller than other types of
skyrmions.Giess (1980); Yu  _et al._ (2012b); Miao  _et al._ (2014); Jiang
_et al._ (2015); Büttner  _et al._ (2015); Chen  _et al._ (2015); Boulle  _et
al._ (2016); Woo  _et al._ (2016) Second, the skyrmion phase is a rigid,
hexagonally-ordered periodic lattice, resulting in an equal spacing between
neighboring skyrmions across the entire sample.Adams  _et al._ (2011); Nagaosa
and Tokura (2013) As a consequence, in a racetrack-like memory scheme,Parkin,
Hayashi, and Thomas (2008) no extra effort is needed to assure that they keep
their distance, as the control of the skyrmion-skyrmion distance is otherwise
experimentally rather challenging.Romming  _et al._ (2013); Jiang  _et al._
(2015); Woo  _et al._ (2016)

The skyrmion lattice phase in noncentrosymmetric helimagnets is usually a
long-range-ordered state, with the correlation length reaching hundreds of
micrometers.Adams  _et al._ (2011) This largely limits the applicability of
this type of skyrmion order for device applications. Therefore, the formation
and manipulation of skyrmion lattice domains, which break the long-range
order, is an indispensable step towards skyrmion-based racetrack
memories.Tomasello  _et al._ (2014) We have recently reported the observation
of a multidomain skyrmion lattice state on the surface of Cu2OSeO3 in a
magnetic diffraction experiment, created by tilting the magnetic field away
from the major crystalline axis.Zhang  _et al._ (2016a, b) However, several
key questions remained unanswered. Most importantly, the size, shape, and
distribution of the domains remained unknown, which are crucial pieces of
information needed for designing skyrmion devices.

In this Letter, we use an imaging technique based on resonant elastic x-ray
scattering (REXS) to map out the lateral distribution of skyrmion lattice
domains. Moreover, we demonstrate that by tuning the tilt angle of the applied
field, the size of the domains can be efficiently manipulated, which is an
important step towards future applications.

Figure 1: (a) Schematic of the REXS setup, showing the field tilt angle
$\gamma$, defined with respect to the surface normal. (b) Photograph of the
sample and its orientation. The blue square marks the real-space region
($1\times 1$ mm2) mapped using magnetic diffraction imaging, shown in Fig. 3.
(c) Sketch of the magnetic phase diagram of Cu2OSeO3, as obtained by REXS. (d)
Magnetization patterns and (e) corresponding REXS diffraction patterns in
reciprocal space $(hk1)$ plane (experimental data) for the conical (50 K, 32
mT, $\gamma=90^{\circ}$), skyrmion (57 K, 32 mT, $\gamma=0^{\circ}$), and
helical (15 K, 0 mT) phase, respectively. The sampling area is $500\times
500~{}\mu\mathrm{m}^{2}$.

Figure 1(a) shows a schematic of the REXS setup used for the characterization
of the magnetically ordered phases of Cu2OSeO3 as a function of applied
magnetic field and temperature. A $\omega$-$2\theta$ geometry was used for the
experiments, where $\omega$ is the angle of incidence of the x-rays and
$2\theta$ the scattering angle. The diffracted x-rays are detected with a
charge-coupled device (CCD) camera in the ultrahigh vacuum scattering chamber
RASOR on beamline I10 at the Diamond Light Source.Beale  _et al._ (2010) A
magnetic field was applied to the sample whose strength and tilt angle,
$\gamma$, with respect to the surface normal can be varied. The incident,
$\sigma$-polarized x-rays were at the resonance with the $L_{3}$ edge of Cu
(931.25 eV), resulting in a wavelength of 13.3 Å. For Cu2OSeO3 with its
relatively large lattice constant of 8.925 Å, the (001) Bragg peak can be
reached at $2\theta\approx 96.5^{\circ}$.

In noncentrosymmetric $P2_{1}3$ helimagnets, a ‘universal’ magnetic phase
diagram is observed that shows helical, conical, or ferrimagnetic order below
the transition temperature $T_{\mathrm{c}}$ as a function of increasing
field.Buhrandt and Fritz (2013) Below $T_{\mathrm{c}}$ of $\sim$57.5 K for
Cu2OSeO3, at finite fields, the skyrmion phase can be found. Figure 1(c) shows
a schematic of the magnetic phase diagram for the Cu2OSeO3 bulk crystal as
observed by REXS.Zhang  _et al._ (2016a) The magnetization patterns
corresponding to the helical, skyrmion, and conical phase are shown in Fig.
1(d), and the experimental REXS results in (e), respectively (from top to
bottom). The modulation wavevector for all three phases has a length of
0.0158-0.0162 reciprocal lattice units (r.l.u.), corresponding to a real-space
modulation pitch of $\sim$56 nm, in agreement with the values reported in the
literature.Adams  _et al._ (2012); Seki  _et al._ (2012a, b); White  _et al._
(2012); Langner  _et al._ (2014); Rajeswaria  _et al._ (2015) In the helical
state, the weak cubic anisotropy locks the propagation wave vector along the
$\langle$100$\rangle$ direction. At 57 K and in an applied magnetic field of
32 mT, $\gamma=0^{\circ}$, the sharp six-fold-symmetric diffraction pattern
emerges, which is a signature of the single-domain, long-range-ordered
skyrmion lattice state. One of the skyrmion wave vectors is along [010] due to
the cubic anisotropy.Zhang  _et al._ (2016a)

By titling the magnetic field to $\gamma=17^{\circ}$, the skyrmion lattice
breaks up into domains, resulting in a necklace-like diffraction pattern, as
shown in Fig. 2(a). The formation of domains is the result of the competing
magnetic anisotropies.Zhang  _et al._ (2016a) Note that the diffraction
pattern was obtained with an incident x-ray beam focused to an area of 300
$\times$ 300 $\mu$m2 on the sample. Once the beam is skimmed down to an area
of 20 $\mu$m in diameter (using a pinhole), the single domain state with its
six-fold-symmetric pattern is recovered [cf. Fig. 2(b)]. This means that the
domains are $>$$100\pi~{}\mu$m2 in area, and that this beam spot can be used
to map out the real-space domain pattern.

Figure 2: Reciprocal space map of the skyrmion lattice plane that is
perpendicular to the external field, reached by field cooling down to 57 K in
a field of 32 mT with $\gamma=15^{\circ}$. Area sampled by the beam: (a)
$300\times 300~{}\mu\text{m}^{2}$ and (b) $20~{}\mu\text{m}$ in diameter. The
scattering intensity is in arbitrary units. Figure 3: Scanning REXS images.
(a) Definition of the lattice rotation angle $\Psi$ in the skyrmion
diffraction pattern ($\Psi\in[0^{\circ},60^{\circ}]$ as indicated by the color
bar). The case of $\Psi=30^{\circ}$ is shown. (b-h) 1 mm $\times$ 1 mm raster
scans showing in-plane rotational domain maps for the indicated field angle
$\gamma$.

The sample shape and geometry is shown Fig. 1(b), in which the raster-scanned
area ($1\times 1$ mm2) is indicated by the blue square. The skyrmion lattice
state is reached by field-cooling from 65 K (paramagnetic phase) down to 57 K
in a field of 32 mT with the $\gamma$ angle as indicated in Fig. 3(b-h),
followed by a 15 min waiting period. Each area scan image is composed of
$50\times 50$ pixels. For most of the pixels, a six-fold-symmetric, single
domain diffraction pattern is observed. Its rotational state is characterized
by the in-plane rotation angle $\Psi$, as shown Fig. 3(a). An angle of
$\gamma=0^{\circ}$ corresponds to two of the six diffraction spots being
aligned along the $q_{x}$-axis. For some pixels, especially for larger field
tilt angles $\gamma$, a multidomain state is found at the domain boundaries.
The evolution of the domain pattern as a function of field tilt angle is shown
in Fig. 3(b-h). For $\gamma=0^{\circ}$, a perfect long-range-ordered single
skyrmion lattice domain is observed [cf. Fig. 3(b)], locked in one direction
determined by the anisotropies. When the field is slightly tilted
($\gamma=5^{\circ}$), two domains start to emerge, which do not differ very
much in $\Psi$. As the field is further tilted, the domains become randomly
oriented and the average size decreases (most strongly for
$\gamma=15^{\circ}\rightarrow 19^{\circ}$). For $\gamma=22^{\circ}$, a mosaic
pattern is observed at the bottom-left corner of Fig. 3(g). This results from
a lack of resolution, as determined by the x-ray beam size. As the domain size
decreases to less than the beam size, a multidomain diffraction pattern is
observed across each pixel. Note that the pinhole diameter can not be reduced
much below 20 $\mu$m in our experimental configuration, as this will result in
Fresnel diffraction (near-field condition giving cylindrical wave fronts),
i.e., the plane wave approximation can no longer be applied for modeling the
scattering process.

The shape of the domains is generally rather irregular, and their distribution
random, suggesting that the domain formation is spontaneous and not governed
by defect-pinning. The domain pattern obtained for each $\gamma$ is stable
over time, as confirmed by multiple scans of the same area. Note, however,
that for the same $\gamma$, if the temperature is increased above
$T_{\mathrm{c}}$ and then lowered down to the skyrmion phase again, the domain
image has no resemblance to the previous pattern. The orientation of
neighboring domains is similar and the system generally prefers the
orientation of neighboring domains to differ by less than $10^{\circ}$. At
higher tilt angles ($\gamma\geq 15^{\circ}$), the domain boundaries are sharp,
however, for smaller angles the transition between domains becomes almost
continuous, as can be seen in Fig. 3(d).

This ‘polycrystalline’ appearance of a skyrmion lattice reflects the delicate
balance of the system’s competing interactions. The isotropic exchange
interaction, the anisotropic exchange (DMI) term, and the Zeeman energy
compete about the principal magnetic order, while anisotropy, demagnetization,
and possibly the ferroelectric effect in the multiferroic material Cu2OSeO3
serve as perturbations which are responsible for the fine adjustment of the
system’s structure. Skyrmions behave like quasi-particles and prefer to keep
their close-packed order, i.e., less ferromagnetic-like space in-between them
is preferred. However, if defects are introduced into the system, some region
would have to become ferromagnetic, unless the skyrmions adapt their shape and
size. As a result, the skyrmions in the defect zone can become elliptically
distorted, as observed using Lorentz transmission electron microscopy (LTEM)
on thinned-down FeGe1-xSix bulk samples.Matsumoto  _et al._ (2016) Moreover,
the close-packed ordering of the skyrmion lattice can have defects at the
domain boundaries, e.g., ‘5-7 defects’ where five or seven skyrmions are
surrounding a reference skyrmion, instead of six.Rajeswaria  _et al._ (2015)
In this case, a line defect can appear, forming the domain boundary. Depending
on its configuration, the neighboring domains can change their orientations
and take an arbitrary angle, depending on the width of the defect-containing
boundary. The thicker this boundary, the larger the relative rotation. If we
relate this to our domain observations, it points towards the fact that for
small $\gamma$, the domain boundary contains thin defect lines, leading to
small relative rotations. With increasing $\gamma$, the number of defects
increases across the domain boundary, giving rise to sharp rotational
transitions. Note that LTEM imaging, despite being immensely useful for
studying the local defects of the skyrmions, is not able to map out the large-
scale skyrmion structures like skyrmion lattice domains. Moreover, for systems
with such a delicate energy balance as Cu2OSeO3, LTEM sample preparation-
induced defects may affect the intrinsic surface domain structure (essentially
a property of the bulk crystal). Therefore, the type and density of defects,
and their influence on the observed domains, can be significantly different
between bulk and thinned-down LTEM samples.

In summary, we have presented a study of the domain structure in the skyrmion
lattice phase of Cu2OSeO3 using REXS imaging. This method provides important
information on the domain distribution, shape, size, and formation. By tuning
the field tilt angle, in-plane rotational domains in the hexagonally ordered
skyrmion lattice phase can be generated. The lateral dimensions of the domains
are for small tilt angles $>$20 $\mu$m. The domain transition is abrupt,
expect for very small tilt angles, where the change is almost gradual. The
study of skyrmion domains may enable potential device applications making use
of the skyrmion lattice state. In skyrmion lattice-based memory, the single-
domain state must be broken up into domains, which each encode information in
a spatially separated manner. This was demonstrated using the field tilt angle
as an additional handle.

The REXS experiments were carried out on beamline I10 at the Diamond Light
Source, UK, under proposals SI-11784 and SI-12958. We thank the EPSRC for
support under grant EP/N032128/1. S.L.Z. and T.H. acknowledge financial
support by the Semiconductor Research Corporation. A.B. and C.P. acknowledge
financial support through DFG TRR80 and ERC AdG (291079, TOPFIT).

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