# Dipole-dipole-interaction-induced entanglement between two-dimensional
ferromagnets

D. Wuhrer  Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany
Authors to whom correspondence should be addressed: dennis.wuhrer@uni-
konstanz.de and<EMAIL_ADDRESS>N. Rohling  Fachbereich
Physik, Universität Konstanz, D-78457 Konstanz, Germany  W. Belzig
Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany  Authors
to whom correspondence should be addressed<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

We investigate the viability of dipole-dipole interaction as a means of
entangling two distant ferromagnets. To this end we make use of the Bogoliubov
transformation as a symplectic transformation. We show that the coupling of
the uniform magnon modes can be expressed using four squeezing parameters
which we interpret in terms of hybridization, one-mode and two-mode squeezing.
We utilize the expansion in terms of the squeezing parameters to obtain an
analytic formula for the entanglement in the magnon ground state using the
logarithmic negativity as entanglement measure. Our investigation predicts
that for infinitely large two-dimensional ferromagnets, the dipole-dipole
interaction does not lead to significant long-range entanglement. However, in
the case of finite ferromagnets, finite entanglement can be expected.

††preprint: AddPreprint/here

Magnetic materials exhibit a wide range of magnetic structures Dyson (1956);
Néel, M. Louis (1948); Bogdanov and Yablonskii (1989); Bode  _et al._ (2007),
ranging from simple collinear magnetic structures including ferromagnets (FMs)
Dyson (1956) and antiferromagnets (AFMs) Néel, M. Louis (1948) to non-
collinear structures like skyrmion lattices Bogdanov and Yablonskii (1989);
Garst, Waizner, and Grundler (2017); Díaz, Klinovaja, and Loss (2019); Díaz
_et al._ (2020). These magnetically ordered materials host collective
excitations of coupled magnetic moments. These excitations can be described
classically using spin waves, but are interpreted as particles - known as
magnons - when treated quantum mechanically Dyson (1956); Kittel (2004);
Nolting and Ramakanth (2009).

The interest in magnons rose in recent years Yuan  _et al._ (2022); Barman,
Gubbiotti, and Ladak et al. (2021) as they provide a platform for easy-to-
manipulate long-distance transport with low dissipation Cornelissen  _et al._
(2015a); Lebrun  _et al._ (2018). They may provide an alternative to the
Joule-heating-plagued electrons in information transport and processing Chumak
_et al._ (2015), as a key computational device, transistors, can be realized
by magnetic structures Chumak, Serga, and Hillebrands (2014). Moreover, it is
possible to link magnetic materials to conventional electronic devices using
the (inverse-)spin-Hall effect Cornelissen  _et al._ (2015b); Lebrun  _et al._
(2018), enabling the integration of magnonic devices with state of the art
technology. This is highly significant due to the reduction in computational
technology size and the emergence of two-dimensional materialsNovoselov  _et
al._ (2005, 2016); Mak  _et al._ (2010); Radisavljevic  _et al._ (2011); Elias
_et al._ (2019); Novoselov  _et al._ (2004); Mayorov  _et al._ (2011), notably
Van der Waals (VdW) materialsHaigh  _et al._ (2012); Geim and Grigorieva
(2013); Castellanos-Gomez  _et al._ (2022); Yang  _et al._ (2022). These
materials provide the potential to produce engineered synthetic systems with
desired attributes by layering different materials together. Two-dimensional
(2D) magnetic layersGibertini  _et al._ (2019); Burch, Mandrus, and Park
(2018); Gong  _et al._ (2017); Huang  _et al._ (2017); Schulz  _et al._
(2023); Ma  _et al._ (2023); Gong  _et al._ (2023); Xu  _et al._ (2022); Zhong
_et al._ (2023); Cheng  _et al._ (2022) represent just one of many potential
building blocks. Therefore, it is necessary to enhance our understanding of
the excitations of 2D magnetic materials, along with the impact of coupling
between distinct magnetic materials, considering the potential for squeezing
and entanglement within magnetic systems.

Similar to photons Walls (1983); Wu  _et al._ (1986), quantum squeezing and
entanglement for magnons have been predicted theoretically Kamra, Belzig, and
Brataas (2020); Kamra  _et al._ (2019); Wuhrer, Rohling, and Belzig (2022);
Kamra and Belzig (2016) and observed in experiments Zhao  _et al._ (2004,
2006). However, in contrast to photons, squeezing is an inherent property of
magnetic materials. Squeezing for magnons refers to quadrature squeezing. This
means decreasing the variance of one observable while increasing the variance
of a canonically conjugate observable, such as the spin components in the $x$
and $y$-direction in a squeezed FM with the $z$-axis as easy axis. This is
because squeezed magnon states are a superposition of non-squeezed magnon
Fock-states. Consequently, the system already holds a substantial number of
non-squeezed magnons even in the ground state. This superposition has been
proposed for exciting multiple quantum dots simultaneously and in the process
entangle them Skogvoll  _et al._ (2021) or being probed by quantum dots
adjacent to the magnetRömling  _et al._ (2023). It was also discussed that the
entanglement due to magnons in VdW materials is switchable by electronic and
magnetic means leading to electrical controllable entanglement of distant
qubitsYuan  _et al._ (2021). As a result, magnons make an interesting tool for
quantum computing Wehner, Elkouss, and Hanson (2018); Milburn and Braunstein
(1999); Bennett, Brassard, and Mermin (1992); Yin  _et al._ (2020).

Our work explores the feasibility of long-distance magnon entanglement through
dipole-dipole interaction in two separated FMs, facilitating the entanglement
of remote systems or by magnetic materials involved in the formation of VdW
materials. By ’long range,’ we denote a distance greater then decay length of
the Heisenberg exchange, which experiences an exponential decay.

Figure 1: (a) System of two FMs parallel to the $x$-$y$-plane separated by a
distance $l$ along the $z$-axis. The magnified area depicts the interactions
between the spins of the different FMs. (b) Phase diagram showing the
orientation of the spins in FM $A$ and $B$ depending on the distance $l/a$ and
the dipole-dipole interaction strength $D/K_{z}$. For small distances or small
dipole-dipole interactions spins in both FM point out of plane forming a
ferromagnetic ordering between the FMs (green, OOP FM). For large distances
and large dipole-dipole interactions spins in both FM lie in-plane forming an
antiferromagnetic ordering between the FMs (yellow, IP AFM). The border
between both phases (black solid line) is given by Eq. (7). The black dashed
line shows the phases depending on the distance for $D=0.5K_{z}$. The
orientation of the spins inside each FM is depicted by the arrows with the
figure-plane being the $x$-$z$-plane.

As a model system we consider two two-dimensional (2D) square lattice FMs
separated by a distance $l$, see Fig. 1(a). The spins inside each FM are
subjected to ferromagnetic Heisenberg interaction and two uniaxial
anisotropies: one perpendicular to the plane of the 2D FMs ($z$-direction) and
the other along one of principle axis of the FMs ($x$-direction). Moreover, we
consider the long range interaction between the spins to be transmitted by
dipole-dipole interaction. The Hamiltonian describing the interactions of
spins inside FM $A$ is given by

$\displaystyle\hat{H}_{A}$
$\displaystyle=\hat{H}_{\text{ex}}^{A}+\hat{H}_{\text{an}}^{A}+\hat{H}_{\text{D}}^{A}\,,$
(1)

with a similar expression for FM $B$. The Heisenberg exchange is given by

$\displaystyle\hat{H}_{\text{ex}}^{A}$
$\displaystyle=-\sum_{\begin{subarray}{c}\bm{r}_{i},\bm{r}_{j}\in{A}\\\
\bm{r}_{i}\neq\bm{r}_{j}\end{subarray}}J_{ij}\bm{\hat{S}}_{i}\cdot\bm{\hat{S}}_{j}\,,$
(2)

with $J_{ij}=J_{ji}>0$ being the ferromagnetic exchange strength between spins
$\bm{\hat{S}}_{i}$ and $\bm{\hat{S}}_{j}$ of spin length $S$ at sites
$\bm{r}_{i}$ and $\bm{r}_{j}$ in FM $A$ preferring a parallel configuration of
spins.

The anisotropy term in the Hamiltonian, Eq. (1), is given by

$\displaystyle\hat{H}_{\text{an}}^{A}$
$\displaystyle=-K_{z}\sum_{\bm{r}_{i}\in
A}\left(\hat{S}^{z}_{i}\right)^{2}-K_{x}\sum_{\bm{r}_{i}\in
A}\left(\hat{S}^{x}_{i}\right)^{2}\,,$ (3)

where $K_{z}$ ($K_{x}$) is the strength of the uniaxial anisotropy along the
$z$-axis ($x$-axis). We assume $K_{z}\gg K_{x}>0$ such that the spins prefer
to align parallel to the $z$-direction.

Finally, we consider dipole-dipole interaction between magnetic moments given
by

$\displaystyle\hat{H}_{\text{D}}^{A}=\sum_{\begin{subarray}{c}\bm{r}_{i},\bm{r}_{j}\in{A}\\\
\bm{r}_{i}\neq\bm{r}_{j}\end{subarray}}D_{ij}\left[\bm{\hat{S}}_{i}\cdot\bm{\hat{S}}_{j}-3\left(\bm{e}_{ij}\cdot\bm{\hat{S}}_{i}\right)\left(\bm{e}_{ij}\cdot\bm{\hat{S}}_{j}\right)\right]\,.$
(4)

Here $\bm{e}_{ij}$ is the unit vector along the connection line of spins at
$\bm{r}_{i}$ and $\bm{r}_{j}$. $D_{ij}$ is the strength of dipolar coupling
between two spins at $\bm{r}_{i}$ and $\bm{r}_{j}$ which is of the form

$\displaystyle
D^{A}_{ij}=\frac{a^{3}D^{A}}{\left|\bm{r}_{i}-\bm{r}_{j}\right|^{3}}=\frac{\mu_{0}g^{A}_{1}g^{A}_{2}\mu_{B}^{2}}{\hbar^{2}}\frac{1}{|\bm{r}_{i}-\bm{r}_{j}|^{3}}\,,$
(5)

with $\mu_{0}$ being the vacuum magnetic permeability, $g_{i}^{A}$ the
gyromagnetic ratio of the interacting magnetic moments, $\mu_{B}$ the Bohr
magneton, $\hbar$ the reduced plank constant and $a$ the lattice constant.
Throughout this work, we will assume that $\hbar,a=1$, which means that all
the interaction parameters ($J_{ij}$, $K_{x}$, $K_{z}$, $D^{A}$) will be given
in the unit of energy.

The dipole-dipole interaction decays with the inverse of the third power of
the distance between the spins and thus is classified as long range.

The coupling between both FMs is also conveyed through dipolar interaction

$\displaystyle\hat{H}_{\text{int}}$
$\displaystyle=\\!\sum_{\begin{subarray}{c}\bm{r}_{i},\in A\\\ \bm{r}_{j}\in
B\end{subarray}}D^{\text{int}}_{ij}\left[\bm{\hat{S}}_{i}\cdot\bm{\hat{S}}_{j}-3\left(\\!\bm{e}_{ij}\cdot\bm{\hat{S}}_{i}\\!\right)\left(\\!\bm{e}_{ij}\cdot\bm{\hat{S}}_{j}\\!\right)\right]\,.$
(6)

We assume similar FMs. Thus, Eq. (5) yields similar dipole constants for the
interaction inside each FM and between both FMs and we assume
$D^{A/B}=D^{\text{int}}=D$.

The linear spin wave theory explains the excitation within magnetic systems as
harmonic excitations, spin waves, by expanding the spin operators around the
classical ground state using magnonic creation and annihilation operators. The
classical configuration for each isolated FM is ferromagnetic, because we
assume that the exchange interaction is dominant inside each FM. The
anisotropies restrict the direction of magnetization to lie within the
$x$-$z$-plane. The exact orientation is determined by the strength of the
dipole-dipole interaction inside each and between both FMs. Thus the classical
ground state depends on the dipole interaction strength $D$ and the distance
$l$ between both FMs. Only two orientations are taken by the FMs, see Fig.
1(b).

For short distances a ferromagnetic out-of-plane configuration (OOP FM)
between both FMs is taken due to the dominance of the dipole interaction
between the systems, which favours a parallel, ferromagnetic alignment along
the connecting line ($\bm{e}_{z}$). For large separations and weak dipole-
dipole interaction also the OOP FM configuration is taken. This is due to the
fact that with $D$ not only the interaction between the FMs is varied, but
also the dipole-dipole interaction between spins inside each FM. Therefore,
for weak dipole-dipole interaction the spins prefer to align with $K_{z}$.

For significant distances, the spins align ferromagnetically in-plane due to
the dominance of the dipole-dipole interaction within each FM over the
anisotropy. Minimising the interaction term when the spins are perpendicular
to the connection line results in an AFM ordering between the two
ferromagnets. This is referred to as in-plane AFM (IP AFM) ordering.

In this work we investigate square lattice FMs, where the phase boundary
between OOP FM and IP AFM phase, seen as black solid line in Fig. 1(b), is
given by

$\displaystyle K_{z}-(K_{x}+3D_{x|\bm{0}})$
$\displaystyle=D_{\bm{0}}^{\text{int}}-\frac{3}{2}\left(D_{x|\bm{0}}^{\text{int}}+D_{z|\bm{0}}^{\text{int}}\right)\,.$
(7)

The coefficient(s) $D_{\alpha|\bm{0}}$
$(D_{\bm{0}}^{\mathrm{int}},D_{\alpha|\bm{0}}^{\mathrm{int}})$ is (are) the
sum(s) of the dipole interaction inside (between) the FMs, with $\alpha$
specifiying the direction of the connection vector. The exact definition is
given in the supplementary partWuhrer, Rohling, and Belzig .

Figure 2: Energy of the $\bm{k}=\bm{0}$ squeezed magnon modes depending on the
distance $l/a$ between both FMs in units of the energy of the eigenmodes in
non-interacting FMs, $\tilde{\varepsilon}_{\bm{0}}$. At the phase change,
$l/a\approx 0.33$, the energy $\varepsilon_{\beta,\bm{0}}$ vanishes. For large
distances $\varepsilon_{\alpha,\bm{0}}$ and $\varepsilon_{\beta,\bm{0}}$
approach $\tilde{\varepsilon}_{\bm{0}}$ (black, dashed line). Parameters are
chosen as $D=0.5\cdot 10^{-4}|J_{1}|$, $K_{z}=10^{-4}|J_{1}|$ and
$K_{x}=10^{-6}|J_{1}|$.

(a)  (b)

Figure 3: Dependence of the Bogoliubov matrix elements on the distance $l$ for
the $\bm{k}=\bm{0}$ mode with $D=0.5\cdot 10^{-4}|J_{1}|$,
$K_{z}=10^{-4}|J_{1}|$ and $K_{x}=10^{-6}|J_{1}|$. Divergences and jumps occur
at the phase change $l/a\approx 0.33$ (a) Elements of $U_{\bm{0}}$. (b)
Elements of $V_{\bm{0}}$.

The classical configuration is represented by the right handed eigensystems

$\displaystyle\bm{e}_{i,1}=\begin{pmatrix}\cos(\vartheta_{i})\\\ 0\\\
-\sin(\vartheta_{i})\end{pmatrix}\,$
$\displaystyle,\qquad\bm{e}_{i,2}=\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}\,$
$\displaystyle\bm{e}_{i,3}=$
$\displaystyle\begin{pmatrix}\sin(\vartheta_{i})\\\ 0\\\
\cos(\vartheta_{i})\,,\end{pmatrix}$ (8)

for $i\in\\{A,B\\}$, where $\vartheta_{A}$ ($\vartheta_{B}$) is the angle
between the classical spin configuration of FM $A$ ($B$) and the
$z$-direction. Applying the linear Holstein-Primakoff transformationHolstein
and Primakoff (1940) yields for spins in FM $A$

$\displaystyle\hat{S}_{i,1}\approx\hbar\sqrt{\frac{S}{2}}\left(\hat{a}_{i}+\hat{a}^{\dagger}_{i}\right)\,,\qquad\hat{S}_{i,2}\approx-i\hbar\sqrt{\frac{S}{2}}\left(\hat{a}_{i}-\hat{a}^{\dagger}_{i}\right)\,,$
$\displaystyle\hat{S}_{i,3}\approx\hbar\left(S-\hat{a}^{\dagger}_{i}\hat{a}^{\phantom{\dagger}}_{i}\right)\,,$
(9)

and in FM $B$

$\displaystyle\hat{S}_{i,1}\approx\hbar\sqrt{\frac{S}{2}}\left(\hat{b}_{i}+\hat{b}^{\dagger}_{i}\right)\,,\qquad\hat{S}_{i,2}\approx-i\hbar\sqrt{\frac{S}{2}}\left(\hat{b}_{i}-\hat{b}^{\dagger}_{i}\right)\,,$
$\displaystyle\hat{S}_{i,3}\approx\hbar\left(S-\hat{b}^{\dagger}_{i}\hat{b}^{\phantom{\dagger}}_{i}\right)\,,$
(10)

with $\hat{S}_{i,\alpha}=\bm{e}_{A/B,\alpha}\cdot\bm{\hat{S}}_{i}$.

We perform the Fourier transformation to take account for the periodic
structure of the system

$\displaystyle\hat{a}_{\bm{k}}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{\bm{r}_{i}\in
A}e^{-i\bm{r}_{i}\cdot\bm{k}}\hat{a}_{i}\,,$ $\displaystyle\hat{b}_{\bm{k}}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{\bm{r}_{i}\in
B}e^{-i\bm{r}_{i}\cdot\bm{k}}\hat{b}_{i}\,,$ (11)

where we assume $N_{A}=N_{B}=N$. Inserting Eqs. (9-11) into Eqs. (1+6) the
Hamiltonian takes the form

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\bm{k}\in\text{BZ}^{+}}\bm{\hat{a}}^{\dagger}_{\bm{k}}\underbrace{\begin{pmatrix}E_{A,\bm{k}}&\mu_{1,\bm{k}}^{\ast}&2\xi_{A,\bm{k}}^{\ast}&\mu_{2,\bm{k}}^{\ast}\\\
\mu_{1,\bm{k}}&E_{B,\bm{k}}&\mu_{2,\bm{k}}^{\ast}&2\xi_{B,\bm{k}}^{\ast}\\\
2\xi_{A,\bm{k}}&\mu_{2,\bm{k}}&E_{A,\bm{k}}&\mu_{1,\bm{k}}\\\
\mu_{2,\bm{k}}&2\xi_{B,\bm{k}}&\mu_{1,\bm{k}}^{\ast}&E_{B,\bm{k}}\end{pmatrix}}_{H_{\bm{k}}}\bm{\hat{a}}_{\bm{k}}\,,$
(12)

where we defined
$\bm{\hat{a}}_{\bm{k}}=\left(\hat{a}_{\bm{k}},\hat{b}_{\bm{k}},\hat{a}^{\dagger}_{-\bm{k}},\hat{b}^{\dagger}_{-\bm{k}}\right)^{\top}$
and $\text{BZ}^{+}$ labels all $\bm{k}$ in the Brillouin zone with a positive
(or zero) $k_{x}$ component. The exact definition of each parameter contained
by the Hamiltonian is given in the supplementaryWuhrer, Rohling, and Belzig .

The meaning of the different parameters, if only they together with the
diagonal elements $E_{i,\bm{k}}$ would appear in the Hamiltonian, is outlined
below. Energy of a bare magnon mode $\hat{a}_{\bm{k}}$ ($\hat{b}_{\bm{k}}$)
inside FM $A$ ($B$) is given by $E_{A,\bm{k}}$ $\left(E_{B,\bm{k}}\right)$,
while $\mu_{1,\bm{k}}$ results in a hybridisation of the magnon modes of the
different FMs. This does not change the vacuum state and corresponds to a
rotation in the phase space spanned by the $A$ and $B$ magnons.

Squeezing inside FM $A$ ($B$) is mediated by $\xi_{A,\bm{k}}$
($\xi_{B,\bm{k}}$) entangling modes with wave vectors $\pm\bm{k}$ inside each
FM, while squeezing of $\pm\bm{k}$ modes of the different FMs is a result of
$\mu_{2,\bm{k}}$. Squeezing alters the vacuum state as the squeezed magnon
states are superpositions of populated $A$ and $B$ magnon states Wuhrer,
Rohling, and Belzig (2022). If we combine different of these parameters a
clear distinction between squeezing and hybridization for each parameter
cannot be given anymore.

In order to diagonalize the Hamiltonian we perform a four-dimensional (4D)
Bogoliubov transformation Bogoliubov (1958); Valatin (1958) which is a
symplectic transformation converting our bare magnon operators
$\hat{a}_{\bm{k}}$ and $\hat{b}_{\bm{k}}$ into squeezed magnon operators
$\hat{\alpha}_{\bm{k}}$ and $\hat{\beta}_{\bm{k}}$. The Bogoliubov
transformation can be written as

Figure 4: Dependence of the squeezing parameters
$\theta,\theta_{1},\theta_{2},\theta_{3},\theta_{4}$ on the distance $l$ for
the $\bm{k}=\bm{0}$ mode. Divergences occur at the phase change $l/a\approx
0.33$. Parameters are $D=0.5\cdot 10^{-4}|J_{1}|$, $K_{z}=10^{-4}|J_{1}|$ and
$K_{x}=10^{-6}|J_{1}|$.

$\displaystyle\bm{\hat{a}}_{\bm{k}}$
$\displaystyle=\begin{pmatrix}\hat{a}_{\bm{k}}\\\ \hat{b}_{\bm{k}}\\\
\hat{a}^{\dagger}_{-\bm{k}}\\\
\hat{b}^{\dagger}_{-\bm{k}}\end{pmatrix}=\begin{pmatrix}U_{\bm{k}}&V_{\bm{k}}\\\
V_{\bm{k}}^{\ast}&U_{\bm{k}}^{\ast}\end{pmatrix}\begin{pmatrix}\hat{\alpha}_{\bm{k}}\\\
\hat{\beta}_{\bm{k}}\\\ \hat{\alpha}^{\dagger}_{-\bm{k}}\\\
\hat{\beta}^{\dagger}_{-\bm{k}}\end{pmatrix}=\mathcal{U}_{\bm{k}}\bm{\hat{\alpha}}_{\bm{k}}\,,$
(13)

with

$\displaystyle
U_{\bm{k}}^{\phantom{\dagger}}U_{\bm{k}}^{\dagger}-V_{\bm{k}}^{\phantom{\dagger}}V_{\bm{k}}^{\dagger}$
$\displaystyle=1\,,$ $\displaystyle
U_{\bm{k}}^{\phantom{\top}}V_{\bm{k}}^{\top}-V_{\bm{k}}^{\phantom{\top}}U_{\bm{k}}^{\top}=0\,.$
(14)

The elements of $U_{\bm{k}}$ are responsible for hybridization and the
elements of $V_{\bm{k}}$ for squeezing, with

$\displaystyle U_{\bm{k}}$
$\displaystyle=\begin{pmatrix}u_{1,\bm{k}}&u_{2,\bm{k}}\\\
u_{3,\bm{k}}&u_{4,\bm{k}}\end{pmatrix}\,,$ $\displaystyle V_{\bm{k}}$
$\displaystyle=\begin{pmatrix}v_{1,\bm{k}}&v_{2,\bm{k}}\\\
v_{3,\bm{k}}&v_{4,\bm{k}}\end{pmatrix}\,.$ (15)

Solving the eigenvalue equation of the Hamiltonian yields the elements of the
Bogoliubov matrix $\mathcal{U}_{\bm{k}}$ and the eigenenergies
$\varepsilon_{\alpha/\beta,\bm{k}}$ of the squeezed magnon modes

$\displaystyle\varepsilon_{\alpha/\beta,\bm{k}}$
$\displaystyle=\sqrt{\left|E_{\bm{k}}\pm\mu_{1,\bm{k}}\right|^{2}-\left|\mu_{2,\bm{k}}\pm
2\xi_{\bm{k}}\right|^{2}}\,,$ (16)

where we used that we cope with similar FMs yielding
$E_{A,\bm{k}}=E_{B,\bm{k}}=E_{\bm{k}}$ and
$\xi_{A,\bm{k}}=\xi_{B,\bm{k}}=\xi_{\bm{k}}$ and that we regard either the OOP
FM or the IP AFM phase. Fig. 2 shows the energies of magnons
$\hat{\alpha}_{\bm{0}}$ and $\hat{\beta}_{\bm{0}}$ depending on the distance.
We see that for large distances both converge towards the energy
$\tilde{\varepsilon}_{\bm{0}}=\sqrt{\varepsilon_{\bm{0}}^{2}-4|\xi_{\bm{0}}|^{2}}$
of the squeezed magnons in the case of non interacting FMs. At the phase
change $\varepsilon_{\alpha,\bm{0}}$ shows a jump, while
$\varepsilon_{\beta,\bm{0}}$ vanishes.

From several two-mode squeezed cases we expect the largest squeezing and
entanglement for the uniform $\bm{k}=\bm{0}$ mode Wuhrer, Rohling, and Belzig
(2022); Wuhrer  _et al._ (2023). Thus, in the following, we will limit our
considerations to the $\bm{k}=0$ mode and will drop the wave vector index for
clarity. For the uniform mode all parameters in Eq. (12) are real yielding
real Bogoliubov parameters of the form

$\displaystyle u_{1}$
$\displaystyle=\\!\sqrt{\\!\frac{E\\!+\\!\mu_{1}\\!+\\!\varepsilon_{\alpha}}{4\varepsilon_{\alpha}}},$
$\displaystyle u_{2}$
$\displaystyle=\\!-\sqrt{\\!\frac{E\\!-\\!\mu_{1}\\!-\\!\varepsilon_{\beta}}{4\varepsilon_{\beta}}}\,,$
(17)

with the remaining matrix elements given in the supplementary materialWuhrer,
Rohling, and Belzig . Fig. 3 shows the dependence of the Bogoliubov matrix
elements on the distance of both FMs. For distances $l/a<0.33$, in the OOP FM
phase, we have non-zero values for all elements of $U$ and close to zero value
for all elements of $V$. Thus, the ferromagnetic structure is dominated by
hybridisation ($U$) with little to no squeezing ($V$). Due to the vanishing
magnon energy of the $\hat{\beta}$ magnons at the phase change $l/a\approx
0.33$, we see divergences in $u_{2}$, $u_{4}$, $v_{2}$ and $v_{4}$.

In the IP AFM configuration ($l/a>0.33$) we see an increase in all elements of
$\mathcal{U}$ compared to the OOP FM configuration with the elements
responsible for squeezing ($V$) being comparable in magnitude. This is due to
the large inherent degree of squeezing of AFMs Wuhrer, Rohling, and Belzig
(2022).

The Bogoliubov transformation for the system at hand is an element of the ten
dimensional group of 4D symplectic matrices $\mathrm{Sp}(4,\mathds{R})$ Colas,
Grain, and Vennin (2022). The limitation to the $\bm{k}=0$ mode leaves us with
real a Bogoliubov matrix only which can be parametrised by four parameters. We
connect the Bogoliubov matrix to the generators of the group and introduce the
four-mode squeezing operator

$\displaystyle\hat{S}_{4}(\bm{\theta})$
$\displaystyle=\exp{\left(\sum_{i=1}^{4}\theta_{i}\hat{\Phi}_{i}\right)}\,,$
(18)

with $\theta_{i}$ being the real parameter corresponding to the generator
$\hat{\Phi}_{i}$ which in quantum representation are given by

$\displaystyle\hat{\Phi}_{1/2}$
$\displaystyle=\hat{a}^{\dagger}\hat{a}^{\dagger}-\hat{a}\hat{a}\mp\left(\hat{b}^{\dagger}\hat{b}^{\dagger}-\hat{b}\hat{b}\right)\,,$
(19) $\displaystyle\hat{\Phi}_{3}$
$\displaystyle=\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger}+\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger}\,,$
(20) $\displaystyle\hat{\Phi}_{4}$
$\displaystyle=\hat{a}^{\dagger}\hat{b}^{\dagger}-\hat{a}\hat{b}+\hat{a}^{\dagger}\hat{b}^{\dagger}-\hat{a}\hat{b}\,.$
(21)

Each of these (pair of) generators can be identify as a one or two-mode
operation. $\theta_{1}\hat{\Phi}_{1}+\theta_{2}\hat{\Phi}_{2}$ results in one-
mode squeezing of modes inside FM $A$ and $B$ with squeezing parameters
$r^{A/B}=\theta_{2}\pm\theta_{1}$ for FM $A$ ($B$). $\hat{\Phi}_{3}$
($\hat{\Phi}_{4}$) results in hybridization (two-mode squeezing) of the magnon
modes of the different FMs. Thus, isolated each of the parameters $\theta_{i}$
can be identified with either one-mode, two-mode squeezing or hybridisation.
However, this clear separation cannot be made if multiple parameters are
involved, as the different generators do not commute. Ongoing we will refer to
all $\theta_{i}$ as "squeezing parameters".

We use the properties of the symplectic group to connect the elements of the
Bogoliubov matrix to the squeezing parameters:

$\displaystyle\bm{\hat{\alpha}}=\mathcal{U}^{-1}\bm{\hat{a}}=\hat{S}_{4}(\bm{\theta})\bm{\hat{a}}\hat{S}_{4}^{-1}(\bm{\theta})\,,$
(22)

which yields

$\displaystyle u_{1}$
$\displaystyle=\cosh(\theta_{2})\cos(\theta)+\frac{\theta_{1}}{\theta}\sin{(\theta)}\sinh{(\theta_{2})}\,,$
(23) $\displaystyle u_{2}$
$\displaystyle=\frac{\theta_{4}}{\theta}\sin{(\theta)}\sinh{(\theta_{2})}+\frac{\theta_{3}}{\theta}\sin{(\theta)}\cosh{(\theta_{2})}\,,$
(24)

with $\theta=\sqrt{(\theta_{3})^{2}-(\theta_{1})^{2}-(\theta_{4})^{2}}$. The
remaining matrix elements expressed in terms of the squeezing parameters are
given in the supplementary materialWuhrer, Rohling, and Belzig .

Fig. 4 shows the dependence of the squeezing parameters on the distance. They
show a similar behaviour as the elements of $\mathcal{U}$. The parameters
identified with intra FM ($\theta_{1},\theta_{2}$) and inter FM ($\theta_{4}$)
squeezing are close to zero in the OOP FM phase and finite in the IP AFM
phase, while the parameter identified with inter FM hybridization
($\theta_{5}$) is of a finite value for the whole parameter range. This
behaviour supports our identification based on the generators $\Phi_{i}$. The
divergence of the squeezing parameters is a result of vanishing energy of the
$\hat{\beta}_{\bm{0}}$ magnon modes at the phase change.

To determine the entanglement between the $\bm{k}=0$ magnon modes $\hat{a}$
and $\hat{b}$ in the squeezed vacuum state

$\displaystyle\left|0\right>_{\text{Sq}}=\hat{S}_{4}(\bm{\theta})\left(\left|0\right>_{A}\otimes\left|0\right>_{B}\right)\,,$
(25)

of the system, we use the logarithmic negativity Vidal and Werner (2002)

$\displaystyle E_{N}$
$\displaystyle=\max{\\{0,-\ln{\left(2\eta^{-}\right)}\\}}\,,$ (26)

as entanglement measure, with $\eta^{-}$ being the lowest symplectic
eigenvalue of the symmetric covariance matrix $V$ given by

$\displaystyle V_{ij}$
$\displaystyle=\frac{1}{2}\left<\left\\{\hat{R}_{i},\hat{R}_{j}\right\\}\right>\,.$
(27)

The anticommutator is represented by $\left\\{.,.\right\\}$ and $\bm{\hat{R}}$
is the phase space vector with

$\displaystyle\bm{\hat{R}}$ $\displaystyle=\begin{pmatrix}\hat{q}_{A}\\\
\hat{q}_{B}\\\ \hat{p}_{A}\\\
\hat{p}_{B}\end{pmatrix}=\underbrace{\frac{1}{\sqrt{2}}\begin{pmatrix}\mathds{1}_{2\times
2}&\mathds{1}_{2\times 2}\\\ -i\mathds{1}_{2\times 2}&i\mathds{1}_{2\times
2}\end{pmatrix}}_{M}\bm{\hat{a}}=M\mathcal{U}\bm{\hat{\alpha}}\,,$ (28)

where we used Eq. (13). The lowest symplectic eigenvalue in terms of the
squeezing parameters is given by

$\displaystyle\eta^{-}=\frac{1}{2}$
$\displaystyle\left|2\left|\frac{\theta_{4}}{2\theta}\sin{\left(2\theta\right)}-\frac{\theta_{3}\theta_{1}}{\theta^{2}}\sin^{2}{\left(\theta\right)}\right|-\sqrt{1+4\left|\frac{\theta_{4}}{2\theta}\sin{\left(2\theta\right)}-\frac{\theta_{3}\theta_{1}}{\theta^{2}}\sin^{2}{\left(\theta\right)}\right|^{2}}\right|\,,$
(29)

which is the central result of this work.

Figure 5: Entanglement of the $\bm{k}=\bm{0}$ magnon modes in FM $A$ and FM
$B$ for infinite (blue, solid), $100\times 100$ (orange, dotted) and
$500\times 500$ (green, dashed) sites FMs. We used $D=0.5\cdot
10^{-4}|J_{1}|$, $K_{z}=10^{-4}|J_{1}|$ and $K_{x}=10^{-6}|J_{1}|$.

From Eq. (29) we see that all squeezing parameter influence the entanglement
except for $\theta_{2}$, because $\Phi_{2}$ commutes with all other
generators. To further investigate the entanglement we concentrate on limiting
cases. If only squeezing inside each FM is present
($\theta_{3},\theta_{4}=0,\,\theta=i|\theta_{1}|$) entanglement between modes
of the different FM vanishes, which is expected as there is no interaction
between the FMs.

Only hybridisation between the FMs
($\theta_{1},\theta_{2},\theta_{4}=0,\,\theta=|\theta_{3}|$) results also in a
vanishing entanglement as the vacuum for hybridised modes is equal to the
vacuum of the bare modes $\hat{a}$ and $\hat{b}$ and thus is a separable
state.

Two-mode squeezing between both FMs
($\theta_{1},\theta_{2},\theta_{3}=0,\,\theta=i|\theta_{4}|$) results in
entanglement linear in the squeezing parameter $E_{N}=2|\theta_{4}|$ similar
to the entanglement in case of simple two-mode squeezing Yuan  _et al._
(2022). Therefore, known results are reproduced by the analytic formula given
in Eq. (29).

The distance dependence of the entanglement can be seen in Fig. 5 where we
regard finite and infinite 2D FMs.Fig. 5 shows similar behaviour of the
entanglement for all lattice sizes for $l/a\leq 1$. The phase transition from
an OOP FM configuration to an IP AFM configuration appears around $l/a\approx
0.33$, in the case of the infinite lattice, and entails a divergence of the
entanglement. This is shifted to larger distances for finite lattices. The
divergence once again arises due to the vanishing magnon energy at the phase
change.

Aside from the divergence the entanglement takes it maximum value at
approximately $l/a\approx 0.5$, rapidly decreases afterwards and, in the
infinite case, vanishes already at $l/a\approx 2$. The origin of the maximum
is still up to debate and subject to future work. The finite cases with
$100\times 100$ and $500\times 500$ lattices do show a plateau in the
entanglement around $l/a\approx 2$, which is higher the smaller the lattice
is. Increasing the distance up to the order of the system diameters ($100a$ or
$500a$ respectively) results in a decrease of entanglement that is
proportional to $1/l^{3}$. This behaviour is derived from the source of
interaction: the dipole-dipole interaction, Eq. (6).

We infer that no long range entanglement between infinite large magnets is
present due to dipole interaction, while for finite magnets the entanglement
can take a finite value. Thus, for large magnetic structures dipole
interaction alone will not be enough to establish a long range entanglement
between magnetic materials and other interactions or structures need to be
considered. We remark that we always assumed periodic boundary conditions for
magnons in finite systems. Also, for small distances ($l/a<1$), other
interactions between the magnetic moments may be of significance which require
investigation in future work.

At this point we would like to emphasise that we have studied an
experimentally easily realizable system of two distant FMs. While the exact
structure may differ from a simple cubic lattice structure, the main point,
the interaction between the FMs via the dipole-dipole interaction, is present
in any magnetic structure. The importance of this work is further emphasized
by the possibility of using the entangled magnons in finite size magnets to
establish long-range entanglement between spin-qubits, as proposed by Skogvoll
et al. Skogvoll  _et al._ (2021) or Yuan et al. Yuan  _et al._ (2021) This is
important for the future experimental realization of quantum computing Wehner,
Elkouss, and Hanson (2018); Milburn and Braunstein (1999); Bennett, Brassard,
and Mermin (1992); Yin  _et al._ (2020).

We investigated the entanglement of magnon modes in two 2D ferromagnets
coupled by dipole-dipole interaction by representing the Bogoliubov matrix as
an element of the symplectic group to represent its matrix elements in terms
of four squeezing parameters, each identified with either two mode squeezing
or hybridisation.

Investing the Bogoliubov matrix elements and the squeezing parameters shows a
clear dominance of hybridization over squeezing in the OOP FM phase and a
significant contribution of squeezing in the IP AFM phase, which is in
agreement with already known results. The system shows a finite hybridization
for all distances due to the long range character of the dipole-dipole
interaction, while the squeezing vanishes for large distances.

Using the logarithmic negativity, we derived an analytic expression for the
entanglement in terms of the squeezing parameters and were able to reproduce
known limiting cases and showed that besides squeezing between both FMs also
squeezing inside one FM paired with hybridization between both FMs is enough
to ensure entanglement between both systems, while hybridization alone is not
enough. This clearly shows the importance of squeezing for the entanglement of
magnon modes. For infinite systems, the entanglement already becomes
insignificant at $l/a=2$ implying no long range entanglement. However, for
finite systems we see a finite plateau in the entanglement before it tends
towards zero for distances of the order of the system diameter. The value of
this plateau decreases with increasing system size.

Concerning potential applications we note that the current high interest in 2D
van-der-Waals magnets Gibertini  _et al._ (2019) are explored experimentally
worldwide. These systems are usually fabricated in small flakes and our
prediction might be relevant to discuss the entanglement of two (finite-size)
flakes of two-dimensional ferromagnetic materials separated by an insulating
layer, thus avoiding a direct electronic contact. These entanglement
properties may have applications in quantum computingSkogvoll  _et al._
(2021). Future investigations could concentrate on the possible tuning of
entanglement between two close FM structures by an applied magnetic field,
which changes the configuration taken by the system.

## Supplementary

We offer supplemental materialWuhrer, Rohling, and Belzig  in which extended
calculations regarding the classical ground state, quantum ground state and
squeezing parameters are given.

###### Acknowledgements.

This work was financially supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) via the Collaborative Research Center SFB
1432 project no. 425217212 and project no. 417034116.

## Authors declaration

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Authors contributions

Dennis Wuhrer: Formal analysis (lead); software (lead); writing – original
draft (lead); vizualization (lead); writing – review and editing (equal);
methodology (equal); Niklas Rohling: Formal analysis (supporting); software
(supporting); supervision (supporting); writing – original draft (supporting);
writing – review and editing (equal). Wolfgang Belzig: Conceptualization
(lead); Funding acquisition (lead); methodology (equal); project
administration (lead); Supervision (lead); writing – original draft
(supporting) writing – review and editing (equal).

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