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.

Magnetic materials exhibit a wide range of magnetic structures,^{1–4} ranging from simple collinear magnetic structures including ferromagnets (FMs)^{1} and antiferromagnets (AFMs)^{2} to non-collinear structures like skyrmion lattices.^{3,5–7} 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.^{1,8,9}

The interest in magnons rose in recent years^{10,11} as they provide a platform for easy-to-manipulate long-distance transport with low dissipation.^{12,13} They may provide an alternative to the Joule-heating-plagued electrons in information transport and processing,^{14} as a key computational device, transistors, can be realized by magnetic structures.^{15} Moreover, it is possible to link magnetic materials to conventional electronic devices using the (inverse-)spin-Hall effect,^{12,13} 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 (2D) materials,^{16–22} notably Van der Waals (VdW) materials.^{23–26} These materials provide the potential to produce engineered synthetic systems with desired attributes by layering different materials together. Two-dimensional (2D) magnetic layers^{27–36} 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,^{37,38} quantum squeezing and entanglement for magnons have been predicted theoretically^{39–42} and observed in experiments.^{43,44} 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^{45} or being probed by quantum dots adjacent to the magnet.^{46} 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 qubits.^{47} As a result, magnons make an interesting tool for quantum computing.^{48–51}

While electric and magnetic dipole–dipole interactions were discussed in the context of entanglement previously for other platforms,^{52–57} 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 than the decay length of the Heisenberg exchange, which experiences an exponential decay.

*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 the principle axes 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

*B*. The Heisenberg exchange is given by

*S*at sites $ r i$ and $ r j$ in FM

*A*preferring a parallel configuration of spins.

*K*(

_{z}*K*) is the strength of the uniaxial anisotropy along the

_{x}*z*-axis (

*x*-axis). We assume $ K z \u226b K x > 0$ such that the spins prefer to align parallel to the

*z*-direction.

*μ*

_{0}being the vacuum magnetic permeability, $ g i A$ the gyromagnetic ratio of the interacting magnetic moments,

*μ*the Bohr magneton, $\u210f$ the reduced Planck constant, and

_{B}*a*the lattice constant. Throughout this work, we will assume that $ \u210f , a = 1$, which means that all the interaction parameters (

*J*,

_{ij}*K*,

_{x}*K*, and

_{z}*D*) will be given in the unit of energy.

^{A}The dipole–dipole interaction decays with the inverse of the third power of the distance between the spins and thus is classified as the long range.

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 favors a parallel, ferromagnetic alignment along the connecting line ( $ 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. Minimizing the interaction term when the spins are perpendicular to the connecting line results in an AFM ordering between the two ferromagnets. This is referred to as in-plane AFM (IP AFM) ordering.

*α*specifying the direction of the connection vector. The exact definition is given in the supplementary material.

*A*(

*B*) and the

*z*-direction. Applying the linear Holstein–Primakoff transformation

^{58}yields for spins in FM

*A*,

*B*,

*k*component. The exact definition of each parameter contained by the Hamiltonian is given in the supplementary material.

_{x}The meaning of the different parameters, if only they together with the diagonal elements $ E i , k$ would appear in the Hamiltonian, is outlined below. The energy of a bare magnon mode $ a \u0302 k$ ( $ b \u0302 k$) inside FM *A* (*B*) is given by $ E A , k ( E B , k )$, while $ \mu 1 , k$ results in a hybridization 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 , k$ ( $ \xi B , k$) entangling modes with wave vectors $ \xb1 k$ inside each FM, while squeezing of $ \xb1 k$ modes of the different FMs is a result of $ \mu 2 , k$. Squeezing alters the vacuum state as the squeezed magnon states are superpositions of populated *A* and *B* magnon states.^{41} If we combine different of these parameters, a clear distinction between squeezing and hybridization for each parameter cannot be given anymore.

^{59,60}which is a symplectic transformation converting our bare magnon operators $ a \u0302 k$ and $ b \u0302 k$ into squeezed magnon operators $ \alpha \u0302 k$ and $ \beta \u0302 k$. The Bogoliubov transformation can be written as

*J*. We see that for large distances, both converge toward the energy $ \epsilon \u0303 0 = \epsilon 0 2 \u2212 4 | \xi 0 | 2$ of the squeezed magnons in the case of non-interacting FMs. At the phase transition, $ \epsilon \alpha , 0$ shows a jump, while $ \epsilon \beta , 0$ vanishes.

From two-mode squeezed cases for an AFM,^{41} a harmonic spin spiral^{61} and most important for a FM with dipole interaction, we expect the largest squeezing, which is proportional to the entanglement, for the uniform $ k = 0$ mode. An investigation of the system at hand shows the maximal squeezing for the $ k = 0$ mode as discussed in the supplementary material. Thus, in the following, we will limit our considerations to the $ k = 0$ mode and will drop the wave vector index for clarity.

*U*and close to zero value for all elements of

*V*. Thus, the ferromagnetic structure is dominated by hybridization (

*U*) with little to no squeezing (

*V*). Due to the vanishing magnon energy of the $ \beta \u0302$ magnons at the phase transition $ l / a \u2248 0.33$, we see divergences in

*u*

_{2},

*u*

_{4},

*v*

_{2}, and

*v*

_{4}. These divergences lead to a breakdown of the linear Holstein–Primakoff approximation, Eqs. (9) and (10), in a narrow region around the phase transition as the number of $ a \u0302 k$ and $ b \u0302 k$ magnons is connected to $ | v i , k | 2$. Thus, in this region, the results should be treated with caution.

In the IP AFM configuration ( $ l / a > 0.33$), we see an increase in all elements of $U$ 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.^{41}

^{62}The limitation to the $ k = 0$ mode leaves us with a real Bogoliubov matrix, which can be parameterized by four parameters. We connect the Bogoliubov matrix to the generators of the group and introduce the four-mode squeezing operator,

*θ*being the real parameter corresponding to the generator $ \Phi \u0302 i$, which in quantum representation are given by

_{i}*A*and

*B*with squeezing parameters $ r A / B = \theta 2 \xb1 \theta 1$ for FM

*A*(

*B*). $ \Phi \u0302 3$ ( $ \Phi \u0302 4$) results in hybridization (two-mode squeezing) of the magnon modes of the different FMs. Thus, each isolated parameters

*θ*can be identified with either one-mode, two-mode squeezing, or hybridization. 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

_{i}*θ*as “squeezing parameters.”

_{i}The analytic treatment in terms of a manageable number of squeezing parameters [ $ N ( 2 N + 1 )$] strongly depends on the dimension of the Bogoliubov transformation (2*N*), Eq. (13). This number could be increased by coupling of different $k$ modes, for example in finite magnets, making translational invariance crucial for the analytic treatment of the problem at hand.

Figure 4 shows the dependence of the squeezing parameters on the distance. They show a similar behavior as the elements of $U$. The parameters identified with intra FM ( $ \theta 1 , \theta 2$) and inter FM (*θ*_{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 (*θ*_{3}) is of a finite value for the whole parameter range. This behavior supports our identification based on the generators $ \Phi i$. The divergence of the squeezing parameters is a result of vanishing energy of the $ \beta \u0302 0$ magnon modes at the phase transition.

^{63}

From Eq. (29), we see that all squeezing parameters influence the entanglement except for *θ*_{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 hybridization between the FMs ( $ \theta 1 , \theta 2 , \theta 4 = 0 , \theta = | \theta 3 |$) results also in a vanishing entanglement as the vacuum for hybridized modes is equal to the vacuum of the bare modes $ a \u0302$ and $ b \u0302$ 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 the case of simple two-mode squeezing.^{10} Therefore, known results are reproduced by the analytic formula given in Eq. (29). Further, we see that squeezing, be it intra- or intermagnet squeezing, is necessary to establish entanglement, which is in agreement with earlier results.^{64,65}

The distance dependence of the entanglement can be seen in Fig. 5 where we regard finite and infinite 2D FMs. Figure 5 shows similar behavior of the entanglement for all lattice sizes for $ l / a \u2264 1$. The phase transition from an OOP FM configuration to an IP AFM configuration appears around $ l / a \u2248 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 transition.

Aside from the divergence, the entanglement takes it maximum value at approximately $ l / a \u2248 0.5$, rapidly decreases afterward, and, in the infinite case, vanishes already at $ l / a \u2248 2$. The origin of the maximum is still up to debate and subject to future work. The finite cases with 100 × 100 and 500 × 500 lattices do show a plateau in the entanglement around $ l / a \u2248 2$, which is the higher, the smaller the lattice is. Increasing the distance up to the order of the system diameters (100*a* or 500*a*, respectively) results in a decrease in entanglement that is proportional to $ 1 / l 3$. This behavior is derived from the source of interaction: the dipole–dipole interaction, Eq. (6).

We infer that no ground-state 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 ground-state 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 emphasize 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.*^{45} or Yuan *et al.*^{47} This is important for the future experimental realization of quantum computing.^{48–51}

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 hybridization.

Investigating 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 toward 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^{27} is 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 computing.^{45} 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 or the effect of different orientations and lattice types of the two FMs.

## SUPPLEMENTARY MATERIAL

See the supplementary material for extended calculations regarding the classical ground state, quantum ground state, and squeezing parameters.

This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Collaborative Research Center SFB 1432 Project Nos. 425217212 and 417034116.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**D. Wuhrer:** Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **N. Rohling:** Formal analysis (supporting); Software (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (equal). **W. Belzig:** Conceptualization (lead); Funding acquisition (lead); Methodology (equal); Project administration (lead); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

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

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