Quadrature skyrmions in two-dimensionally arrayed parametric resonators

Skyrmions are topological solitons in two-dimensional systems and have been observed in various physical systems. Generating and controlling skyrmions in artificial resonator arrays lead to novel acoustic, photonic, and electric devices, but it is a challenge to implement a vector variable with the chiral exchange interaction. Here, we propose to use quadrature variables, where their parametric coupling enables skyrmions to be stabilized. A finite-element simulation indicates that a acoustic skyrmion would exist in a realistic structure consisting of a piezoelectric membrane array.

Quadrature skyrmions in two-dimensionally arrayed parametric resonators Hiroshi Yamaguchi, Daiki Hatanaka, and Motoki Asano NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan Skyrmions are topological solitons in two-dimensional systems and have been observed in various physical systems.Generating and controlling skyrmions in artificial resonator arrays lead to novel acoustic, photonic, and electric devices, but it is a challenge to implement a vector variable with the chiral exchange interaction.Here, we propose to use quadrature variables, where their parametric coupling enables skyrmions to be stabilized.
A finite-element simulation indicates that a acoustic skyrmion would exist in a realistic structure consisting of a piezoelectric membrane array.Topological solitons are fundamental excitations observed in various nonlinear systems [1].The soliton is the phase boundary between two identical but topologically distinguished domains and can be created by spontaneous breaking of translational symmetry [2,3].One of the most important two-dimensional (2D) solitons is the skyrmion, which has recently attracted much attention especially in the field of ferromagnetic systems [4][5][6][7][8][9][10].A skyrmion has a chiral spin texture and can be induced by an antisymmetric exchange interaction together with a symmetric exchange coupling [11][12][13].Skyrmions are stable because of their topological nature and a scheme to control their motion has been proposed [14,15] which may lead to applications of skyrmions in memory [16,17], logic [18], and microwave devices [19][20][21].Skyrmions have been shown to appear in numerous physical systems, including quantum Hall systems [22][23][24] and Bose-Einstein condensate [25][26][27] and are considered universal phenomena.
The concept of topological insulators has been recently utilized in artificial 2D lattice systems of photonics [28][29][30], microwave [31], and acoustics [32][33][34].The generation and control of skyrmions in these artificial systems are expected to similarly lead to new device and material technologies.However, there have been few reports on the skyrmions generated by spontaneous breaking of translational symmetry in artificial structures.One of the major challenges is how to construct the 2D array of threecomponent vector variables, with both symmetric and antisymmetric exchange interactions.Various efforts have been made using velocity fields [35] and hybrid displacements [36] in phononics, and also evanescence fields in photonics [37][38][39], but in the majority of the studies, skyrmion-like textures were created not by spontaneous symmetry breaking but by interference between external drive signals.Therefore, in those studies the position of each skyrmion is externally specified and no motion can be induced without changing the external drive.Thus, finding artificial lattice systems with a chiral exchange interaction to induce the spontaneous breaking remains an important research target in the field of skyrmion physics.
In this letter, we propose an alternative approach that utilizes quadrature variables [40], instead of three real-space vector components, in a doubly degenerate 2D harmonic resonator array.This approach has three significant features.First, the skyrmion texture can be generated with the array of only two-component variables.Second, the periodic temporal perturbation, which so far has been discussed mainly in terms of linear Floquet systems [41][42][43], is also a useful scheme in nonlinear dynamics.Finally, as in the case of topological insulators, the stability of the texture induced by the topological properties can be utilized to develop novel devices and materials technologies.The approach can be applied to various artificial 2D-arrayed resonators, such as microwave circuits and photonic and acoustic metamaterials, holding out the prospect of extensive applications of skyrmion physics to device and material technologies.( 0 sin  0  ,  0 cos  0 )/√2, respectively.
We consider a square array with doubly degenerate resonance modes A and B (Fig. 1(a)).The mode variables   () at a particular site on the lattice is expressed as   () =   () cos  0  +   () sin  0  , ( = , ), under the rotating frame approximation, where the cosine and sine oscillation amplitudes,   and   , are called quadratures. 0 is the angular frequency of the two modes.We introduce a parametric excitation at twice the frequency, 2 0 , and consider the case that the excitation changes sign between the two orthogonal modes.This is the essential assumption under which to assign stable parametric oscillation states to z-polarized vector states.The physical implementation will be described later.Here,   = k is the canonically conjugate momentum of   .We have assumed isotropic cubic nonlinearity with a strength of  and Γ is the parametric excitation intensity.We also assume identical effective masses for both modes and use the unit to make it unity for simplicity.The new Hamiltonian ℎ 0 is expressed as and the equation of motion is given by k =  0 −1 ∂ℎ 0 / ∂  and k = − 0 −1 ∂ℎ 0 / ∂  (See the Supplemental Materials (S.M.)).The steady-state solution, k = k = 0 , corresponds to a local minimum (or maximum) of ℎ 0 [40,44] and consists of four oscillation states,  ± and  ± , given by (  ,   ,   ,   ) = √Γ/ (0, ±1, ±1,0) and √Γ/ (0, ±1, ∓1,0), i.e.
±:   () = ± 0 sin  0  ,   () = ± 0 cos  0  ±:   () = ± 0 sin  0  ,   () = ∓ 0 cos  0  ( 0 = √Γ/). (4) As will be shown later,  ± and  ± are clockwise and anti-clockwise circularly polarized oscillation states, where the sign corresponds to the parametric oscillation phases.This Hamiltonian picture provides a good approximation even for a realistic case that has a finite damping, because it assumes that the parametric actuation Γ is sufficiently larger than the oscillation threshold determined by the damping.Because of the system nonlinearity and the  phase difference in parametric excitation between modes A and B, the two modes are equally mixed with a /2 phase difference and form stable circularly polarized oscillations.
Next, we define a three-component quadrature moment using four variables,   ,   ,   ,   .We employ two definitions in which the moment is linear with respect to the quadratures,  ⃑ = (  ,   ,   ) ≡ (  ,   ,   +

√2
) ,  ⃑ = (  ,   ,   ) ≡ (  ,   ,   −   √2 ). ( These definitions do not satisfy the Poisson-bracket algebra of angular momentum but are ideal for our purpose of creating skyrmions.We can easily confirm that the four parametric oscillation states  ± and  ± correspond to z-polarized quadrature moments (See Fig. 1(b) and S.M.).It should be noted that ℎ 0 is invariant under the replacement, (     ) → (   −  ), which corresponds to a π/2 rotation in the  plane as confirmed by the definition (5).
Next, as a preliminary step before discussing the detailed device structures, we mathematically construct a Dzyaloshinskii-Moriya (DM) interaction for the quadrature moments as well as the symmetric exchange coupling and numerically confirm that it leads to a skyrmion texture under parametric excitation.The symmetric exchange coupling between two nearest-neighbor sites is given by the isotropic interaction, ℎ  = −  ( 1  2 +  1  2 ) with   > 0, where the indexes 1 and 2 specify adjacent site positions.The rotating frame approximation leads to Then, we introduce a chiral exchange interaction to create a skyrmion texture.The DM interaction between two nearest-neighbor magnetic moments  ⃗⃗ 1 and  ⃗⃗ 2 is expressed [11,12].Here,  ⃗ 12 is the DM vector and it can be chosen as, for example, a unit vector directed from site 1 to site 2 for generating a Blochtype skyrmion [45].For a resonator pair parallel to the x-axis,  ⃗ 12 =   ≡ (1,0,0) and . Similarly, for the y-axis,  ⃗ 12 =   ≡ (0,1,0) and ) .If we replace  ⃗⃗ by  ⃑ or  ⃑ , we obtain the DM interaction from Eq. ( 5) as follows: Here, the plus (minus) sign corresponds to the L (R) mode.The terms mixing sine and cosine quadratures need a phase-shifted coupling.We can eliminate these terms by making a linear combination, For a π/2 rotation of the quadrature moments, i.e. (     ) → (   −  ) , this interaction sustains symmetry for  ± states and antisymmetry for  ±.This difference induces different skyrmion textures, Bloch skyrmions for  ± but antiskyrmions for  ±, as shown below.Although this DM interaction seems artificial, we will later find that it can be implemented in a realistic mechanical resonator array.
We then performed a numerical calculation to find a stable texture of quadrature moments.Hereafter, we will employ the unit  0 = 1 for simplicity.The full interaction Hamiltonian is given by Here, the suffix (, ) denotes the site position.A stable solution that minimizes the total When   is smaller than the onset value for skyrmion formation (  < 0.009), the skyrmion texture is unstable (U-phase) and all of the moments are directed downward, i.e.  ⃑ (,) = (0,0, −√2 0 ) for all (, ).Above the onset, a single skyrmion texture forms (S-phase) whose structure is nearly independent of   .The calculated time evolution shows that the texture is metastable because a perturbation of 5% of the oscillation amplitude destabilizes it, whereas it remains stable against the smaller perturbation.
Above the upper bound (  ≥ 0.0105), the transition to an extended texture (T-phase) emerges, where the skyrmion deconfines and occupies the whole lattice in an extremely anisotropic shape, and finally the texture of a skyrmion lattice (SL-phase) forms at   ≥ 0.0125 .This texture is independent of the initial guess in the calculation in contrast to the S-phase, so that the lattice remains stable, at least within our numerical calculations.We examined the topological features of the obtained textures by calculating the skyrmion number.Instead of the generally used expression for a continuous system,  = ∫  ⃗ • (   ⃗ ×    ⃗ ) /4, summing up the solid angles formed by all sets of three adjacent pseudo moments makes the skyrmion number unity as expected for a topological soliton (S.M.). Figure 2(b) and (c) show the phase diagrams of the texture as a function of ,   , and   .The S-U boundary in Fig. 2(c) suggests that the parameter   2 /  governs the transition [45].We can see that the skyrmion and its lattice textures occur over a wide parameter range.Similar calculations using  ⃑ generated an antiskyrmion texture (Fig. 3 (b)).We also generated a Neel-type skyrmion by using the different DM vector,  ⃗ 12 =   for the x-axis and −  for the y-axis (Fig. 3(c)) [45], indicating that various skyrmion textures are possible.
Next, let us discuss a realistic device model implementing the DM interaction.
Although our concept can be applied to various physical systems, we here envision an implementation in a nanomechanical resonator array consisting of coupled piezoelectric circular membranes [47].The unit structure, shown in Fig. 4  The symmetric exchange coupling, Eq. ( 6), is mediated by the elastic coupling through the overlap (Fig. 4(d)), where we need to consider two coupling constants,  1 and  2 (Fig. 4(e)).A comparison of ( 8) with (6) indicates that a AB mixing interaction is required to induce the quadrature DM interaction.Accordingly, we employed a misalignment between the piezoelectric and membrane-lattice axis (Fig. 4(e)).After redefining the even-site resonator amplitude with reversed sign, the mixing term is given by Here,  is the misalignment angle and   = ( 1 +  2 )/2 (see full expression in S.M.).
Although the AB mixing is introduced, this interaction is identical between the x-and yaxis and unable to introduce the chiral interaction.This is reasonable because both   ′ and   ′ are coupled in the same way through   ′ .To induce chirality, we need to introduce an interaction that independently couples   ′ and   ′.This can be done by using the parametric coupling proposed for another topological system [30].We here show that combining this parametric coupling together with the parametric excitation induces a stable skyrmion texture.A modulation,   () =  0 +   cos(2 0 ) , leads to the following  mixing exchange interaction, Here, the signs of   are opposite to those in (10) so that applying a different signed 2 0 modulation between the x-and y-axis coupling creates a chiral interaction.This can easily be done in our device configuration by applying opposite signed alternating voltages.The total  mixing part becomes Here, . In the case of   = −2 0 for the x-axis and +2 0 for the y-axis, we obtain These equations are identical to the DM interaction, Eq. ( 8).Therefore, the combined effects of the misaligned piezoelectric axis and 2 0 modulation of the symmetric exchange coupling can induce the chiral interaction required to form a skyrmion.We can also generate a Neel-type DM interaction by changing the sign of the parametric coupling (S.M.), so that the kind of creating skyrmion can be externally controlled.
We performed numerical calculations to verify the above findings.First, we performed a simulation using the finite element method (FEM) using COMSOL Multiphysics® to numerically determine the coupling constants.Finally, let us discuss the robustness of skyrmions against external fluctuations.
Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations.We used a model employing the ideal DM interaction with Eq. ( 9) and the steady-state textures for various resonance frequency fluctuation  = √ ( 0(,) −  0 ) 2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ / 0 are calculated (S.M.).When  ≤ 0.02 , the skyrmion texture remains metastable against the frequency fluctuation.For a larger , skyrmion-shaped excitations were randomly generated, indicating instability.When the fluctuation is larger than   , ferromagnetic ordering is not maintained.In addition, the degeneracy lifting between A and B modes should be within the resonance linewidth to maintain circularly polarized modes.Nonetheless, since these fabrication tolerances are easily obtained [50], it should be possible to demonstrate acoustic skyrmions in actual devices.
In conclusion, we proposed an approach to generating various skyrmions that utilizes four quadrature variables in 2D parametric resonator arrays.This is the first numerical demonstration of skyrmions in systems formed by only two dynamical variables through temporal periodic perturbation.The concept can be applied to other kinds of parametric resonator, including photonic and microwave resonator arrays.
To find the steady-state solution, we define , ,   and   as   =  cos   ,   =  sin     =  cos   ,   =  sin   Then, We assume that Γ > 0 and  > 0; then ℎ 0 → +∞ for ,  → ±∞.Therefore, a stable solution is found at the minimum of ℎ 0 .For given  and  , the last term becomes smallest when Then, When  2 ≥  2 , the coefficients of  4 and  2 are always positive so that the energy has a minimum value when  = 0 and s = 0. Therefore,  2 <  2 .Then,  = 0 and This expression reaches a minimum at On the other hand, from sin 2  = ±1 , we have   = . Thus, the four solutions are   = 0,   = 0,   =  cos   = ± 0 ,   =  sin   = ± 0 .
Here, the sign ± can be chosen independently.The above correspond to the solutions of eq. ( 4).The expression using the quadrature moments  ⃑ and  ⃑ is shown in Table S1, indicating that  ± and  ± correspond to z-polarized quadrature moments.

Stability test of skyrmion textures
We perform a two-step calculation to investigate the stability of the skyrmion solution.
First, a solution providing the local minimum of ℎ 0 is calculated.Then, small perturbations are added to the quadratures,  (,) →  (,) + ∆ (,)  (,) →  (,) + ∆ (,) at  = 0, and their time evolution is calculated, where the equations of motion are modified to include the effect of weak damping, where  is the quality factor of the resonator (assumed to be 10 4 ). Figure S1  .The texture is maintained for  ≤ 6%, but it disappears for  ≥ 7%.This result confirms that the skyrmion is locally stable but a large perturbation breaks the texture, causing the system to fall into the lowest-energy single ferromagnetic domain.
Therefore, the isolated skyrmion is not energetically lowest but metastable, corresponding to the locally minimum pseudo-energy state.We compared the textures using various initial guess, changing the position, radius, and shape of oppositely polarized domains and no significant change in phase diagram was observed.

Calculation of skyrmion number
The skyrmion number of a continuous system is calculated using the formula,  (mode B and C).We can also confirm that a frequency shift of about 1.5% is obtained at   = 1 , which is large enough to induce the required parametric excitation of Γ = 0.01.

Stability against the resonance frequency fluctuation
We discuss the robustness of skyrmions against external fluctuations.Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations.Here, let us introduce a frequency detuning δ (,) = ( 0(,) −  0 )/ 0 for each resonator obeying a Gaussian distribution.We used a model employing the ideal DM interaction with Eq. ( 8) and the steady-state textures for various fluctuation  = √  (,) 2 ̅̅̅̅̅̅̅ are calculated (Fig. S6).When  ≤ 0.02, the skyrmion texture remains stable against the frequency fluctuation.For a larger  , skyrmion-shaped excitations were randomly generated, indicating instability.When the fluctuation is larger than   , ferromagnetic ordering is not maintained.Nonetheless, since fabrication tolerances of 2% are easily obtained, it should be possible to demonstrate acoustic skyrmions in actual devices.

Transition textures
In the parameter region between S-phase and SL-phase, we obtained highly isotropic texture shown in Fig. S7.In this parameter regions, the skyrmion deconfines and occupies the whole lattice in an extremely anisotropic shape.
(a), has doubly-degenerate modes, A and B (Fig. 4(b)), and the circular edge is fixed.Because of its  31piezoelectric component[48,49], the applied voltage induces a different sign of tension between two orthogonal directions, [110] and [11 ̅ 0] (Fig.4(a) and S.M.).This anisotropy changes the sign of parametric excitation required to derive the third term in Eq. (2) and enables four circularly polarized parametric oscillation states to be excited (Eq.(4) and Fig.4(c)).

FIG. 4 .
FIG. 4. (a) Schematic drawing of circular electromechanical membrane resonator and (b) two vibration modes, A and B. (c) Four parametric oscillation states.L/R indicates the circular polarization and +/− indicates the oscillation phase.(d) Schematic drawing of the device configuration.An alternating voltage at 2 0 is applied to the center electrode for parametric excitation and to the junction electrode for parametric coupling.(e) Schematic illustration of the coupling constants.Two different constants,  1 and  2 , are introduced and the coupling between the A and B modes is established by misaligning the piezoelectric and membrane axis.

Figure 5 (
FIG. 5. (a) Calculated static coupling constants as a function of gate voltage using FEM for the geometry indicated at the top.The resonator thickness is 100 nm and the elastic and piezoelectric parameters of GaAs are used in the calculation.(b) The texture of Neel skyrmion in terms of  ⃑ (,) calculated from the parameters given by FEM and using a voltage modulation of 4.0 V  and  = 20 ∘ .
plots the time evolution of the skyrmion area, i.e. the average number of positively z-polarized quadrature moments for different amounts of perturbation  = √ (∆ (,) )

Fig. S1 .
Fig. S1.Time evolution of skyrmion area with a typical texture.The unit of time is the oscillation cycle, i.e.  0 /2.

Fig. S2 shows
Fig. S2 shows the displacement distribution of the membrane resonator array calculated from the steady-state quadrature moments.A movie showing the time evolution is in a separate file.

Fig. S2 .
Fig. S2.Snapshot of displacement distribution of membrane resonator array calculated from the steady-state quadrature moments of the Bloch skyrmion shown in Fig. 3(a).The green dashed line roughly indicates the boundary of the skyrmion.

Figure S4 .
Figure S4.Schematic drawing of the coupling of two modes  ̂ and  ̂, which are defined as the (a) x-and (b) y-aligned linearly polarized modes.

Fig. S5 .
Fig. S5.Calculated flexural mode frequency for a circular membrane GaAs resonator as a function of gate voltage applied between the top and bottom electrodes.The top electrode and membrane radius are 6 m and 15 m, respectively, and the bottom electrode covers the back surface of the membrane.We set fixed boundary conditions along the edge of the membrane.