Emergent magnonic singularities in anti parity-time symmetric synthetic antiferromagnets

We study the impact of the transition across the anti-parity-time (anti- PT ) symmetry breaking point on the low-energy spin excitations in a synthetic antiferromagnet (SyAFM) that consists of two magnetic layers coupled antiferromagnetically via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. When varying the interlayer (RKKY) coupling strength a degeneracy point (exceptional point) is reached beyond which the system enters the anti- PT symmetry-broken phase. This phase is marked by the emergence of a finite net magnetization and low-lying excitations beyond the Goldstone modes in the anti- PT symmetry-preserved phase. For systems hosting an interfacial Dzyaloshinskii–Moriya interaction, we find a bound state in the continuum with a maximal coherent superposition of SyAFM excitations without any radiation. Analytical results for linearized models are corroborated with full numerical micromagnetic simulations endorsing the robustness and generalities of the theory predictions.


Introduction
Antiferromagnetic (AFM) materials with zero net magnetization offer an exciting platform for information handling that is robust against magnetic field perturbations and magnetic crosstalk between neighboring devices [1]. Here, we focus on synthetic antiferromagnetic (SyAFM) structures which are fabricated out of two ferromagnetic (FM) layers and are coupled via Ruderman-Kittel-Kasuya-Yosid (RKKY) interaction. SyAFMs allow for electrical manipulation of AFM ordering and are convenient to investigate using well-developed techniques for ferromagnets [2]. Recent research on their collective excitations, the spin waves (with magnons as quanta of excitations) [3][4][5][6][7], demonstrated their potential for logic gates, information processing, and sensory devices with low power operation [8][9][10]. Unlike FM systems with only right-handed spin wave modes, AFMs have two degenerate spin wave eigenmodes with opposite circular polarization, referred to as right-and left-handed magnons depending on the precessional handiness of the Néel vector [3,11,12] (cf figure 1). Any polarization state can be produced by a combination of these two eigenmodes, providing a way to encode information based on the spin wave polarization as well as on the amplitude and phase. Adverse however is that, the two-fold degeneracy is protected by the combined symmetry of time-reversal and sub-lattice exchange, and we have thus to seek controlled interactions that break either or both of the two symmetries. A way to achieve that is presented in this study.
The magnetic dynamic is generically non-unitary due to the ubiquitous magnetic damping. Hence, the Hamiltonian governing the low energy (spin wave) modes can be intrinsically non-Hermitian, and may become defective at special conditions (exceptional points, EPs) depending on the external parameters. At these (degeneracy) points the number of eigenvalues (algebraic multiplicity) exceeds the number of eigenvectors (geometric multiplicity). For FM structure, various facets of how EPs affect the magnetic dynamics were discussed. We are concerned with SyAFMs. Such (interacting) systems are suitable for studying anti-parity-time (anti-PT ) symmetry behavior (in contrast, FM structures with balanced gain and loss magnetic damping are related to PT symmetry [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). Different interlayer coupling (RKKY) strengths J correspond to different free-energy densities of the SyAFMs. At a certain value of the interaction strength J = J c a degeneracy (or EP) in the relevant low-energy modes occur and the modes undergo a symmetry change: for J < J c the anti-PT symmetry is broken (APTB phase), while for J > J c the anti-PT symmetry (APT) is preserved. The degree of degeneracy is related to the order of the EP, which in turn can be tuned by structural design [31]. The value of J c depends on other determining parameters of the free energy density such as the magnetic anisotropy and exchange energy as well as on magnetic damping and is given explicitly in the next section. In the APT phase the system has an imaginary spectrum. In the APTB phase the spectrum is complex [32][33][34][35][36][37][38][39][40].
The anti-PT phase transition point J c can be reached by scanning J or the other parameters in the free energy density. Experimentally, electric tuning of J were realized in SyAFM structures, e.g. FeCoB/Ru/FeCoB and (Pt/Co) 2 /Ru/(Pt/Co) 2 [41,42], in which not only the amplitude but also the sign of RKKY interaction can be tuned by changing gate voltages. Thus, when we discuss below properties with varying J we implicitly mean that there is some external fields (such as gate voltage) which are varied affecting the respective changes in J (and hence the free energy density). Furthermore, the magneto-electric interactions upon interfacing an FM layer with a ferroelectrics allows for an electrical control of the magnetization damping [43][44][45] and the magnetic anisotropy [46,47].
Considering SyAFMs with an interfacial Dzyaloshinskii-Moriya interaction (DMI), we find that a magnonic bound state in the continuum (BIC) is formed. This hybrid BIC cannot radiate away and consists of a maximally coherent superposition of modes of the two FM layer. The above statement are inferred analytically from a linearized model and confirmed with full numerical micromagnetic simulations.

Results
Anti-PT symmetric SyAFM dynamics. The magnetic dynamics in a SyAFM is describable by coupled Landau-Lifshitz-Gilbert (LLG) equations [48]. For two coupled layers we write Here m n (with n = 1,1) denote the magnetization direction for two FM sublayers, γ n is the gyromagnetic ratio, and α n is the Gilbert damping constant. The effective magnetic field acting locally on m n reads where A n is the Heisenberg exchange coupling constant within each m n -layers, K z is a magnetic anisotropy along the easy z-axis, and J is the RKKY interaction between the two sublayers. When J = 0, the two layers are decoupled; each magnetization m n prefers locally right-handed precessions around its own easy-direction, as sketched in figure 1(a).
A positive RKKY interaction (J > 0) tends to align the layers magnetization antiparallel, enforcing m 1 and m1 to process in the same global manner, i.e. right-or left-handed precessions around the Néel vector n = (m 1 − m1)/2 (cf figure 1(b)). To show that the low-energy magnetic dynamics is anti-PT symmetric, we linearize the LLG equations by writing m n = m 0 nêz + δm n with m 0 n ≈ ±1 and |δm n | 1 and obtain the Schrödinger-type equation for the dynamical excitations of δm n with renormalizedγ n = 2γ n /(1 + α 2 n ) andα n = α nγn . For J = 0, the magnetization dynamics of uncoupled layers are governed by their own Hamiltonian H n = −A n ∇ 2 + K z + J.
For SyAFMs with A 1 = A1 = A, γ 1 = γ1 = γ, and the same damping rates α 1 = α1 = α, H reduces to where H k = Ak 2 + K z + J in the long-wavelength limit of a plane-wave ansatz Ψ ∼ e i(k·r−ωt) , where k stands for the wave number of propagating spin waves [49]. Symmetry considerations. The eigenvalues of H are where (a) For purely imaginary ω ± (when |ξ k | > 1) the eigenvectors read where tanh ϕ = 1 − 1/ξ 2 k and tan η = α. A parity operationP corresponds to applying the Pauli operator σ x = 0 1 1 0 . The time-reversalT corresponds to i → −i, t → −t and r → r. The states, given by equation (6), Ψ ± APT are also eigenvectors of thePT operator [50], satisfyingPT Ψ ± APT = λ ± Ψ ± APT with |λ ± | 2 = 1. The anticommutator {H ,PT } vanishes. Thus, H of a coupled SyAFM possesses an anti-PT symmetry. This symmetry is preserved till J reaches where the EP occurs at J = H k / √ 1 + α 2 (cf figure 1(c)). Physically this means, the dynamic motions of m 1 and m1 in the two magnetic layers are equator modes, sharing the same amplitude. No net dynamic magnetization is generated. (b) For smaller J < J c the system enters anti-PT symmetry-broken (APTB) phase with complex eigenvalues (|ξ k | < 1). The corresponding eigenvectors [51] are given by where tanh φ = ξ k . Considering cosh 2 φ 2 > sinh 2 φ 2 , one infers that m 1 and m1 precesses with different cone angles in two FM sublayers. The left-and right-circularity polarized modes Ψ ± APTB around the Néel vector n are dominated by the precession in the upper and lower sublayers. The precessional dynamics is accompanied by the emergence of a small intrinsic magnetization m = (m 1 + m1)/2 whose dynamics is governed by the temporal evolution of the Néel order, m ∝ (n × ∂ t n) [52]. Thus, the emergent magnetization in collinear SyAFM systems signals a breaking of anti-PT symmetry.
Generically, |ξ k | < 1 and the anti-PT symmetry is broken. For isotropic SyAFMs (no magnetic anisotropy K z = 0) and for a frequency at/below the ferromagnetic resonance (FMR) point H k=0 = J, no spin waves are excited, leading to ξ k = √ 1 + α 2 and the eigenvalues ω + = 0 and ω − = −2iJα. This APT phase persists in the presence of a small anisotropy, as long as K z /J < α 2 /(1 + √ 1 + α 2 ). Both eigenvalues become imaginary and their absolute values depend monotonically on K z : |ω − | (|ω + |) approaches increasingly (decreasingly) the Gilbert damping constant α, as K z increases. No precessional motions are found even at FMR point in the micromagnetic simulations and m n decays with time exponentially to the easy-axis.
Generally, one may argue that due to the uniaxial magnetic anisotropy, the SU(2) spin rotational symmetry is reduced to SO(2)×Z 2 . The low-lying excitations (Goldstone modes) are associated with the continuous SO(2) symmetry. Above the energy gap which is determined by Z 2 symmetry, the excitation energy of the Goldstone mode should vanish when k → Q, where Q is the wavevector of magnetic ordering, for example, Q = 0 for the SyAFMs. This is indeed confirmed by figures 2(a) and (b)) (for α = 0 or J = 0). However, as the Gilbert damping constant α and/or RKKY interaction J increase in a way that √ 1 + α 2 − 1 > K z /J is satisfied, an EP emerges and the APT phase is spread out over the k-space. As a result, the real eigenfrequency of spin waves reaches its minimum value (zero) far away from the Q-point, as shown by the results in figure 2 which follow from micromagnetic simulations using the OOMMF package. A one-dimensional SyAFM system consisting of two 1000 × 1 × 1 nm 3 FM layers is considered with a mesh size of 1 × 1 × 1 nm 3 . The ground state equilibrium antiparallel magnetization is oriented along the z-axis. To launch the magnetization excitations, we apply the rf magnetic fields,h x (t) = h 0 sinc(ω c t)ê x with the amplitude h 0 = 10 mT and the cut-off frequency ω c = 60 GHz. The dispersion relation given by the fast Fourier transform (FFT) clearly indicates that not all momenta k can be excited into spin waves as long as ξ k > 1 is fulfilled within a certain k-range, where the system is in the APT phase and the eigenfrequencies are pure imaginary. It is clear that such low-energy magnetization excitations are beyond the traditional Goldstone modes and can be realized by adjusting not only the Gilbert damping α but also the RKKY interactions J (cf the right column of figure 2). These full numerical simulations endorse the theoretical analysis as well as the experimental feasibility and relevance of the anti-PT symmetry breaking in SyAFM systems.
BIC controlled by DMI. Besides EP, BIC is a (standing) wave that remains perfectly confined without any radiation even though it falls within the continuum spectrum. BICs were originally proposed by von Neumann and Wigner for electronic systems [53] and have been observed experimentally in electromagnetic, acoustic, water and elastic waves [54][55][56][57][58][59][60][61][62][63]. The unique properties of BICs have led to numerous applications, including slow light, sensors, filters, and quantum memory. Here, we show that magnonic BICs can be generated by introducing the asymmetric DMI. This new kind of singularity maybe useful for SyAFM magnonics.
For systems hosting interfacial DMI, we incorporate in the theory the interaction where∇ =ê n × ∇ withê n being the normal direction of DMI surfaces. For the one-dimensional SyAFM along the x-axis, the DMI contribute to the energy density with H n = −A∇ 2 + K z + J − iD n ∇ x . Note that e n is the normal direction of the upper or the lower surface of each FM sublayers, depending on where the DMI is dominantly generated. For the sake of discussion, we have assigned an effective value D n to each magnetic layers. Two distinct magnonic BICs are then found in SyAFMs depending on whether they are anti-PT symmetric or not:  Néel-type AFM ordering is preserved in a narrow window beyond D 2 4AK z for the geometrically constrained system discussed here. Unlike the above BICs in asymmetric SyAFMs, these anti-PT symmetry-protected BICs are equator modes that stem from the maximal coherent superposition of magnetization excitations with 50-50 contributions from the upper and lower FM layers, as demonstrated by equation (6). They are further confirmed by the dynamic magnetic susceptibility (χ(k, ω)) or dynamics spin correlations (S n (k, ω) = S − n (k)S + n (−k) ω ). Based on the linear response theory with S − 1 = δm x 1 − iδm y 1 and S + 1 = δm x 1 + iδm y 1 , one has As shown in figure 4, the imaginary part of the susceptibility Iχ(k, ω) diverges at the wavevector k BIC , implying an infinite lifetime and a zero leakage rate of BICs. These unique hybridized modes are robust against the detuning parameters in the APT phase.

Conclusions
As a typical dissipative system, the low-energy magnetization excitations in the balanced SyAFMs are shown to be eigenstates of the anti-PT symmetry operator for certain intrinsic parameters in the free energy density. Such parameters can be tuned by external probes or by material engineering, and the system can be driven to a phase with the low-energy modes are no longer eigenstates of the anti-PT operator. The phase transition between the symmetry-preserved and the symmetry-broken APT phase is accompanied with the emergence of a finite magnetization and bound states in the continuum with maximally coherent superposition for spin waves. These tunable singularities can be relevant to transport of magnons and imply numerous cross applications, such as magnon entanglement, slow spin waves, and magnonic quantum-information devices.