Steering magnonic dynamics and permeability at exceptional points in a parity–time symmetric waveguide

Tuning the magneto optical response and magnetic dynamics are key elements in designing magnetic metamaterials and devices. This theoretical study uncovers a highly effective way of controlling the magnetic permeability via shaping the magnonic properties of coupled magnetic waveguides separated by a nonmagnetic spacer with strong spin–orbit interaction (SOI). We demonstrate how a spacer charge current leads to enhancement of magnetic damping in one waveguide and a decrease in the other, constituting a bias-controlled magnetic parity–time (PT) symmetric system at the verge of the exceptional point where magnetic gains/losses are balanced. We find phenomena inherent to PT-symmetric systems and SOI-driven interfacial structures, including field-controlled magnon power oscillations, nonreciprocal propagation, magnon trapping and enhancement as well as an increased sensitivity to perturbations and abrupt spin reversal. The results point to a new route for designing magnonic waveguides and microstructures with enhanced magnetic response.


Technical details of the numerical simulations
The LLG equation is numerically solved employing a fth-order Runge-Kutta scheme with a xed time step of 0.2 ps. We discretize the two coupled waveguides with a mesh cell size of 4 nm. In the simulation, we rst let the magnetization relax to a stationary state (+y direction), and then add the spin wave excitation in WG1 or WG2. For the spin wave excitation we apply a microwave magnetic eld h(t) = h a xsin(2πf t) with an amplitude h a = 100 A/m and a variable frequency. The eld is applied locally to the region of the waveguides that we take as the origin x = 0. The spin waves propagate along the x− direction. The spin wave amplitude is related to the oscillation amplitudes of the x component m x (x, t) and z component m z (x, t) of the magnetization. To suppress the spin-wave reection from geometric boundaries, a suciently large (40µm) extension along x is adopted in the simulation.

Dynamic magnetic permeability in separated waveguides
To prove that the system exhibits a PT-symmetric behavior we calculate the transverse dynamic permeability in two separate waveguides (meaning the coupling between two waveguides are dropped). The accumulated spin density polarizations at WG1 and WG2 are T 1 = y and T 2 = −y. Around the equilibrium state (m 0 = y), we linearize the LLG equation and obtainṁ Here, ω H = γH 0 , ω c = γc J , ω ex = 2γAex µ 0 Ms , h x,z is the microwave magnetic eld acting on the magnetization. +ω c and −ω c correspond to WG1 and WG2, respectively.
The eects of residual magnetic damping From the above dynamic magnetic susceptibilities, we extract the eective damping constant α 1,2 eff = α ± ω c /ω k in the two uncoupled waveguides. With the parameters used in the main article, we have ω c /ω k = 0.17 under k x = 0.1 nm −1 and ω c = κ = 22.1 GHz. The Gilbert damping constant α = 0.004 is subsidiary in comparison to ω c /ω k . To justify the inequalities involving the α related terms in the main article (i.e., ω 0 αω J and ω J αω 0 ), we note that for ω J = κ = 22.1 GHz and ω 0 = 149 GHz under k x = 0.1 nm −1 , we have αω J = 0.09 GHz and αω 0 = 0.6 GHz with α = 0.004.
For a further demonstration of the inuence of damping, we calculate the two eigenvalues assuming a larger α = 0.08. The results are displayed in Fig. 1. Below the threshold ω J /κ < 1, a slight dierence between the two imaginary parts of the eigenvalues is identied.
At the exceptional point ω J /κ = 1, the two eigenvalues also become identical. Above the exception point ω J /κ > 1, the real parts of the two eigenvalues are separated by a suciently large damping term.
the two waveguides (see Fig. 1(d) in the main text). If ω J increases but is still below the EP, the phase angle changes from the initial value θ = 0 reaching eventually θ = π/2 at the EP. In this range, the superposition of two asymmetric spin wave modes leads to the non-reciprocal wave propagation, where the spin wave distribution at the output end can be entirely dierent by exchanging the input from one waveguide to the other ( Fig. 1(e) in the main text). At EP, the two spin waves modes coalesce to the same mode, and spin waves in the two waveguides travel simultaneously ( Fig. 1(f) in the main text).
To understand the inuence of damping, we expand Eq. (4) in series by neglecting α 2 and higher order terms (as α is usually much smaller than 1) which yields From the above equations, we infer that the two complex eigenvalues still merge at the same degenerate EP ω J /κ = 1. The existence of EP in coupled dissipative dynamical systems has been discussed in Ref. [ 1]. Below EP (ω J /κ < 1), the separation of the imaginary parts of the two eigenvalues is very weak due to the smallness of α. The separation suddenly becomes obvious above EP. Similarly, the real parts of ω ± are obviously distinct below EP, and their dierence is very weak above EP. With α = 0.004, the results in our main article conrm this conclusion. Furthermore, we nd the α dependent term does not aect the eigen-vectors ψ ± of the two spinwaves modes. This shows that ω J induces non-reciprocal propagation below EP and simultaneous propagation at EP, as shown in the main article. Also, we studied the case of a very large damping α = 0.08. The results in Fig. 1 follow the above analysis.

Enhanced sensitivities
In the main text we reported on an enhanced sensitivity to slight changes in magnetic elds at EP. This feature is benecial for sensing variations in the magnetic environment or for increasing the magnetic response in photonic applications. To quantify this sensitivity increase we apply a perturbation 1 κ and 2 κ in WG1 and WG2, where the perturbations 1,2 1 negligibly aect the coupling. In this case, the perturbed waveguide equations become, Here, ω 1 = ω 0 + 1 κ and ω 2 = ω 0 + 2 κ. Assuming the perturbation aects only WG1, at the EP ω J = κ, the obtained eigenfrequencies can be expanded perturbatively using a Newton-Puiseux series 2 that begins with a square-root element, and the rst three terms of this series are These expressions indicate that, the real parts of the eigenfrequencies bifurcate with a squareroot dependence on the applied perturbation, i.e.

Excitations of spin waves
To realize spin waves excitation experimentally, we suggest the structure shown in Fig.   2. Putting a stripe antenna perpendicular to x axis, the dynamic magnetic eld from the injected microwave current excites locally propagating spin waves in one waveguide. The input spin waves propagate to the middle region in Fig. 2 where PT symmetry phase transition is eective, and the output spin wave strength can be pick up to the right end via conventional spin-wave detection techniques.
To circumvent diculties in exciting spin waves with a large wave vector, one can combine the method proposed in Ref. [ 3]. Via microwave magnetic elds from antennas, large- T < T c ). In addition, the emergent net ferroelectric polarization couples (through the magneto-electric coupling which mimics a dynamical DM interaction) to an external electric eld. Thus, the applied nonuniform electric eld is similar to a nonuniform DM term and leads to a particular type of torque called inhomogeneous electric torque. 4 The expression of the inhomogeneous electric torque is similar to the spin-transfer torque l E m × (m × p E ).
The vector p E = x × e i , e i=x,y,z points to the direction of the electric eld, and the electric Depending on the distance between the two ferromagnetic waveguides, the RKKY interaction J RKKY which is responsible for the interlayer coupling can vary from positive to negative. 5 Let us consider situations with dierent J RKKY . We rst consider a smaller positive J RKKY which allows operating at a lower frequency. This is advantageous for an experimental realization, as the excitation of spin waves with high frequency is challenging. In Fig.   3, we use a smaller J RKKY = 9 × 10 −6 J/m 2 and an external magnetic eld H 0 = 2 × 10 4 A/m. Still, the EP is present at ω J = κ, and the spin waves can travel simultaneously in both waveguides at EP. With a smaller J RKKY the spin wave has a frequency around 2 GHz, which is much smaller than the frequency studied in the main text (around 20 GHz). In this section, we extend our study to coupled metallic ferromagnetic waveguides, in the calculations we use permalloy. The electric current owing in the Pt spacer causes spin orbit torques (SOTs) on the two coupled waveguides γc J Ms M p × T p × M p . 6,7 In addition, the electric current ows in the metallic ferromagnetic waveguides possibly causing an inplane spin transfer torque (STT) b J ∂ x M p . 8 Here, c J = T θ SH Je 2µ 0 e tpMs and b J = gµ B P Je 2eMs , g is the Landé factor, µ B is the Bohr's magneton, P is the spin-polarization eciency, e is the electron charge, t p is the thickness of pth waveguide, θ SH is the spin Hall angle,and M s is the saturation magnetization. Including the SOT and STT in the LLG equation, and following the same derivation as in the main text, we obtain the eigenfrequencies for the optical and acoustic magnon modes, From the above equation, one can see that, in the metallic coupled waveguides, two magnons modes still merge at the EP ω J = κ. The existence of the in-plane spin transfer torque (b J term) leads a weak asymmetry in the magnon dispersion, as shown by the calculations in Fig. 5.
Above EP, the increase of b J can further enhance ω (Fig. 5(c-d)). The existence of EP is further evidenced by inspecting the magnon propagation in Fig. 5(e-f).

Inuence of the dipole-dipole interaction
For an experimental realization it is important to address the role of the dipole-dipole interaction and how it may inuence PT-symmetry-related eects. To this end we conducted micromagnetic simulations and analytical calculations and summarize the results in this section. Generally, for coupled nano-strip waveguides with a nite size accounting for the inuences of demagnetization eld, we nd that the dipole-dipole interaction between the magnons in the two waveguides slightly increases the value of EP in the low frequency range.
The main conclusions concerning the SOT driven PT-symmetry-behavior are however unaltered.
For a numerical implementation let us adopt a similar model, as discussed in the main text. Two stripe waveguides are coupled via the RKKY interaction and the dipole-dipole interaction across the non-magnetic spacer. The demagenetization eld enters the eective eld of LLG equation as, For calculating the spin wave dispersion relations we use a two-dimensional fast Fourier Here, y i is the i-th cell along y axis, and N y is the total cell number along y axis.
The results for the spin wave dispersion relation in the coupled waveguides are shown in Figs. 6(a-b). The combination of RKKY coupling and the dipolar coupling between the two waveguides leads to the formation of the two spinwave modes (acoustic and optical modes). The dispersion of the two modes clearly splits when ω J = γc J = 0. At the lower frequencies, the gaps between the two modes are larger as the dipolar coupling becomes more important. Applying SOT at EP ω J = κ = γ J RKKY µ 0 Mstp , the two spin wave modes become identical in the high frequency range, while the gap between the two spin wave modes in lower frequency range becomes much smaller. Also, we analyzed the spin wave transmission.
Without SOT (ω J = 0), the spin waves injected at one end in one of the two waveguides oscillates between WG1 and WG2 during propagation. Applying SOT at EP ω J = κ, we nd the spin waves travel simultaneously in both waveguides. All results here conrm our ndings about PT symmetry and the presence of EP in coupled waveguides the main text.
Thus, we may conclude that for the investigated case the dipole-dipole interaction does not alter qualitatively the predicted PT-symmetry-related phenomena.
To provide more details on the magnon propagation in the coupled waveguides with the dipole-dipole interaction, we develop a simple analytical model. The magnetization vectors with small derivations in the two waveguides (p = 1, 2) have the form m p (r, t) = m 0,p + m s,p e i(ks·r−ωt) . Here, m 0,p = y is the static equilibrium magnetization parallel to y axis, and the small deviation from the equilibrium is m s,p = (δm x,p , 0, δm z,p ) with δm x(z),p 1. The wave vector k s is the sum of the in-plane wave vector k = k x x + k y y and the perpendicular (to the x-y plane) wave vector k z . Then, from the Fourier representation of the linearized LLG equation, we derive the following expression for the spin wave, Here, p, q = 1, 2 enumerates two waveguides, and the tensorΩ pq has the form, Here, we introduce ω 0 = γH 0 + 2γAexk 2 µ 0 Ms + κ, κ = γ J RKKY µ 0 Mstp , and ω M = γM s . The wave vector k is equal to k 2 x + k 2 y , the distance between the two waveguides d 12 = t + σ, t is the waveguide thickness, and σ is the gap between the waveguides. The dynamic magnetodipolar interaction is described by the tensorF: 912 The "shape amplitude" D p (k z ) = t 0 m(z)e −ikzz dz describes the inuence of the nite thickness t of the thin waveguide. The width prole of the SW mode in the waveguide is usually nonuniform (m(z) ∼ cos(k p z z)) due to the pinning eect from the geometric boundaries. In general, if the thickness of the waveguide is close to or smaller than the material exchange length or the eective boundary condition are free, the SW prole is almost uniform, i.e. m(z) = 1. Then, setting ψ ± p = δm x,p ± iδm z,p , we obtain the spin wave equation ωψ =Ĥψ with ψ = (ψ + correspond to right-hand precessions around their ground states (the other two negative frequencies are for left-hand precessions). The two positive eigenfrequencies are, With the above analytical expression, we insert the same material parameters used in the simulations, and calculate the real parts (i.e., the dispersion relations) of the eigenfrequencies, see Fig. 7. The calculated spinwave dispersion is in a good agreement with the simulation results. Scanning ω J /κ, the two separated real parts of eigenfrequencies merge at the same value at the EP, and the two imaginary parts are obviously separated after EP. Comparing the calculations for dierent wavevectors k x , the EP with a lower k x has a larger EP due to the nature of the dynamic dipole-dipole interaction. The small dierence between the simulated and the analytical results can be attributed to the approximations made in the analytical model.