Parity‐Time Symmetry in Magnetic Materials and Devices

Non‐Hermitian Hamiltonians may still possess real eigenvalues in case of the existence of parity‐time (PT) symmetry. Exceptional points (EPs) occur at the phase transition from real to complex eigenvalues due to PT‐symmetry breaking in the parameter space. Magnonic devices use magnons to carry, transport, and process information, which have the advantages of low energy dissipation, wave‐based computing, and nonlinear data processing. The combination of PT‐symmetry and magnonics may lead to novel physics as well as unprecedented functional device applications. Recently, the research of PT‐symmetry in magnetism has developed rapidly. In this review, the theoretical predictions as well as experimental findings of PT‐symmetry in magnetism are summarized. First, a brief introduction to PT symmetry, EPs, and anti‐PT symmetry is presented. Second, the theoretical and experimental progress of magnonic PT symmetry are summarized. Third, the theoretical predictions of higher‐order EPs and anti‐PT symmetry in magnonic systems are given. Finally, the study concludes by discussing the future challenges and research prospects in magnonic PT‐symmetry, and proposals for experimental observations of magnonic higher‐order EPs and anti‐PT symmetry are suggested.


DOI: 10.1002/aelm.202300674
non-Hermitian operators mean that they do not satisfy the conditions for selfconjugation, they do not necessarily have a real spectrum.[6][7] In recent years, the research on non-Hermitian systems has been widely extended in optics, [8][9][10][11][12][13] acoustics, [14,15] Bose-Einstein condensates, [16] electronics, [17] and other aspects. [18]There has been an increasing focus on non-Hermiticity in magnon systems, [19][20][21][22][23][24] particularly in the control of the gain and loss of magnets in PTsymmetric systems. [19,25]In this review, we first introduce the basic concept of PTsymmetry, and then we highlight the recent theoretical and experimental developments of magnonic PT-symmetry.Finally, we conclude and provide a guide for future research on magnonic PTsymmetry toward device applications.

Physics of Non-Hermitian Hamiltonian 1.PT-Symmetry
PT-symmetry is an essential point when people study non-Hermitian Hamiltonian in physics.In 1998, Carl M. Bender and Stefan Boettcher proved that the eigenvalues of the PTsymmetric Hamiltonian can be completely real even if the Hamiltonian is not Hermitian. [6]This means that the physical system satisfying PT-symmetry can still obtain the real spectrum when it has nonconservative parameters.The phase transition of PTsymmetry in physics systems can be simply explained as follows: Consider two identical closed and isolated boxes whose positions are mirrored symmetrically relative to the center of the frame of reference, with coordinates of x = − a and x = + a respectively, as shown in Figure 1.The particles in the box at x = − a and x = + a act as the signal receiver and emitter sources, respectively, and the whole system is PT-symmetric compliant.When a parity transformation P is operated on the system, the left and right particles swap positions, and the two sources exchange their roles after a time reversal T operation.Note, however, that if the two boxes are weak or even zero coupled, the system is not in equilibrium after P and T operations.Instead, the energy on the left decays to zero, and the energy on the right increases to infinity, corresponding to the PT-symmetry broken case.On the other Figure 1.Schematic diagram of the model for convenient understanding of PT-symmetry in physics system: the left box contains particles as receivers at x = − a, and the right box contains particles as emission sources at x = + a (radiation rates were equal).After the space reflection P operation, the two particles exchange positions.After the time reversal T operation, the particle as the emission source becomes the receiver, and the other particle as the receiver becomes the emission source.Reproduced with permission. [5]Copyright 2013, American Journal of Physics.
hand, if the two boxes are strongly coupled, the system can be rebalanced after P and T operations.In this case, the PT symmetry will not be destroyed, and the energy eigenvalue is also real. [5]onsider a 2 × 2 matrix Hamiltonian Ĥ to describe the above example: where g is a real number representing the coupling strength of the two boxes, r represents coordinates, and  is the counterclockwise angle between the complex vector and the real axis.
Considering the parity operator P described above, it can actually be understood as: r → − r, which can be regarded as a Pauli matrix in matrix operation  x,y, z , so P operator satisfies P = P † , while time inversion T can be understood as a complex conjugate transformation. [6]

Exceptional Points
Exceptional points (EPs) correspond to the phase transition between the unbroken phase and broken phase of PT-symmetry, which play a very important role in both classical physics and quantum mechanics.EPs have been demonstrated in optics, [26][27][28][29][30][31] acoustics, [32] etc.In recent years, the research on EPs has been booming in magnetism [21,33] and spintronics. [34]ear the EPs in the parameter space, if a small perturbation ɛ is added to the equilibrium system, the value of ɛ much less than 1 can make the eigenvalues diverge according to  1∕N .Here N is the order of EPs. [35,36]It is considered that this characteristic of EP for non-Hermitian systems can be used to enhance the sensitivity of the system to external information, which can be applied in device sensing. [30,37]milar to the example mentioned above, [5] a 2 × 2 matrix Hamiltonian is used to describe the general coupling of a simple two-level magnetic physical system: [38] Ĥ = where  1,2 represent the resonance frequencies of the two magnon modes,  1,2 are the dissipative or gain parameters of the two modes, and g represents the coupling strength parameter of the system.The eigenvalue  of Ĥ is obtained as: It can be seen that  1,2 have two eigenvalues and are mutually repulsive due to the different signs.EP is the point where the two eigenvalues merge, i.e.,  1 =  2 = .The above formula is rewritten as follows: Based on the equation provided above, it is evident that a particular  value merges, indicating the merging of two eigenvalues, when 4 g 2 = ( 1 −  2 ) 2 .This condition signifies the occurrence of an EP.

Anti-PT symmetry
In fact, the Hamiltonian with PT-symmetry satisfies the commutation relation of PT Ĥ = Ĥ PT, therefore the eigenfunction of the operator PT is also the eigenfunction of Ĥ and the eigenvalue  of the Hamiltonian is naturally real.While in PT-symmetric broken Reproduced with permission. [19]Copyright 2015, American Physical Society.
phase, the commutation relationship is not satisfied.The anti-PT symmetry mentioned below satisfies the anti-commutative relationship as PT Ĥ = − ĤPT.Similar to PT-symmetry, anti-PT symmetry also is non-Hermitian, and if all the eigenfunctions of Hamiltonian Ĥ are also eigenfunctions of anti-PT, then the eigenvalue of Ĥ is the pure imaginary, i.e., E i = −E * i .In this case, the Hamiltonian Ĥ is in the state of anti-PT symmetric phase unbroken. [39]In 2013, Ge et al. proposed a class of refractive index that is antisymmetric under combined PT operation with balanced positive refractive index materials (PIMS) and negative index materials (NIMS). [40,41]It is noted that to realize the anti-PT symmetric system, the conditions of dissipative coupling must be met to ensure the eigenvalue is an imaginary number.In 2020, Yang et al. designed an anti-PT symmetric cavity magnon system with precise eigenspace controllability. [42]The two different singularities have been observed in the system, with one EP generated by adjusting the magnon damping and another from the dissipative coupling of the two anti-resonances, i.e., the bound state in the continuum (BIC).Up to now, anti-PT symmetry has been theoretically studied and predicted in photonics, [43] topology, [44] acoustics, [45] waveguide systems, [46] and magnetism. [42,47]However, further extensive experimental exploration is still needed in this aspect.

Theoretical Research Progress in PT-Symmetric Magnetic Nanomaterials
Nowadays, low dimensional nano-magnetic materials [48] have not only been widely applied in storage devices, magnetic logic circuits, and various magnetic sensor devices, [49][50][51] but also aroused fundamental theoretical research interest in solid-state physics. [52]In addition to the extensive research of PT-symmetry in optics, [10,11] electronic circuits, [53] etc., it has also been explored in the field of spintronics [34] and magnetism recently. [54]The reason why PT-symmetry is so attractive in magnetism is because the eigenvalue can be transformed between real and complex by adjusting parameters in parameter space, such as magnetic gain/loss parameters. [55]n 2015, Lee et al. introduced PT-symmetry into magnetic nanostructures with balanced magnetic gain and loss, which ini-tiated the research of PT-symmetry in magnetism. [19]Considering a magnetic system with two layers of ferromagnetic films coupled by dipole-dipole interaction, [56][57][58] as shown in Figure 2a.
The magnetic system can be described by a pair of coupled LLG equations: [59]  ⃗ where  is the gyromagnetic ratio, K is the interlayer coupling constant,  is the Gilbert damping constant, H 1,2 and M 1,2 are the effective fields and saturation magnetization of the two ferromagnetic layers, respectively.This system satisfies PTsymmetry, since under parity P operation, M 1 ↔ M 2 , simultaneously H 1 ↔ H 2 , while for the third dissipation term, under the operation of time reversal T, t ↔ − t. Figure 2b presents the eigenfrequency  of the system as a function of gain/loss parameter  in the parameter space, which reflects the characteristics of the PT-symmetry system.The eigenfrequency completes a transition from pure real number to complex number with increasing magnetic gain/loss , corresponding to the transition from PT-symmetric phase to PTsymmetric broken phase.The transition point is EP in the magnetic system. [18,60]The model has been further extended by considering the limited interchange coupling within the layer, which enabled the system to study spin wave (magnon) excitation. [25]A new first-order ferromagnetic-antiferromagnetic phase transformation can be achieved by tuning the gain/loss parameters in the newly modified LLG equation, as well as considering the wave vectors k for the different spin waves. [25][63] The study unveils the potential for a PT-symmetry phase transition driven by STT in magnetic nanomaterials and devices.The theory has been further advanced by using the topological properties of EPs to parameterize the Riemann surface in which EP exists, [64] enabling the nonreciprocal temporal evolution of classical spins. [34]igure 3. Nonspin polarized current is input from left to right a) and right to left b).The spin of the current is controlled by spin torque and the effective magnetic field experienced in the nonuniformly magnetized material.The red and blue arrows represent spin accumulations due to the spin Hall effect.Spin torque is applied to incident current in a medium (gray) with a spatially varying magnetic field (orange arrows).At  = 0 c), the system is in a Diabolical point (DP) with a Hermitian Hamiltonian.At  = 1 d,e), the Hamiltonian becomes non-Hermitian, and the DP splits into two EP points.The correct parameter evolution f-h) shows clockwise and anticlockwise encircling of the EP, leading to a sharp jump between the two spectral surfaces.Reproduced with permission. [34]Copyright 2019, The Author(s), licensed under a Creative Commons Attribution (CC BY) License.
Considering a linear non-Hermitian Hamiltonian to describe the dynamics of a classical spin under a time-dependent effective magnetic field (Figure 3a,b), [65] the Hamiltonian is expressed as Equation (7).Equation (8) represents the complex eigenvalues of the Hamiltonian, and  is the dimensionless amplitude of the spin torque applied to the system. [22]The asymmetric spin-filter effect can be achieved by injecting a nonpolarized electron current into the medium that is spatially nonuniform magnetized and biased by spin torque, as shown in Figure 3. [64,[66][67][68][69] Here, h x and h y are the external magnetic field components along the x and y axes, respectively, and  x,y represents the Pauli matrices.At  = 0, the system Hamiltonian corresponds to the Diabolical point (DP) of the real energy spectrum, and the Hamiltonian is Hermitian, as shown in Figure 3c.When  = 1, as shown in Figure 3d, the Hamiltonian enters a non-Hermitian state, and the DP splits into two EPs. Figure 3e shows the imag-inary part of the eigenvalues.The red line predicts a loop on the Riemann surface composed of parameters, but it is usually incorrect for the dynamic encircling of the EP at a limited speed for control parameters.The correct parameter evolution is shown in Figure 3f-h: the purple and green loops represent clockwise and anticlockwise encircling of the EP, respectively, and it is evident that there is a sharp jump between the two spectral surfaces.The final state of the system after a 2-rotation depends on the direction of parameter evolution.Figure 3f-h depicts scenarios of parameter evolution paths around a single EP, the area where the eigenvalues are real near the EP, and the situation ≈2 EPs, respectively.
In 2020, Wang et al. conducted a theoretical study on the non-Hermitian characteristics of spin-wave propagation. [70][73] By modulating the charge current in the spacer layer, enhanced magnetic damping has been achieved in one waveguide and reduced magnetic damping in the other, creating a magnetic PT-symmetric system with bias control near the EP. [74]In their model, the two identical magnetic waveguides  [70] Copyright 2020, The Author(s), licensed under a Creative Commons Attribution (CC BY) License.
(WG1 and WG2) are made of Yttrium Iron Garnet (YIG) and coupled by Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction [75][76][77] (see Figure 4a).The spin wave propagates along the x-axis parallel to magnetic waveguides, and Pt thin film is adopted as the spacer layer due to its large spin Hall angle. [78]he magnetization dynamics of the system were described by the LLG equations with including a SOT term, which is expressed as follows: [63,[79][80][81] Here T p represents spin-orbit torque, p is the indices of the two waveguides (p = 1, 2).c J = T SH ℏJ e 2 0 et p M s is the strength parameter of SOT, where  0 is the vacuum permeability, T is the transparency at the interface, e is the elementary charge, J e is the charge current density,  SH is the spin Hall angle, t p is the corresponding waveguide thickness, and M s is the corresponding waveguide's saturation magnetization.In this magnetic system, the effective field includes the internal exchange field, interlayer RKKY coupling field, and external magnetic field.The LLG equations are linearized and lead to a vector equation eigenvectors  p = m x,p + im z,p , where m x,p and m z,p represent the small time-varying components of the magnetic moments in the x and z directions for the two waveguides.Thus, the magnon coupling waveguide equation with SOT is obtained as here ) is the natural frequency of a given waveguide, where H 0 is the applied external magnetic field, A ex is the internal exchange interaction parameter, J RKKY is the interlayer RKKY exchange interaction parameter, and k x is the wave vector of the spin wave along the x direction. J = c J 1+ 2 is the gain-loss term related to SOT, which is the core of this PTsymmetric system.q = J RKKY (1+i) 0 M s t p is coupling strength, and q =  = J RKKY  0 M s t p at zero intrinsic damping.By solving Equations (10)   and (11), the dispersion relation of the control mode  is given by: where ± separates the eigenfrequencies  into acoustic and optical modes, respectively. [82]][85] When the value under the square root in Equation ( 12) is greater than 0,  corresponds to a real eigenvalue, indicating that the system is in the PT-symmetric phase.Conversely, when the value is less than 0,  becomes a complex eigenvalue, indicating that the system enters the PT-symmetric broken phase. [5]At the value under the square root equals 0, the two eigenmodes merge, indicating the presence of an EP, [11,29,31,84,[86][87][88] as shown in Figure 4b.The input signal in WG1 would generate periodic energy exchange in the two waveguides due to the interference of the two magnon modes, resulting in a non-uniform spin wave amplitude as shown in Figure 4c.The current input in the Pt interlayer  J can be modulated to make the system reach EP, i.e.,  J = , meanwhile, the periodic oscillation gradually disappears as the system parameters get closer to the EP and finally reach the EP point with the symmetric propagation of the amplitudes of the two waveguides.The SOT intensity can be modulated by the current passing through the spacer.The permeability study shows that the sensitivity of the system to the magnetic environment near EP was significantly higher than that under other conditions.This sensitivity enhancement is a unique behavior of PT-symmetric systems near EPs in the parameter space, which has been discussed and implemented in many systems. [21,35,89,90]urthermore, as shown in Figure 4d, no matter which waveguide input signal is, the spin wave signal is always output at the end of WG2 with negative effective damping.This provides a way to realize the nonreciprocal propagation of spin waves in PT-symmetric magnetic systems.In addition, the different configurations of YIG waveguide and Pt current transmission layer have also been studied theoretically.They further proved that the considered configuration, near an EP, enables the switching of spin-wave power in a waveguide through slight variations in the substrate's current density.This capability holds promising applications in sensing or logic gates. [91]

Experimental Research Progress of PT-Symmetric Magnetic Nanomaterials
For PT-symmetric magnetic nanodevices, the most challenging problem is how to realize tunable coupling of the devices and magnetic gain/loss. [92]In 2019, Liu et al. designed and manufactured a ferromagnetic three-layer structure, where two ferromagnetic layers are made of 30 nm Co and 5 nm NiFe, and the spacer layer is made of Pt with tunable thickness, as shown in Figure 5a. [93]][96] A pair of LLG equations has been employed to describe the magnetization dynamics of the system.The passive PT-symmetry has been considered by linearizing the magnetization dynamics equation, [96] as shown in Equation ( 13) where  is the damping parameter, which has the parameter relation of  1,2 =  ± . [96] H and  J respectively represent the external magnetic field and the exchange coupling field, where  J is related to the exchange coupling constant J.After considering the real parameters of the experimental material for the system, the two relative damping parameters  = − Im()/Re() of the collective mode and the eigenfrequency difference Δ  1,2 = Re( 1,2 ) − Re( 0 1,2 ) after subtracting the eigenfrequency of the corresponding uncoupled FM can be obtained, as shown in Figure 5c. 0 1,2 represent the undisturbed ferromagnetic resonance (FMR) frequencies.It can be clearly seen that EP exists when the RKKY interaction J > 0 keeps increase in the parameter space, where the acoustic mode and optical mode branches evolve from crossing to anti-crossing.It should be noted that this phase transition is asymmetric in the range of J > 0 and J < 0 due to the asymmetry of two different magnetic layers.
To experimentally demonstrate the existence of passive PTsymmetry, the eigenfrequency and damping rate of collective The YIG waveguide is grown on the GGG substrate, and the thin film is carved with a local laser ablation system to create an air gap between the two waveguides.The GaAs semiconductor is covered on the two waveguides, and the 830 nm infrared laser is irradiated on the waveguide YIG-S2 to control the conductivity of the GaAs semiconductor, thus causing the loss in the YIG-S2 waveguide.b-h) The two-dimensional plots show the spin wave intensity measured using BLS (Brillouin Light Scattering) technique from a continuous wave infrared laser at different power levels.Figure (b-f) representS the spin waves excited in the first waveguide, while Figure (g,h) show the spin waves excited in the second waveguide.The values of the laser power are indicated on the right side of each plot.Reproduced with permission. [105]Copyright 2022, American Physical Society.
dynamics have been measured by three different techniques, i.e., broadband ferromagnetic resonance (FMR), [97] Brillouin light scattering (BLS), and magneto-optical Kerr effect (MOKE). [98]igure 5d shows the FMR signals of the devices with different Pt layer thicknesses.It can be seen that at the EP (0.8 nm-Pt thickness) the acoustic mode disappears, and only the optical mode is observed since the two modes merge.When the Pt thickness is larger, the acoustic mode and the optical mode exist simultaneously, but the optical mode is suppressed.This is a classic picture of a PT-symmetry system in its phase transition.This phenomenon can be more obviously seen in the signal detection of BLS for devices with different thickness spacers, where the acoustic and optical modes gradually merge when they approach the EP, as shown in Figure 5e. Figure 5g,h summarizes the eigenfrequencies and damping constants obtained experimentally, which follow the theoretical model shown in Figure 5c and reveal the phase transition phenomenon between PT-symmetry and PTsymmetry breaking.
In passive PT-symmetric magnetic devices with Co films of different thicknesses, the magnon absorbs microwave photons and inspires coherent magnon currents.The magnon current can be converted into charge currents in the nonmagnetic layer by inverse spin Hall effect (ISHE), [99] as shown in Figure 5f.Compared with the conventional magnet Co, the passive PTsymmetric device can absorb more microwave photons in the broken PT-symmetric phase.It also confirms that the system near EP is highly sensitive to external magnetic environment disturbances, [21,35,100] such as microwave photon irradiation mentioned above.
For the dipole-dipole interaction dominating synthetic ferromagnet film materials, [101] the macro-spin model cannot accurately describe the physical effects in the real magnetic structure, [19,22,70] since the dipole-dipole interaction will cause the magnetic layer to tend to AFM configuration even without the interlayer exchange interaction.Recently Jeffrey et al. discussed a synthetic ferromagnet based on permalloy with dipole-dipole interaction in micromagnetic simulations. [102,103]he occurrence of EP and the high parameter sensitivity of ferromagnetic resonance frequency amplitude near the EP are reported. [102]t should be stressed that in the above experimental work, the damping parameters are greater than 0, [92] inconsistent with the natural parameters of the material.Recently, the negative Gilbert damping can be generated by the magneto-optical interaction of three orthogonal circularly polarized laser beams with submicron magnets placed in an optical cavity. [104]Under the far blue detuning, the nonresonant coupling between the driving laser and the cavity photons can produce an accurate Gilbert-type magnetic torque.The formula of optical torque on spin is given, and the fardetuning condition of the torque in Gilbert form is accurately determined.The realization of EP with laser-modulating magnetic loss and gain has also been studied.In 2022, Sadovnikov et al. proposed a method to control the spin wave loss in one of the YIG waveguides covered by GaAs semiconductor layer by infrared laser irradiation, as shown in Figure 6. [105]The infrared laser irradiates the GaAs-covered waveguide YIG-S2, which makes the charge carriers in the GaAs semiconductor layer transition from the valence band to the conduction band, [106] increasing the conductance of the semiconductor. [107]This changes the boundary conditions of the interface of YIG-S2, and finally leads to the change of spin wave dispersion and increases the loss in YIG-S2 waveguides. [108,109]he dispersion and coupling of spin waves in waveguides can be established by considering LLG equation, Maxwell equation, and electromagnetic boundary conditions at the same time. [59,70]y combining all the equations, a set of Schrödinger type equations for amplitude can be obtained. [110]Assuming that the distribution of the spin waves does not change along the waveguide and the spin wave propagates along the x-axis, the amplitude of the spin wave follows the following Schrödinger equation: where the matrix M = ( iq 1 i i iq 2 ) describes the propagation and interaction of waveguide modes, and  is a phenomenological dipole-dipole coupling parameter.q 1,2 = k 1,2 + ig 1,2 is the complex wavenumber of uncoupled waveguides, and the imaginary part of the complex wavenumber g 1,2 is related to Gilbert damping.The eigenstate of the matrix M is the waveguide mode of the coupled waveguide system, so the complex wavenumber of the coupled waveguide system can also be obtained as: where k and ḡ are the average values of the real and imaginary parts of the complex wavenumber in the two waveguides, respectively.Δk and Δg are the differences between the real and imaginary parts of the complex wavenumber for the two waveguides.It can be seen that the complex wavenumber of the coupled waveguide system has the classical form of EP. [28,31,111] When the value under the root passes through 0, the system transits between PT-symmetric phase and PT-symmetric broken phase by going through EP. [19,70,93] The spatial distribution of magnetization oscillation intensity in the waveguide is measured experimentally by BLS technique.With increasing the IR laser power, the phase transition across EP is observed in BLS data, as presented in Figure 6b-h.Since the imaginary part g of the wavenumber inevitably leads to the change of the real part k, the root term of the complex wavenumber Q 1,2 cannot be strictly equal to 0, and EP cannot be found accurately.In the aid of numerical analysis, the EP has been confirmed. [112]The researchers simulated the system at both zero and nonzero IR laser powers and observed characteristic PT-symmetric behaviors of the complex wave number difference around a critical value of IR laser power.This work provides an important empirical reference for experimental analysis of EP problems in PT-symmetric systems.One of the important applications of magnetic nanodevices is magnetic memory devices, through the magnetoresistance effect and spin torque effect. [113]The PT-symmetry in magnetic nanodevices has been demonstrated to control two coupled spin torque nano-oscillators (STNOs). [114]As potential nextgeneration microwave signal generators, [115,116] STNOs are very suitable for studying the EPs since they have both the loss related to magnetic damping and the gain provided by the spin torque effect during the spin-polarized current injection. [117,118]Wittrock et al. studied the collective dynamic behavior of the two coupled STNO, whose magnetizations are in a dynamic vortex state and referred to as STVOs (spin-torque vortex oscillators).The fixed layer below the two STNOs is connected with wires, and there is a feed strip antenna above them.Each antenna is connected with the free layer located on the top of the other STNO through wires.The microwave current I rf generated by the spin torque ef-fect in one oscillator will be transmitted to the strip antenna on the top of the other one to generate radiofrequency (rf) field H rf , as shown in Figure 7a.
Thiele-like theory is conducted to explain the experimental results, which can well describe the system with a magnetic vortex structure. [119]According to the complex state function z n = x n + iy n related to the positions of x-axis and y-axis of the n-th vortex core, the linearized Thiele equation controlling the small oscillation state of the vortex core near the stationary position is as follows: where the matrix A = and d l is the damping constant (l = 1, 2), I l are the injected current,  l represent the angular frequencies of vortex-free oscillations, C l is the parameter that determines the efficiency of spin torque effect, and the dissipation term and conservative term of the coupling are described by the coefficients k d and k c , respectively.At this time, we can see the classical form of the system with EPs, and  is a function of the current, so we can adjust the current to make the system reach EP.The complex state function z n is transformed into the typical time form e it by Euler formula, and then the eigenfrequency can be obtained as: [11,19,31] The condition of the occurrence of EPs is that the value under the root in Equation ( 17) equals zero.Experimentally, the dispersion spectrum of frequency with respect to I 2 can be obtained by fixing the magnitude of I 1 and adjusting the value of I 2 , as demonstrated in Figure 7b-e.At different fixed values of I 1 , EPs occurs with increasing the value of I 2 , accompanied by the amplitude death. [120]By configuring I 1,EP and I 2,EP according to the current at EP, the function i 1,2 with respect to I 2 can be obtained, as shown in Figure 7f-i.Consequently, the decrease in gain effect and the disappearance of amplitude around EP can be observed in the STNOs system.It is noteworthy that as the system approaches I 1,EP , its sensitivity to external thermal noise is enhanced.This is evident by the presence of a noisy vibration signal at frequencies where amplitude death should occur, as depicted in Figure 7e.Moreover, it has been predicted that higher-order EP could be reached by increasing the number of synchronous oscillators. [117]s a unique topological magnetic exciton, skyrmions have also been discussed in relation to the PT-symmetry. [25,115,116,121]t has been demonstrated that in ferromagnetic bilayers with balanced gain and loss, antiferromagnetic skyrmions can arise in the broken PT-symmetry phase. [25]In addition, skyrmions can be used to supplant conventional vortices in STNOs with better performances, i.e., antiferromagnetic skyrmion-based STNO can produce wider tunable frequency; [115,116] while ferrimagnetic domain and skyrmions can be stabilized without magnetic field at room temperature; [122] the repulsion between domain walls and skyrmions provides an alternative possibility to achieve synchronization of two STNOs. [123]igure 7. a) Two coupled spintronic nano-oscillators.The feeding strip-line antennas above each oscillator form a symmetrical coupling, which is realized by the microwave current I rf generated by spin torque dynamics in another oscillator.This produces a RF field H rf and results in effective coupling.b-e) In the coupling system, the frequency spectrum of the current I 2 is measured in STNOV2 under different currents I * 1 of STNOV1.f-i) Real and imaginary part of i 1 (red) and i 2 (blue) at fixed current I 1 and varying current I 2 .The gray line represents the same amount calculated when the coupling constant is set to 0. The experimental and theoretical diagrams in the same column correspond to the same parameters.Reproduced with permission. [114]Copyright 2021, The Author(s).

High Order EPs of Magnetic Nanomaterials
In addition to parameter modulation of EPs as discussed above, the physical mechanisms within the parameter range near EP are also of great interest.Particularly intriguing is the observation that the sensitivity to external perturbations can be further enhanced upon the realization of higher-order EPs. [21,35]As we mentioned in the Section 1, the eigenfrequency splitting Δ follows  1∕N -dependence, and N represents the order of EPs. [36,100]This is related to the fact that in the parameter space, both the eigenvalue and the eigenvector will merge at the correlation point. [93,100]The higher-order non-Hermitian EP has been realized theoretically and experimentally in the fields of optomechanical systems, [15,124] coupled-resonatornetworks, [125] acoustics, [126] coupled optical cavity systems, [100] etc.In the field of magnetism, it has also been realized in cavity magnon coupling systems based on magnon-photon coupling, [33] and the study of higher-order non-Hermitian EP based on pure magnetic system is indispensable for spintronic applications.
Recently, there have been ongoing theoretical studies addressing this issue.In analogy to the model in Figure 2a, magnetic thin film structure with third-order EP has been established by adding a magnetic neutral layer between the magnetic gain and attenuation layers, as shown in Figure 8a. [36]The three ferromagnetic layers are stacked and spin waves are excited by interlayer exchange coupling.The magnetization dynamics of the system can be described by a set of coupled LLG equations, e.g., Equation (18-20), and the eigenfrequencies can be calculated by their linearization.It can be seen that due to the existence of EP3 (see Figure 8b), the intrinsic frequency appears at different inherent damping parameters .
When a perturbation ɛ is applied at the top of the device, [100] and at EP, the eigenfrequency varies as  1∕3 with respect to the perturbation, as shown in Figure 8c-h.Figure 8d,f,h show the logarithmic scale of the eigenfrequencies with respect to the different orders of EPs on the left.For FMR, EP2 and EP3.As a consequence, linearity (ɛ 1 ), ɛ 1/2 and ɛ 1/3 responses have been observed, respectively.This work serves as a theoretical foundation for the practical utilization of magnetic layered devices in high-order EP scenarios, enabling the design of devices with nonlinear sensitivity.
The prerequisites of the above-mentioned theoretical research are lossless interlayer, and positive and negative damping parameters for the upper and lower layers, which are not practical in reality. [92]However, spin transfer torque and parametric drive open windows for damping or anti-damping the excitation of the magnon, [19] thus modulating the PT-symmetry of the system. [70]The high-order EP of the system can also be realized by using STT to modulate the damping parameters., i.e., Wang et al. proposed a feasible structure with three YIG layers as waveguides, [127] and a Pt layer embedded between each of two adjacent YIG layers as spacing layers to transmit charge and generate spin-orbit torque.The top layer and bottom layer are composed of fixed layers, and a spacing layer is added between them and the YIG layer so that WG1, WG3, and the spacing layer are coupled through RKKY interaction.The collective magnetization dynamics are governed by three coupled LLG equations.
The effective field consists of an internal exchange field, an interlayer RKKY coupling field, and an external magnetic field.In contrast to Equations ( 5) and (6), in this case, the additional consideration of the spin-orbit torque T p is included.And T p is controlled by the charge current in the Pt spacer.This term leads to the magnetic loss and gain in WG1 and WG3 respectively, while the SOT in WG2 is neutralized.After considering the The eigenfrequency of the system evolves with the eigenvalue of the gain-loss parameter .It can be observed that there are two critical  in the parameter space due to the existence of EP3.c) The function of the FMR frequency of a single-layer ferromagnet with the ɛ, in which the perturbation ɛ is applied at the top of the device by a magnetic field.e) is the change of eigenfrequency near EP2 with ɛ in the parameter space of double-layer ferromagnet system, and (g) is the change of eigenfrequency near EP3 with ɛ in the parameter space of three-layer ferromagnet system.d,f,h) are logarithmic coordinate graphs corresponding to the frequency shift and frequency splitting of (c,e,g), respectively.The slopes are 1,0.5 and 0.33 respectively, proving linearity (ɛ 1 ), ɛ 1/2 and ɛ 1/3 response corresponds to FMR, EP2 and EP3, respectively.Reproduced with permission. [36]Copyright 2020, American Physical Society.
above conditions, the coupled LLG equations can be linearized to obtain the Hamiltonian of the system: where 1+ 2 is the gain-loss term related to SOT.  1,2, 3 =  ex + 2 + ɛ 1,2, 3 includes the effective magnetic field  ex , the coupling strength , and the weak magnetic perturbations ɛ 1,2, 3 applied to the three waveguides.And by calculating the eigenvalues of the above Hamiltonian, we can get the eigenfrequency of the system: It can be clearly seen that EP exists when  J = √ 2, [19,93] and the three eigenfrequencies are merged.The enhancement of fre-quency amplitude can be observed at EP, compared with the situation without STT.It should be noted that only one EP is observed, which is different from that in Figure 8b.After adding the perturbation (adding the perturbation ɛ magnetic field at WG1), the eigenfrequency changes in  1∕N relation with the perturbation, as shown in Figure 9c,d.Moreover, if the number of stack layers increases, i.e., adding a group of spacer layers and waveguide layers on the bottom layer as WG3 and WG4, the fourth-order EP can be achieved.The fourth-order eigenvalueperturbation responses are shown in Figure 9e,f.This work has realized high-order EP magnetic systems in film structure through micromagnetic simulation, which promotes the development of high-order EP magnetic sensitivity in application to a certain extent.

Discussion on Anti-PT Symmetry
In the study of PT-symmetry by non-Hermitian Hamiltonian, anti-PT symmetry (APT), as the counterpart of PTsymmetry, [39,42,128] has been briefly introduced in Section 1.3 Below we will discuss the theoretical mechanism of Hamiltonian of anti-PT symmetric system.The anti-PT symmetry satisfies the anti-commutation relation PT Ĥ(PT) (−1) = − Ĥ39 in a coupling system, and the coupling between the two pure states is Figure 9. a) Schematic of the experimental structure shows that there are two hard magnetic fixed layers at the top and bottom, and the middle WG1, WG2, WG3 are made of YIG.Between each two waveguides, and between the waveguide and the fixed layer, are separated by a spacer layer, so that they are coupled by an interlayer RKKY.The spacing layer made of the Pt layer between each two waveguides can be inputted current to generate a SOT, the SOT damping and antidamping WG1 and WG3, respectively, and the SOT in WG2 is neutralized.The magnon is emitted at the left end of the waveguide, that is, at x = 0. b) The real part (Top) and imaginary part (Bottom) of the calculated system eigenfrequency after considering the actual parameters.c) The nonlinear response of the eigenfrequency of the system to the perturbation ɛ after adding the perturbation at WG1. d) The relationship between the value of Re( T2 −  T3 )/ = Δ/ and ɛ, and the relationship in logarithmic coordinates.The logarithmic behavior of the curve shows that the slope is 1/3.(e) and (f) correspond to (c,d) with a four-layer waveguide structure under consideration.In the inset of (f), the slope of the logarithmic behavior of the curve is 1/4.Reproduced with permission. [127]Copyright 2021, American Physical Society.
expressed by i. [42,129]Similar to the PT-symmetric systems represented by Equation ( 2), the anti-PT symmetry system can be described by the following matrix Hamiltonian of 2 × 2 (here, for convenience, we consider the case of  1 =  2 = ): where  1,2 represents the eigenmodes of the two coupled systems,  represents intrinsic damping, and  represents the dissipative coupling term.The eigenvalue of this system can be obtained as: To simplify the analysis, assume  1 =  2 =  and the above formula can be written as follows: [93] When  2 −  2 > 0, the eigenvalue is a pure imaginary and the system is in an anti-PT symmetric phase, while with  2 −  2 < 0, the eigenvalue is complex and the system is in an anti-PT symmetric broken phase.Therefore, the parametric case of the eigenvalues in the anti-PT symmetric system has opposite properties to those in the PT-symmetric system.It is challenging to realize the pure dissipative coupling condition needed by an anti-PT symmetric magnetic system.However, in 2020, Zhao et al. proposed a design of coupled magnoncavity-magnon polaritons, utilizing two YIG spheres in a cavity for dissipative coupling through an antenna.This work provided compelling evidence for the realization of anti-PT symmetry in cavity magnon systems, [130] Nevertheless, further investigations on magnetic film materials are still necessary for spintronic applications.
In 2022, Chao et al. theoretically proposed the anti-PT symmetry in synthetic antiferromagnet (SyAFM). [47]The two magnetic layers are antiferromagnetically exchanged coupled by RKKY interaction, and the system experiences the transition from anti-PT symmetric broken phase to anti-PT symmetric phase by increasing the antiferromagnetic interaction.For SyAFM, LLG can also be used to describe its magnetic dynamics, and the effective field is expressed as Here J represents the inter layer RKKY antiferromagnetic coupling coefficient, A i is the Heisenberg exchange coupling constant within each magnetic layer, and K z is magnetic anisotropy along the easy z-axis.When J = 0, the two magnetic layers are decoupled, and the magnetization vectors of both layers process in the right-handed direction around their own easy magnetization directions, as illustrated in Figure 10a.Performing the parity transformation P operation results in the exchange of the two Néel vectors, while the time inversion T operation multiplies the two magnetic vectors by −1.It can be seen that the overall magnetization state remains unchanged after the PT operation. [93]When J > 0, it indicates the presence of antiferromagnetic coupling with one layer processing in the right-handed direction and the other in the left-handed direction.According to the theory, in this scenario of J > 0, the low-energy magnetization dynamics of the system exhibit anti-PT symmetry.After applying the same PT transformation as shown in Figure 10a, the overall magnetization state becomes the opposite of the initial state, as depicted in Figure 10b.In Figure 10c, it can be observed that when J > 0 and J < J C , the system is in the anti-PT symmetry broken When J < J C , the real parts of the eigenfrequency of the magnon will split into two branches with opposite signs.In this case, the system is in the anti-PT symmetry broken phase.When J > J C , the coupling strength exceeds the requirements for EP, and the system enters the anti-PT symmetry phase with the two eigenvalues merging.Reproduced with permission. [47]Copyright 2022, The Author(s).Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.
phase, whereas the system enters the anti-PT symmetry phase when J > J C with the two real eigenfrequencies merging.Here, J C represents the critical point, or EP, that separates the anti-PT symmetry broken phase from the anti-PT symmetry phase in the parameter space.This theoretical study presents a concept for utilizing anti-PT symmetry in magnetic thin film materials.

Conclusion and Outlook
The study of PT symmetry in magnetic materials remains an active and exciting research area.Various approaches have been explored, ranging from simple parameter modulations such as changing the inherent damping parameters, [19] to the influence of STT and SOT on the gain/loss of the magnetic system, and the exploration of phase transitions in PT-symmetric systems. [22,23,34,70]It has been theoretically predicted and experimentally demonstrated that remarkable features can be achieved in coupled magnetic waveguides within non-Hermitian systems and through PT-symmetry parameter modulation in multilayer magnetic films.For instance, by tuning the system near EP in the parameter space, the nonreciprocal propagation of spin waves in magnon waveguides can be significantly enhanced. [21,70,91]his research is beneficial for enhancing the performance of magnon circuits, [131] magnetic logic gates, [132,133] and magnetic logic memory devices. [51,134]It also aids in the design of tunable magnetic sensors [135][136][137] capable of exhibiting enhanced magnetic susceptibility near the PT-symmetry phase transition, thereby achieving higher sensitivity.
Recent studies in antiferromagnetic materials satisfying PTsymmetry have revealed the persistent presence of a secondorder nonlinear Hall effect generated by the collaboration of skew scattering and Berry curvature. [138]This unique phenomenon, known as the "anomalous skew-scattering nonlinear Hall effect (ASN)," results from the interplay of spin-dependent anomalous velocity and skew-scattering spin-dependent distribution.
ASN can produce helicity-dependent photocurrents, setting it apart from other PT-symmetric-related nonlinear effects in metals that remain unresponsive to optical helicity. [139]In addition, spin current can be generated by the nonlinear spin Hall effect in PT-symmetric antiferromagnetic systems, [140] which does not require any spin-orbit coupling (SOC), uniform magnetization, or spin-split band structures, promoting its applications in spintronics. [141,142]or more practical magnetic nanomaterials that can be used in device fabrication, further research is needed to explore their performance in non-Hermitian systems.Several aspects which are worth further investigating in this field are listed as follows:

High Susceptibility near EP
Near the special parameter point known as EP, the system becomes highly sensitive to external disturbances.[145][146] The nonlinear response characteristics of higher-order EP magnetic systems to external stimuli have been extensively studied through theoretical simulations.However, recent theoretical studies have shown that quantum fluctuations near EP can be significant. [147]This suggests that achieving parameter control points with high susceptibility near EP may be challenging in practical experiments.Therefore, further experimental research is needed to explore the possibility of achieving ultra-high susceptibility near EP.

Magnetic Anti-PT Symmetry
There are also intriguing topics that warrant further discussion in the investigation of magnetic non-Hermitian systems, such The two magnetic layers are antiferromagnetically and the influence of an external magnetic field causes the magnetic moments to process, resulting in the generation of a pumped spin current.b) High-order EPs ferromagnetic thin films coupled through RKKY interlayer exchange interaction.The ferromagnetic layers, separated by nonmagnetic interlayers, are subjected to modulation of interlayer thickness and magnetic material parameters to establish the conditions for EP attainment in the system.c) By employing an IG gating process, the RKKY interlayer exchange interaction strength can be electrically controlled, thereby achieving abrupt changes in the coupling parameters.This enables the controllable PT-symmetry phase transition of the device.
as anti-PT symmetry and the magnetic skin effect in bulk magnetic materials. [148,149]Researchers have discovered many fascinating phenomena when investigating the pure imaginary coupling in magnetic systems.These include the discovery of additional phenomena such as bound states in the continuum (BIC), [150] in addition to EP. BIC was located inside the continuum with fully localized, no leakages, and zero-line width.Currently, BIC has found applications in various fields, including optics, [151,152] quantum memory, [153] sensors, [154][155][156] spectral filtering, [157] microwave, and acoustics. [158]The coexistence of BIC and EP has been observed in the cavity magnon coupling system. [42]It is predicted that the anti-PT symmetric system with double exceptional points (EPs) can be more practical than the PT-symmetric system. [42][161][162] We provide an alternative possibility to achieve anti-PT symmetry in magnetic films based on the spinpumping effect.Considering a three-layer structure of nonmagnetic layer sandwiched by two magnetic layers, the exchange coupling between the two magnetic layers is antiferromagnetic type from RKKY interaction.When the magnon in one magnetic layer is excited, the angular momentum is transmitted through the nonmagnetic layer and reaches the other magnetic layer due to the spin-pumping effect. [159]The two magnetic layers can establish dynamic dissipative coupling in addition to RKKY interaction, allowing for the realization of a system where anti-PT symmetry can be discussed.This configuration is illustrated in Figure 11a.
Currently, spin pumping, as a method for generating spin currents, has been extensively studied in various materials, including traditional magnetic materials, [160,161,163] nonmagnetic materials, [164][165][166] semiconductors, [167] organic semiconductors, [168,169] spin glasses, [170] and graphene materials. [171,172]It is possible to design spin spin-pumpingbased based anti-PT symmetric system with valuable applica-tions.Therefore, there is still ample opportunity for further exploration in both theory and experimentation in anti-PT symmetric systems.

Nonlinear Sensing Based on PT-Symmetry
STNOs are well-suited for studying coupled non-Hermitian magnetic systems since STNOs possess two crucial attributes of magnetic loss and gain generated by the spin-polarized current injection. [113,114]In recent years, there have been advances in realizing higher-order network systems using STNOs. [117]The STNOs exhibit a nonlinear perturbation response to the external magnetic environment, which holds significance in the development of arrayed high-order EP devices with nonlinear sensing capabilities.

RKKY Interaction Modulated High-Order EP
The magnetic gain/loss parameter has always been tuned to reach EP in previous theoretical analysis, however, the RKKY exchange interaction is another important parameter for the exploration of magnetic PT-symmetry.We have designed a multilayer system featuring the potential for high-order EPs coupled through RKKY exchange interaction, as illustrated in Figure 11b.The three-layer ferromagnetic structure is separated by two nonmetallic interlayers.The magnetization dynamics of the system can be described by considering three coupled LLG equations, taking into account the influence of the RKKY exchange interaction. [93]The high-order EP in Figure 8 may also be achieved by varying the nonmagnetic interlayer thicknesses to tune the RKKY exchange interaction, which is expected to significantly broaden the experimental scope for achieving high-order EP in magnetic film materials. [36]The modulation of RKKY exchange interaction is not only limited to adjusting the thickness but can also be achieved by the application of a gated voltage using ionic gel (IG) [173] or ionic liquid (IL). [174]The Fermi level of synthetic antiferromagnetic heterostructures [175,176] can be influenced by the accumulation of charge.This allows us to achieve an electrically controlled PT-symmetry phase transition in the device, as illustrated in 11c.A similar approach can be employed to control the temperature dependence of the RKKY interaction. [177,178]

Figure 2 .
Figure 2. a) Two coupled ferromagnetic films when an external magnetic field is applied along the z-axis.b) (Top) The intrinsic frequency of the PTsymmetric magnetic system that is shown in (a) varies with the dissipation parameters .(Bottom) The magnitude of the ratio between the y components of the normal modes and their related phase difference ΔΨ l = 1,2 y

Figure 4 .
Figure 4. a) Schematic diagram of coupled magnetic waveguide separated by nonmagnetic spacers with strong SOI.The two waveguides are made of YIG, and the non-magnetic spacers are made of Pt with a large spin Hall angle.Driving the charge current j Pt along the spacer (x direction) will result in the spin Hall torque acting on the magnetic waveguide.b) The spatial distribution of the amplitude of the propagating spin wave under different loss/gain balances.c) The color change from blue to red corresponds to a linear amplitude change from 0 to the maximum input signal.d) When  J = 3 the spatial distribution of x component of time magnetized M x at t = 40ns.Reproduced with permission.[70]Copyright 2020, The Author(s), licensed under a Creative Commons Attribution (CC BY) License.

Figure 5 .
Figure 5. a) The geometry of the magnetic passive PT-symmetric system and the external field is applied along the z-axis direction.b) The schematic of a three-layer system in which two coupled ferromagnetic (FM) layers are separated by a nonmagnetic layer.The coupling strength can be tuned by changing the thickness of the nonmagnetic (NM) layer.c) The top is the eigenfrequency difference between the eigenfrequency calculated from the theoretical parameters of the materials used experimentally after subtracting the eigenfrequency of the two FM layers with exchange coupling, and the bottom is the corresponding damping constant .d) Two magnon modes, acoustic (A) and optical (O), were measured by FMR in three layers with different Pt interlayer thickness d. e) Typical BLS spectra of the magnon in three layers with different Pt thickness d.The corresponding arrows point to the acoustic (pink) and optical (blue) modes.f) ISHE current (black square) as a function of Pt thickness d.The red circle is the conventional Co (30 nm)/Pt (x nm) double-layer ISHE current as a reference.g,h) Eigenfrequency changes and Gilbert damping parameters of the magnon plotted as a function of the RKKY interlayer coupling strength measured experimentally.Reproduced with permission.[93]Copyright 2019, The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science.No claim to original U.S. Government Works.Distributed under a Creative Commons Attribution-NonCommercial License 4.0 (CC BY-NC).

Figure 6 .
Figure 6.a) Schematic of the structure of the experimental device.The YIG waveguide is grown on the GGG substrate, and the thin film is carved with a local laser ablation system to create an air gap between the two waveguides.The GaAs semiconductor is covered on the two waveguides, and the 830 nm infrared laser is irradiated on the waveguide YIG-S2 to control the conductivity of the GaAs semiconductor, thus causing the loss in the YIG-S2 waveguide.b-h) The two-dimensional plots show the spin wave intensity measured using BLS (Brillouin Light Scattering) technique from a continuous wave infrared laser at different power levels.Figure(b-f) representS the spin waves excited in the first waveguide, while Figure(g,h) show the spin waves excited in the second waveguide.The values of the laser power are indicated on the right side of each plot.Reproduced with permission.[105]Copyright 2022, American Physical Society.

Figure 8 .
Figure 8. a) Schematic of ferromagnetic heterostructure with gain layer, neutral layer, and loss layer.b)The eigenfrequency of the system evolves with the eigenvalue of the gain-loss parameter .It can be observed that there are two critical  in the parameter space due to the existence of EP3.c) The function of the FMR frequency of a single-layer ferromagnet with the ɛ, in which the perturbation ɛ is applied at the top of the device by a magnetic field.e) is the change of eigenfrequency near EP2 with ɛ in the parameter space of double-layer ferromagnet system, and (g) is the change of eigenfrequency near EP3 with ɛ in the parameter space of three-layer ferromagnet system.d,f,h) are logarithmic coordinate graphs corresponding to the frequency shift and frequency splitting of (c,e,g), respectively.The slopes are 1,0.5 and 0.33 respectively, proving linearity (ɛ 1 ), ɛ 1/2 and ɛ 1/3 response corresponds to FMR, EP2 and EP3, respectively.Reproduced with permission.[36]Copyright 2020, American Physical Society.

Figure 10 .
Figure 10.a,b) PT-symmetry property with SyAFM structure.a) The case of J = 0. b) The case of J > 0. The red and black arrows represent the two magnetic moments mn and − mn of the layer.|R〉 and |L〉 represent local right-handed and left-handed precession around the easy magnetization direction of the equilibrium.The vectors mn and − mn are exchanged under the parity operation P. The time reversal operation T causes mn to − mn , while the precession direction (|R〉 or |L〉) remains unchanged under the time reversal T. c) When J > 0, the phase diagram of anti-PT symmetry of SyAFM structure.When J < J C , the real parts of the eigenfrequency of the magnon will split into two branches with opposite signs.In this case, the system is in the anti-PT symmetry broken phase.When J > J C , the coupling strength exceeds the requirements for EP, and the system enters the anti-PT symmetry phase with the two eigenvalues merging.Reproduced with permission.[47]Copyright 2022, The Author(s).Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.

Figure 11 .
Figure11.a) Schematic representation of the anti-PT symmetry magnetic nanofilm device achieved through the spin pumping effect of two magnetic layers.The two magnetic layers are antiferromagnetically and the influence of an external magnetic field causes the magnetic moments to process, resulting in the generation of a pumped spin current.b) High-order EPs ferromagnetic thin films coupled through RKKY interlayer exchange interaction.The ferromagnetic layers, separated by nonmagnetic interlayers, are subjected to modulation of interlayer thickness and magnetic material parameters to establish the conditions for EP attainment in the system.c) By employing an IG gating process, the RKKY interlayer exchange interaction strength can be electrically controlled, thereby achieving abrupt changes in the coupling parameters.This enables the controllable PT-symmetry phase transition of the device.