Non-Hermitian Topological Magnonics

I would like to thank the editors of Physics Reports for giving me the opportunity to introduce the emerging field Non-Hermitian Topological Magnonics on behalf of my theoretical and experimental collaborators at Hong University of Science and Technology of China.

This is the outline. I will briefly explain the concept of magnon and its general non-Hermitian topology. Then I will explain several experimental and theoretical findings in the context of non-Hermitian topology of magnons.

Non-Hermitian Topological Magnonics is a field that combines Magnonics, non-Hermiticity, and Topology. A magnon is the quantum of spin waves, which are waves of magnetic moments propagating in magnetic material with a specific dispersion relation. The dispersion relation in the figure displays how spin waves propagate perpendicular to the magnetization plane in a magnetic field.

The ferromagnetic resonance of the thermal mode occurs with a mode of zero wave vector. When the wavelength is long or the wave vector is small, the long-range dipolar interaction controls the movement of magnetic moments. Surface spin waves are associated with these modes, which are localized to the two surfaces of the magnetic material. As the wavelengths decrease, the short-range exchange interaction becomes dominant, leading to spin waves with frequencies as high as several terahertz. This recent discovery suggests that magnons can be utilized for information transport and storage due to various advantages. For instance, there is no Joule heating when the magnet moves through the magnetic insulator.

The term becomes non-Hermitian when it interacts with the environment, as they exchange energy. The transfer of energy to the environment causes damping, such as Gilbert damping of magnons in insulators from magnon interaction. Another source of damping is spin pumping, where the angular momentum of magnons is transferred to electrons, photons, and photons in the substrate. This damping affects the coupling between two magnons, assuming there are two magnons, β one and β two, interacting with the same environment.

The environment acts as a bus with many traveling modes that can mediate the coupling between β one and β two. The coupling is not Hermitian and is called dissipative coupling, as it partially takes away energy without returning it. If the bus contains a standing mode, the energy can return, leading to coherent coupling between β one and β two. However, if the standing wave mode is coupled by sources, it can induce TCP two coupling, causing interesting non-Hermitian topological phenomena in Magnonics.

To theoretically describe the magnon subsystem when interacting with the environment, we consider two terms: HS, the spin or magnetic system of interest, and HE, the environment they interact with through coupling HSE. HSE is responsible for energy and particle exchange between the systems. The evolution of the whole system is described by the density matrix ρSE and the Liouville equation governing its dynamics.

The goal is to find an equation describing the magnon subsystem dynamics by integrating out information of the environment. By expressing the density matrix and performing the necessary integrations, we arrive at a simplified equation equivalent to the original one. Tracing out the environment leaves only information about the magnonic system.

Assuming weak coupling between magnon baths, we can simplify the equation by neglecting memory effects and considering the Markov approximation, where the density matrix of the magnonic system is a function of S with an integration over S. This simplification allows for a more manageable solution to the equation.

After making these observations, we will arrive at a relatively simple form of the master equation. This form is:

The general properties of the equation will not be addressed in depth. It is typically used to perform the rotating wave approximation, resulting in the linear master equation. This equation consists of two main structures: the density matrix of the magnonic system based on the effective Hamiltonian, which includes renormalization from the environments, and the quantum jump effect from the environment represented by the quantum jump operator. The effective Hamiltonian is no longer Hermitian, and there is a probability for the state to quantum jump before evolving according to the effective Hamiltonian. In condensed matter physics, the Green function approach is commonly used to study the interaction between magnons, leading to self-energizing and renormalized self-energizing. The effective Hamiltonian obtained through this approach is also no longer Hermitian. While the Green function approach does not account for the quantum jump effect, the master equation approach is equivalent to the Green function approach when the quantum jump effects are disregarded. In this presentation, the effective Hamiltonian approach will be utilized. Let us now examine the effective Hamiltonian.

When the effective Hamiltonian is not Hermitian, the resulting energy spectra are generally not real, containing both real and imaginary parts. This structure allows for a new characterization of topology. In the Hermitian case, wave functions are used to characterize topology, known as wave function topology. For example, Berry phase is used to calculate Berry curvature and the Chern number. However, in the non-Hermitian case, there is some freedom in the energy spectrum that can also play a role. This freedom can be used to define spectral quantities. By evolving some parameters in this space, energy spectra can form structures. This allows for the definition of spectral winding numbers or energy vorticity to characterize non-Hermitian topology. I will provide examples to explain this point later.

Based on non-Hermitian topology, many exciting phenomena have been discovered.

In my presentation on Magnonics, I will discuss examples such as exceptional points, the non-Hermitian nodal phase, the non-Hermitian SSH model, and the non-Hermitian skin effect.

I will begin by explaining exceptional points and their significance in non-Hermitian systems. An example of the interaction between magnons and photons can be described by a two by two Hamiltonian, where Omega P and κ P represent the frequency and damping of the photon, and Omega M and κ M represent the frequency and damping of magnons, respectively. The coupling between them is denoted by G A and G B, which may not necessarily be conjugates due to the coupling that I will discuss.

This Hamiltonian results in two eigenvalues, Omega plus and Omega minus, which can coalesce at specific parameter values. This coalescence leads to the two eigenvectors becoming parallel, defining an exceptional point. By adjusting parameters such as damping, coherent coupling, gain, and loss, one can achieve this coalescence.

It is believed that exceptional points can enhance the sensitivity of a system, making them a valuable tool in Magnonics research.

The exceptional point is much larger than other places. An example will be shown where sensitivity is enhanced at an exceptional point. A magnetic sphere is placed on a coplanar waveguide to support traveling microwaves. The experiment measures microwave transmission by changing the external magnetic field magnitude. The external magnetic field tunes the Kittel mode of the ferromagnetic resonance of the magnetic sphere. This data process is called a probe. A microwave generator, called P, can generate microwaves to drive the magnetic sphere. This setup is a P-probe experiment. When the microwave frequency matches the ferromagnetic resonance, microwaves are absorbed by the magnetic sphere, resulting in a dip in transmission shown by the red curve. Turning on the pump reveals an anticrossing curve in the transmission spectra. The microscopic origin of this phenomenon is still unclear. The anticrossing occurs at the pump frequency, suggesting that the pump effectively drives a mode at the pump frequency, which then interacts with the parametric resonance to form the anticrossing phenomenon. The coupling strengths can be measured by this curve.

The coupling strength between the pump induced mode and the ferromagnetic resonance depends on the pump power. Increasing the power of the pump results in the appearance and increase of the anticrossing gap, as shown in the figure. This phenomenon is roughly scaled to the power of the pump, approximately 14. It only occurs when the applied magnetic field is sufficiently low; if the magnetic field is increased, the phenomenon disappears.

This platform is ideal for realizing the exceptional point. The Hamiltonian for this phenomenon is the effective Hamiltonian. The term α represents the ferromagnetic resonance damping rate, while the term β represents the bandwidth of the pump induced mode. The coupling strength J depends on the pump power, as shown on the last page. The frequency can be tuned significantly by adjusting the power due to this dependence. The frequency of the pump induced mode is a function of the pump frequency, allowing us to identify two modes as the ferromagnetic resonance. When G equals minus β divided by two times the square root of one, the two modes coalesce into one, revealing the exceptional points. The microwave transmission spectrum shows the realization of the exceptional point, where the two modes are connected and sensitivity is enhanced.

Frequency combs involve generating a series of signals with equal frequency differences. For example, if input f1 and f2 are given, the output signal will have a frequency difference of f1 minus f2. The figure demonstrates that the number of tones in the comb is enhanced at the exceptional point due to increased sensitivity to perturbations.

Next, I will explain how exceptional points can be achieved in magnetic heterostructures. For instance, exceptional points have been suggested in measurements of ferromagnet-normal metal and ferromagnet-heterostructures.

The realization of exceptional points in this setup requires fine-tuning parameters, which can be challenging in heterostructures. For instance, adjusting damping and coupling strengths in a single device can be difficult. To address this issue, we suggest that an exceptional point can be consistently achieved in a normal metal ferromagnet and normal metal heterostructure. The magnon mode in the ferromagnet is well comprehended, while the focus here is on understanding the photon mode in the heterostructure.

The ferromagnet has been replaced by an insulator, creating a structure with two normal metals separated by an insulator, resembling a plane-parallel capacitor. The dynamics of the electromagnetic field in this structure are now being analyzed. The electric field can be either in-plane or out-of-plane. Let's first consider the out-of-plane component. The out-of-plane electric field (E_z) causes charge accumulation on the upper and lower normal metal layers, generating a voltage across the insulator. This charge accumulation results in the electric field only existing in the insulator layer, vanishing in the normal metal. As for the in-plane magnetic field, it is continuous at the interface, allowing it to penetrate into the normal metals and drive current within them. The blue curve represents the distribution of the electric field. The total current in one normal layer can be found using the equation: (I = ∫ B \cdot dl), connecting the total current to the magnetic field. The electric field can induce a voltage across the insulator and also generate total current in the normal metal.

The voltage and current dynamics are interdependent and interact with each other through electromagnetic forces.

The change in the electric field generates a magnetic field flux. This relation shows the current I and the (E_x) are proportional to the voltage, establishing the first relation between voltage and current, defining capacitance as (C). Faraday's law states that a change in magnetic flux can create electromotive force, leading to the second relation between voltage and current, defining inductance. Inductance is complex, with (\delta) representing penetration depth in normal metal. Capacitance and inductance can be used to define an effective Hamiltonian for the LC circuit, allowing for the calculation of the frequency of the photon mode, which is inversely proportional to the square root of (L) times (C). The frequency contains both real and imaginary components. This quantum approach can be used to analyze the photon mode in the normal metal ferromagnet and normal metal structure.

The magnon mode in the ferromagnet is coupled with the photon mode, resulting in a non-Hermitian Hamiltonian. The non-Hermitian Hamiltonian is defined by the coupling constants. After the hybridization of the magnon and photon modes, two frequencies are obtained: omega up and omega down. These frequencies are determined by a specific equation. The question arises as to why the square roots can be zero, and if so, we can identify the nodal magnon-photon parameter.

To understand the nodal magnon-photon parameter, we will examine the mode propagating towards the saturated magnetization, which is in the z direction in the figure. The evolution of the real and imaginary components of ω_R and ω_I is shown in the movie as the thickness of the ferromagnetic layer decreases. It is observed that at a certain thickness, both the real and imaginary components coincide.

In this paper, I demonstrate the exact coalescence with a film thickness of around 50 nanometers. Both the real and imaginary components are coalescent. We determine and fix the thickness of the ferromagnetic field.

I searched for a coalescent point when the wave vector changes. This movie demonstrates the evolution of the real and imaginary components of omega up and omega l when the direction of propagation changes. It is observed that for certain propagation directions, both the real and imaginary components coalesce. The exact coalescent points are shown.

In this figure, the real and imaginary components are coalescent with the propagation direction ( k_C ). Let's plot a phase diagram with the X axis representing the thickness of the ferromagnetic layer and the Y axis representing the propagation direction. The red curve shows the coalescence of the two mode frequencies.

When the thickness of the ferromagnet is larger than the critical value dF zero, the exceptional point always appears in some propagation directions. This is the persistent emergence of a nodal magnon-polariton in the heterostructure. In the next part, I will introduce the concept as an example of the non-Hermitian skin effect of magnons. The non-Hermitian effect can be easily understood by the high-dimensional model.

In the high analysis model, the hopping from left to right (tL) and from right to left (tR) differs from tI, making the Hamiltonian non-Hermitian. When tL is greater than tR, modes accumulate at the left edge of the chain. Conversely, when tR is greater than tL, modes accumulate at the right edge. This is a typical example demonstrating how the energy spectrum characterizes non-Hermitian topology, analyzed under periodic boundary conditions. Non-Hermitian effects are evident in the energy spectrum, with the real part on the X-axis and the imaginary part on the Y-axis. The non-Hermitian effect occurs when the spectra form a loop, and the evolution direction of the wave vector in the first Brillouin zone (clockwise or anti-clockwise) corresponds to accumulation at the left or right edge of the chain.

We propose a model to demonstrate polarity in hopping with different TL and Titr in magnonics. We consider a plane where the stray field emitted by magnons propagating normal to the plane's magnetization has chirality. Magnons propagating to the right generate a stray field above the field, while those propagating to the left generate a stray field below the field. This chirality is illustrated in a cartoon figure showing a snapshot of the magnetic moment. When spin waves propagate to the right, the stray field connects the north pole to the south pole, enhancing the field above and canceling it below. The opposite is true for spin waves propagating to the left.

If other magnets are placed above the magnetic field, they interact via dipolar interaction only with spin waves propagating to the right, acting as carriers. This phenomenon is demonstrated in a microwave transmission experiment using two cobalt magnetic wires fabricated above an ETM AO magnetic field. When one wire is excited with microwaves, the emitted microwaves are detected by the other wire.

The signal is a microwave transmission. Transmission S21 means exciting the left wire and detecting at the right one, while transmission S12 means exciting the right wire and detecting at the left one. With a magnetic bias, only transmission at S21 is observed, with no transmission at S12 due to dipolar interaction. The wire only interacts with spin waves propagating to the right, meaning excitation of the left wire will excite the right wire mediated by the spin wave. This indicates that the left wire can influence the right wire, but not vice versa, showing a chiral interaction between the two wires.

If an area of magnetic wire is fabricated on top of the magnetic field, chiral coupling is observed with different coupling constants γ R connecting the left wire to the right wire and γ L connecting the right wire to the left wire. This results in a current in the hopping right, leading to the construction of a hopping model similar to the Hatano-Nelson model with long-range coupling between the two wires.

Spin waves can propagate over long distances, allowing for coupling to remote wires. In this structure, it is observed that when γ_L is greater than γ_R, all states accumulate at the right edge of the area.

When γ_R is smaller than γ_L, all states accumulate at the left edge of the area. This is a non-Hermitian effect of magnons. By calculating the energy spectra under periodic boundary conditions, we can understand the non-Hermitian skin effect.

Constructing the periodic boundary condition for long-range interactions can be complicated. It is necessary to repeat the area for an infinite system and define a large Hamiltonian. Despite these challenges, we were able to find an analytical solution for the energy. When the wave vector evolves from minus π to π, the energy vector forms a closed circle, as shown in a movie. This evolution of the wave vector from minus π to π results in a closed circle, indicating an integer value number and no significant Hermitian effect. In our further exploration, we extended the one-dimensional array to a two-dimensional case with various shaped magnets. We anticipate that the magnon states may accumulate under the edge of the column.

The topology of the states can be characterized topologically by considering the accumulation of edges and corners. This can be done under periodic boundary conditions, where energy is a function of two real wave vectors, κ one and κ two. A winding number and value tuple can be defined in this context. For example, to calculate W one, κ two is fixed and the energy vector is a function of κ one. By evolving the wave vector of κ, W one can be determined in the first Brillouin zone. Similarly, to calculate W two, κ one is fixed and κ two evolves in the first Brillouin zone. This approach allows for the identification of edge accumulation.

The first integer is W one and the second is zero. This is the evolution of the energy spectra when κ two is fixed by κ one. The curve is closed but follows a complicated path, eventually closing. This is the evolution of the energy spectrum when κ one is fixed and κ two is evolving. It is not closed, starting from a terminal point and then returning. This indicates W one is an integer and W two is zero. For the corner accumulation, both W one and W two are integers. The evolution of the spectrum forms closed paths.

Finally, I will briefly mention the anomalous transport enabled by Cato damping, which is a source of non-Hermitian Hamiltonians. When a normal metal is placed near spin waves, the eddy current driven by the dial field from the spin waves causes damping. In a measurement, the NV center in the diamond observed the propagation of the spin wave beneath the normal metal (yellow region) and found that the damping below the normal metal is larger than outside of it. Recently, it was discovered that this damping is chiral.

The damping of spin waves is affected by the current of the strip field. Chiral damping results in larger damping for spin waves propagating to the right and smaller damping for spin waves propagating to the left. This leads to more depletion for waves propagating to the right. However, there is no depletion for waves propagating to the left. This asymmetry has surprising effects on spin transport. In contrast to electrons, where transmission across a barrier is symmetric regardless of direction, spin waves exhibit different behavior depending on their direction of propagation.

In the presence of chiral damping, the transmission of spin waves becomes direction-dependent. Spin waves propagating to the right are totally reflected with no scattering across the barrier. This phenomenon is universal when propagating to the left, using data functions to determine the potential. Assuming chiral damping for spin waves, the transmission is closer to one in one direction. This result pertains to magnon transmission in ferromagnet-normal metal heterostructures, where the potential is created by a local magnetic field. It was found that transmission across the barrier in the minor direction is nearly unity, while transmission in the propagation direction is strongly suppressed.

If we consider the chiral gating effect, it raises the question of whether it is related to Maxwell's demon.

A Maxwell demon is able to control the movement of molecules between two reservoirs, allowing molecules to move from the left reservoir to the right reservoir while preventing movement in the opposite direction. In thermal equilibrium, molecules move randomly in both directions, but more molecules moving to the right results in the right reservoir becoming hotter. This same principle applies to the chiral gating effect, where thermal noise can move from the left reservoir to the right reservoir, but not in the opposite direction, leading to thermal rectification. However, it is important to note that the second law of thermodynamics cannot be violated.

The chirality diverts thermal flow into the gate, demonstrating that the second law of thermodynamics cannot be violated. A model was set up to show this effect in the magnetic field, with the magnetic wire at X and the gate interacting with the magnetic field in a chiral manner when the magnetization is tuned. Complications arise due to the wire also being a magnet, but when the magnetization between the wire and the film is parallel, the coupling is not chiral. However, chirality appears when the magnetization is anti-parallel, leading to asymmetric transmission of spin waves as shown by S parameters S12 and S21.

The surprise effect occurs when the current is turned on, with the spin current injected into the gate depending on the direction of the film magnetization. By considering this spin injection, it is found that the second law of thermodynamics is not violated. This summary confirms that the second law of thermodynamics cannot be violated.

We demonstrated that chirality diverts thermal flow into the gate. A model was set up to show this effect in a magnetic field, with a magnetic wire at X and a gate. The interaction between the magnetic wire and the magnetic field can be tuned to be chiral when the magnetization is considered. Complications arise because the wire is also a magnet, but when the magnetization between the wire and the film is parallel, the coupling is not chiral. This is evident in the symmetric transmission of spin waves from left to right and right to left, as seen in S12 and S21. However, chirality appears when the magnetization is anti-parallel, leading to asymmetric transmission of spin waves, as shown in S21, S12, and S21.

A surprising effect occurs when the current is turned on. The Y-axis represents the spin current injected into the gate, while the X-axis shows the direction of the film magnetization. The magnetization of the magnetic wire is fixed due to strong coupling. In the anti-parallel configuration with current, spin current is injected into the gate, which does not occur when there is no current. By considering this spin injection, we find that the second law of thermodynamics is not violated.