Non-hermitian Systems Enable Real Energy Spectra with Pseudo-Hermitian Magnon Dynamics

The behaviour of energy within physical systems is fundamentally defined by the Hamiltonian, traditionally represented as a Hermitian operator ensuring predictable energy levels. Jamal Berakdar of Martin-Luther Universität Halle-Wittenberg, alongside Xi-guang Wang from Central South University, and their colleagues, investigate a departure from this established principle, exploring the dynamics of ‘pseudo-Hermitian’ systems. These systems, while not strictly Hermitian, can still exhibit real energy spectra despite representing open systems where energy can flow in and out. This research is significant because it details how this deviation from Hermitian physics impacts low-energy excitations , specifically, spin waves known as magnons , within magnetically ordered materials, revealing phenomena such as mode amplification and unconventional energy transfer. The study offers a comprehensive overview of these effects across various magnetic structures, including those with engineered properties, and highlights the potential for manipulating magnon behaviour in novel ways.

Non-Hermitian Magnon Dynamics and Energy Eigenvalues A defining

Specifically, the research focuses on the implications of pseudo-Hermiticity for the stability and temporal evolution of magnon modes, revealing novel behaviours not captured by standard Hermitian treatments. Numerical simulations and analytical calculations demonstrate that the introduction of parity-time (PT) symmetry, a specific form of pseudo-Hermiticity, can lead to exceptional points in the energy spectrum. These exceptional points signify a qualitative change in the system’s response and offer potential for manipulating magnon dynamics in unconventional ways. The findings presented demonstrate that pseudo-Hermitian approaches provide a powerful tool for understanding and controlling spin wave excitations in a broader range of physical scenarios. This framework allows for the investigation of systems where energy is not necessarily conserved, opening avenues for exploring novel functionalities in spintronic devices and magnetic technologies. Ultimately, this research contributes to a deeper understanding of non-equilibrium phenomena in condensed matter physics and expands the possibilities for designing advanced magnetic materials.

Hamiltonian, Pseudo-Hermiticity and Magnetic Excitations

Researchers began their investigation by framing the energy of a physical system through the Hamiltonian, employing a Hermitian operator to ensure a real energy spectrum and complete basis set decomposition. The study then pivoted to explore pseudo-Hermitian systems, acknowledging their open nature while retaining the potential for real energy spectra, though with non-orthogonal eigenmodes, a key distinction from traditional Hermitian physics. This work specifically examines these principles as they apply to low-energy excitations, particularly spin waves and their constituent quanta, magnons, within magnetically ordered materials. To model these magnetic excitations, the team utilized micromagnetic simulations, a continuum approach where magnetic moments are coarse-grained into a magnetization vector field, m(r, t).

This method is valid when investigating phenomena occurring at wavelengths significantly larger than the atomic lattice constant. The temporal evolution of these spin waves is mathematically governed by the Landau-Lifshitz-Gilbert equation, a partial differential equation incorporating material properties, geometry, external fields, and damping mechanisms quantified by the Gilbert damping parameter, α, and the gyromagnetic ratio, γ. The effective magnetic field, Heff, central to this equation, is derived from a functional representing the magnetic energy density, Em, which accounts for exchange energy, Zeeman energy, magnetic anisotropy energy, and magnetostatic interactions. Calculating the demagnetizing field, Hd, a crucial component of Heff, involved integrating volume and surface charge densities using established formulas, effectively capturing the long-range magnetostatic interactions.

Equilibrium magnetization was then determined by minimizing the energy functional, leading to a torque equation. To analyze magnon excitations, the researchers introduced small deviations from the equilibrium state into the LLG equation, resulting in a linear equation of motion describing the frequency and attenuation of these spin waves. This yielded a dispersion relation, ω = (1 −iα)(ωH + ωexk2), where the negative imaginary component signifies attenuation and the real component defines the parabolic dispersion characteristic of exchange-dominated spin waves. Further refinement incorporated the Herrings-Kittle formula to account for dipolar interactions in extended ferromagnetic media, revealing anisotropic behaviour dependent on the wavevector’s orientation.

The study then extended this analysis to confined geometries, specifically thin ferromagnetic films, where boundary effects generate surface charges and modify the dipolar field. A modified dispersion relation, incorporating a film-specific dipolar field approximation, was derived, demonstrating how confinement influences spin wave propagation and ultimately, the material’s magnetic properties. This detailed methodology allows for precise prediction of magnon behaviour in complex magnetic systems.

Magnon Dynamics and Exceptional Points in Pseudo-Hermitian Systems

Scientists are reporting significant advances in the study of pseudo-Hermitian physics, particularly as it applies to the low-energy excitations of magnetically ordered materials. This work focuses on long wavelength spin excitations, known as magnons, and explores systems ranging from ferromagnets to hybrid structures with engineered couplings. Experiments reveal that these pseudo-Hermitian systems, while open, can still exhibit real energy spectra, differing from traditional Hermitian physics through non-orthogonal eigenmodes and unique steady-state behaviours. The research details how these differences manifest in phenomena like mode amplification and non-reciprocal propagation.

The team measured instances of exceptional points (EPs) in various physical systems, demonstrating their potential for enhancing light-matter interaction and nonlinear optical processes. Specifically, saturable gain within a PT-symmetric dimer resulted in intensity-dependent gain-loss balance, leading to non-reciprocal transitions for differing input beam directions. Floquet PT-symmetric photonics, utilising time-periodic driving, achieved asymmetric energy transfer between modes, while the merging of pseudo-Hermiticity with topological band structures enabled the enhancement and protection of topological edge states through gain-loss arrangements. Data shows that a photonic Su, Schrieffer, Heeger chain with alternating gain and loss maintained robust topological interface states even while attenuating continuum modes.

In mechanical systems, researchers achieved pseudo-Hermitian behaviour by coupling oscillators with active gain and dissipative loss elements. Carl Bender and colleagues demonstrated this with coupled pendula, one driven by a motor and the other damped by a brake, observing a characteristic PT phase transition and EP by adjusting the gain/loss levels. Acoustic experiments, overcoming the challenge of achieving acoustic gain, employed piezoelectric elements to create negative resistance, resulting in an acoustic sensor that perfectly absorbed incoming sound without reflection at a specific frequency. Electrical circuits provided a straightforward platform for simulating pseudo-Hermitian Hamiltonians, utilising active and passive components in geometrically symmetric configurations.

Schindler et al. demonstrated PT symmetry in coupled RLC resonant circuits, one with a negative resistor for gain and the other with a positive resistor for loss. Measurements confirm the dynamic control of pseudo-Hermitian phase transitions through electrical parameter modulation, and the emulation of higher-order EPs by cascading resonant LC cells with alternating gain and loss. Furthermore, anti-PT symmetric RLC resonators were shown to exhibit EP transitions conserving the energy difference between the two resonators. Metamaterials, engineered for unique optical features, proved to be an ideal platform for realising and controlling non-Hermitian phenomena. Researchers incorporated gain and loss elements, active components, and symmetries within the metamaterial structure to achieve pseudo-Hermiticity, enabling functionalities such as unidirectional wave propagation, enhanced sensing, and unconventional laser sources. These findings demonstrate the versatility of pseudo-Hermitian physics across diverse physical domains, including photonic, acoustic, and superconducting systems.

Magnon Control via Pseudo-Hermitian Physics

This work reviews recent advances in pseudo-Hermitian physics as applied to the study of low-energy excitations, specifically magnons, in magnetically ordered materials. Researchers have demonstrated that incorporating pseudo-Hermitian concepts allows for novel manipulation of magnonic systems through external fields and environmental effects, effectively controlling gain and loss in magnon amplitudes. These investigations reveal phenomena such as mode amplification, non-reciprocal propagation, and even magnon cloaking, highlighting the unique behaviours arising from this non-Hermitian approach. The study establishes magnetic and magnonic platforms as intrinsically non-linear systems, with the potential for controllable nonlinearity via external fields, and spanning a broad frequency range from THz to GHz. While acknowledging challenges in creating magnonic systems with precisely engineered damping and gain, the authors suggest future research should focus on integrating low-damping materials with those offering strong spin-orbit torque. Further exploration of nonlinear regimes, including multiple magnon-magnon scattering, and expansion into two and three-dimensional magnonic crystals, coupled with plasmonic and metamaterials, are also proposed as promising avenues for investigation.