🔬 mesoscale physics

Magnons and excitons from Bethe-Salpeter equation and its application to graphene-based magnets

This paper presents a unified Bethe-Salpeter equation framework beyond the Tamm-Dancoff approximation that accurately computes both excitons and gapless magnons in graphene-based magnets, resolving previous theoretical inconsistencies regarding Goldstone modes and clarifying the roles of on-site and non-local interactions.

Original authors: Simran Chourasia, Maurício F. C. Martins Quintela, J. J. Palacios

Published 2026-10-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Simran Chourasia, Maurício F. C. Martins Quintela, J. J. Palacios

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of solid materials, electrons are not merely solitary travelers; they are social creatures that constantly interact, repel, and attract one another. When these interactions are strong, the electrons can organize themselves into collective states that behave like new, distinct particles. Two of the most important of these collective behaviors are excitons and magnons. An exciton is a bound pair formed when an electron jumps up to a higher energy level, leaving behind a vacancy, or "hole," that it is attracted to. It is a spin-conserving dance where the electron and hole stay together, creating a neutral package that can carry energy without carrying a net electric charge. Magnons, on the other hand, are ripples of magnetism. They occur when the spin of an electron flips, sending a wave of magnetic disturbance through the material. While excitons are crucial for understanding how materials absorb and emit light, magnons are the key to understanding magnetic properties and how information might be stored and moved in future magnetic devices. For decades, scientists have had powerful tools to calculate the behavior of excitons, but their methods for calculating magnons have been less reliable, often producing results that contradict the fundamental laws of physics regarding how magnets should behave at very low energies.

A team of researchers has now bridged this gap, demonstrating a way to calculate both excitons and magnons using the same mathematical framework while ensuring the results obey the strict rules of magnetism. The study focuses on a specific, highly controllable system: graphene, the single layer of carbon atoms that is famous for its strength and conductivity, but which is normally non-magnetic. By introducing hydrogen atoms onto the graphene surface, the researchers created a scenario where the material becomes magnetic, acting as a perfect testbed for their new method. The core of their work involves solving a complex equation known as the Bethe-Salpeter equation, which describes how particles interact. Historically, scientists have used a simplified version of this equation that ignores certain interactions to make the math easier. However, the authors found that this simplification creates a fatal flaw when studying magnets: it predicts that it takes a specific amount of energy to create a magnetic ripple, even when the laws of physics say it should take zero energy. This artificial energy gap is a mathematical error that hides the true nature of the material.

To fix this, the researchers applied the full, unsimplified version of the equation to their model of magnetic graphene. They worked with two specific types of magnetic order: ferromagnetism, where all the atomic spins point in the same direction, and antiferromagnetism, where neighboring spins point in opposite directions. In their simulations, they observed that when they used the simplified method, the magnetic ripples appeared with a false energy gap, making the material seem more rigid than it truly is. When they switched to the full equation, which includes the coupling between different types of particle movements, this artificial gap vanished completely. The magnetic ripples now correctly showed zero energy at the start, exactly as the Goldstone theorem—a fundamental principle in physics—demands. This theorem states that whenever a continuous symmetry is broken, such as when a material chooses a specific magnetic direction, a massless mode must appear. The researchers showed that only by including the full complexity of the interactions could they recover this zero-energy mode.

The study also revealed how different types of forces within the material create these particles. They found that the formation of excitons relies heavily on interactions between electrons that are not right next to each other, while the formation of magnons depends on interactions between electrons on the same atom. By turning these specific interactions on and off in their computer models, they could watch excitons disappear while magnons remained, or vice versa, confirming exactly which forces are responsible for which phenomenon. In the case of the ferromagnetic graphene, the corrected magnetic ripples showed a quadratic dispersion, meaning their energy increased with the square of their momentum, while the antiferromagnetic version showed a linear dispersion, where energy increases directly with momentum. These distinct behaviors were only visible when the full equation was used; the simplified version failed to capture the linear nature of the antiferromagnetic ripples entirely, mistaking them for a different type of wave.

The implications of this work extend beyond just understanding graphene. The researchers demonstrated that their approach is robust enough to handle the complex, real-world interactions found in two-dimensional magnetic materials. They confirmed that for excitons, the simplified method remains a very good approximation, producing results nearly identical to the full calculation. However, for magnons, the full treatment is non-negotiable if one wants to understand the true, gapless nature of magnetic waves. This distinction is vital for the future design of spintronic devices, which rely on the manipulation of electron spins rather than just their charge. By proving that these two very different quasiparticles can be described within a single, consistent framework without violating fundamental physical laws, the team has provided a clearer path for theorists and experimentalists alike. Their work ensures that when scientists model new magnetic materials, they are not looking at a distorted image created by mathematical shortcuts, but at the genuine, gapless behavior of the magnetic world.

Technical Summary: Magnons and Excitons from Bethe-Salpeter Equation and its Application to Graphene-based Magnets

Problem Statement
The theoretical study of two-dimensional semiconducting magnetic materials requires a unified framework to describe both spin-conserving electron-hole bound pairs (excitons) and spin-flip collective excitations (magnons). While the Bethe-Salpeter equation (BSE), combined with Hedin's equations, is the standard formalism for calculating excitons, its application to magnons has been historically problematic. Specifically, standard approximations—particularly the Tamm-Dancoff approximation (TDA)—often fail to reproduce the gapless Goldstone mode required by Goldstone's theorem for systems with broken continuous symmetry. Instead, these methods frequently yield a spurious gapped Goldstone mode. Previous attempts to resolve this using many-body perturbation theory (e.g., BSE/GW) have been hindered by inconsistencies between dynamical screening in quasiparticle self-energies and static screening in the interaction kernel.

Methodology
The authors propose a unified approach to compute both excitons and magnons within the same Bethe-Salpeter formalism by strictly adhering to the equation-of-motion (EOM) derivation and the Goldstone theorem. The methodology involves:

Model Hamiltonian: The study utilizes the extended Hubbard model applied to hydrogenated graphene (representing both ferromagnetic and antiferromagnetic phases). This model includes on-site ( $U$ ) and non-local ( $J$ ) interactions.
Mean-Field Reference: Calculations begin with an unrestricted Hartree-Fock (UHF) ground state. The authors acknowledge that UHF states typically suffer from spin contamination (where $S^2$ is not a good quantum number), which is a known source of errors in excitation spectra.
Full BSE vs. TDA: The core methodological shift is the evaluation of the full Bethe-Salpeter equation (BSE/RPA) rather than the Tamm-Dancoff approximation (BSE/TDA).
- BSE/TDA: Considers only the resonant block ( $H^{(r,r)}$ ) of the Hamiltonian, neglecting anti-resonant excitations and their coupling.
- BSE/RPA: Includes the resonant block, the anti-resonant block ( $H^{(a,a)}$ ), and the coupling block ( $H^{(r,a)}$ ).
Consistency: The interaction kernel used in the BSE is derived from the exact same interaction terms used to generate the UHF ground state, ensuring consistency between the mean-field and the excitation calculation.

Key Contributions

Resolution of the Goldstone Gap: The paper demonstrates that the spurious gap in the magnon Goldstone mode arises from the truncation of the BSE to the TDA. By including the anti-resonant block and its coupling to the resonant block (BSE/RPA), the exact zero-energy Goldstone mode is recovered, even when the reference UHF ground state possesses spin contamination.
Unified Framework: The authors establish that excitons and magnons can be computed on the same footing within the BSE formalism without requiring ad-hoc corrections (such as manual "scissor cuts") to enforce gaplessness.
Interaction Analysis: The work dissects the roles of on-site ( $U$ $U$ ) and non-local ( $J$ $J$ ) interactions. It identifies that:
- Excitons (spin-conserving) rely on non-local interactions for binding. In the standard Hubbard model (purely on-site $U$ ), the direct term in the kernel vanishes, preventing exciton formation; non-local interactions are required to generate the binding direct term.
- Magnons (spin-flip) rely on on-site interactions for binding. For spin-flip excitations, the exchange term in the kernel vanishes, and the binding is driven by the direct term, which is non-zero only when on-site interactions are present. Consequently, magnons form in the standard Hubbard model but disappear if only non-local interactions are considered.

Results
The methodology was applied to ferromagnetic and antiferromagnetic hydrogenated graphene:

Ferromagnetic Graphene:
- BSE/TDA: Produced a gapped Goldstone mode (gap $\approx 0.14$ eV at $\Gamma$ ) and quadratic dispersion.
- BSE/RPA: Successfully recovered the exact zero-energy Goldstone mode with the expected quadratic dispersion ( $\hbar\omega \propto |Q|^2$ ).
- Excitons: The inclusion of anti-resonant terms had a negligible effect on the excitonic dispersion and linear optical conductivity, confirming that TDA is sufficient for excitons in these systems.
Antiferromagnetic Graphene:
- BSE/TDA: Failed to capture the correct physics, yielding a gapped mode with quadratic-like dispersion, which is qualitatively incorrect for antiferromagnets.
- BSE/RPA: Correctly reproduced two exact zero-energy modes at $\Gamma$ with linear dispersion ( $\hbar\omega \propto |Q|$ ), consistent with the type-A Goldstone modes expected for antiferromagnets.
Optical Properties: The linear optical conductivity calculations showed that while magnons are optically dark, excitons exhibit spin-selective bright transitions. The distinction between TDA and RPA results was significant for magnons but minimal for excitons.

Significance
The paper claims that the primary significance of this work lies in demonstrating that the gapless Goldstone mode for magnons can be rigorously recovered within the BSE formalism by going beyond the Tamm-Dancoff approximation. This resolves a long-standing inconsistency in theoretical descriptions of magnetic excitations. The authors note that while their results are derived using model Hamiltonians, the logical extension to first-principles calculations (e.g., BSE/GW) is clear, provided that the interaction kernel and the ground state are treated consistently. The work clarifies the specific roles of local and non-local interactions in the formation of these quasiparticles and identifies the consistency requirements necessary for extending this approach to ab initio studies of 2D magnets.

More like this