Quantum Geometry and Topology of Bulk Plasmons in Weyl Metals

Technical Summary: Quantum Geometry and Topology of Bulk Plasmons in Weyl Metals

Problem Statement
Plasmons, traditionally described as semiclassical collective particle-hole excitations of the Fermi surface, possess an internal quantum structure arising from the quantum geometry of the Hilbert space. While quantum geometric effects have been identified in 2D topological insulators (producing plasmon Berry dipoles) and flat-band systems, bulk plasmons in Weyl metals have not been recognized as intrinsically topological objects. Furthermore, in known Weyl metals, the spectral weight is dominated by non-topological Fermi surfaces, making the experimental isolation of Weyl fermion plasmons challenging. This work addresses the nature of plasmons in topologically nontrivial Fermi surfaces enclosing a monopole charge (Chern number $C_W$), specifically investigating whether these plasmons encode the quantum geometry and topology of the underlying fermions.

Methodology
The authors model a single-valley Weyl metal using a Hamiltonian comprising a linear Weyl dispersion and a normal-ordered Coulomb interaction. The analysis proceeds through the following steps:

1. Formulation in Energy Basis: The fermion operators are transformed into the energy basis (conduction and valence bands, $\alpha = \pm$). The plasmon state is defined as a superposition of electron-hole pairs with center-of-mass (COM) momentum $\mathbf{Q}$ and relative momentum $\mathbf{q}$.
2. Bethe-Salpeter Equation: The plasmon dispersion and envelope function are determined by solving the Bethe-Salpeter equation at the Random Phase Approximation (RPA) level. This involves finding the zeroes of the inverse retarded correlation function of electron-hole pair operators.
3. Expansion of Form Factors: The form factor matrix, which encodes the quantum geometry of the Weyl fermions, is expanded to second order in the COM momentum $\mathbf{Q}$. This expansion reveals the role of the Berry connection tensor ($\hat{A}$) and the quantum metric tensor ($g$).
4. Topological Classification: The envelope function components are analyzed using the theory of spin-weighted spherical harmonics and monopole spherical harmonics (MSH). This allows for the classification of the "vorticity" ($\zeta$) of the plasmon wavefunction in relative momentum space.
5. Optical Response: The coupling between the plasmon modes and an external linearly polarized light field is calculated using linear response theory. The optical selection rules are derived by evaluating the plasmon effective electric dipole moment.

Key Contributions and Results

• Topological Nature of the Envelope Function: The study demonstrates that while intraband plasmon components exhibit a nodal plane orthogonal to the COM momentum (vorticity $\zeta = 0$), the interband components of the envelope function possess a distinct monopole structure. These interband components exhibit a finite vorticity in the relative momentum coordinates given by $\zeta = 2C_W$, where $C_W$ is the Chern number of the Fermi surface enclosing the Weyl point. Specifically, for a single Weyl node, the vorticity is $\zeta = \pm 2$.
• Plasmon Dispersion: The plasmon frequency $\omega_Q$ in the limit $Q \to 0$ is derived, including corrections from interband processes. The interband processes introduce a logarithmic red-shift to the plasmon frequency, proportional to $\ln(\Lambda/k_F)$, where $\Lambda$ is the ultraviolet cutoff. The authors note that for small Fermi surfaces, this correction can suppress plasmon formation if the Fermi momentum falls below a critical value.
• Quantum Geometric Dipole: A central result is the identification of a finite effective electric dipole moment ($\mathbf{d}$) for these topological plasmons. This dipole moment arises from the quantum geometry (specifically the interband Berry connection) and points strictly along the direction of the plasmon COM momentum ($\hat{\mathbf{Q}}$).
• Optical Selection Rules: Due to the alignment of the dipole moment with $\hat{\mathbf{Q}}$, the plasmons exhibit a unique optical selection rule: they selectively couple to light linearly polarized parallel to the plasmon momentum ($\mathbf{E} \cdot \hat{\mathbf{Q}} \neq 0$) and are transparent to light polarized perpendicularly. This contrasts sharply with conventional plasmons, which couple equally to light in any polarization direction.
• Symmetry Breaking and Nonlinear Effects: The paper suggests that in multivalley Weyl metals, the hybridization of valley-resolved density oscillations can lead to collective modes. If global inversion symmetry is broken, these modes may contribute to second-harmonic generation and DC photocurrents, although the detailed theory of these nonlinear currents is reserved for future work.

Significance and Claims
The authors claim that this work establishes bulk plasmons in Weyl metals as intrinsically topological objects that encode the quantum geometry of the fermion bands. The primary significance lies in the prediction of distinctive optical properties—specifically the anisotropic coupling to linearly polarized light and the existence of a quantum geometric dipole—that differentiate topological plasmons from conventional ones. The paper posits that these unique properties may provide a pathway for the first experimental observation of Weyl fermion plasmons, which have previously been obscured by the dominance of non-topological Fermi surfaces in known materials. The work does not propose specific experimental setups or materials but suggests that the derived selection rules and nonlinear optical responses (like second-harmonic generation) serve as potential signatures for detection.