Advances in Phonons: From Band Topology to Phonon Chirality

TL;DR

This review surveys the emergence of topology and chirality in phonons, detailing how Berry-phase and Chern-number concepts extend from fermionic systems to bosonic lattice vibrations. It contrasts phonon dynamics with electronic motion, outlines the harmonic framework and experimental probes, and then develops a comprehensive topological phonon taxonomy that includes gapped phases (Chern and Stiefel–Whitney insulators) and gapless excitations (Dirac/Weyl points, nodal lines). It connects topological invariants to observable surface states and to circular polarization via phonon angular momentum and pseudo-angular momentum, illustrating the deep link between topology and chirality in phonons through concrete 1D/2D models and 3D bulk materials. The review also highlights experimental realizations (IXS, INS, EELS, RIXS) and the growing database of topological phonon materials, while outlining open questions on phonon–electron coupling, quantum geometry, and potential quantum-technological applications. Overall, topological and circularly polarized phonons offer a rich platform for robust phononic transport, engineered surface modes, and spectroscopic diagnostics in crystalline materials.

Abstract

Phonons are ubiquitous quasiparticles in solid state systems describing the quantized vibrations of a crystal lattice. Phonons play a central role in a wide range of physical phenomena, from transport to symmetry-breaking orders, such as charge density waves and superconductivity. {Traditionally treated as spin-0 bosons that obey Bose-Einstein statistics,} phonons have recently emerged as a fertile ground for exploring topological physics, spurred by the rapid development of topological band theory initially formulated for fermionic systems. {It is now understood that the phonon eigenstates, characterized by their eigenvalues and eigenvectors, can carry nontrivial topological invariants, including the Berry phase and Chern number. This new understanding opens up avenues to investigate the interplay between lattice dynamics, topology, and chirality in bosonic systems. In this article, we review recent theoretical and experimental advances in the field of topological phonons and circularly polarized phonons. We introduce foundational concepts, including the classification of phononic band structures, symmetry-protected topological phases, and the definition of topological invariants in bosonic systems. We emphasize the concept of phonon angular momentum and its fundamental connection to Weyl phonons in $\mathcal{PT}$-breaking systems. Key experimental progresses on topological and circularly polarized phonons are discussed. We also outline outstanding challenges and promising directions for future research, such as the role of topology in phonon-mediated quasiparticle interactions and the manipulation of phonon angular momentum for potential applications in quantum technologies.}

Figures (39)

• Figure 1: Comparison bewteen electrons and phonons. Electrons have multiple degrees of freedom, such as charge, spin, orbital, etc. Thus, electrons can be modulated by external fields (e.g., electric fields and magnetic fields) and thereby realizing relevant applications like spin devices. However, phonons are charge neutral, and are traditionally considered to be spin zero and orbital free. Therefore, new degrees of freedom are needed to be introduced to effectively modulate relevant physical properties.
• Figure 2: Schematics of photon scattering processes. (a) depicts the IR spectroscopy. When the incident photon energy, $\hbar\omega$, matches the phonon excitation energy, $E_{ph}$, the incident light will be absorbed, resulting reduced reflection or transmission light intensity. (b) shows the Raman scattering process. The incident photon drives the initial state to a virtual excited state. The system then relax to a final state and emit a photon. Following the energy and momentum conservation, $E_{ph}=\hbar\omega-\hbar\omega'$ and $\boldsymbol{Q}_{ph}=\hbar\boldsymbol{k}-\hbar\boldsymbol{k}'$, where $\hbar\boldsymbol{k}$ and $\hbar\boldsymbol{k}'$ are incident and scattered photon momentum. (c) illustrates the direct RIXS process. Similar to the Raman scattering, RIXS is also a scattering process. The incident photon energy is tuned to a element specific resonant energy. The core electron is excited to an intermediate state. The core hole is then filled by an electron from the conduction band.
• Figure 3: Illustration of the integral curved surface for the Chern number. (a) Depicts a typical Brillouin zone for a two-dimensional system. (b) and (c) Show two configurations of integrated surfaces that are topologically equivalent to the BZ in (a). This equivalence arises due to the periodic boundary conditions of the Brillouin zone, allowing for alternative representations of the integration domain. (d) Illustrates the integrated surface used to calculate the Chern number for Weyl phonons. This surface is typically a sphere enclosing the Weyl phonon, capturing the topological charge associated with it. (e) and (f) display the Berry curvature distributions for Weyl points with Chern numbers $C$ = +1 and $-1$, respectively. The integration of the Berry curvature over the enclosing surface yields the Chern number, which characterizes the topological nature of the Weyl phonon. (e) and (f) are adapted from Ref. weng2015weyl
• Figure 4: (a) and (b) depict the massive and massless Dirac phonon spectra, respectively. In both cases, the Chern number is zero, indicating the absence of nontrivial topological charge. (c) Weyl phonons with opposite Chern number and chirality. The massless Dirac pononon is composed of two Weyl phonons with $C=\pm1$. (d) Topological surface states of Weyl phonons are in helicoid shapes. These surface states share the same chirality as the Chern numbers of the corresponding Weyl phonons, highlighting the direct connection between bulk topology and surface topology.
• Figure 5: (a) Surface states and surface arcs for Weyl phonons. The Chern number of Weyl phonons can be calculated either on a sphere enclosing the Weyl phonons or the 2D colored plane shown in (a1). In (a2), when two Weyl phonons with opposite chirality project onto different momenta in the surface Brillouin zone (BZ), a surface arc connecting the two Weyl phonons emerges. (a3) depicts the topological surface states linking two Weyl phonons, which exhibit different chiralities near each Weyl phonon. In (a4), when two Weyl phonons with opposite chirality project onto the same momentum in the surface BZ, the surface arc disappears. (b1-b3) Topological surface states for Weyl phonons with Chern number of $C$ = +1, +2 and +4, respectively, all of them are in helicoid/spiral shapes. (a) is adapted from Ref. zhang2022z, (b1)-(b3) are adapted from Ref. zhang2023parallel.