Topological Quantities and Properties of Phonons
Topological Geometry
Topology studies geometric properties that remain invariant under continuous transformations, reflecting global characteristics.
For example, the Gaussian curvature $K$ of a surface describes local geometric properties and is related to global topological characteristics through the Gauss-Bonnet theorem:
where $\text{genus}$ represents the number of holes in a geometric object. For example, the genus of a sphere is $0$, while the genus of a torus (like a donut) is $1$.
Topological Physics
Topological physics primarily investigates the topological properties in momentum space. In periodic lattices, electronic states can be described by the time-independent Schrödinger equation:
\[H|\varphi\rangle = E|\varphi\rangle,\]where $H$ is the Hamiltonian of the system, $E$ is the energy eigenvalue of the electron, and $\vert\varphi\rangle$ is the corresponding eigenstate. By solving this equation, one can obtain the band structure, i.e., the relationship between energy and momentum of electrons.
Traditional band theory focuses on the energy eigenvalues of electrons, while the significant contribution of topological physics lies in uncovering the hidden topological information within wavefunctions, i.e., the topology of electronic wavefunctions. For example, in the quantum Hall effect, the time-reversal symmetry of the system is broken, and the Chern number \(C \neq 0\). Based on different symmetries, researchers have developed many topological material systems, such as the valley Hall effect, topological crystalline insulators, and gapless topological semimetals, including Dirac states, Weyl states, nodal lines, and triple degeneracy points.
Topological Phonons
Phonons, lacking charge and spin properties, entered topological research relatively late. Traditionally, phonon studies primarily focused on their particle-like properties (dispersion and scattering) and wave-like properties (coherence effects). Inspired by the analogy between phonons and electrons, researchers have recently observed novel transport phenomena in phononic systems, such as the phonon thermal Hall effect.
In phononic systems, breaking time-reversal symmetry is challenging. For crystals with time-reversal symmetry, atomic vibrations can be described by the equations of motion based on Newton’s laws. Under the harmonic approximation, the phonon characteristic equation can be written as:
\[\boldsymbol{D}(\boldsymbol{k}) \boldsymbol{u}(\boldsymbol{k}) = \omega^2(\boldsymbol{k}) \boldsymbol{u}(\boldsymbol{k}),\]where $\boldsymbol{u}(\boldsymbol{k})$ is the phonon eigenstate, and $\omega(\boldsymbol{k})$ is the phonon frequency. Traditional studies only focus on eigenvalues, yielding phonon dispersion relations and density of states, while topological phonon research further explores the hidden topological information within eigenstates.
Topological Quantities in Phononic Systems
Using the wavefunctions of phonon eigenstates, topological quantities can be defined:
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Berry Connection:
\[A_{s,\boldsymbol{k}} = \text{i} \langle u_{s\boldsymbol{k}} | \nabla_{\boldsymbol{k}} | u_{s\boldsymbol{k}} \rangle,\]describing the geometric properties of wavefunctions in momentum space.
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Berry Curvature:
\[\Omega_{s,\boldsymbol{k}} = \nabla_{\boldsymbol{k}} \times A_{s,\boldsymbol{k}},\]the curl of the Berry connection, characterizing the local topological structure of wavefunctions in momentum space.
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Berry Phase:
\[\gamma_s = \int_L A \cdot d\boldsymbol{k},\]the integral of the Berry connection along a specific path or loop.
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Chern Number:
\[C = \frac{1}{2\pi} \oint \Omega_{s,\boldsymbol{k}} d^2\boldsymbol{k}.\]
In three-dimensional systems, the Chern number is defined as the integral over a closed surface in momentum space:
Since phonons do not have spin $1/2$ properties, the theory of quantum spin Hall states cannot be directly applied. Quantum spin Hall states in electronic systems can be seen as two quantum Hall states with opposite Chern numbers, resulting in a total Chern number of $0$ under time-reversal symmetry. In phononic systems, lattice symmetry can be used to achieve double or multiple degeneracies, constructing pseudo-spin-like properties and providing new approaches to topological phonon research.
In classical phonon transport research, lattice symmetry is used to analyze phonon scattering channels and simplify force constant calculations. From the perspective of topological phonons, lattice symmetry is primarily used to determine phonon degeneracy and classify phonon branches.
At the boundaries or interfaces of materials, topological phonons can form special boundary states. These boundary states originate from inherent topological invariants of materials (such as Chern numbers or Berry curvature) and can propagate along scattering-free channels. This stability is known as the topological protection mechanism.
The dispersion relations of topological phonons also exhibit unique characteristics. In certain frequency ranges, phonon propagation is more efficient, while in other ranges, phonons may be suppressed.