Theory of elastic continuum

Acoustic waves in crystals can be well described by the elastic continuum theory. The elastic continuum theory^33,36 is an excellent approximation in the long wavelength limit, where the variation in the displacement u occurs over a length scale of 10⁻⁶ cm or larger for typical crystals. When a crystal is time-reversal symmetric, the elastic continuum is described by the Lagrangian density, L[u,u˙]=T[u˙]−U[u], where T[u˙] and U[u] denote the kinetic energy density and elastic energy density, respectively. The kinetic energy is given by T[u˙]=12ρu˙2 where ρ is the mass density. The elastic energy density is proportional to the square of strain tensor uij=12(∂iuj+∂jui) where ∂_i = ∂/∂x_i: U[u]=12λijlkuijukl, where λ_ijl is the elastic modulus tensor. Then, the equation of motion is given by ρu¨i=∂j(λijklukl). Fourier transformation of this equation yields D(k)ijuj(k)=ωk2ui(k), which is nothing but the eigenvalue equation for the phonon spectrum ω_k. (Further details on the elastic continuum can be found in Supplementary Note ³.) Here, the dynamical matrix D(k)_ij is defined as D(k)_ij = ρ⁻¹λ_iljmk_lk_m and it is referred to as the “Hamiltonian” H_k in the main text.

From this dynamical matrix, we see that for ATPs formed by acoustic phonons in a crystal with time-reversal symmetry, the topological charge q can be defined for both centrosymmetric and non-centrosymmetric elastic crystals as long as the elastic continuum limit is considered. Recall that a necessary condition to define q is that the Hamiltonian must be a real-symmetric 3 × 3 matrix, where the reality condition can be satisfied if there is PT symmetry. Since we are assuming that T is a symmetry of the crystal, the above statement holds if P is a symmetry of the elastic continuum Hamiltonian, and this is precisely the case even for non-centrosymmetric crystals.

Now, let us explain why this statement is true, since it is essential for characterizing the acoustic phonons with q. First, notice that under P, we have u(k) → −u(−k). Then, the constraint on the D(k) from P is D(k) = D(−k), which is obviously true because D(k) is quadratic in k, even in non-centrosymmetric crystals. Consequently, PT is a symmetry of the elastic continuum theory, and this allows us to define q. Notice that this argument is true whether or not the crystal is centrosymmetric or non-centrosymmetric. On the other hand, P-breaking terms are allowed when we consider terms that are higher-order in k. Nevertheless, such higher-order terms are negligible for the description of acoustic phonons with long wavelengths. To conclude, the topological charge q can be defined as local property of acoustic phonons in time-reversal symmetric crystals.

Next, let us comment if T is broken in the phonon Hamiltonian, it may not be restored in the elastic continuum limit, which is in contrast to the behavior of P. Typically, the time-reversal breaking terms in the phonon Hamiltonian is modeled by terms such as the Raman spin-phonon coupling⁶⁴, whose leading contribution is constant in k, and the Mead–Truhlar term in the Born–Oppenheimer approximation^65,66, whose leading contribution is quadratic in k. Because these terms do not vanish in the elastic continuum limit, T is not restored in the elastic continuum limit.

To summarize, P (and therefore PT) is a symmetry of the elastic continuum Hamiltonian in T-symmetric crystals, but T symmetry is generally not restored as a symmetry in the elastic continuum Hamiltonian in T-broken crystals.