Topological Nature of the Phonon Hall Effect

We provide a topological understanding of the phonon Hall effect in dielectrics with Raman spin-phonon coupling. A general expression for phonon Hall conductivity is obtained in terms of the Berry curvature of band structures. We find a nonmonotonic behavior of phonon Hall conductivity as a function of the magnetic field. Moreover, we observe a phase transition in the phonon Hall effect, which corresponds to the sudden change of band topology, characterized by the altering of integer Chern numbers. This can be explained by touching and splitting of phonon bands.

Figure 1

(a) Phonon Hall conductivity vs magnetic field for different temperatures. The dotted, dashed, and solid lines correspond to $T=50$, 100, and 300 K, respectively. The inset shows $h$ dependence of ${\kappa }_{xy}$ at low temperatures: $T=10$ (solid line), 20 (dashed line), and 40 K (dotted line). (b) $d{\kappa }_{xy}/dh$ as a function of $h$ at different temperatures: $T=50$ (dotted line), 100 (dashed line), and 300 K (solid line); here $N_L=400$. The inset in (b) shows the $h$ dependence of $d{\kappa }_{xy}/dh$ for different sizes $N_L$ at $T=50\mathrm{K}$, around $h\approx 25.5\mathrm{rad}/\mathrm{ps}$; from top to bottom, $N_L=80$, 320, and 1280, respectively.

Figure 2

(a)–(d) The contour map of Berry curvatures for bands 1–4 at $h_{c-}=h_c-{10}^{-2}\mathrm{rad}/\mathrm{ps}$; (e)–(h) The contour map of Berry curvatures for bands 1–4 at $h_{c+}=h_c+{10}^{-2}\mathrm{rad}/\mathrm{ps}$. For (a)–(h), the horizontal and vertical axes correspond to wave vector $k_x$ and $k_y$, respectively. (i) $\Omega$ at different magnetic fields. The solid and dashed lines correspond to ${\Omega }^2$ and ${\Omega }^3$ at $h_{c-}$, respectively, while dotted and dash-dotted lines correspond to those at $h_{c+}$. (j) Chern numbers of four bands: $C^1$ (solid line), $C^2$ (dashed line), $C^3$ (dotted line), and $C^4$ (dash-dotted line). (k) The dispersion relation of bands 2 and 3 at different magnetic fields in the vicinity of $h_c$. The dashed, solid, and dotted lines correspond to the bands at $h_{c-}$, $h_c$, and $h_{c+}$, respectively. The lower three and upper three correspond to bands 2 and 3, respectively. $k_y=0$ in (i) and (k).