Topological Magnons with Nodal-Line and Triple-Point Degeneracies: Implications for Thermal Hall Effect in Pyrochlore Iridates

We analyze the magnon excitations in pyrochlore iridates with all-in-all-out (AIAO) antiferromagnetic order, focusing on their topological features. We identify the magnetic point group symmetries that protect the nodal-line band crossings and triple-point degeneracies that dominate the Berry curvature. We find three distinct regimes of magnon band topology, as a function of the ratio of Dzyaloshinskii-Moriya interaction to the antiferromagnetic exchange. We show how the thermal Hall response provides a unique probe of the topological magnon band structure in AIAO systems.

Figure 1

(a) Spin configuration of the AIAO state on the pyrochlore lattice. (b) Nearest-neighbor DM vectors (blue), ${\widehat{d}}_{14}={\widehat{d}}_{{14}^{\prime }}=(-1,1,0)/\sqrt{2}$. The unit-length DM vectors $\{{\widehat{d}}_{ij}\}$ are determined by lattice symmetry [91]. The numbers (1–4) denote the four sublattices of the pyrochlore lattice. (c) Three regimes (I,II,III) of magnon band topology separated by transition points $D_{t1}$ and $D_{t2}$, showing magnon triple points (blue: $A$ type, red: $B$ type) in the $k_z=0$ plane of the Brillouin zone [Fig. 2]. Black arrows indicate the directions in which the triple points move with increasing $D$.

Figure 2

Magnon band structures for (a) $D=0.18J$, (b) $0.33J$, and (c) $0.8J$. Two types of the triply degenerate band crossings are marked by blue ($A$-type TDC) and red ($B$-type TDC) dots. The cyan line denotes the doubly degenerate band protected by the $\mathrm{\Theta }·S_4$ antiunitary symmetry. The other doubly degenerate bands along $XW$ (orange) correspond to the nodal-line band crossings protected by the nonsymmorphic $G_d$ glide and $C_2$ rotation. (d) Brillouin zone of the pyrochlore lattice with high symmetry points [$\mathrm{\Gamma }:(0,0,0)$, $X:(\pi /2,0,0)$, $W:(\pi /2,\pi /4,0)$, $L:(\pi /4,\pi /4,\pi /4)$]. (e)–(f) Locations of the triple points (e) and nodal lines (f) in the Brillouin zone, with the same color code as in (a)–(c). The table summarizes the associated symmetries protecting the band crossings. See the Supplemental Material [93] for the definitions of the symmetry operations.

Figure 3

Thermal Hall conductivity ${\kappa }_{\mu \nu }$ under a small magnetic field $h=0.02J$ along the [110] direction. (a),(b) ${\kappa }_{xy}$ as a function of $T$ and $D$. The vertical line in (b) corresponds to $D_{t1}$. (c),(d) ${\kappa }_{xy}$ (red) and ${\kappa }_{yz}={\kappa }_{zx}$ (blue) for $D=0.18J$ and $D=0.33J$. The arrow in (c) indicates the crossing between ${\kappa }_{xy}$ and ${\kappa }_{yz}$ for $k_{B}T\approx D$.

Figure 4

Momentum-resolved thermal Hall conductivity $K_{xy}(\mathbf{k})$ at $k_{B}T=0.3J$ under an applied field $h=0.02J//[110]$. Left: $F={\int }_{-\pi /2}^{\pi /2}d{k}_x{\int }_{-\pi /2}^{\pi /2}d{k}_y{K}_{xy}(\mathbf{k})$ as a function of $k_z$ for (a) regime I and (f) regime II. Right: color map of $K_{xy}$ for different $k_z$ slices for (b)–(e) $D=0.18J$ and (g)–(j) $D=0.33J$. In each map, the horizontal and vertical axes represent $k_x/\pi$ and $k_y/\pi$, respectively. See the text for an explanation of the shaded gray regions and orange circles.

Figure 5

Field-direction dependence of ${\kappa }_{\mu \nu }$ for $D=0.18J$ (a)–(c) and $D=0.33J$ (d)–(f). The horizontal axis represents the field direction $\widehat{h}$ which changes from the [100] to [110] and then [111] direction, with the fixed magnitude $h=0.02J$.