Non-Hermitian topology of spontaneous magnon decay

Spontaneous magnon decay is a generic feature of the magnetic excitations of anisotropic magnets and isotropic magnets with noncollinear order. In this Rapid Communication, we argue that the effect of interactions on one-magnon states can, under many circumstances, be treated in terms of an effective, energy-independent, non-Hermitian Hamiltonian for the magnons. In the vicinity of Dirac or Weyl touching points, we show that the spectral function has a characteristic anisotropy arising from topologically protected exceptional points or lines in the non-Hermitian spectrum. Such features can, in principle, be detected using inelastic neutron scattering or other spectroscopic probes. We illustrate this physics through a concrete example: a honeycomb ferromagnet with Dzyaloshinskii-Moriya exchange. We perform interacting spin-wave calculations of the structure factor and spectral function of this model, showing good agreement with results from a simple effective non-Hermitian model for the splitting of the Dirac point. Finally, we argue that the zoo of known topological protected magnon band structures may serve as a nearly ideal platform for realizing and exploring non-Hermitian physics in solid-state systems.

Figure 1

Spectral function, $A(\mathit{k},\omega )$, of the honeycomb ferromagnet with Dzyaloshinskii-Moriya interactions [see Eq. (8)], on constant energy slices as a function of wave vector near the $K$ point (arbitrary linear scale from zero). The magnon lifetime is shortest in the lower band on one side of the $K$ point and, in the higher energy band, on the opposite side of $K$. Energy slices of the full zone are also shown.

Figure 2

(a),(b) Constant slices of the real part of the quasinormal modes, $E_{\mathit{k}}$, as a function of momentum for the touching point in (a) 2D, showing the arc joining exceptional points (EPs) and (b) 3D showing the surface bounded by exceptional lines (ELs). The inverse of the imaginary part of the modes, $1/{\mathrm{\Gamma }}_{\mathit{k}}$, is also shown. (c)–(h) Comparison of the spectral function, $A(\mathit{k},\omega )$, evaluated for $D/J_1=0.125$ near the Dirac touching for the [(c)–(e)] analytic effective model (with the arc and exceptional points indicated) using ${\mathit{\Sigma }}^{-+}(K,{\omega }_0)$ [Eq. (11)] and [(f)–(h)] the result from nonlinear spin-wave theory (NLSWT), with a full solution of Dyson's equation (arbitrary linear scale from zero).

Figure 3

(a) The structure factor, $S(\mathit{k},\omega )$, computed within NLSWT for $D/J_1=0.125$ along a path in momentum space (arbitrary linear scale from zero). The linear spin-wave dispersions (dashed) and the bottom of the two-magnon continuum (dotted) for $D/J_1=0$ are indicated. (b)–(d) Evolution of the neutron scattering intensity at a fixed $D/J_1=0.125$ as the energy is varied through the Dirac node, from (b) $\omega /J_1=2.525$ to (c) $\omega /J_1=2.55$ to (d) $\omega /J_1=2.575$.