Scattering Theory of Chiral Edge Modes in Topological Magnon Insulators

TL;DR

This paper addresses how chiral edge magnons in topological magnon insulators scatter when interactions are present, while edge modes remain confined to the boundary. The authors model a 2D array of coupled XXZ chains under an Aharonov-Casher phase, show that edge-edge scattering is renormalized by resonances with bulk bound states, and extract a two-magnon edge Hamiltonian $\mathcal{H}_{\text{2-eff}}=\varepsilon(k_1)+\varepsilon(k_2)+U_{2\text{-eff}}(k_1,k_2)$. They relate the edge scattering phase $\phi(k_1,k_2)$ to a moving-frame Lieb-Liniger description with $\phi_{\mathrm{LL}}(K,q)=-2\arctan\left(\frac{k_1-k_2}{m^{*}(p)c(p)}\right)$ and provide a prescription to determine $a(p)$ and $U_{2-\text{eff}}$, enabling a systematic many-body theory for edge states. The work also proposes a real-time inelastic STS protocol to probe edge interactions, highlighting how bulk spectra shape edge dynamics and outlining pathways to magnon-based quantum devices and interferometric concepts in topological magnets.

Abstract

Topological magnon insulators exhibit robust edge modes with chiral properties similar to quantum Hall edge states. However, due to their strong localization at the edges, interactions between these chiral edge magnons can be significant, as we show in a model of coupled magnon-conserving spin chains in an electric field gradient. The chiral edge modes remain edge-localized and do not scatter into the bulk, and we characterize their scattering phase: for strongly-localized edge modes we observe significant deviation from the bare scattering phase. This renormalization of edge scattering can be attributed to bound bulk modes resonating with the chiral edge magnons, in the spirit of Feshbach resonances in atomic physics. We argue that the scattering dynamics can be probed experimentally with a real-time measurement protocol using inelastic scanning tunneling spectroscopy. Our results show that interaction among magnons can be encoded in an effective edge model of reduced dimensionality, where the interactions with the bulk renormalize the effective couplings. Our work introduces a systematic way to determine the many-body effective theory for edge states in topological magnon insulators.

Figures (7)

• Figure 1: Topological magnons in coupled wires. a) Sketch of the model \ref{['eq:Hamiltonian-Chain']}: we consider a two-dimensional system made of an array of spin$-1/2$ chains. The presence of a linear Aharonov-Casher (AC) phase induced by a constant electric field gradient in the $y$-direction $\vec{E} = \Delta E \vec{y}$, gives rise to localized bulk modes and chiral edge modes. b) Cartoon scattering of two chiral magnons at the same edge. Our results demonstrate that the scattering of edge modes is elastic (no scattering into the bulk), but resonances with bulk modes strongly renormalize effective interactions, manifesting in scattering shifts $d_{1}, d_{2}$ of the individual edge modes.
• Figure 2: Real time scattering of chiral magnons. a) Single magnon spectrum with $\delta\varphi=2\pi/5$ for the case of uncoupled wires, consisting of multiple copies of single-wire bands shifted in the momentum space (gray lines). Inter-wire hopping $J_y\ne 0$ causes band repulsion, and the spectrum shifts to stationary bulk modes and chiral edge bands (within gray box). Location of the individual bands is indicated by color coding. b) Real-time simulation of a scattering event of two chiral magnons with total momentum $K=-\frac{\pi}{8}$ and relative momentum $q=-\frac{\pi}{16}$, and parameters $(J_{y}, \Delta)=(0.5, 0.4)$. We track the bulk-integrated magnon distribution $\vert\psi(x)\vert^2$, with $x$ the relative distance. For a better visualization, we consider a comoving frame with the average velocity of the pair of magnons. The gray dashed lines are the trajectories of the initial position of the wavepackets, evolved with the initial velocities (free evolution). c) The state remains localized at the edge during the scattering, as shown in the average position in the transverse direction $\left\langle y \right\rangle$, where $\left\langle y \right\rangle\!=\!1$ corresponds to magnons being located in the lowest chain. d) We compare the free evolution (gray dashed line) with the peak positions (blue line), showing displacements $d_{1}, d_{2}$ after scattering.
• Figure 3: Scattering phases. a) Left: Allowed energy-momentum window for scattering magnons at the lower edge (blue) embedded in the two-magnon continuum (gray area). The asymptotic energy region (blue) overlaps with non-dispersive bulk bands associated to bound states of two neighboring magnons (orange). The parameters are $(J_{y}, \Delta)\!=\!(0.5, 0.0)$. Right: Momentum-integrated two-magnon density of states capturing the low density of states within the bulk gap (lightblue shaded region). b) Extracted scattering phases for asymptotic states of different relative momenta $q$ for fixed total momenta $K\in\{-0.43, 0, 0.43\}$ (left, middle, right column) for selected $\Delta\in\{0.1, 0.3, 1.0\}$. The one-dimensional XXZ prediction (gray line) is provided for comparison, highlighting a strong renormalization for positive $K$ and small $\Delta$. c) Comparison of equal-momenta effective interaction strength $U_\text{2-eff}(K/2,K/2)$ extracted from the scattering phase and chiral dispersion $\varepsilon(k)$ for numerically extracted results and analytical predictions for one-dimensional XXZ (gray lines).
• Figure 4: Resonances. a) Pictorial representation of scattering resonances. Upon changing $\Delta$, a set of bulk BS bands of total momentum $K$ gets tuned through a continuum of asymptotic states with variable relative momentum $q$, triggering resonances. b) Tracing one particular bound mode in the bulk of fixed total momentum $K=0.32$ through the continuum (blue region) of asymptotic states yields resonances with different chiral states. c) This is resolved in the participation of the wavefunction at the edge. For our finite size simulations we find three resonance peaks. d) - f) Resonances of the bulk BS with asymptotic states are reflected in the magnon distribution $\vert \psi(x,y) \vert^{2}$ in relative space $x$ and the bulk $y$. In contrast to the unhybridized case d) the wavefunction contains strong contributions from the edge at the resonances (e,f).
• Figure 5: Chiral Bound States and Scattering Phase Analysis. a) Bound state bands separated in energy from the two magnon continuum for anisotropy $\Delta=1.25$ and uncoupled wires. b) - d) Increasing hopping strength between the wires $J_{y}\in \{0.25, 0.50, 0.75\}$ opens a bulk gap and stabilizes modes of chiral bound states. e) Analysis of the scattering phase consists of (1.) - (3.) selecting eigenstates within the chiral energy window for a given momentum sector $K$, (4.) fitting the states against a plane wave ansatz and (5.) compute the scattering matrix $S$. f) - h) Comparison of extracted (top row) and XXZ predicted (bottom row) scattering phases for different $K$ momenta and anisotropies $\Delta\in\{0.0, 0.4, 1.0\}$. The associated relative momenta are highlighted by color coding. Deviations are evident at low momenta ($\Delta=0.0$) and especially for less localized states.