Quantum decay of magnons in the unfrustrated honeycomb Heisenberg model

TL;DR

The paper investigates whether the one-magnon excitation in the unfrustrated honeycomb Heisenberg model remains a sharp quasiparticle or decays into a multi-magnon continuum. It combines quantum Monte Carlo with stochastic analytic continuation, continuous similarity transformations, and high-order series expansions to map the magnon spectrum across the Brillouin zone and to quantify the spectral weight of the magnon peak. The results show that near the $K$-point the magnon peak loses weight and decays into a continuum, a finding corroborated by SE (with quantitative agreement except near $K$) and CST (which exhibits a divergent flow indicating breakdown of the single-magnon picture). The study highlights the role of strong magnon-magnon interactions and bound-state formation in driving decay, demonstrating that quantum fluctuations alone can destroy magnons in nonfrustrated lattices and informing high-energy magnon dynamics relevant to experiments.

Abstract

We investigate the physical properties of elementary magnon excitations of the ordered antiferromagnetic Heisenberg model on the honeycomb lattice using quantum Monte Carlo (QMC) simulations, series expansions (SE), and continuous similarity transformations (CST). The stochastic analytic continuation method is used to determine the dynamic structure factor from correlation functions in imaginary time obtained by QMC. In contrast to the "roton minimum" of the square lattice Heisenberg antiferromagnet, we find that magnons on the honeycomb lattice completely decay in the corner of the Brillouin zone ($K$-point); the entire weight is shifted into the continuum. These findings are fully supported by SE and CST in momentum space. The extrapolated one-magnon dispersion obtained from SE about the Ising limit quantitatively agrees with the extracted QMC excitation energies except around the $K$-point, where large uncertainties in the extrapolation indicate the magnon decay. This quantum decay is further confirmed and understood by the CST, which yields a divergent flow when enforcing a magnon quasi-particle picture. The divergence originates from strong attractive magnon-magnon interactions leading to a bound state and thereby to a three-magnon continuum overlapping with the one-magnon state. This has the magnon quasi-particle picture break down at high energies on the honeycomb lattice.

Figures (5)

• Figure 1: Panel (a) shows a sketch of the honeycomb lattice and the elementary lattice vectors $\vec{a}_1 = (3/2, \sqrt{3}/2)$ and $\vec{a}_2 = (3/2, -\sqrt{3}/2)$. In (b) a sketch of the Brillouin zone with reciprocal lattice vectors $\vec{b}_1 = (\pi, \sqrt{3} \pi)$, $\vec{b}_2 = (\pi, -\sqrt{3} \pi)$ and a high-symmetry path (blue) from the $\Gamma$-point $(0,0)$ to the $M$-point $(\pi,0)$ to the $K$-point $(\pi, \pi/\sqrt{3})$ is shown.
• Figure 2: In the restricted sampling procedure the ansatz for $S(\omega)$ contains a delta peak with fixed amplitude $S_0$ followed by a continuum of equal amplitude delta functions. The ansatz for $S(\omega)$ is schematically shown in panel (a). Comparing $\chi^2$ for different simulations at fixed values $S_0$ the optimal $S_0$ can be determined. In panel (b), $\chi^2(S_0)$ for the $L=36$ Heisenberg model on the honeycomb lattice at the $M$-point is shown.
• Figure 3: The dynamic structure factor $S_{\vec{k}}(\omega)$ of the Heisenberg antiferromagnet on a $L=36$ honeycomb lattice on the high-symmetry path $\Gamma \rightarrow M \rightarrow K \rightarrow \Gamma$ (see \ref{['fig:honeycomb_sketch']}) is shown in panel (a). The spectrum is calculated using the restricted SAC sampling scheme. The corresponding optimal amplitudes $S_0$ of the first delta peak in the simulation are depicted in panel (b) below. In (c) and (d), cuts through the dynamic structure factor at the M- and K-point with their optimal $S_0$ are shown.
• Figure 4: Scaling of the optimal $S_0$ (a) and corresponding position of the first delta peak $\omega_0$ (b) with different system sizes $L$ at the $K$- and $M$-point. The simulation and optimization is repeated for different sampling temperatures $\theta$ and independent sets of QMC data to receive reasonable error bars.
• Figure 5: Panel (a) shows the comparison between the one-magnon dispersion $\omega_\mathrm{SE}$ from SE and the lower band edge of the dynamic structure factor $\omega_\mathrm{QMC}$ from QMC. The error bars of $\omega_\mathrm{SE}$ reflect the uncertainty of the averaged Padé extrapolants. Panel (b) shows the dispersions from different evaluations in the framework for a linear system size of ${L=18}$ with periodic boundary conditions. The initial mean-field dispersion before the flow (mfSWT) is shown as a dashed green line. The data stems from the $0n$ flow (blue markers) and the flow stopped at $\ell=1$ (red markers), both with ($\tilde{\omega}$) and without ($\omega$) a re-diagonalization in the one-three magnon space. The red tripod markers mark the two lowest eigenvalues $\epsilon_{1,2}$ of the two-magnon subspace with $S^z=0$ calculated from the stopped flow. For comparison, the QMC dispersion $\omega_{\mathrm{QMC}}$ is also displayed.