Disentangling spin excitation continua in classical and quantum magnets using 2D nonlinear spectroscopy

TL;DR

The paper addresses the challenge of distinguishing quantum spin-liquid continua from continua arising in classical or large-unit-cell magnetic states observed in INS. It employs two-dimensional coherent spectroscopy (2DCS) and compares classical molecular dynamics simulations with the quantum Kitaev model, extending to the KGammaGamma' regime to contrast quantum spin liquids with ordered ensembles. A key finding is that the quantum Kitaev model exhibits flux-gap- and selection-rule–driven 2DCS signatures that are absent in the classical limit, while in the KGammaGamma' region 2DCS can reveal discrete magnon peaks versus INS continua, offering a powerful diagnostic to identify the underlying excitations. Temperature strongly suppresses classical 2DCS features, highlighting the potential importance of quantum coherence in experimental settings and providing a framework for interpreting 2DCS data in candidate Kitaev materials such as alpha-RuCl3 under magnetic fields.

Abstract

Inelastic neutron scattering (INS) has traditionally been one of the primary methods for investigating quantum magnets, particularly in identifying a continuum of excitations as a hallmark of spin fractionalization in quantum spin liquids (QSLs). However, INS faces severe limitations due to its inability to distinguish between such QSL signatures and similar excitation continua arising from highly frustrated magnetic orders with large unit cells or classical spin liquids. In contrast, two-dimensional coherent spectroscopy (2DCS) has emerged as a powerful tool to probe nonlinear excitation dynamics, offering insights into the underlying mechanisms behind these broad spectral features. In this paper, we utilize classical molecular dynamics (MD) techniques to explore the 2DCS responses of frustrated magnets with dominant Kitaev interactions. Comparing the classical and quantum versions of the pure Kitaev model our results indicate both clear similarities, in the form of sharp line features, and clear distinctions, in the locations of these features and in selection rules. Moreover, in the extended $KΓΓ'$ model, we show that the 2DCS response of the Kitaev spin liquid is completely distinct from that of large unit cell magnetic orders, despite both generating a broad continuum in INS. Additionally, we demonstrate the extreme sensitivity of classical 2DCS to thermal fluctuations and discuss the potential significance of quantum coherence in experimental settings. Overall, our work illustrates the potential of 2DCS in resolving the complex physics underlying ambiguous spin excitation continua, thereby enhancing our understanding of the dynamics in these frustrated systems.

Figures (6)

• Figure 1: A, Schematic of the 2DCS experiment with incident pulses $\textbf{B}_A$ and $\textbf{B}_B$ separated by time $\tau$. The signal field $\textbf{B}_{NL}=\textbf{B}_{AB}-\textbf{B}_{A}-\textbf{B}_{B}$ is measured at a time $t$ after the second pulse. The incident pulses are polarized in the local $z$ direction, depicted in B. The spin axis $(S_x, S_y, S_z)$ is coming out of the plane and the global axis $(a,b,c)$ is shown in the basis of the local coordinates.
• Figure 2: A, Illustrative example of the flux selection rules for the $\chi^{(3),z}_{zzz}(t,0,\tau)$ nonlinear susceptibility. Each $\hat{\sigma}^z$ operator creates or destroys a flux pair across the $z$-bond. The red pulse corresponds to the incident A pulse, while the two purple pulses correspond to B pulses that interact twice. B, Sketch of the two-dimensional Fourier spectrum of the third-order susceptibility $\chi^{(3),z}_{zzz}(t,0,\tau)$. The contributions of each four-point correlation functions obtained by the flux selection rules are indicated. Note that $R^{(3),z}_{zzz}(t,0,\tau)=R^{(2),z}_{zzz}(t,0,\tau)=R^{(2,3),z}_{zzz}$. On the axis ticks, $E_{ab}=E_a - E_b$. Only the flux energies and not the matter fermion energies are shown here for brevity.
• Figure 3: Dynamical spin structure factor (A,B) and two-dimensional coherent spectroscopy (C-N) for the pure Kitaev models with $K=-1$ and $K=1$. A and B are replotted from zhang_spin_2023, and were performed at $T/|K|=0.001$. $|M_{NL}^{x}(\omega_t, \omega_\tau)|$ (C-L), $|M_{NL}^{y}(\omega_t, \omega_\tau)|$ (D-M), and $|M_{NL}^{z}(\omega_t, \omega_\tau)|$ (E-N) are shown for $T/|K|=0$ and $T/|K|=0.001$. All plots were normalized to their respective maximum intensities. Before taking the Fourier transform, a Gaussian filter of $e^{-\eta(t^2+\tau^2)}$ was applied, with $\eta=10^{-6}$.
• Figure 4: A, Even and B, odd decomposition of the 2DCS for the pure $K=-1$ model at zero temperature. $\textbf{M}^{+}_{NL}$ ($\textbf{M}^{-}_{NL}$) refers to the nonlinear response with two pulses polarized in the $+\hat{\textbf{z}}$ ($-\hat{\textbf{z}}$) direction. Before taking the Fourier transform, a Gaussian filter of $e^{-\eta(t^2+\tau^2)}$ was applied, with $\eta=10^{-6}$.
• Figure 5: Field dependence of the dynamical spin structure factor (A-D) and two-dimensional coherent spectroscopy (E-L) for $K=-1$, $\Gamma=0.25$, and $\Gamma'=-0.02$. A-D are replotted from zhang_spin_2023, and were performed at $T/|K|=0.001$. $|M_{NL}^{z}(\omega_t, \omega_\tau)|$ is shown for $T/|K|=0$ (E-H) and $T/|K|=0.001$ (I-L). Panel F includes black solid, dashed, and dotted lines at energies corresponding to the three high intensity peaks within the $\Gamma_0$ continuum in B. Panel G includes a solid black line at the energy corresponding to the high intensity peak in panel C at $\Gamma_0$. In panels H and L, TR=Terahertz Rectification, 2Q=2 Quantum, NR=Non-Rephasing, R=Rephasing, PP=Pump Probe. $\mathbf{M}$, The zero temperature classical phase diagram at each field. ZZ=Zig-Zag, 32=32-site, 50=50-site, 18=18-site, PM=Polarized paramagnet.