Classical and quantum spin dynamics of the honeycomb $\mathrm{\Gamma }$ model

Quantum-to-classical crossover is a fundamental question in dynamics of quantum many-body systems. In frustrated magnets, for example, it is highly nontrivial to describe the crossover from the classical spin liquid with a macroscopically degenerate ground-state manifold to the quantum spin liquid phase with fractionalized excitations. This is an important issue, as we often encounter the demand for a sharp distinction between the classical and quantum spin liquid behaviors in real materials. Here we take the example of the classical spin liquid in a frustrated magnet with novel bond-dependent interactions to investigate the classical dynamics, and critically compare it with quantum dynamics in the same system. In particular, we focus on signatures in the dynamical spin structure factor. Combining Landau-Lifshitz dynamics simulations and the analytical Martin-Siggia-Rose approach, we show that the low-energy spectra are described by relaxational dynamics and highly constrained by the zero mode structure of the underlying degenerate classical manifold. Further, the higher energy spectra can be explained by precessional dynamics. Surprisingly, many of these features can also be seen in the dynamical structure factor in the quantum model studied by finite-temperature exact diagonalization. We discuss the implications of these results and their connection to recent experiments on frustrated magnets with strong spin-orbit coupling.

Figure 1

Trace of the dynamical magnetic structure factor $S\left(\mathit{Q},\omega \right)={\sum }_{\alpha }{S}^{\alpha \alpha }\left(\mathit{Q},\omega \right)$ for (a) AFM ($\mathrm{\Gamma }>0$), (b) FM ($\mathrm{\Gamma }<0$) $\mathrm{\Gamma }$ models along the Brillouin zone path, $\mathit{K}-\mathbf{\Gamma }-\mathit{M}-\mathit{Y}-\mathit{X}-\mathit{K}-\mathit{M}$, as depicted in the inset.

Figure 2

(a) Elastic component of the trace of the magnetic structure factor $S(\mathit{Q},\omega =0)$. (b)–(d) $S(\mathit{Q},\omega )$, integrated over finite energy cuts: (b) $\omega /\mathrm{\Gamma }=[0.1,0.3]$, (c) $\omega /\mathrm{\Gamma }$ = [0.8, 1.0], and (d)$\omega /\mathrm{\Gamma }$ = [1.8, 2.0].

Figure 3

$S(\mathit{r},\omega =0)$ as a function of $\left|\mathit{r}\right|/a$ (where $a$ is the lattice parameter) at $T={10}^{-5}\mathrm{\Gamma }$.

Figure 4

Static spin-spin correlation function, ${\tilde{S}}^{\alpha \beta }=〈S_{\mathit{o}}^{\alpha }{S}_{\mathit{r}}^{\beta }〉=\int S^{\alpha \beta }\left(\mathit{r},\omega \right)d\omega$, mapped on the real-space lattice with respect to an arbitrary origin denoted by the cross ($\times$). The temperature of the simulation is $T={10}^{-5}\mathrm{\Gamma }$.

Figure 5

Six-sublattice decomposition of the honeycomb lattice. We also show the three different decompositions of the lattice into isolated hexagonal plaquettes.

Figure 6

$S\left(\mathit{r},\omega \right)$ as a function of temperature and energy ($\hslash \omega$) for AFM ($\mathrm{\Gamma }>0$) $\mathrm{\Gamma }$ model. Panels (a), (d), and (e) show $-S\left(\mathit{r},\omega \right)$ for $\mathit{r}$ connecting first, third, and fifth nearest-neighbor sites, respectively. Panel (b) shows $S\left(\mathit{r},\omega \right)$ for $\mathit{r}$ connecting nearest-neighbor sites. Panel (c) shows that $S\left(\mathit{r},\omega \right)=0$ for $\mathit{r}$ connecting second-nearest-neighbor sites.

Figure 7

Temperature evolution of $S(\mathit{Q}=\mathbf{\Gamma },\omega )$ for AFM ($\mathrm{\Gamma }>0$) $\mathrm{\Gamma }$ model.

Figure 8

Feynman diagrams for the MSR formalism: (a) bare response function, (b) bare correlation function, (c) precession vertex, (d) self-energy diagram, (e) perturbative expansion of the dressed correlation function, in powers of the self-energy $\mathrm{\Sigma }$, see Eq. (15).

Figure 9

Dynamic structure factor as obtained from Eq. (15), for (a) the AFM $\mathrm{\Gamma }>0$, and (b) the FM $\mathrm{\Gamma }<0$ models. Here we have used $\gamma =0.12, \mathrm{\Delta }=2.05\mathrm{\Gamma }$, and $g^{2}T=0.04\mathrm{\Gamma }$.

Figure 10

Finite-size honeycomb clusters with 24 spins and periodic boundary conditions. Bonds along the three different directions are labeled as the $x, y$, and $z$ bond, which are along $-{60}^{\circ }, {60}^{\circ }$, and horizontal directions, respectively.

Figure 11

Specific heat $C/N$ of the AFM $\mathrm{\Gamma }$ model on the $N=24$-site cluster [24], calculated by the typical pure quantum states approach [30, 31]. There are two maxima in the temperature dependence of $C/N$. The error bars are the standard errors estimated by 32 random initial vectors.

Figure 12

Equi-energy slices of the dynamical spin structure factors of the $\mathrm{\Gamma }>0$ model. The momentum dependence of the equi-energy slices is shown by changing temperature and frequency. From the top row to the bottom row, the equi-energy slices of the dynamical spin structure factors are shown at $T/\mathrm{\Gamma }=1$, 0.5, 0.2, and 0.1. The equi-energy slices are prepared by averaging the spectra within an energy window whose width is 0.1. For visibility, the dynamical spin structure factors at discrete momenta obtained by the simulation are interpolated. Here, the broadening factor $\eta /\mathrm{\Gamma }=0.02$ is used (see Appendix pp4 for the definition of $\eta$).

Figure 13

Temperature evolution of $S(\mathit{Q}=\mathrm{\Gamma },\omega )$ and $S(\mathit{Q}=M,\omega )$ for the $\mathrm{\Gamma }>0$ model.

Figure 14

Dynamical spin structure factors of the $S=1/2$ AFM $\mathrm{\Gamma }$ model at (a) $T=0.5$, (b) $T=0.2$, (c) $T=0.1$, and (d) $T=0$, along symmetry lines. For visibility, the broadening factor $\eta /\mathrm{\Gamma }=0.02$ is used (see Appendix pp4 for the definition of $\eta$).

Figure 15

Real-space static spin-spin correlation function, $\langle S_{\mathit{0}}^{\alpha }{S}_{\mathit{r}}^{\beta }\rangle (\alpha ,\beta =x,y,z)$, of the $S=1/2\mathrm{\Gamma }>0$ model on the 24-site cluster with periodic boundary conditions, at $T=0$. The location of the origin $\text{0}$ is denoted by the open circle ($\circ$). The radiuses of the closed circles at $\mathit{r}$ represent the amplitude of $|\langle S_{\mathit{0}}^{\alpha }{S}_{\mathit{r}}^{\beta }\rangle |$, while the color of the closed circles shows $\langle S_{\mathit{0}}^{\alpha }{S}_{\mathit{r}}^{\beta }\rangle$. Within numerical accuracy, there is no spin-spin correlation represented by a circle with a radius smaller than the width of the solid lines representing the bonds.

Figure 16

Temperature dependence of static spin-spin correlations of the AFM $\mathrm{\Gamma }$ model on the 24-site cluster. Here, the error bars, which are smaller than or comparable to the symbol size, are the standard errors estimated by several random initial vectors.

Figure 17

(a) Two-sublattice decomposition of the honeycomb lattice. (b) Four-sublattice decomposition of the honeycomb lattice.

Figure 18

Dynamic structure factor as obtained in Eq. (C6), for (a) the AFM Kitaev $K>0$ and (b) the FM Kitaev $K<0$ models. Here we used $\gamma =0.25, \mathrm{\Delta }=1.05K$, and $g^{2}T=0.1K$.

Figure 19

Site indices used in calculations of spin correlations.