Dynamics of a Two-Dimensional Quantum Spin Liquid: Signatures of Emergent Majorana Fermions and Fluxes

We provide a complete and exact theoretical study of the dynamical structure factor of a two-dimensional quantum spin liquid in gapless and gapped phases, as realized in Kitaev’s honeycomb model. We show that there are direct signatures—qualitative and quantitative—of the Majorana fermions and gauge fluxes emerging in this model. These include counterintuitive manifestations of quantum number fractionalization, such as a neutron scattering response with a gap even in the presence of gapless excitations, and a sharp component despite the fractionalization of electron spin. Our analysis identifies new varieties of the venerable x-ray edge problem and explores connections to the physics of quantum quenches.

Figure 1

(a) The structure of the model on the honeycomb lattice with two sublattices (labeled $A$ and $B$) and three bond directions (denoted $x$, $y$, $z$). The calculation of the dynamical response can be mapped to a local quantum quench, in which two adjoining $Z_2$ fluxes, shown in blue, are inserted. (b) The ground state and dynamical phase diagrams of the model: the phase with gapless fermion excitations fills the central triangle while gapped phases occupy the three outer triangles. The dynamical response $S^{zz}(\mathbf{q},\omega )$ includes a contribution sharp in $\omega$ in the dark red region, but not in the light green region.

Figure 2

Total dynamical structure factor $S(\mathbf{q},\omega )=\sum _a{S}^{aa}(\mathbf{q},\omega )$ and the inequivalent components $S^{aa}(0,\omega )$, as would be measured in inelastic neutron scattering and electron spin resonance, respectively, evaluated for three points in the phase diagram: ($\mathrm{a}+\mathrm{b}$) the symmetric point ($J_x=J_y=J_z$); ($\mathrm{c}+\mathrm{d}$) a gapless asymmetric point ($J_x=J_y,J_z=0.7J_x$); and ($\mathrm{e}+\mathrm{f}$) a gapped point ($J_x=J_y=0.15J_z$). Top: $S(\mathbf{q},\omega )$ on a logarithmic color scale as a function of $\omega$ along the cut $\mathrm{M}\mathrm{\Gamma }\mathrm{K}\mathrm{M}$ through the Brillouin zone. Bottom: dynamical susceptibility $S^{aa}(0,\omega )$ for $a=z,x$ at the same values of the exchange. Comparison with the adiabatic response as explained in the main text is given in panel b (black dashed line). The dashed line in (f) indicates the delta-function contribution to the response, present only in the region of the dynamical phase diagram colored red in Fig. 1. The insets to panels (b) and (d) show the density of states of the matter fermions.

Figure 3

The integrated structure factor $\mu (\omega )={\int }_0^{\omega }d{\omega }^{\prime }\sum _a{S}^{aa}(0,{\omega }^{\prime })$ in the gapless $J_x=J_y=J_z$ (red) and the gapped $J_x/J_z=J_y/J_z=0.15$ (black) phases. The sharp step in the black curve at low $\omega$ at the two-flux gap $\mathrm{\Delta }\simeq J_x^4/8J_z^3$ is due to the $\delta$-function contribution of the strong bond correlator ($a=z$). Inset: the dependence of the ground state overlap ${|⟨{\tilde{\lambda }}_0|M_0⟩|}^2$, proportional to the $\delta$-function contribution, on interaction strength, along the line $J_x/J_z=J_y/J_z$, see Supplemental Material [22] for details.