Quantitative determination of the confinement and deconfinement of spinons in the anomalous spectra of antiferromagnets via the entanglement entropy

We introduce an entanglement entropy analysis to quantitatively identify the confinement and deconfinement of the spinons in the spin excitations of quantum magnets. Our proposal is implemented by the parton construction of a honeycomb-lattice antiferromagnet exhibiting high-energy anomalous spectra. To obtain the quasiparticles of spin excitations for entanglement entropy calculations, we develop an effective Hamiltonian using the random phase approximation. We elaborate quantitatively the deconfinement-to-confinement transition of spinons in the anomalous spectra with the increase of the Hubbard interaction, indicating the avoided fractionalization of magnons in the strong interaction regime. Meanwhile, the Higgs mode at the ${\mathrm{\Gamma }}^{\prime }$ point is fractionalized into four degenerate spinons, although it appears as a sharp well-defined peak in the spectra. Our work extends our understanding of the deconfinement of the spinon and its coexistence with the magnon in quantum magnets.

Figure 1

(a) Illustration of Eq. (1) in the honeycomb lattice. A and B denote the two sublattices of the honeycomb lattice. $t$ is RVB hopping parameter and $m$ denotes the order parameter of the AF order. (b) High symmetry points in the Brillouin zone.

Figure 2

Spectral functions of the (a)–(c) transverse and (d)–(f) longitudinal spin excitations along the $\mathrm{\Gamma }\text{−}M\text{−}{\mathrm{\Gamma }}^{\prime }\text{−}K\text{−}{K}^{\prime }\text{−}{\mathrm{\Gamma }}^{\prime }$ path denoted in Fig. 1.

Figure 3

On the left: transverse spectral functions at (a) $K$ and (c) $M$. On the right: scalings with system size $N$ of the EEs of the lowest transverse modes at (b) $K$ and (d) $M$. The legends in the figures indicate the increasing of $a$ by different colors. The dashed lines denote the EEs of degenerate modes.

Figure 4

On the left: longitudinal spectral functions at (a) ${\mathrm{\Gamma }}^{\prime }$ and (c) $M$. On the right: scalings of the EEs of the lowest longitudinal modes at (b) ${\mathrm{\Gamma }}^{\prime }$ and (d) $M$.