Spinons and Spin-Charge Separation at the Deconfined Quantum Critical Point

Abstract

Using quantum Monte Carlo and numerical analytic continuation methods, we study the dynamic spin structure factor and the single-hole spectral function of a two-dimensional quantum magnet ($J$-$Q$ model) at its quantum phase transition separating Néel antiferromagnetic and spontaneously dimerized ground states. At this putative deconfined quantum-critical point, we find a broad continuum of spinon excitations that can be accounted for by the fermionic $π$-flux state; a known mean-field model for deconfined quantum criticality. We find that the best description of the two-spinon continuum is with a version of the model with a $2\times 2$ unit cell, reflecting non-trivial mutual statistics of spinons and anti-spinons. The single-hole spectral function can be described by the same spinon dispersion relation and an independently propagating holon. Thus, the system exhibits spin-charge separation and will likely evolve into an extended holon metal phase at finite doping.

Figures (7)

• Figure 1: Dynamic spin structure factor $S({\bf k},\omega)$ of the $J$-$Q$ model at the AFM--VBS transition. Results are shown for system sizes $L=16$ and $L=32$ at four different momenta $(k_x,k_y)$. The static structure factor $S({\bf k})$ is used to normalize all spectra to unity; for $L=32$, $S(\pi,\pi/2)=1.142$, $S(\pi,0)=0.516$, $S(\pi/2,\pi/2)=0.564$ and $S(\pi,\pi)=13.4$. The ${\bf k}=(\pi,\pi)$ results are multiplied by $3$.
• Figure 2: Location of the edge in $S({\bf k},\omega$) versus the momentum along a path in the BZ, based on results such as those in Fig. \ref{['fig1']} for $L=16$ and $L=32$. The black curve shows a fit to the $\pi$-flux dispersion relation Eq. (\ref{['epsilon_s']}) with velocity $v=3.73$.
• Figure 3: Heat map of $S({\bf k},\omega)$ for $L=32$. There is no spectral weight in the white area, while weight below $10^{-2}$ above the edge is shown as as black. The black dashed curve is the same fit to the edge as in Fig. \ref{['fig2']}. The red dashed curve shows the upper bound of the two-spinon continuum predicted from this dispersion relation with the diamond shaped BZ of size $1/2$ of the original square lattice BZ. The white dashed curve corresponds to the further reduced BZ with ${\bf k} \in [0,\pi]^2$.
• Figure 4: Single-hole spectral function $A({\bf k},\omega$) of the $t$-$J$-$Q$ model with $t=J=1$, $Q=0.6667$ for lattices with $L=16$ and $L=32$ at four different momenta $(k_x,k_y)$.
• Figure 5: Heat map of $A({\bf k},\omega$). The upper bound shown as the white dashed curve is obtained on the basis of the SAC data points for the lower edges of the two-spinon (Fig. \ref{['fig3']}) and spinon-holon (this figure) continua. The black dashed lines represent spinons added to holons with momentum fixed at one of the three peak locations. There is no spectral weight in the white area below the edge, while above the edge any weight below $10^{-2}$ is represented as black.