Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet

We study the dynamic spin structure factor of the spin-$1/2$ square-lattice Heisenberg antiferromagnet and of the $J$-$Q$ model (with 4-spin interactions $Q$ and Heisenberg exchange $J$). Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with QMC simulations, we can treat the sharp ($\delta$-function) contribution from spinwaves (magnons) and a continuum at higher energy. The results for the Heisenberg model agree with neutron scattering experiments on Cu(DCOO)$_2$$\cdot$4D$_2$O, where a broad spectral-weight continuum at $q=(\pi,0)$ was interpreted as deconfined spinons. Our results at $(\pi,0)$ show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at $q=(\pi/2,\pi/2)$ (as also seen experimentally). Turning on $Q$, we observe a rapid reduction of the $(\pi,0)$ magnon weight to zero, well before the deconfined quantum phase transition into a spontaneously dimerized state. We re-interpret the picture of deconfined spinons at $(\pi,0)$ in the experiments as nearly deconfined spinons---a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile $(\pi,0)$-magnon in the Heisenberg model and its depletion in the $J$-$Q$ model, we introduce an effective model in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy with the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model reproduces the $(\pi,0)$ and $(\pi/2,\pi/2)$ features of the Heisenberg model. It can also account for the rapid loss of the $(\pi,0)$ magnon with increasing $Q$ and a remarkable persistence of a large magnon pole at $q=(\pi/2,\pi/2)$ even at the deconfined critical point.