Dynamic structure factor of a spin-1/2 Heisenberg chain with long-range interactions

We study the dynamic structure factor $S(k,\omega)$ of the spin-1/2 chain with long-range, power-law decaying unfrustrated (sign alternating) Heisenberg interactions $J_r \sim (-1)^{r-1} r^{-\alpha}$ by means of stochastic analytic continuation (SAC) of imaginary-time correlations computed by quantum Monte Carlo calculations. We do so in both the long-range antiferromagnetic (AFM, for $\alpha \lesssim 2.23$) and quasi-long-range-ordered (QLRO, for $\alpha \gtrsim 2.23$) ground-state phases, employing different SAC parametrizations of $S(k,\omega)$ to resolve sharp edges characteristic of fractional quasi-particles and sharp peaks expected with conventional quasi-particles. In order to identify the most statistically accurate parametrization, we apply a newly developed cross-validation method as a ``model selection'' tool. We confirm that the spectral function contains a power-law divergent edge in the QLRO phase and a very sharp (likely $\delta$-function) magnon peak in the AFM phase. From our SAC results, we extract the dispersion relation in the different regimes of the model, and in the AFM phase we extract the weight of the magnon pole. In the limit where the model reduces to the conventional Heisenberg chain with nearest-neighbor interactions, our $S(k,\omega)$ agrees well with known Bethe ansatz results. In the AFM phase the low-energy dispersion relation is known to be nonlinear, $\omega_k \sim k^z$, and we extract the corresponding dynamic exponent $z(\alpha)$, which in general is somewhat above the form obtained in linear spin-wave theory. We also find a significant continuum above the magnon peak. This study serves as a benchmark for SAC/QMC studies of systems with a transition from conventional to fractionalized quasi-particles.