Exact Dynamical Correlations of Hard-Core Anyons in One-Dimensional Lattices

The dynamical correlations of a strongly correlated system is an essential ingredient to describe its non-equilibrium properties. We present a general method to calculate exactly the dynamical correlations of hard-core anyons in one-dimensional lattices, valid for any type of confining potential and any temperature. We obtain exact explicit expressions of the Green's function, the spectral function, and the out-of-time-ordered correlators (OTOCs). We find that the anyonic spectral function displays three main singularity lines which can be explained as a double spectrum in analogy to the Lieb-Liniger gas. The dispersion relations of these lines can be given explicitly and they cross at a \emph{hot point} $(q_m,\omega_m)$, which induces a peak in the momentum distribution function at $q_m$ and a power-law singularity in the local spectral function at $\omega_m$. We also find that the anyonic statistics can induces spatial asymmetry in the Green's function, its spectrum, and the OTOC. Moreover, the information spreading characterized by the OTOCs shows light-cone dynamics, asymmetric for general statistics and low temperatures, but symmetric at infinite temperature. Our results pave the way toward studying the non-equilibrium dynamics of hard-core anyons and experimentally probing anyonic statistics through spectral functions.