Fig. 1: Static properties and density evolutions for N = 2 particles with different statistical angles θ and interaction strength U.

a Bosons (θ = 0) with zero interaction, with (a1) eigenenergies of the model under periodic boundary conditions (PBCs) (red) and open boundary conditions (OBCs) (blue); (a2) particle distribution ρ(x) of all many-body eigenstates (pink); and (a3) density evolution for two particles evenly distributed at the center of the 1D chain. (b1–b3), (c1–c3), (d1–d3) displayed the same quantities for systems with different statistical angles and interaction strengths, as labeled on top of each panel. e The almost identical average density \({\bar{\rho }}_{x}\) for all states in (a2–d2), represented by cyan star, red square, black line, and blue triangular, respectively. It is seen that anyonic statistics have little effect on the distribution of eigenstates, even though they cause distinguished dynamics. Other parameters are J_L = e^−α, J_R = e^α, α = 0.1 and L = 30. In each of (b1), (d1), some eigenenergies form a loop separated from the others, corresponding to two-particle bound states induced by the Hubbard interaction (see Supplementary Note 1).