Parastatistics in Interacting Periodic Chains Revealed by Peierls Phase Twists and Shifted Conformal Towers

We consider interacting paraparticle chains with a constant R matrix where the Hamiltonian sums over the internal degrees (flavors) of the paraparticles. For such flavor-blind Hamiltonians, we show a general factorization of the Hilbert space into occupation and flavor parts with the Hamiltonian acting nontrivially only on the former. For open boundaries, the spectrum therefore coincides with that of the occupation Hamiltonian H_{occ} with the flavor part merely adding degeneracies. For periodic boundaries, a cyclic reordering of the flavors leads to a separation of H_{occ} into flux sectors at fixed particle number, thus making the parastatistics directly observable in the energy spectrum. For important exemplary cases, H_{occ} reduces to the XXZ chain with flux, allowing for an exact solution. In the gapless regime, this solution shows flux-shifted c=1 conformal towers in the low-energy spectrum and a temperature-dependent chemical potential in the bulk thermodynamics.