The 40-year-old Calogero-Sutherland (CS) model remains a source of inspirations for understanding 1d interacting fermions. At
β=1,or0, the CS model describes a free non-relativistic fermion, or boson theory, while for generic
β, the system can be interpreted either as interacting fermions or bosons, or free anyons depending on the context. However, we shall show in this letter that the fermionic picture is advantageous in diagonalizing the CS Hamiltonian. Comparing to the previously known multi-integral representation or the Dunkl operator formalism for the CS wave functions, our method depends on the (upper or lower) triangular nature of the fermion interaction, which is resolved in perturbation theory of the second quantized form. The eigenstate is constructed from a multiplet of unperturbed states and the perturbation is of finite order. The full construction is a similarity transformation from the free fermion theory, in the same spirit as the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means quasi-particles or anyons can be represented in terms of free fermion modes (or bosonic modes via bosonization). The method is applicable to other (higher than one space dimension) systems for which the adiabatic theorem applies.
We analyse the discrete-time dynamics of a model of non-interacting fermions coupled to an infinite reservoir formed by a bosonic quantum walk on
Z. This dynamics consists of consecutive applications of free evolutions of the fermions and bosons followed by a local coupling between them. The unitary operator implementing this coupling accounts for energy exchanges between the system and reservoir while it preserves the number of fermions. The free fermion evolution is given by a second-quantized single-particle unitary operator satisfying some genericity assumptions. The free boson evolution is given by the second-quantized shift operator on
Z. We derive explicitly the Heisenberg dynamics of fermionic observables and obtain a systematic expansion in the large-coupling regime, which we control by using spectral methods. We also prove that the reduced state of the fermions converges in the large-time limit to a mixture of infinite-temperature Gibbs states in each particle sector.