Spin-incoherent Luttinger liquid of one-dimensional SU($\kappa$) fermions

We investigate spin-incoherent one-dimensional (1D) SU($\kappa$) fermions in a harmonic trap. Specifically we focus on Tonks-Girardeau gas limit where its density is sufficiently low that effective repulsions between atoms become infinite. In such case, spin exchange energy of 1D SU($\kappa$) fermions vanishes and all spin configurations are degenerate, which automatically puts them into spin-incoherent regime. In this limit, we can write down the spatial wave functions by the conventional Slater determinant, and furthermore we are able to express the single-particle density matrices in terms of those of anyons. This allows us to numerically simulate the number of particles up to $N=32$. We numerically calculate single-particle density matrices for (1) equal populations for each components (balanced) and (2) all Sz manifolds included. We find their momentum distributions are broadened due to highly degenerate spin configurations, a signature of spin-incoherent regime. We then compare numerically calculated high momentum tails of momentum distributions with analytical predictions which are proportional to $1/p^{4}$, in good agreement. Thus, our theoretical study provides a direct comparison with experiments of repulsive multicomponent alkaline earth fermions.