Abstract:
Interacting bosons in one dimension (1D) are described by the exactly solvable
Lieb-Liniger (LL) model, which satisfies the Bethe Ansatz equations. The dynamical
structure factor (DSF) is given by the Fourier transform of the time-dependent
density-density correlation function. This study aims to obtain a simplified analytical
expression for the density form factor (DFF), which is currently unavailable
due to the complexity of the calculation related to the density-density correlation
function. The DFF is a function that describes how energy eigenstates are distributed
in space. As a first step, the excited momentum of the boson particle in
the ground state was approximated up to 1/c, where c is the coupling constant,
based on the momentum of the ground state. The DFF was then expanded using
the Lehmann representation, and each term in the expansion was simplified separately
using mathematical techniques such as the Cauchy determinant, in terms
of energy eigenstates. Finally, a simplified analytical expression was derived for
the DFF, and the result was confirmed by numerical results and the dispersion
relation. The dispersion relation for 1D strongly interacting bosons in the LL
model was obtained considering a large system with constant density. Furthermore,
this study helps to gain knowledge about the significant excitation of the
system based on the sum rule.
