Exponents of the spectral functions and dynamical structure factor of the 1D Lieb–Liniger Bose gas

We study the (k,ω)(k,ω)-plane finite-energy line shape of the zero-temperature one-boson removal spectral function (ω<0)(ω<0), one-boson addition spectral function (ω>0)(ω>0), and charge dynamical structure factor (ω>0)(ω>0) of the 1D Lieb–Liniger Bose gas with repulsive boson interaction c>0c>0. Our analysis of the problem focuses on the line shape at finite excitation energies in the vicinity of these functions spectrum upper (ω<0)(ω<0) or lower (ω>0)(ω>0) threshold. Specifically, we derive the exact momentum, interaction, and density dependences of the exponents controlling such a line shape in each of the N=1,2,3,…N=1,2,3,… momentum subdomains k∈[(N−1)2πn,N2πn]k∈[(N−1)2πn,N2πn]. Here n=N/Ln=N/L is the boson density, NN the boson number, and LL the system length. In the thermodynamic limit considered in our study nearly all spectral weight of the dynamical correlation functions is for large values of n/cn/c contained in the N=1N=1 momentum subdomain k∈[0,2πn]k∈[0,2πn]. As n/cn/c decreases a small fraction of that weight is transferred to the remaining set of N=2,3,4,…N=2,3,4,… momentum subdomains, particularly to the N=2N=2 subdomain. In the case of the momentum subdomain k∈[0,2πn]k∈[0,2πn], our exact results agree with those of previous studies. For that subdomain the above exponents are plotted as a function of the momentum for several n/cn/c values. Our derivation of the line shapes of the three dynamical correlation functions relies on the use of a simplified form of the pseudofermion dynamical theory of the fermionic 1D Hubbard model suitably modified in this paper for the 1D Bose gas.