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

Abstract

We study the (k, ω)-plane finite-energy line shape of the zerotemperature one-boson removal spectral function (ω < 0), one-boson addition spectral function (ω > 0), and charge dynamical structure factor (ω > 0) of the 1D Lieb–Liniger Bose gas with repulsive boson interaction c > 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) or lower (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, . . . momentum subdomains k ∈ [(N − 1)2πn,N2πn]. Here n = N/L is the boson density, N the boson number, and L 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/c contained in the N = 1 momentum subdomain k ∈ [0, 2πn]. As n/c decreases a small fraction of that weight is transferred to the remaining set of N = 2, 3, 4, . . . momentum subdomains, particularly to the N = 2 subdomain. In the case of the momentum subdomain 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/c values. Our derivation of the lineshapes 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.