Time and temperature-dependent correlation function of an impurity in one-dimensional Fermi and Tonks–Girardeau gases as a Fredholm determinant

Abstract

We investigate a free one-dimensional spinless Fermi gas, and the Tonks–Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or attractive impurity-gas δ -function interaction potential. It includes, as particular cases, the representations obtained for zero temperature and arbitrary repulsion in (Gamayun et al 2015 Nucl. Phys. B http://dx.doi.org/10.1016/j.nuclphysb.2015.01.004 892 http://dx.doi.org/10.1016/j.nuclphysb.2015.01.004 ), and for arbitrary temperature and infinite repulsion in (Izergin and Pronko 1998 Nucl. Phys. B http://dx.doi.org/10.1016/S0550-3213(98)00182-5 520 http://dx.doi.org/10.1016/S0550-3213(98)00182-5 ).

Keywords

• impurity dynamics
• Fredholm determinants
• one-dymensional systems
• quantum gases
• Bethe ansatz