Kinetics of Mobile Impurities and Correlation Functions in One-Dimensional Superfluids at Finite Temperature

We examine the hydrodynamic approach to dynamical correlations in one-dimensional superfluids near integrability and calculate the characteristic time scale, $\tau$, beyond which this approach is valid. For time scales shorter than $\tau$, hydrodynamics fails, and we develop an approach based on kinetics of fermionic quasiparticles described as mobile impurities. New universal results for the dynamical structure factor relevant to experiments in ultracold atomic gases are obtained.

Figure 1

Typical support in the ($q$, $\omega$) plane of the dynamical structure factor $S(q,\omega )$ in different regimes. The dominant feature is the ballistic behavior linear in momentum $q$ as in Eq. (3). The hydrodynamic result Eq. (2) shown in the inset is only valid for momenta $q<1/c\tau$, where the collision time $\tau$ is given by Eq. (4).

Figure 2

Upper panel: vertex corresponding to fermion-phonon interactions, Eq. (17). Lower panel left: diagram for the self-energy Keldysh matrix of right fermions, ${\mathrm{\Sigma }}_{+}(k,\epsilon )$. Lower panel right: diagram for the self-energy Keldysh matrix of right phonons, ${\mathrm{\Pi }}_{+}(q,\omega )$ (see the Supplemental Material [24]). The solid line represents the fermionic propagator defined through Eq. (16), and the wavy lines represent the right ($+$, black) or left ($-$, red) bosonic propagator defined through Eq. (8).