Edge Singularities and Quasilong-Range Order in Nonequilibrium Steady States

The singularities of the dynamical response function are one of the most remarkable effects in many-body interacting systems. However in one dimension these divergences only exist strictly at zero temperature, making their observation very difficult in most cold atomic experimental settings. Moreover the presence of a finite temperature destroys another feature of one-dimensional quantum liquids: the real space quasilong-range order in which the spatial correlation functions exhibit power-law decay. We consider a nonequilibrium protocol where two interacting Bose gases are prepared either at different temperatures or chemical potentials and then joined. We show that the nonequilibrium steady state emerging at large times around the junction displays edge singularities in the response function and quasilong-range order.

Figure 1

Nonequilibrium protocol to observe edge singularities in the NESS. Top: two quantum gas at different density $n$ are prepared and then joined together, Bottom. After waiting some time that the system reaches its steady state close to the junction (see plot of the density $n(x)$ on the right) a Bragg pulse is shined around the center of the cloud to probe its dynamical correlations.

Figure 2

Density plot of the DSF (in unit of $n/{\omega }_F$) computed on the NESS (left) and on the ground state (right) of a gas, with the density $n=n_{\mathrm{NESS}}$ and the coupling strength $c=10$ ($k_F=\pi n$, ${\omega }_F=k_F^2$). The NESS is obtained by joining a left gas with $T_L=1$, $n_L=1$ and a right gas with $T_R=1$, $n_R=0.1$. The coupling strength $c=10$ of both gases is the same (density $n_{\mathrm{NESS}}=0.54$). In the NESS and for $k>0$, the DSF is divergent close to the particle excitations ${\omega }_{+}(k)\sim k^2/2m_0$, and it has a zero in proximity of the hole edge ${\omega }_{-}(k)\sim -k^2/2m_0$. For a negative $k$, the situation is reversed. In the ground state, the DSF has a pole (zero) around the particle (hole) edge ${\omega }_{\pm }(k)\sim v_s|k|\pm k^2/2m_F$ instead, see for example [25, 113].

Figure 3

DSF (9) of the NESS (red line) and on a thermal state with $T=1$, $n=n_{\mathrm{NESS}}$, and $c=4$. Plots on top display data with $k=k_F$ and the ones on the bottom with $k=-k_F$ ($k_F=\pi n$). The NESS is the same as the one computed in Fig. 2.

Figure 4

The distribution function $\vartheta (\lambda )$ for the ground state (top) and the NESS state (bottom). For a fixed momentum $k$, a particle-hole excitation of the ground state can be created only for a finite interval of energies ${\omega }_{+}(k)-{\omega }_{-}(k)$. The edge excitations correspond to a configuration where either particle or hole is right at the edge of the Fermi sea ${\lambda }_F$. On the contrary, over the NESS state, particle-hole pairs of arbitrary energy can be created; however, there are again two edge modes where either particle or hole is placed at the discontinuity ${\lambda }_0$.