Spectral Properties of One-Dimensional Fermi Systems after an Interaction Quench

We show that the single-particle spectral properties of gapless one-dimensional Fermi systems in the Luttinger liquid state reached at intermediate times after an abrupt quench of the two-particle interaction are highly indicative of the unusual nonequilibrium nature of this state. The line shapes of the momentum-integrated and -resolved spectral functions strongly differ from their ground state as well as finite temperature equilibrium counterparts. Using an energy resolution improved version of radio-frequency spectroscopy of quasi-one-dimensional cold Fermi gases, it should be possible to experimentally identify this nonequilibrium state by its pronounced spectral signatures.

Figure 1

(a) The momentum distribution functions of the TLM in the ground state, the intermediate-time nonequilibrium steady state following the interaction quench with $\widehat{g}=2$, and the thermal equilibrium state with temperature $T/(v_F{q}_c)\approx 0.2804$ for which the average energy is equal to the one quenched into the system. (b) The corresponding momentum-integrated spectral functions. The smooth features at $\omega /(v_F{q}_c)$ of order 1 are nonuniversal and depend on the chosen potential. The inset of (b) shows the nonuniversal $2\pi v_F{\rho }_{\text{ss}}^{<}(0)$ as a function of the interaction $\widehat{g}$ for the Gaussian potential, an exponential one $\widehat{g}(q)=\widehat{g}{e}^{-|q/q_c|}$, and a box potential $\widehat{g}(q)=\widehat{g}\mathrm{\Theta }(q_c-|q|)$.

Figure 2

The momentum-resolved spectral functions of the TLM in the ground state, the intermediate-time nonequilibrium steady state following the interaction quench with $\widehat{g}=2$, and the thermal equilibrium state with temperature $\tau =T/(v{q}_c)\approx 0.2168$ for which the average energy is equal to the one quenched into the system. The steady-state function shows a characteristic peak at positive energies for $k-k_F>0$. For this case, thermal curves with two additional $\tau$ are shown for comparison.

Figure 3

The momentum-integrated spectral function for the lattice model of spinless fermions with nearest-neighbor interaction $U$ after the inverse quench. (a) ${\rho }_{\text{ss}}^{<}(\omega )$ of the translational-invariant model. Inset: $|{\rho }_{\text{ss}}^{<}(\omega )-{\rho }_{\text{ss}}^{<}(0)|$ on a log-log scale indicating the power-law behavior with ${\gamma }_{\text{gs}}$ for $|\omega |\to 0$. The dashed lines are the expected power laws with $K$ taken from Bethe ansatz [26, 27]. The upper bound over which the power law manifests depends on $U$ [45] and $\mathrm{sgn}(\omega )$. At small $|\omega |$ the power law is cut off by ${t^{\prime }}^{-1}$. (b) Spectral function at the first site ${\rho }_t^{<}(1,\omega )$ for open boundaries with $U/J=1$ at different $t$. Inset: $d\ln |{\rho }_{\text{ss}}^{<}(1,\omega )-{\rho }_{\text{ss}}^{<}(1,0)|/d\ln |\omega |$, that is the effective exponent, in the steady state. The dashed lines indicate the expected exponents