Time evolution during and after finite-time quantum quenches in Luttinger liquids

Abstract

We consider finite-time quantum quenches in the interacting Tomonaga-Luttinger model, for example time-dependent changes of the nearest-neighbour interactions for spinless fermions. We use the exact solutions for specific protocols including the linear and cosine ramps (or, more generally, periodic pumping). We study the dynamics of the total and kinetic energy as well as the Green functions during as well as after the quench. For the latter we find that the light-cone picture remains applicable, however, the propagating front is delayed as compared to the sudden quench. We extract the universal behaviour of the Green functions and in particular provide analytic, nonperturbative results for the delay applicable to quenches of short to moderate duration but arbitrary time dependency.

Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids

“…to determine analytically the explicit space dependence of the sound velocity. Similar curved light-cones have also been observed in time-dependent quenches [30][31][32][33], where the velocity depends on time but not position. In the present work we focus on space-dependent velocities instead.…”

Time evolution during and after finite-time quantum quenches in the transverse-field Ising chain

“…An additional simplification is the fact that we may use translational invariance of the system to write S ij = S(j − i), Q ij = Q(j − i) and G ij = G(j − i). It is then evident that the corresponding matrices in (24) and (25) are block-Toeplitz matrices, with entries on each descending diagonal in a block identical, which reduces the computational effort. The behaviour of these functions depends on the form and duration of the quench and will be discussed in subsequent sections.…”

Finite-time quantum quenches in the XXZ Heisenberg chain

“…We have compared this delay with analytical results obtained for finite-time quenches in the Luttinger model. 42 We find very good agreement for short and intermediate quench durations, with deviations only showing up for slow, strong quenches. We conclude that, despite the non-equilibrium nature of the quantum quench setup, the Luttinger model can be used to adequately describe features of the time evolution in the XXZ chain after a finite-time quantum quench.…”

“…For the above mentioned Luttinger model this finitetime quench protocol has been studied in several works. 31,[34][35][36][37][38][39][40][41][42][43][44] In particular, the light-cone spreading in two-point correlation functions was found 35,42 to be delayed as compared to the light cone after sudden quenches, with the delay being related to the length and form of the finite-time quench protocol. Comparisons between the Luttinger model predictions and numerical simulations for interacting microscopic models have been limited so far, 31,35 with a detailed analysis of the lightcone spreading and aforementioned delay for the Heisenberg chain still missing.…”