Escort distribution function of work done and diagonal entropies in quenched Luttinger liquids

We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing $a$, which is the order of the escort distribution, and depends on the universal combination $\left[\right|K_i-K_f{|/(K_i+K_F)]}^a$, with $K_i$, $K_f$ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal Rényi entropies and the many-body inverse participation ratio (IPR) are determined to evaluate the information content of the time-evolved wave function in terms of the eigenstates of the final Hamiltonian. The hierarchy of overlaps is dominated by that of the ground states. The IPR exhibits a crossover from Gaussian to power-law decay with increasing interaction quench parameter.

Figure 1

The escort PD function of work is plotted for $L/2\pi \alpha =10$ and ${tanh}^{2a}\left(\theta \right)=0.1$ (blue), 0.3 (red), 0.5 (black), 0.7 (green), and 0.9 (magenta) with peak position from left to right. The vertical arrow at $W=0$ denotes the Dirac delta function with spectral weight $P_a^{ad}$, given in Eq. (14). Inset: The evolution of the ${tanh}^{2a}\left(\theta \right)=0.5$ case for $L/2\pi \alpha =1$ (blue), 10 (red), 40 (black), 100 (green), and 200 (magenta). For the last four, the spectral weight of the Dirac $\delta$ peak is practically zero.

Figure 2

The Rényi entropies as a function of $K=K_f/K_i$ for $a=1/4$, 1/2, 1, 2, 4, and $\infty$ (from top to bottom) are plotted.