Inhomogeneous Quantum Quenches

TL;DR

The authors study inhomogeneous quantum quenches in one dimension using both massless conformal field theory and free-field theory, deriving general analytic expressions for energy flow, correlations, and entanglement entropy. They map the problem to a variable-width Euclidean strip and reveal a horizon structure with wave-like propagation, then validate and extend these results via perturbative free-field calculations for massive cases. A quasiparticle interpretation explains the light-cone spreading and entropy growth, and a careful comparison shows qualitative and quantitative consistency with a notable non-diffusive evolution unlike the thermal diffusion analogue. The work clarifies how initial inhomogeneities affect late-time local observables and highlights fundamental differences between quantum and thermal relaxation in inhomogeneous quenches.

Abstract

We study the problem of a quantum quench in which the initial state is the ground state of an inhomogeneous hamiltonian, in two different models, conformal field theory and ordinary free field theory, which are known to exhibit thermalisation of finite regions in the homogeneous case. We derive general expressions for the evolution of the energy flow and correlation functions, as well as the entanglement entropy in the conformal case. Comparison of the results of the two approaches in the regime of their common validity shows agreement up to a point further discussed. Unlike the thermal analogue, the evolution in our problem is non-diffusive and can be physically interpreted using an intuitive picture of quasiparticles emitted from the initial time hypersurface and propagating semiclassically.

Figures (5)

• Figure 1: Schematic representation of the transformation from the VWS to the CWS.
• Figure 3: Evolution of the entanglement entropy in the case of a step distribution, with the strip width smoothly varying over a distance of order $\epsilon=1$, from 2 on the left of the origin to 4 on the right. The actual transformation used is $f(z)=\frac{1}{2}\log(1+e^{2z})$ in which case the inverse transformation can be found analytically. The subsystem $A$ under consideration is of length $l=10$ and placed in 3 different positions with respect to the origin: with its middle at $x_{m}=-30$ (i), 0 (ii) and 30 (iii), all in units of $\epsilon=1$. We notice that in each case $S_{A}$ first saturates to the homogeneous saturation value that corresponds to the local (i,iii) or average width (ii), at time $t=l/2=5$. Note the different slopes and offset values before the saturation. In cases (i,iii) and at time $t=25$, i.e. the distance of the closest to the origin boundary of $A$, $S_{A}(t)$ starts changing again, since the first quasiparticles from the opposite half of space enter $A$. After time equal to $l=10$, the entropy saturates again to its final value which is common for all positions and equal to the homogeneous value that corresponds to the average width. In the plot we used (\ref{['ent3']}), (\ref{['tr5']}) and (\ref{['F0d']}) which are exact not only for infinitesimal transformations.
• Figure 4: Physical explanation of (\ref{['cor-f-dq1']}). The connected two-point correlation function is determined by the values of the initial distribution at the overlap of their horizons (red thick line).
• Figure 5: Illustration of the physical interpretation of the entanglement entropy evolution using the concept of entangled quasiparticles. Pairs of entangled quasiparticles emitted from the same point on the $t=0$ hypersurface. One of the quasiparticles of the pair denoted by dark red colour is inside subsystem $A$ at some time $t$, while the other is in its complement $B$. Therefore this pair contributes to the entanglement $S_{A}(t)$ between $A$ and $B$ at that time. In contrast, pairs like those denoted by light grey colour whose quasiparticles are both in the same subsystem, either $A$ or $B$, do not contribute to $S_{A}(t)$.The red thick lines denote the regions on the $t=0$ hypersurface where entangled quasiparticles that contribute to $S_{A}(t)$ come from. For $t<l/2$ these regions are $[x_{1}-t,x_{1}+t]$ and $[x_{2}-t,x_{2}+t]$, while for $t>l/2$ they are $[x_{1}-t,x_{2}-t]$ and $[x_{1}+t,x_{2}+t]$.
• Figure 6: Contour integration for the derivation of $f(z)$.