Absence of Orthogonality Catastrophe after a Spatially Inhomogeneous Interaction Quench in Luttinger Liquids

We investigate the Loschmidt echo, the overlap of the initial and final wave functions of Luttinger liquids after a spatially inhomogeneous interaction quench. In studying the Luttinger model, we obtain an analytic solution of the bosonic Bogoliubov–de Gennes equations after quenching the interactions within a finite spatial region. As opposed to the power-law temporal decay following a potential quench, the interaction quench in the Luttinger model leads to a finite, hardly time-dependent overlap; therefore, no orthogonality catastrophe occurs. The steady state value of the Loschmidt echo after a sudden inhomogeneous quench is the square of the respective adiabatic overlaps. Our results are checked and validated numerically on the $XXZ$ Heisenberg chain.

Figure 1

The low-energy dynamics of the $XXZ$ Heisenberg chain is faithfully represented by the Luttinger model. An inhomogeneous interaction quench in the Luttinger model is equivalent to the continuum limit of a harmonic chain, sketched in the figure, where the spring constants (and/or masses) of a region of size $l$ are altered abruptly at $t=0$.

Figure 2

The long time limit of the LE after a local interactions quench for $L{q}_c=4000$ is evaluated for the Luttinger model from Eq. (10). The inset depicts the time evolution for $K_{0}gl{q}_c/{\mathsf{v}}_F=-1$ and 2 from Eq. (12).

Figure 3

The long time limit of the LE for the Luttinger model with $L{q}_c=2000$ is shown for $l{q}_c=40$ (blue curve), 120 (red curve), and 500 (black curve) evaluated from Eq. (12). The latter is indistinguishable from that after a homogeneous quench. The insets show the time dependence of the LE for the Luttinger model for various $K$’s and $l{q}_c=40$ (blue solid line), 80 (red dash-dotted line), and 120 (black dashed line) from Eq. (10).

Figure 4

The inhomogeneous Loschmidt echo of the $XXZ$ chain for $L=300$ from tDMRG, starting from the $XX$ point to several final $J_z/J$ for $l=10$ (blue circles), 40 (red squares), and 120 (black triangles), and the green dashed line is Eq. (14). The inset shows the time evolution of the LE for $J_z=0.5J$ ($K=3/4$) and $-0.5J$ ($K=3/2$) for several $l$’s, as indicated by the legend. The pentagons indicate the long time asymptotes of $\ln [\mathcal{L}(t\to \infty )]/L$ after a homogeneous quench from Ref. [31] in the $L\to \infty$ limit.