Eigenstate Thermalization in a Locally Perturbed Integrable System

Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-$1/2$ $XXZ$ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.

Figure 1

Diagonal matrix elements of $\widehat{T}$ [(a), (b)] and $\widehat{K}$ [(c), (d)] in the eigenstates of ${\widehat{H}}_{XXZ}$ [(a), (c)] and ${\widehat{H}}_{\mathrm{SI}}$ [(b), (d)] ($\mathrm{\Delta }=0.55$). The black lines show microcanonical averages (within windows with $\delta {\epsilon }_n=0.008$) in ${\widehat{H}}_{XXZ}$ for the largest chain ($N=20$). The insets in (a) and (c) show the equivalence of the microcanonical predictions in both models for each observable, while the insets in (b) and (d) show the $(N\mathcal{D}{)}^{-1/2}$ and ${\mathcal{D}}^{-1/2}$ scalings, respectively, of $\overline{|\delta O_{nn}|}≔\overline{|O_{nn}-O_{n+1n+1}|}$ (the dashed lines are $\propto x^{-1/2}$), where we average over the central 20% of the eigenstates in chains with $N=10,12,\dots ,20$.

Figure 2

[(a), (b)] Off-diagonal matrix elements of $\widehat{T}$, and the corresponding coarse-grained average [continuous (black) line], plotted vs $\omega$ for chains with $N=18$. [(c), (d)] Coarse-grained averages of $T_{nm}$, including the ones reported in (a) and (b), for different chain sizes. [(e), (f)] Coarse-grained averages of $J_{nm}$ for different chain sizes (the insets show results at low $\omega$, see also Fig. 4). The left panels [(a), (c), and (e)] show results for ${\widehat{H}}_{XXZ}$, while the right ones [(b), (d), and (f)] show results for ${\widehat{H}}_{\mathrm{SI}}$ ($\mathrm{\Delta }=0.55$). The matrix elements were computed within a small window of energy around $\overline{E}\approx 0$ (center of the spectrum) of width $0.05ϵ$ ($0.075ϵ$ for the insets), where $ϵ≔E_{\max }-E_{\min }$. The coarse-grained averages were computed using a window $\delta \omega =0.1$ [$\delta \omega =0.075$ and $\delta \omega =0.01$ for the insets in (e) and (f), respectively].

Figure 3

${\mathrm{\Gamma }}_{\widehat{O}}(\omega )$, see Eq. (8), for the total kinetic energy per site [(a), (b)] and for the current operator [(c), (d)], in the $XXZ$ [(a), (c)] and single-impurity [(b), (d)] models ($\mathrm{\Delta }=0.55$). The horizontal line in (b) and (d) marks $\pi /2$. The matrix elements were computed using the same energy window as in Fig. 2, while the coarse-graining parameter is $\delta \omega =0.05$.

Figure 4

Scaled variances of the off-diagonal matrix elements of $\widehat{J}$ in the eigenstates of ${\widehat{H}}_{XXZ}$ (a) and ${\widehat{H}}_{\mathrm{SI}}$ (b) plotted vs $N\omega$. The main panels (insets) show results for $\mathrm{\Delta }=0.55$ ($\mathrm{\Delta }=1.1$). The matrix elements were computed within a small window of energies around $\overline{E}\approx 0$ of width $0.075ϵ$. For the binned averages, we used $\delta \omega =0.075$ in (a) and $\delta \omega =0.01$ in (b).