Edge States Effects in Quantum Work Statistics

TL;DR

The paper investigates how boundary edge states in a solvable open-boundary Ising-like chain affect quantum work statistics under a local $\delta(t)$-kick at an impurity. By exactly solving the model and employing a two-time energy measurement framework, it shows that edge states strongly reshape the work distribution $P(w)$, producing edge singularities with exponents that depend on the impurity strength $\mu$ and bulk field $h$, and even generating mid-energy features or delta peaks when two edge modes are present. The approach highlights a direct link between boundary physics and energetic cost of local control, providing analytic predictions for low-, mid-, and high-energy fingerprints that are testable in platforms like Rydberg quantum simulators. The study thus advances understanding of nonequilibrium boundary phenomena and their thermodynamic signatures in quantum many-body systems, with potential generalizations to finite-quench durations and Floquet driving.

Abstract

Motivated by the objective of quantifying the energetic cost of accessing boundary phases through local control, we investigate here a simple, analytically tractable quantum impurity model. This model exhibits a rich boundary phase diagram, characterized by phases with different numbers of edge states. By considering a local quench protocol that drives the system out of equilibrium, we calculate exactly the resulting quantum work distribution across these phases. Our results show that the presence of edge states strongly alters this distribution. In particular, we analytically determine key fingerprints of these states both near the low-energy threshold and in the high-energy region.

Figures (6)

• Figure 1: $\matholdcal P(w)$ in the region $\sqrt{1-1/h}\leq\mu\leq\sqrt{1+1/h}$ with $h=1.5$. Insets highlight the low- and high-energy edge singularities of $\matholdcal P(w)$, where the dashed and dotted black lines correspond to Eqs. (\ref{['edge_sing_deloc_deloc']}) and (\ref{['edge_sing_boundary_h_right']}) (see also the text below this equation). The system spectrum is indicated in the middle of the figure.
• Figure 2: $\matholdcal P(w)$ for $\mu=0.2$ and $h=1.5$ (in the region $\mu<\sqrt{1-1/h}$ and $h>1$). The blue line represents the pure bulk states contribution (times $10^2$), while the orange one has the edge state contribution (their sum gives $\matholdcal P(w)$; see the green line in the insets). The insets, plotted on a log-log scale, highlight the low- and high-energy edge singularities of $\matholdcal P(w)$, where the dashed and dotted black lines correspond to Eq. (\ref{['edge_sing_mode2']}). The system spectrum is indicated in the middle of the figure, where the dashed line marks the energy of the edge state.
• Figure 3: $\matholdcal P(w)$ for $\mu = 2.0$ and $h = 1.5$ (in the region $\mu > \sqrt{1 + 1/h}$ and $h > 1$). The same color scheme as in Fig. \ref{['fig_WorkDist_h1.5_mu_mode2']} is used. The dashed and dotted black lines in the insets correspond to Eq. (\ref{['edge_sing_mode2_2']}). The system spectrum is shown in the middle of the figure, with the dashed line indicating the energy of the edge state.
• Figure 4: $\matholdcal P(w)$ for $\mu=1.5$ and $h=0.5$ (in the region $\mu<\sqrt{1+1/h}$ and $h<1$). The same color scheme as in Fig. \ref{['fig_WorkDist_h1.5_mu_mode2']} is used. The dashed and dotted black lines in the insets correspond to Eq. (\ref{['edge_sing_mode1']}). The system spectrum is indicated in the middle of the figure, where the dashed line marks the exponentially small energy of the edge state in this case.
• Figure 5: $\matholdcal P(w)$ for $\mu=\sqrt{1+1/h}$ and $h=0.5$. The same color scheme as in Fig. \ref{['fig_WorkDist_h1.5_mu_mode2']} is used. The dashed, dotted and dashed-dotted black lines in the insets correspond to Eq. (\ref{['edge_sing_mode1_boundary']}). Again, the system spectrum is indicated in the middle of the figure.