Defect-Driven Nonlinear and Nonlocal Perturbations in Quantum Chains

Abstract

Transport and localization in isolated quantum systems are typically attributed to spatially-extended disorder, leaving the influence of a few controllable defects largely unexplored despite their relevance to engineered quantum platforms. We introduce an analytic framework showing how a single defect profoundly reshapes wave-function spreading on a finite isolated and periodic tight-binding lattice. Adapting the defect technique from classical random-walk studies, we obtain exact time-resolved site-occupation probabilities and several observables of interest. Even one defect induces striking nonlinear and nonlocal effects, including non-monotonic suppression of transport, enhanced localization at distant sites, and strong sensitivity to the initial particle position at long times. These results demonstrate that minimal perturbations can generate unexpected long-time transport signatures, establishing a microscopic defect-driven mechanism of quantum localization.

Figures (6)

• Figure 1: MSD $\Delta_2(t)$ versus $t$ without defect; $\gamma=1$, and for $N=150$ and $200$, and initial site $n_0=75$ and $100$, respectively. The dashed line denotes the steady-state value $\overline{\Delta}_2$, Eq. \ref{['steadystateMSD_defectfree']}. For the timescale $t^\star$ over which oscillations appear, the inset shows $t^\star \sim N/\gamma$.
• Figure 2: Steady-state probability $\overline{P}_n^{(d)}$ versus $n$ for defect strength $q = 0.3,0.5,1.5,10,20$ on a lattice of size $N=50$ and $\gamma=1$. In panel (a), the initial site coincides with the defect site, $n_d = n_0 = 2$. Inset: Probability at $n_d$ increases monotonically with $q$ and asymptotically reaches $1$ as $q \to \infty$. In panel (b), the initial site is located away from the defect site: $n_d = 4$, $n_0 = 2$. Inset: Non-monotonic dependence of the probability at sites $n_0$ and $n_d$ on the defect strength; the probability at both sites asymptotically reaches values given by Eq. \ref{['eq:Pnd-qinf']}, see the dashed lines.
• Figure 3: Steady-state mean displacement $\overline{\Delta}^{(d)}_1$ (panel (a)) and MSD $\overline{\Delta}^{(d)}_2$ (panel (b)) versus $q$ with varying defect location $n_d=2,3,4,5,6$, initial location $n_0=2$, system size $N=200$, $\gamma=1$. The lines correspond to analytical results obtained using the steady-state version of Eqs. \ref{['eq:average-Mean-q']} (panel (a)) and \ref{['eq:average-MSD-q']} (panel(b)), while points denote numerical results from unitary evolution of the dynamics. In the inset of panel (b), for the case of $n_d=4$, we show $\overline{\Delta}^{(d)}_2$ asymptotically approaching the $q\to\infty$ value (dashed line).
• Figure 4: $\overline{P}_n^{(d)}$ versus $n$ for $q \to \infty$ and for even and odd $N$. Panel (a): $N=50$, $n_d=25$, $n_0=22$; Panel (b): $N=55$, $n_d=25$, $n_0=30$. Insets: $\overline{P}_n^{(d)}$ versus $n$ for different initial locations, $n_0=15$ (a) and $n_0=40$ (b). In all cases, $\gamma=1$.
• Figure 5: Defect-free mean displacement and MSD, showing agreement between theory (Appendix A) and numerical results from unitary evolution of the dynamics ($\gamma=1, N=50$).