Dephasing-induced relaxation in tight-binding chains with linear and nonlinear defects

Abstract

We investigate thermalization in a tight-binding chain with an on-site defect subject to local dephasing noise implemented as random phase kicks. For a single linear defect of strength $ε$, we obtain an exact analytical description of the system spectrum and formulate the dephasing-induced dynamics in the eigenstate basis. We derive an approximate kinetic equation for mode populations that describes a continuous-time random walk in action space. The walk transition rates are defined by the overlap matrix encoding the spatial structure of eigenstates that can be computed exactly. Analyzing the spectral properties of the equation, we show that defect-induced localized modes act as bottlenecks that strongly slow down relaxation, with rates scaling as $ε^{-2}$ for strong defects. Using large-deviation theory, we characterize rare dynamical trajectories and identify distinct relaxation pathways associated with low- and high-activity regimes in action space. We provide numerical evidence that the large-deviation function exhibits a dynamical phase transition in the limit $ε\to \infty$. We then extend our analysis to the nonlinear case, considering a single nonlinear defect embedded in either a linear or a fully nonlinear discrete Schrödinger equation. Numerical simulations reveal a qualitatively faster approach to equilibrium driven by the amplitude-dependent weakening of the defect. Our results provide a unified framework for understanding thermalization, rare fluctuations, and relaxation pathways in stochastic tight-binding systems.

Figures (10)

• Figure 1: Spectrum of the defective Hamiltonian \ref{['eq:H_defect']} for a chain of size $N=21$ with a defect located at site $M=10$, and $C=2$. (a) Eigenvalues $E_\nu$ denoted by circles as a function of the defect strength $\epsilon$. The gray region denotes the energy band, with dashed black lines marking the band edges at $E=\pm 2C$. (b) Spatial distribution $|\xi_n^\nu|^2 \equiv |\braket*{n}{\xi^\nu}|^2$ of two representative eigenstates $\ket*{\xi^\nu}$ at $\epsilon=10$, marked by the blue ($E_\nu \approx -10.77$) and red ($E_\nu \approx -3.95$) crosses on panel (a). The blue curve (corresponding to the blue cross) depicts the defect-localized mode, while the red curve (corresponding to the red cross) represents an extended mode.
• Figure 2: (a) Dependence of the rates $\mu_i$ on the defect strength $\epsilon$. The points are obtained from Eq. \ref{['eq:mui-def']} with $\gamma=1$, using numerically computed eigenvalues $\lambda_i$ of the overlap matrix $W$ for $N=21$ and $C=2$. The gray dashed line highlights the relaxation rate $\mu_2$. (b) Trace of $W$ as a function of the lattice size $N$ for different defect strengths $\epsilon$, with $C=20$. In both panels, the matrix $W$ is constructed from Eq. \ref{['eq:W-def']} using numerically obtained eigenvectors $\ket*{\xi^\nu}$ of the defective Hamiltonian \ref{['eq:H_defect']}.
• Figure 3: Relaxation in trimer system with $N=3$. (a) Scaled relaxation rate $\mu_2$ as a function of $\epsilon$ showing a maximum value 0.75 at $\epsilon=C=2$. The line is obtained using Eq. \ref{['eq:3lev-tau']} with $\mu_2 = \tau^{-1}_{\mathrm{relax}}$, while the points denote $(1-\lambda_2)$, where $\lambda_2$ is obtained by numerically computing the eigenvalues of $W$. (b) Evolution of scaled total energy as a function of time $t$ for different values of $\epsilon$ and $C=2$. The lines are obtained from $2^{15}$ stochastic realizations of the dephasing dynamics of the linear model \ref{['eq:H_defect']} with $\kappa=2$, $\beta = 1$ so that $\gamma = 2/3$. Note that relaxation is the fastest for the red curve with $\epsilon=C$.
• Figure 4: (a) Scaled relaxation rate $\mu_2$ as a function of $\epsilon$ for different values of $N$. Lines are obtained by numerically computing the eigenvalue $\lambda_2$ of $W$ with $C=2$. (b) Evolution of total energy as a function of time $t$ for different values of $N$. The points are obtained from $2^{15}$ stochastic realizations of the dephasing dynamics of model \ref{['eq:H_defect']} with $C=\epsilon=\kappa=2$ and $\beta = 1$. The dashed lines denote the analytical form $E(t) = E(\infty) + (E(0) - E(\infty)) e^{-t/\tau_{\mathrm{relax}}}$.
• Figure 5: Relaxation of the absolute value of mode energy $|E_\nu I_\nu|$ in a chain of $N=501$ sites with $C=1$ and $\epsilon=20$. Two types of initial conditions are compared: one in which all the initial energy is fed into the localized mode (red solid line) with $E_\nu \approx -20.09975$ and one where an extended mode with $E_\nu \approx -1.99996$ is initially excited (green solid line). The dashed lines show the steady-state value of the mode energy, i.e., $|E_\nu|/ N$. The inset plot shows the same quantities in the semi-log scale. Trajectories are generated directly in mode space, using Eq.(\ref{['eq:modedyn']}) with uniform $g(\theta)$ so that $\kappa=2$ and an exponential $p(\tau)$ with $\beta=100$, and averaged over 100 stochastic realizations. The black dotted line in the inset depicts the expected exponential decay, $\exp(-\mu_2 t)$, computed as explained in the text.