Dynamics of two particles with quasiperiodic long-range interactions

Abstract

We investigate the dynamics of two identical spinless fermions on a one-dimensional lattice with open boundary conditions (OBC), subject to quasiperiodic long-range interactions. Using numerical exact diagonalization (ED), we study this non-integrable system as a continuous-time quantum walk and uncover a robust correlated dynamical regime. This regime, characterized by an approximately constant inter-particle distance, emerges under sufficiently strong quasiperiodic modulation of the long-range interactions. Further, the study shows that the behavior is determined by the nature of the interaction and the choice of boundary condition. Notably, by tuning the phase of the quasiperiodic modulation, we observe three distinct manifestations of this phenomenon: localization, nearest-neighbor separation oscillations, and next-nearest-neighbor separation transitions -- each arising for specific initial separations. Furthermore, we identify the suppression of entanglement entropy in the system, including instances of oscillatory behavior. Our results highlight how quasiperiodic long-range interactions shape few-body quantum dynamics.

Figures (6)

• Figure 1: Probability density evolution and two-particle correlations. (a, b) Probability density $P(i,t)$ [defined in Eq. \ref{['eq:prob_density']}] up to time $t = 50$ for an initial inter-particle distance of 12, with (a) one particle initially at the boundary and (b) both particles away from the boundary. (c, d) Corresponding two-particle correlation functions $\Gamma(i,j)$ [defined in Eq. \ref{['eq:gamma']}] at $t = 30$ for the same initial configurations as (a) and (b), respectively. $L = 34$, $\Delta = 10$, $\varphi = 0$.
• Figure 2: Expected inter-particle distance and its variance. (a,b) Time evolution of the expected inter-particle distance $\langle r(t) \rangle$ [defined in Eq. \ref{['eq:r_avg']}] up to $t = 300$ for (a) $\Delta = 1$ and (b) $\Delta =10$. (c,d) At $t = 10$: (c) expected distance $\langle r \rangle$ and (d) variance [defined in Eq. \ref{['eq:variance']}] as a function of $\Delta$. Dashed line in (d): variance threshold $0.50 \rightarrow \Delta_0 = 7.58$. $L = 34$, $\varphi = 0$.
• Figure 3: Localization dynamics of two particles. The left (right) column shows initial configurations with one (neither) particle at the boundary. (a,b) For phase $\varphi = 3\pi/8$ and initial separation $9$. (c--f) For the fixed phase $\varphi = \pi/64$, with two distinct initial separations: (c,d) separation $= 2$; (e,f) separation $= 19$. $L = 23$, $\Delta = 10$.
• Figure 4: Nearest-neighbor separation oscillations at the boundaries. Evolution of (a) the probability density distribution and (b) the Loschmidt echo [defined in Eq. \ref{['eq:Loschmidt echo']}] up to $t = 10$. (c) Correlation function at $t = 1.09$, the moment when the Loschmidt echo $L(t)$ reaches its minimum. $L = 29$, $\Delta = 10$, $\varphi = 0$. (d) Angular frequency $\omega$ of the Loschmidt echo oscillation versus $\Delta E_1$. The 28 data points are selected from lattices of size $L = 8$–$71$ with $\Delta = 10$, $\varphi = 0$ or $\pi/2$, where oscillations are pronounced ($\omega \in (0, 2\pi]$).
• Figure 5: Next-nearest-neighbor separation transitions. (a) Interaction energies for initial separations of $17$, $18$, and $19$. (b) Time evolution of the probability density (two particles initially at boundaries) up to $t = 7$. (c) Correlation function at $t = 7$. $L = 20$, $\Delta = 10$, $\varphi = 3\pi/4$.