Characterizing dynamical behaviors in topological open systems with boundary dissipations

TL;DR

This work studies the open-system dynamics of the Su-Schrieffer-Heeger model under boundary dissipation described by Lindblad master equations. It employs the third-quantization framework to map the Liouvillian spectrum to a rapidity spectrum derived from a P matrix, enabling exact analysis across topological phases. A key finding is a dynamical duality between weak and strong dissipation in the topologically nontrivial phase, contrasted with its absence in the trivial phase, along with an exponential versus power-law scaling of the Liouvillian gap when dissipation is applied at a single boundary; the exponential scaling is explained by a dark state localized at the far boundary. These results reveal how topology imprints distinctive dynamical signatures in open quantum systems and guide future experimental investigations of boundary-dissipative topological matter.

Abstract

We investigate the dynamics of the Su-Schrieffer-Heeger model with boundary dissipations described by Lindblad master equations and unravel distinct dynamical features in the topologically different phases of the underlying Hamiltonian. By examining the long-time damping dynamics, we uncover a dynamical duality phenomenon between the weak and strong dissipation region, which exists only in the topologically non-trivial phase, linked to the structure of the Liouvillian spectra,particularly the stripe closest to the steady state. When dissipation is confined to a single boundary, the dynamical duality phenomenon still exists. Under this condition, the Liouvillian gap fulfills an exponential size scaling relation in the topologically non-trivial phase and a power-law size scaling relation in the topologically trivial phase. Within the topologically non-trivial region, we identify the existence of boundary-localized dark states in the thermodynamical limit, which is responsible for the exponential size decay of Liouvillian gap.

Figures (8)

• Figure 1: The Liouvillian spectra, the stripe of Liouvillian spectra closest to the steady state and rapidity spectra with $N=6$, and (a1, a2, a3) $t_{1} =0.1, t_{2}=1$ or (b1, b2, b3) $t_{1} =10, t_{2}=1$. The blue points represent the Liouvillian spectra of the parameters with $\gamma_{l}=\gamma_{r}=0.2$ while the red ones represent that of the parameters with $\gamma_{l}=\gamma_{r}=5$. The eigenvalues of Liouvillian spectra in all panels satisfy $\Re[\lambda]\leq 0$ and the data of Liouvillian spectra are consistent with the ones by exact diagonalization.
• Figure 2: The dynamical evolution of the particle number density with (a1) $N=6$, $t_{1}=0.1$ and $t_{2}=1$, and (a2) $N=50$, $t_{1}=0.1$ and $t_{2}=1$, and (b1) $N=6$, $t_{1}=10$ and $t_{2}=1$, and (b2) $N=50$, $t_{1}=10$ and $t_{2}=1$. Both left and right boundaries are loss.
• Figure 3: The dynamical evolution of the particle number density with (a1) $N=6$, $t_{1}=0.1$ and $t_{2}=1$, and (a2) $N=50$, $t_{1}=0.1$ and $t_{2}=1$, and (b1) $N=6$, $t_{1}=10$ and $t_{2}=1$, and (b2) $N=50$, $t_{1}=10$ and $t_{2}=1$. The left boundary is loss and the right boundary is gain.
• Figure 4: The dynamical evolution of the particle number density with (a1, a2) $N=50$, $t_{1}=0.1$, $t_{2}=1$, and (b1, b2) $N=50$, $t_{1}=10$, $t_{2}=1$.
• Figure 5: The Liouvillian gap with $N=200$ in (a1, a2). Finite size scaling of the Liouvillian gap with (a2) $t_{1}=0.5, t_{2}=1, \gamma=0.2$ and (b2) $t_{1}=0.5, t_{2}=1, \gamma=2$ and (a3) $t_{1}=2, t_{2}=1, \gamma=0.2$ and (b3) $t_{1}=2, t_{2}=1, \gamma=2$. The blue points are the data by strict diagonalization and the red dashed lines are the fit data.