Lifetime of Almost Strong Edge-Mode Operators in One-Dimensional, Interacting, Symmetry Protected Topological Phases

Almost strong edge-mode operators arising at the boundaries of certain interacting one-dimensional symmetry protected topological phases with $Z_2$ symmetry have infinite temperature lifetimes that are nonperturbatively long in the integrability breaking terms, making them promising as bits for quantum information processing. We extract the lifetime of these edge-mode operators for small system sizes as well as in the thermodynamic limit. For the latter, a Lanczos scheme is employed to map the operator dynamics to a one-dimensional tight-binding model of a single particle in Krylov space. We find this model to be that of a spatially inhomogeneous Su-Schrieffer-Heeger model with a hopping amplitude that increases away from the boundary, and a dimerization that decreases away from the boundary. We associate this dimerized or staggered structure with the existence of the almost strong mode. Thus, the short time dynamics of the almost strong mode is that of the edge mode of the Su-Schrieffer-Heeger model, while the long time dynamics involves decay due to tunneling out of that mode, followed by chaotic operator spreading. We also show that competing scattering processes can lead to interference effects that can significantly enhance the lifetime.

Figure 1

Decay rate of $A_{\infty }$ obtained from ED for different $L$ and $J_z$ behaves as $\mathrm{\Gamma }={[\max (J_z,J_y)]}^{L/2}$, where $g\ll J_{z,y}\ll 1$. When $J_z=J_y$, destructive interference between two scattering channels leads to a pronounced increase of the lifetime, as indicated by the cusp.

Figure 2

Top panel: The $b_n$ for the Pauli spins on the first site ${\sigma }_1^{x,y,z}$ for $J_z=0$, $\gamma >0$. The model maps to free fermions and supports an SM with overlap with ${\sigma }_1^x$. The deviation from perfect staggered behavior for $n>10$ is a finite-size effect. Bottom panel: $A_{\infty }$ from ED for ${\sigma }_1^{x,y,z}$. Because of overlap with the SM, ${\sigma }_1^x$ persists up to $t\sim {10}^4$ as opposed to ${\sigma }_1^{y,z}$ that decay by $t=O(1)$.

Figure 3

Top panel: The $b_n$ for increasing $J_z$ with finite-size effects appearing as a plateau for $n>20$. As $L$ is increased [29], the linear ramp is extended, with the $b_n$ plateauing at a larger $n$ and at a larger value. Bottom panel: $A_{\infty }$ shows rapid decrease in lifetime with increasing $J_z$.

Figure 4

Top panel: $A_{\infty }$ from ED for $L=14$, with $J_z=g$ increasing from the top to the bottom of the panel. It is compared with the approximate $A_{\infty }$ obtained from time evolution in a Krylov basis with $\sim 40$ exact $b_n$ [29] and $b_{40<n<n_{\max }}=b_{40}$ held constant, mimicking a reservoir. We choose $n_{\max }=2e5$, which is large enough to capture the decay of the ASM before finite-size effects set in. Bottom panel: Decay rate extracted from both numerical methods. For $1/J_z<4$, the overlapping lines for $L=12$, 14 indicate saturation of the lifetime.