How a Small Quantum Bath Can Thermalize Long Localized Chains

We investigate the stability of the many-body localized phase for a system in contact with a single ergodic grain modeling a Griffiths region with low disorder. Our numerical analysis provides evidence that even a small ergodic grain consisting of only three qubits can delocalize a localized chain as soon as the localization length exceeds a critical value separating localized and extended regimes of the whole system. We present a simple theory, consistent with De Roeck and Huveneers’s arguments in [Phys. Rev. B 95, 155129 (2017)] that assumes a system to be locally ergodic unless the local relaxation time determined by Fermi’s golden rule is larger than the inverse level spacing. This theory predicts a critical value for the localization length that is perfectly consistent with our numerical calculations. We analyze in detail the behavior of local operators inside and outside the ergodic grain and find excellent agreement of numerics and theory.

Figure 1

LIOMs (blue) coupled to an ergodic grain (red) modeled by a Gaussian Orthogonal Ensemble (GOE) matrix.

Figure 2

Probability distributions and average of the adjacent gap ratio $r$ for different coupling strengths and system sizes compared to the distribution obtained for large GOE matrices.

Figure 3

Top: Standard deviation of the eigenvector expectation of the magnetization at site $i$ ($i<0$ is in the bath, and $i\ge 0$ are LIOMs) for various total length $L$. Bottom: Full distribution for $L=16$ and different positions in the chain.

Figure 4

Left panels: Second Rényi entropy $S_2$ for excitation created by the operator ${\sigma }_i^x$ in an eigenstate. Right panels: Evolution of $S_2$ at some given sites as a function of the total length $L$.

Figure 5

Entanglement entropy of the last four l-bits in the chain.