Local ergotropy dynamically witnesses many-body localized phases

Abstract

Many-body localization is a dynamical phenomenon characteristic of strongly interacting and disordered many-body quantum systems which fail to achieve thermal equilibrium. From a quantum information perspective, the fingerprint of this phenomenon is the logarithmic growth of the entanglement entropy over time. We perform intensive numerical simulations, applied to a paradigmatic model system, showing that the local ergotropy, the maximum extractable work via local unitary operations on a small subsystem in the presence of Hamiltonian coupling, dynamically witnesses the change from ergodic to localized phases. Within the many-body localized phase, both the local ergotropy and its quantum fluctuations slowly vary over time with a characteristic logarithmic law analogous to the behaviour of entanglement entropy. This showcases how directly leveraging local control, instead of local observables or entropies analyzed in previous works, provides a thermodynamic marker of localization phenomena based on the locally extractable work.

Figures (8)

• Figure 1: (a) Quantum spin chain divided into the subsystem $S$ (first two spins), and environment $E$ (remaining spins). The initial Néel state is depicted. (b) Entanglement entropy as a function of time, for the $2,(N-2)$ partition, comparing the MBL phase ($J_z/J_{\perp}=0.2$, green triangle) and AL phase ($J_z/J_{\perp}=0.0$, blue circle). Results are obtained by averaging over $10^3$ disorder realizations, with chain length $N=8$, and disorder strength $W/J_{\perp}=5$.
• Figure 2: Local ergotropy as a function of time in the three phases: AL (blue circle) for $J_z/J_{\perp}=0.0$ and $W/J_{\perp}=5$; MBL (green triangle) for $J_z/J_{\perp}=0.2$ and $W/J_{\perp}=5$; and ERG (red diamond) for $J_z=0.2$ and $W=0$. The inset provides a zoomed-in view of the local ergotropy as a function of time for the two localized phases. Results are obtained by averaging over $10^3$ disorder realizations, with chain length $N=8$.
• Figure 3: Local ergotropy $\mathcal{E}_{S}$ (green circle) and switch-off ergotropy $\mathcal{E}_{SO}$ (orange triangle), both in units of $J_{\perp}$, as functions of time for the system in MBL phase. The inset provides a view over the difference between local ergotropy and the switch-off ergotropy $\mathcal{E}_{S}-\mathcal{E}_{SO}$. Results are obtained by averaging over $2.6\cdot 10^3$ disorder realizations, with chain length $N=12$, disorder strength $W/J_{\perp}=5$, and $J_z/J_{\perp}=0.2$.
• Figure 4: Relative quantum fluctuations of the local ergotropy as a function of time in the MBL ($J_z/J_{\perp}=0.2$, green triangle) and AL phase ($J_z/J_{\perp}=0.0$, blue circle). Results are obtained by averaging over $10^3$ disorder realizations for the AL case and approximately $2.6 \cdot 10^3$ disorder realizations for the MBL case, both with chain length $N=8$, and disorder strength $W/J_{\perp}=5$.
• Figure 5: Entanglement entropy for the $N/2,N/2$ partition, as a function of time in the three phases: AL (blue circle, $J_z/J_{\perp}=0$, $W/J_{\perp}=5$), MBL (green triangle, $J_z/J_{\perp}=0.2$, $W/J_{\perp}=5$), and ERG phase (red diamond, $J_z/J_{\perp}=0.2$, $W=0$). The dotted line indicates the saturation value of entanglement entropy for the system. The inset provides a zoomed-in view of the local ergotropy as a function of time in the two localized cases. Results are obtained by averaging over $10^3$ disorder realizations, with chain length $N=8$.