Quantifying unitary flow efficiency and entanglement for many-body localization

We probe the bulk geometry of the Wegner Wilson Flow (WWF) in the context of many-body localization by addressing efficiency and bulk entanglement growth measures through approximating upper bounds on the boundary entanglement entropy. We connect these upper bounds to the Fubini-Study metric and clarify how a central quantity, the information fluctuation complexity, distinguishes bulk unitary rotation from entanglement production. We also give a short new proof of the small incremental entangling theorem in the absence of ancillas, achieving a dimensionindependent, universal factor of $c=2$.

Figure 1

Histogram and box plot of $\dot{E}\left(\rho \right)$ for a two-qubit system, over $2\times {10}^6$ realizations of pure states and Hamiltonians $H, {\left|\right|H\left|\right|}_{\infty }=1$. See main text for details on the sampling. The vertical lines in the box plot reflect interquartile ranges, while the mass of points starting around 0.4 denote outliers. The red line demarcates the bound defined in the main text.

Figure 2

Average WWF arclength $d_{FS}$ against $D_X$ for $L=8$. Colors denote varying disorder strengths; the red diagonal line is a guide to the eye. (Left inset) Median efficiency against disorder strength $W$ for both $L=8$ and $L=10$. Shading denotes standard deviation estimated via bootstrap sampling. (Right inset) Average entanglement entropy against WWF efficiency for $L=8$, with color coding the same as the main plot.

Figure 3

The three bulk entanglement measures $(D_{RS},D_{XY},D_X)$ against $E\left(\rho \right)$. The dashed lines denote $L=8$, while the solid lines denote $L=10$.

Figure 4

(Top) Average (over flow states) IFC ${\overline{\sigma }}_Y$ and eigenstate entanglement entropy against disorder strength $W$ for $L=8$ (left) and $L=10$ (right). Shading denotes standard deviation. (Bottom) Average IFC against $E\left(\rho \right)$ for $L=8$ (left) and $L=10$ (right). The points represent values averaged within a disorder realization, while the red line denotes the trend after averaging over disorder realizations.