Entanglement entropy scaling laws from fluctuations of non-conserved quantities

Abstract

Entanglement patterns reveal essential information on many-body states and provide a way to classify quantum phases of matter. However, experimental studies of many-body entanglement remain scarce due to their unscalable nature. The present work aims to mitigate this theoretical and experimental divide by introducing reduced fluctuations of observables, consisting of a sum of on-site operators, as a scalable experimental probe of the entanglement entropy. Specifically, we illustrate by Density Matrix Renormalization Group calculations in spin chains that the reduced fluctuations exhibit the same size scaling properties as the entanglement entropy. Generalizing previous observations restricted to special systems with conserved quantities, our work introduces experimentally feasible protocol to extract entanglement scaling laws.

Figures (2)

• Figure 1: a): Illustration of fluctuation-entanglement relation in spin chains. The subsystem spin fluctuations within a spin sector (red) are sensitive to bipartite entanglement. In contrast, the subsystem fluctuations between sectors (green), taking place when the total spin is not conserved, give rise to a spurious volume-law term independent on entanglement. b): Scaling of subsystem spin fluctuations and entanglement entropy in a critical chain. While the subsystem spin variance exhibits a volume-law scaling due to non-conservation, the reduced fluctuation $\delta_r^2S^z$, introduced in the present work, exhibits the same scaling as the entanglement entropy $\mathcal{S}_{vN}$. The reduced fluctuations are rescaled by a numerical factor for visualization purposes.
• Figure 2: Numerical results: a) Phase diagram in the $\gamma$--$J_z$ space. The critical lines given by $J_z=\pm(J+\gamma)/2$ are presented as dashed lines. Colorful shapes refer to parameters for which other marked figures are obtained. b) Critical (log) scaling of von Neumann entropy and $S^z$ half system fluctuations in the conserved case ($\gamma=0$). c) Area law scaling of von Neumann entropy and the reduced fluctuations of all spin operators. Inset: half system fluctuations of $S^x$ showing volume law scaling due to being non-conserved. d) Critical (log) scaling of von Neumann entropy and scaled reduced $S^z$ fluctuations. e) Same as d) but without entanglement entropy and showing all spin directions. f) Visibility for all spin operators along a cut in the phase diagram $J_z=0.7J$.