Gumbel statistics for entanglement spectra of many-body localized eigenstates

An entanglement spectrum encodes statistics beyond the entanglement entropy, of which several have been studied in the context of many-body localization. We numerically study the extreme value statistics of entanglement spectra of many-body localized eigenstates. The physical information encoded in these spectra is almost fully carried by the few smallest elements, suggesting the extreme value statistics to have physical significance. We report the surprising observation of Gumbel statistics. Our result provides an analytical, parameter-free characterization of many-body localized eigenstates.

Figure 1

Density of ${\tilde{e}}_{\text{min}}$ for $W=4$ (top) and $W=5$ (bottom) at $L=10,12,14$, combined with the densities $g\left(x\right)$ and $h\left(x\right)$. Note the logarithmic scales on the vertical axes.

Figure 2

Density of ${\tilde{e}}_{\text{min}}$ for $W=2,3,4,5$ at $L=14$, combined with the densities $g\left(x\right)$ and $h\left(x\right)$. Note the logarithmic scales on the vertical axis.

Figure 3

Densities of $r_1$ for $W=2,3,4,5$ at $L=14$, combined with the densities for Poissonian and Wigner-Dyson statistics ($\beta =1$).