Two-Component Structure in the Entanglement Spectrum of Highly Excited States

We study the entanglement spectrum of highly excited eigenstates of two known models that exhibit a many-body localization transition, namely the one-dimensional random-field Heisenberg model and the quantum random energy model. Our results indicate that the entanglement spectrum shows a “two-component” structure: a universal part that is associated with random matrix theory, and a nonuniversal part that is model dependent. The nonuniversal part manifests the deviation of the highly excited eigenstate from a true random state even in the thermalized phase where the eigenstate thermalization hypothesis holds. The fraction of the spectrum containing the universal part decreases as one approaches the critical point and vanishes in the localized phase in the thermodynamic limit. We use the universal part fraction to construct an order parameter for measuring the degree of randomness of a generic highly excited state, which is also a promising candidate for studying the many-body localization transition. Two toy models based on Rokhsar-Kivelson type wave functions are constructed and their entanglement spectra are shown to exhibit the same structure.

Figure 1

Average entanglement spectrum of highly excited eigenstates for a system of size $L=16$, averaged over ten realizations of disorder (plotted in logarithmic scale). Panels (a)–(f) show the spectrum for $h=0.5,1.5,2,2.5,3$, and 6, respectively. The solid lines correspond to the spectrum of a completely random state (derived from a Marchenko-Pastur distribution), and is shown for reference. Insets: scaling of the average entanglement entropy $S^{(1)}$ with system size.

Figure 2

The order parameter defined as the fraction of the universal component in the full entanglement spectrum for the Heisenberg spin model (upper panel) and the QREM (lower panel).

Figure 3

The order parameter for the random-sign RK-type wave functions. Upper panel: $E(\mathit{\sigma })=E_{\mathrm{REM}}(\mathit{\sigma })$. Lower panel: $E(\mathit{\sigma })=-(J/L)\sum _{i<j}{\sigma }_i^z{\sigma }_j^z$ with $J=1$.