Power-Law Entanglement Spectrum in Many-Body Localized Phases

The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.

Figure 1

ES of the highly excited eigenstates of the $XXZ$ spin chain with disorder strength $W=5$. The spectrum has a power law form in the MBL phase and in the vicinity of the delocalization transition.

Figure 2

Power-law exponent $\gamma$, extracted from the fit of the ES, $⟨\ln {\lambda }_k⟩$, increases with disorder $W$. Theoretical prediction refers to $\gamma$ extracted from the scaling of the matrix elements in Ref. [39].

Figure 3

Distribution of ${\lambda }_1$ across the MBL transition for different $L$. Solid lines indicate full distribution, dashed lines show distribution of ${\lambda }_1$ between different disorder realizations. Disorder strength is $W=0.5$ (left), $W=2$ (middle), $W=6.5$ (right).

Figure 4

Number of singular values required to reproduce the EE with fixed precision (99%) decreases with disorder strength $W$ and saturates at strong disorder. The bars represent statistical fluctuations, which are most pronounced near the transition.