Mobility edge in long-range interacting many-body localized systems

As disorder strength increases in quantum many-body systems a new phase of matter, the so-called many-body localization, emerges across the whole spectrum. This transition is energy dependent, a phenomenon known as mobility edge, such that the mid-spectrum eigenstates tend to localize at larger values of disorder in comparison to eigenstates near the edges of the spectrum. Many-body localization becomes more sophisticated in long-range interacting systems. Here, by focusing on several quantities, we draw the phase diagram as a function of disorder strength and energy spectrum, for a various range of interactions. Regardless of the underlying transition type, either second-order or Kosterlitz-Thouless, our analysis consistently determines the mobility edge, i.e., the phase boundary across the spectrum. We show that, long-range interaction enhances the localization effect and shifts the phase boundary towards smaller values of disorder. In addition, we establish a hierarchy among the studied quantities concerning their corresponding transition boundary and critical exponents. Interestingly, we show that deliberately discarding some information of the system can mitigate finite-size effects and provide results in line with the analytical predictions at the thermodynamic limit.

Figure 1

Top: Density of states (DoS) as a function of rescaled energy $\varepsilon$ for different disorder strengths in system with exponents (a) $\alpha =0.5$ and (b) $\alpha =1$. Bottom: DoS as a function of $\varepsilon$ in system with fixed disorder strength $W=3$ and exponents (c) $\alpha =0.5$, and (d) $\alpha =1$. The colored bars represent the domains from them $M=50$ consecutive eigenstates in the vicinity of rescaled energies $\varepsilon \in \{0.2,...,0.8\}$ have been selected. All the results have been obtained for a system of size $N=15$.

Figure 2

(a) the averaged gap ratio $\overline{\langle r\rangle }$, (b) the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$, (c) the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$, and (d) the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$ as a function of disorder strength $W$ for different system sizes. All the quantities are obtained for long-range system with interaction's exponent $\alpha =0.5$ in the midspectrum $\varepsilon =0.5$.

Figure 3

Top: The finite-size scaling analysis in MBL transition as a second-order phase transition. (a) the averaged gap ratio $\overline{\langle r\rangle }$, (b) the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$, (c) the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$, and (d) the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$ as a function of $sN/{\xi }_{{}_{SO}}$ with $s=sgn(W-\omega )$ and ${\xi }_{{}_{SO}}\sim {|W-\omega |}^{-\nu }$. For each desired quantity $O$, the extracted critical disorder ${\omega }_{{}_{SO}}^{{}_O}$, critical exponent ${\nu }^{{}_O}$, and the quality of data collapse $Q$ have been attached on the relevant panel. Bottom: The finite-size scaling analysis in MBL transition as a Kosterlitz-Thouless transition type. (e) the averaged gap ratio $\overline{\langle r\rangle }$, (f) the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$, (g) the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$, and (h) the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$ as a function of $sN/{\xi }_{{}_{KT}}$ with $s=sgn(W-\omega )$ and ${\xi }_{{}_{KT}}\sim exp\left(b\right|W-\omega {|}^{-0.5})$. The extracted critical disorder ${\omega }_{{}_{KT}}^{{}_O}$, fitting parameter $b^{{}_O}$, and the quality of data collapse $Q$ have been attached on the relevant panel. All the data are obtained in a system with interaction's exponent $\alpha =0.5$ in the midspectrum $\varepsilon =0.5$.

Figure 4

The averaged gap ratio $\overline{\langle r\rangle }$ as function of energy $\varepsilon$ and disorder strength $W$ for different interaction ranges from (a) $\alpha =0.5$ to (f) $\alpha \to \infty$. The extracted critical disorders ${\omega }_{{}_{SO}}^{{}_O}$ and ${\omega }_{{}_{KT}}^{{}_O}$ are obtained by considering the transition as second-order and Kosterlitz-Thouless type, respectively. In all the panels, ${\omega }_{{}_{SO,KT}}^{{}_{GR}}$, ${\omega }_{{}_{SO,KT}}^{{}_{EE}}$, ${\omega }_{{}_{SO,KT}}^{{}_{DE}}$, and ${\omega }_{{}_{SO,KT}}^{{}_{SG}}$ are obtained from finite-size scaling analysis for the averaged gap ratio $\overline{\langle r\rangle }$, the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$, the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$, and the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$, respectively.

Figure 5

The critical exponent $\nu$'s [see Eq. (3)] as a function of energy scale $\varepsilon$ for various interaction ranges from (a) $\alpha =0.5$ to (f) $\alpha \to \infty$. These exponents are obtained from finite-size scaling analysis for different observables including the averaged gap ratio $\overline{\langle r\rangle }$ (dark blue triangle), the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$ (red circle), the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$ (magenta square), and the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$ (yellow diamond).

Figure 6

The fitting parameter $b$ for Kosterlitz-Thouless transition type [see Eq. (4)] as a function of energy scale $\varepsilon$ for various interaction ranges from (a) $\alpha =0.5$ to (f) $\alpha \to \infty$. These parameters are obtained from finite-size scaling analysis for different observables including the averaged gap ratio $\overline{\langle r\rangle }$ (dark blue triangle), the normalized entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle }/S_{{}_{EE}}^P$ (red circle), the normalized diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle }/S_{{}_{DE}}^P$ (magenta square), and the Schmidt gap $\overline{\langle \mathrm{\Delta }\rangle }$ (yellow diamond).