Alexander L. Burin

We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $\sim r^{-a}$, and the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems via their interaction with cavity modes. We focus on the dilute dipolar excitation case when the problem can be effectively considered as single-particle with the interaction providing long-range dipolar-like hopping. We show that the spatially homogeneous tilt $β$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion, $a<d$, unlike the models with random dipole orientation. The Anderson transitions are found to occur at the finite values of the tilt parameter $β= a$, $0<a<d$, and $β= a/(a-d/2)$, $d/2<a<d$, showing the robustness of the localization at small and large anisotropy values. Both extensive numerical calculations and analytical methods show power-law localized eigenstates in the bulk of the spectrum, obeying recently discovered duality $a\leftrightarrow 2d-a$ of their spatial decay rate, on the localized side of the transition, $a>a_{AT}$. This localization emerges due to the presence of the ergodic extended states at either spectral edge, which constitute a zero fraction of states in the thermodynamic limit, decaying though extremely slowly with the system size.

We investigate many-body localization in the chain of interacting spins with a transverse power-law interaction, $J_{0}/r^α$, and random on-site potentials, $φ_i \in \left(-W/2,W/2\right)$, in the long-range limit, $α< 3/2$, which has been recently examined experimentally on trapped ions. The many-body localization threshold is characterized by the critical disordering, $W_c$, which separates localized ($W > W_c$) and chaotic ($W < W_c$) phases. Using the analysis of the instability of localized states with respect to resonant interactions complemented by numerical finite size scaling, we show that the critical disordering scales with the number of spins, $N$, as $W_c \approx [1.37 J_{0}/(4/3 - α)]N^{4/3 - α} \ln N$ for $0 < α\leq 1$, and as $W_c \approx [J_{0}/(1-2α/3)]N^{1-2α/3} \ln^{2/3} N$ for $1 < α< 3/2$ while the transition width scales as $σ_{W} \propto W_{c}/N$. We use this result to predict the spin long-term evolution for a very large number of spins ($N = 50$), inaccessible for exact diagonalization, and to suggest the rescaling of hopping interaction with the system size to attain the localization transition at finite disordering in the thermodynamic limit of infinite number of spins.