The transition from a many-body localized phase to a thermalizing one is a dynamical quantum phase transition which lies outside the framework of equilibrium statistical mechanics. We provide a detailed study of the critical properties of this transition at finite sizes in one dimension. We find that the entanglement entropy of small subsystems looks strongly subthermal in the quantum critical regime, which indicates that it varies discontinuously across the transition as the system-size is taken to infinity, even though many other aspects of the transition look continuous. We also study the variance of the half-chain entanglement entropy which shows a peak near the transition, and find substantial variation in the entropy across eigenstates of the same sample. Further, the sample-to-sample variations in this quantity are strongly growing, and larger than the intra-sample variations. We posit that these results are consistent with a picture in which the transition to the thermal phase is driven by an eigenstate-dependent sparse resonant "backbone" of long-range entanglement, which just barely gains enough strength to thermalize the system on the thermal side of the transition as the system size is taken to infinity. This discontinuity in a global quantity --- the presence of a fully functional bath --- in turn implies a discontinuity even for local properties. We discuss how this picture compares with existing renormalization group treatments of the transition.Comment: v2 - published version, discussions clarified in various place

A many-body localized (MBL) state is a new state of matter emerging in a disordered interacting system at high energy densities through a disorder driven dynamic phase transition. The nature of the phase transition and the evolution of the MBL phase near the transition are the focus of intense theoretical studies with open issues in the field. We develop an entanglement density matrix renormalization group (En-DMRG) algorithm to accurately target the entanglement patterns of highly excited states for MBL systems. By studying the one dimensional Heisenberg spin chain in a random field, we demonstrate the high accuracy of the method in obtaining statistical results of quantum states in the MBL phase. Based on large system simulations by En-DMRG for excited states, we demonstrate some interesting features in the entanglement entropy distribution function, which is characterized by two peaks; one at zero and another one at the quantized entropy S = ln2 with an exponential decay tail on the S > ln2 side. Combining En-DMRG with exact diagonalization simulations, we demonstrate that the transition from the MBL phase to the delocalized ergodic phase is driven by rare events where the locally entangled spin pairs develop long-range power-law correlations. The corresponding phase diagram contains an intermediate regime, which has power-law spin-z correlations resulting from contributions of the rare events. We discuss the physical picture for the numerical observations in the intermediate regime, where various distribution functions are distinctly different from results deep in the ergodic and MBL phases for finite-size systems. Our results may provide new insights for understanding the phase transition in such systems.

We study the many-body localization of spin chain systems with quasiperiodic fields. We identify the lower bound for the critical disorder necessary to drive the transition between the thermal and many-body localized phase to be W cl ∼ 1.85, based on finite-size scaling of entanglement entropy and fluctuations of the bipartite magnetization. We also examine the time evolution of the entanglement entropy of an initial product state where we find power-law and logarithmic growth for the thermal and many-body localized phases, respectively. For larger disorder strength, both imbalance and spin glass order are preserved at long times, while spin glass order shows dependence on system size. Quasiperiodic fields have been applied in different experimental systems and our study finds that such fields are very efficient at driving the MBL phase transition.

First-principles total-energy electronic structure calculations based on the full-potential linear-muffin-tinorbital method have been used to study the electronic and mechanical properties of the L1 2-type ordered nickel-based intermetallics Ni 3 X ͑XϭMn, Al, Ga, Si, Ge͒. The calculated values for the equilibrium volume and elastic properties are generally in good agreement with experiments. The large shear anisotropy factor across the series is attributed to the anisotropy of the bonding charge density, which can be described by the combination of charge transfer from X to Ni and strong X p-Ni d ͑Mn d-Ni d in Ni 3 Mn͒ hybridization effect. The more pronounced directional bonding between the Ni and Si atoms compared to that between the Ni and Al atoms, and the small ͑large͒ redistribution of bonding charge in Ni 3 Al ͑Ni 3 Si͒ when the systems are under shear strain result in a stronger resistance to a shear for Ni 3 Si. The bonding charge densities for Ni 3 Ga and Ni 3 Ge are found to be similar to those for Ni 3 Al and Ni 3 Si, respectively. These results suggest that the addition of the extra p electron on the X atom increases the directionality of the bonding. The change of bonding charge directionality in Ni 3 Mn is due to the Mn d-Ni d hybridization. The calculated ratio of bulk to shear modulus of polycrystalline systems, B/G, proposed by Pugh to provide a simple rule of measuring the ease of plastic deformation, is found to correlate well with the absolute difference in the s-orbital electronegativity between the atomic constituents, and the difference in energy, E d ͑Ni͒ϪE p (X) ͓E d ͑Ni͒ϪE d ͑Mn͒ for Ni 3 Mn͔, across the series. ͓S0163-1829͑96͒01443-9͔

We examine the interplay of interaction and disorder for a Heisenberg spin-1/2 ladder system with random fields. We identify many-body localized states based on the entanglement entropy scaling, where delocalized and localized states have volume and area laws, respectively. We first establish the dynamic phase transition at a critical random field strength hc ∼ 8.5 ± 0.5, where all energy eigenstates are localized beyond that value. Interestingly, the entanglement entropy and fluctuations of the bipartite magnetization show distinct probability distributions which characterize different phases. Furthermore, we show that for weaker h, energy eigenstates with higher energy density are delocalized while states at lower energy density are localized, which defines a mobility edge separating these two phases. With increasing disorder strength, the mobility edge moves towards higher energy density, which drives the system to the phase of the full many-body localization.