Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition, a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this Letter, we explore two entanglement properties that are promising for the study of the many-body localization transition: the variance of the half-chain entanglement entropy of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a disordered quantum Ising chain and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a spin-glass transition at large disorder strength and provide evidence for it being a separate transition. We, thereby, give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [Huse et al., Phys. Rev. B 88, 014206 (2013).

We study the ground-state phase diagram of the quantum spin-2 XXZ chain in the presence of on-site anisotropy using a matrix-product state based infinite-system density matrix renormalization group (iDMRG) algorithm. One of the interests in this system is in connecting the highly quantum-mechanical spin-1 phase diagram with the classical S = ∞ phase diagram. Several of the recent advances within DMRG make it possible to perform a detailed analysis of the whole phase diagram. We consider different types of on-site anisotropies, which allows us to establish the validity of the following statements: (1) the spin-2 model can be tuned into a phase, which is equivalent to the "topologically nontrivial" spin-1 Haldane phase, and (2) the spin-2 Haldane phase at the isotropic Heisenberg point is adiabatically connected to the "trivial" large-D phase, with a continuous change of the Hamiltonian parameters. Furthermore, we study the spin-3 XXZ chain to help explain the development of the classical phase diagram. We present details on how to use the iDMRG method to map out the phase diagram and include an extensive discussion of the numerical methods.

In a recent experiment on CoNb2O6, Coldea et al. [1] found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-productstate method. The excitation spectra show bound states characteristic of the weakly broken E8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data. The one-dimensional (1D) quantum Ising model in transverse and longitudinal fields is one of the most studied theoretical models in condensed matter physics. It is a relatively simple model that contains very rich physics; for example, it contains a quantum critical point (QCP) at zero longitudinal field related to the 2D classical Ising model. A remarkable fact is that the integrability present at the critical point remains under addition of a longitudinal field as a mass-generating perturbation. Zamolodchikov conjectured in 1989 an S-matrix describing eight emergent particles whose mass ratios are connected to the roots of the Lie algebra E 8 [2,3]. Recently, Coldea et al. performed neutron scattering experiments on CoNb 2 O 6 (cobalt niobate), a material that to a good approximation can be described by a quantum Ising chain. At low temperatures and in the presence of a strong external transverse magnetic field which tunes the system to near criticality, the observed spectrum shows characteristic excitations of the E 8 symmetry [1].However, a serious problem in comparing theory and experiment is that the real material has additional couplings that strictly speaking invalidate the exact solution, and until recently it was impractical to extend the theory non-perturbatively to include these couplings. In this Letter, we study a theoretical model for CoNb 2 O 6 which includes in addition to the Ising interaction other interactions arising from the lattice structure and the weak coupling between the chains. Using this model, we calculate the dynamical spectral function and compare the results to the observed spectra. Close to the QCP, the model retains features expected from the quantum Ising model, in particular the characteristic particles of the E 8 symmetry.We begin by deriving the theoretical model used to describe the low-energy physics of CoNb 2 O 6 . The spin lattice structure consists of chains of easy axis spins, realizing a two level system, on the Co 2+ ions coupled by a ferromagnetic Ising interaction along the chain direction, see Fig. 1A. We thus start from the quantum Ising chain, described by the Hamiltonian

The rapid development of artificial gauge fields in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentals. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases ν = 1/2 and ν = 2/3 have the edge spectra predicted by the chiral Luttinger liquid theory.Fractional quantum Hall (FQH) phases [1, 2] contain a wide variety of interesting physics, including topologically degenerate ground states, fractional bulk excitations, and gapless chiral edge excitations. They arise at low temperatures when strong magnetic fields are applied to high-quality two-dimensional electron gases with low carrier concentration. Ultracold gases of neutral atoms are being used to investigate several properties of materials which can be hard to control precisely in the solid state. As these systems are charge-neutral, an ordinary magnetic field cannot be used to create the Lorentz force. A synthetic magnetic field can be created by rotation, but technical issues appear to limit this approach to lower field strengths than are necessary for FQH [3], with the exception of a recent experiment with few trapped particles [4]. Much theoretical work has been done for these systems, for a review see [5], including edge spectrum calculations [6].Recently, several theoretical [7-9] and experimental [3,10] proposals have been made for stronger synthetic magnetic fields for ultracold neutral atoms. All of them can be used with optical lattices which enhance interaction effects and give a larger energy gap above the FQH ground state. Theoretical work on lattice systems with an effective magnetic field goes back at least to Hofstader's work [11] on non-interacting particles; the FQH phases are strongly interacting, and Sørensen et al. [7] showed that in the low flux limit and strong interactions the system could be well described by Laughlin's wavefunction [12]. In subsequent work Hafezi et al.[13] concluded that this could be extended to larger fluxes per unit cell by investigating the topological structure of the ground state.The goal of this work is to understand practical experimental conditions for observation of edge states in bosonic lattice FQH systems and compare numerical results for edge excitations in hierarchy states to the prediction of chiral Luttinger liquid theory [14]. Convincing observation of bosonic FQH states will depend on an experimentally viable probe of the topological order; while many multi-part...