We analyze a class of quantum spin models defined on half-spaces in the d-dimensional hypercubic lattice bounded by a hyperplane with inward unit normal vector m ∈ R d . The family of models was previously introduced as the single species Product Vacua with Boundary States (PVBS) model, which is a spin-1/2 model with a XXZ-type nearest neighbor interactions depending on parameters λ j ∈ (0, ∞), one for each coordinate direction. For any given values of the parameters, we prove an upper bound for the spectral gap above the unique ground state of these models, which vanishes for exactly one direction of the normal vector m. For all other choices of m we derive a positive lower bound of the spectral gap, except for the case λ 1 = · · · = λ d = 1, which is known to have gapless excitations in the bulk.
In this paper, we show the that the ground state energy of the onedimensional Discrete Random Schrödinger Operator with Bernoulli Potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓ N , the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as π 2 (ℓN +1) 2 in the sense that the ratio of the quantities goes to one.
We investigate the interplay between disorder and interactions in a Bose gas on a lattice in the presence of randomly localized impurities. We compare the performance of two theoretical methods, namely the simple version of multi-orbital Hartree-Fock and the common Gross-Pitaevskii approach, and show how the former gives a very good approximation to the ground state in the limit of weak interactions, where the superfluid fraction is small. We further prove rigorously that for this class of disorder the fractal dimension of the ground state d * tends to the physical dimension in the thermodynamic limit. This allows us to introduce a quantity called the fractional occupation, which gives useful information on the crossover from a Lifshits to a Bose glass. Finally, we compare the temperature and interaction effects and highlight their similarities and intrinsic differences.
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