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Recently, there has been increasing interest in understanding correlations in quantum lattice systems prompted by applications in quantum information theory and computation [10, 3, 2, 4] and the study of complex networks [5]. The questions that arise in the context of quantum information and computation are sufficiently close to typical problems in… (More)

- Bruno Nachtergaele, Hillel Raz, Benjamin Schlein
- 2008

We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems.

A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+l)-invariant quantum spin-S chains with the interaction — P, where P is the projection onto the singlet… (More)

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice… (More)

One of the folk theorems in quantum lattice models claims the equivalence of the existence of a nonvanishing spectral gap and exponential decay of spatial correlations in the ground state. It has been known for some time that there are exceptions to one direction of this equivalence. There are models with a unique ground state with exponential decay of… (More)

- Tom Michoel, Bruno Nachtergaele
- Physical review. E, Statistical, nonlinear, and…
- 2012

Complex networks possess a rich, multiscale structure reflecting the dynamical and functional organization of the systems they model. Often there is a need to analyze multiple networks simultaneously, to model a system by more than one type of interaction, or to go beyond simple pairwise interactions, but currently there is a lack of theoretical and… (More)

(1.1) i∂tψ(t) = Hψ(t) . For all initial conditions ψ(0) ∈ H, the unique solution is given by ψ(t) = e−itHψ(0), for all t ∈ R. Due to Stone’s Theorem e−itH is a strongly continuous one-parameter group of unitary operators on H, and the self-adjointness of H is the necessary and sufficient condition for the existence of a unique continuous solution for all… (More)

We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed… (More)

We review some stochastic geometric models that arise from the study of certain quantum spin systems. In these models the fundamental properties of the ground states or equilibrium states of the quantum systems can be given a simple stochastic geometric interpretation. One thus obtains a new class of challenging stochastic geometric problems.

Abstract. For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C… (More)