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Amanda Young - One of the best experts on this subject based on the ideXlab platform.
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quasi locality bounds for quantum lattice systems i lieb robinson bounds quasi local maps and spectral flow automorphisms
Journal of Mathematical Physics, 2019Co-Authors: Bruno Nachtergaele, Robert Sims, Amanda YoungAbstract:Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frust...
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quasi locality bounds for quantum lattice systems part i lieb robinson bounds quasi local maps and spectral flow automorphisms
arXiv: Mathematical Physics, 2018Co-Authors: Bruno Nachtergaele, Robert Sims, Amanda YoungAbstract:Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustration-free models satisfying a Local Topological Quantum Order condition, which we present in a sequel to this paper.
Bruno Nachtergaele - One of the best experts on this subject based on the ideXlab platform.
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quasi locality bounds for quantum lattice systems i lieb robinson bounds quasi local maps and spectral flow automorphisms
Journal of Mathematical Physics, 2019Co-Authors: Bruno Nachtergaele, Robert Sims, Amanda YoungAbstract:Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frust...
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quasi locality bounds for quantum lattice systems part i lieb robinson bounds quasi local maps and spectral flow automorphisms
arXiv: Mathematical Physics, 2018Co-Authors: Bruno Nachtergaele, Robert Sims, Amanda YoungAbstract:Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustration-free models satisfying a Local Topological Quantum Order condition, which we present in a sequel to this paper.
Esteban Andruchow - One of the best experts on this subject based on the ideXlab platform.
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metric geometry of partial isometries in a finite von neumann algebra
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Esteban AndruchowAbstract:Abstract We study the geometry of the set I p = { v ∈ M : v ∗ v = p } of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C ∞ submanifold of M in the Norm Topology of M. However, we study it in the strong operator Topology, in which it does not have a smooth structure. This Topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic Norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that ( I p , d g ) is a complete metric space, where d g is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).
Vladimir G. Troitsky - One of the best experts on this subject based on the ideXlab platform.
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unbounded Norm Topology beyond Normed lattices
Positivity, 2018Co-Authors: Marko Kandic, Vladimir G. TroitskyAbstract:In this paper, we generalize the concept of unbounded Norm (un) convergence: let X be a Normed lattice and Y a vector lattice such that X is an order dense ideal in Y; we say that a net $$(y_\alpha )$$ un-converges to y in Y with respect to X if $$\bigl |\bigl ||y_\alpha -y|\wedge x\bigr |\bigr |\rightarrow 0$$ for every $$x\in X_+$$ . We extend several known results about un-convergence and un-Topology to this new setting. We consider the special case when Y is the universal completion of X. If $$Y=L_0(\mu )$$ , the space of all $$\mu $$ -measurable functions, and X is an order continuous Banach function space in Y, then the un-convergence on Y agrees with the convergence in measure. If X is atomic and order complete and $$Y=\mathbb R^A$$ then the un-convergence on Y agrees with the coordinate-wise convergence.
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unbounded Norm Topology beyond Normed lattices
arXiv: Functional Analysis, 2017Co-Authors: Marko Kandic, Vladimir G. TroitskyAbstract:In this paper, we generalize the concept of unbounded Norm (un) convergence: let $X$ be a Normed lattice and $Y$ a vector lattice such that $X$ is an order dense ideal in $Y$; we say that a net $(y_\alpha)$ un-converges to $y$ in $Y$ with respect to $X$ if $\Bigl\lVert\lvert y_\alpha-y\rvert \wedge x\Bigr\rVert\to 0$ for every $x\in X_+$. We extend several known results about un-convergence and un-Topology to this new setting. We consider the special case when $Y$ is the universal completion of $X$. If $Y=L_0(\mu)$, the space of all $\mu$-measurable functions, and $X$ is an order continuous Banach function space in $Y$, then the un-convergence on $Y$ agrees with the convergence in measure. If $X$ is atomic and order complete and $Y=\mathbb R^A$ then the un-convergence on $Y$ agrees with the coordinate-wise convergence.
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Unbounded Norm Topology in Banach lattices
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Marko Kandić, Mohammad A. A. Marabeh, Vladimir G. TroitskyAbstract:Abstract A net ( x α ) in a Banach lattice X is said to un-converge to a vector x if ‖ | x α − x | ∧ u ‖ → 0 for every u ∈ X + . In this paper, we investigate un-Topology, i.e., the Topology that corresponds to un-convergence. We show that un-Topology agrees with the Norm Topology iff X has a strong unit. Un-Topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-Topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B X is un-complete. For a Banach lattice X, B X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
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Unbounded Norm Topology in Banach Lattices
arXiv: Functional Analysis, 2016Co-Authors: Marko Kandić, Mohammad A. A. Marabeh, Vladimir G. TroitskyAbstract:A net $(x_\alpha)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-Topology, i.e., the Topology that corresponds to un-convergence. We show that un-Topology agrees with the Norm Topology iff $X$ has a strong unit. Un-Topology is metrizable iff $X$ has a quasi-interior point. Suppose that $X$ is order continuous, then un-Topology is locally convex iff $X$ is atomic. An order continuous Banach lattice $X$ is a KB-space iff its closed unit ball $B_X$ is un-complete. For a Banach lattice $X$, $B_X$ is un-compact iff $X$ is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
Young Amanda - One of the best experts on this subject based on the ideXlab platform.
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Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
'AIP Publishing', 2019Co-Authors: Nachtergaele Bruno, Sims Robert, Young AmandaAbstract:Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator Norm Topology with the strong operator Topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustration-free models satisfying a Local Topological Quantum Order condition, which we present in a sequel to this paper.Comment: 106 pages; added comments, updated references, minor correction
