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Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

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Many-Electron Approaches in Physics, Chemistry and Mathematics

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound that gives rise to a maximum speed of propagation of correlations in many body systems of locally interacting spins and fermions. Finally, we discuss a number of important consequences concerning the simulation of time evolution and properties of ground states and stationary states.

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Notes

  1. 1.

    As we will later mostly work in the Heisenberg picture it is convenient to denote the Liouvillian in the Schrödinger picture by \(\fancyscript{L}^\dagger \) rather than \(\fancyscript{L}\).

  2. 2.

    In Ref. [6] the bound is given for an arbitrary metric on the vertex set and the Liouvillians are allowed to have interaction range \(a\) in that metric. Our interaction graph distance \(d\) is induced by a metric on \(V\) for which \(a=1\).

  3. 3.

    The spectral gap of a Hamiltonian \(\varDelta E\) is the difference between the ground state energy and the energy of the first exited state.

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Acknowledgments

We thank Earl T. Campbell, Mathis Friesdorf and Albert H. Werner for comments. We acknowledge support from the EU (Q-Essence, Raquel), the BMBF (QuOReP), the ERC (Taq), and the Studienstiftung des Deutschen Volkes.

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Authors and Affiliations

  1. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 , Berlin, Germany

    Martin Kliesch, Christian Gogolin & Jens Eisert

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  1. Martin Kliesch
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Correspondence to Martin Kliesch .

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Editors and Affiliations

  1. Institut für Analysis Und Algebra Carl-Friedrich-Gauss-Fakultät, Technische Universität Braunschweig, Braunschweig, Germany

    Volker Bach

  2. Institute for Mathematics, Freie Universität Berlin, Berlin, Germany

    Luigi Delle Site

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Kliesch, M., Gogolin, C., Eisert, J. (2014). Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems. In: Bach, V., Delle Site, L. (eds) Many-Electron Approaches in Physics, Chemistry and Mathematics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-06379-9_17

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