### Quantum states on Harmonic lattices

**Norbert Schuch, J. Ignacio Cirac, Michael M. Wolf**

Commun. Math. Phys. 267, 65 **, (2006)

**DOI**: 10.1007/s00220-006-0049-6

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*arXiv.org*: quant-ph/0509166

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**Abstract:**We investigate bosonic Gaussian quantum states on an infinite cubic lattice in arbitrary spatial dimensions. We derive general properties of such states as ground states of quadratic Hamiltonians for both critical and non-critical cases. Tight analytic relations between the decay of the interaction and the correlation functions are proven and the dependence of the correlation length on band gap and effective mass is derived. We show that properties of critical ground states depend on the gap of the point-symmetrized rather than on that of the original Hamiltonian. For critical systems with polynomially decaying interactions logarithmic deviations from polynomially decaying correlation functions are found. Moreover, we provide a generalization of the matrix product state representation for Gaussian states and show that properties hold analogously to the case of finite dimensional spin systems.