paper

Quantum states on Harmonic lattices

arXiv:quant-ph/0509166 · doi:10.1007/s00220-006-0049-6

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.

33 pages, 6 figures. Sec. I-VI published in Commun. Math. Phys. 267, 65 (2006). Sec. VII, which introduces Gaussian Matrix Product States, is now available separately at arXiv:1201.3945, and has been published in the Proceedings on the conference on Quantum information and many body quantum systems, edited by M. Ericsson and S. Montangero, pg. 129 (Edizioni della Normale, Pisa, 2008)

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