SIGMA 4 (2008), 007, 14 pages      arXiv:0801.2780      https://doi.org/10.3842/SIGMA.2008.007
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

Teunis C. Dorlas ^a and Wolodymyr I. Skrypnik ^b
a) Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland
^b) Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine

Received October 29, 2007, in final form January 08, 2008; Published online January 17, 2008
Sections 1 and 2 have been rewritten, the main result and the proof have not been changed February 18, 2008.

Abstract
Quantum models on the hyper-cubic d-dimensional lattice of spin-1/2 particles interacting with linear oscillators are shown to have three ferromagnetic ground state order parameters. Two order parameters coincide with the magnetization in the first and third directions and the third one is a magnetization in a continuous oscillator variable. The proofs use a generalized Peierls argument and two Griffiths inequalities. The class of spin-oscillator Hamiltonians considered manifest maximal ordering in their ground states. The models have relevance for hydrogen-bond ferroelectrics. The simplest of these is proven to have a unique Gibbsian ground state.

Key words: order parameters; spin-boson model; Gibbsian ground state.

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