Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 011, 6 pages      math-ph/0601060

Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States

Wolodymyr Skrypnik
Institute of Mathematics, 3 Tereshchenkivs'ka Str., Kyiv 4, 01601 Ukraine

Received October 19, 2005, in final form January 16, 2006; Published online January 24, 2006

A class of general spin 1/2 lattice models on hyper-cubic lattice Zd, whose Hamiltonians are sums of two functions depending on the Pauli matrices S1, S2 and S3, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously for large values of the parameter α playing the role of the inverse temperature. It is shown that the ferromagnetic order in x direction exists for all dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple).

Key words: Gibbsian eigen (ground) states; quantum spin models.

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  1. Dorlas T., Skrypnik W., Two order parameters in quantum XZ spin midels with Gibbsian ground states, J. Phys. A: Math. Gen., 2004, V.37, 6623-6632.
  2. Kirkwood J., Thomas L., Expansions and phase transitions for the ground state of quantum Ising lattice systems, Comm. Math. Phys., 1983, V.88, 569-580.
  3. Matsui T., A link between quantum and classical Potts models, J. Statist. Phys., 1990, V.59, 781-798.
  4. Matsui T., Uniqueness of translation invariant ground state in quantum spin systems, Comm. Math. Phys., 1990, V.126, 453-467.
  5. Alcaraz F., Exact steady states of asymmetric diffusion and two-species annihilation with back reaction from the ground state of quantum spin model, Internat. J. Modern Phys., 1994, V.25-26, 3449-3461.
  6. Alcaraz F., Salinas S., Wrechinsky W., Anisotropic quantum domains, Phys. Rev. Lett., 1995, V.5, 930-933.
  7. Matsui T., On ground state degeneracy of Z2 symmetric quantum spin models, Publ. Res. Inst. Math. Sci., 1991, V.27, 658-679.
  8. Thomas L., Yin Z., Low temperature expansions for the Gibbs states of quantum Ising lattice systems, J. Math. Phys., 1984, V.10, 3128-3134.

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