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

SIGMA 4 (2008), 007, 14 pages      arXiv:0801.2780
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.

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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  1. Kobayashi K., Dynamical theory of the phase transition in KH2PO4-type ferroelectric crystals, J. Phys. Soc. Japan 24 (1968), 497-508.
  2. Villain J., Stamenkovic S., Atomic motion in hydrogen-bond ferroelectrics, Phys. Stat. Sol. 15 (1966), 585-596.
  3. Kurbatov A., Plechko V., Exactly solvable model with order-disorder phase transition for ferroelectrics, Teoret. Mat. Fiz. 26 (1976), 109-116.
  4. Bogolyubov N.N. Jr., Brankov J.G., Kurbatov A.M., Tonchev N.C., Zagrebnov V.A., Method of approximating Hamiltonian in statistical physics, Sophia, 1981.
  5. Kirkwood J., Thomas L., Expansions and phase transitions for the ground state of quantum Ising lattice systems, Comm. Math. Phys. 88 (1983), 569-580.
  6. Matsui T., A link between quantum and classical Potts models, J. Statist. Phys. 59 (1990), 781-798.
  7. Matsui T., Uniqueness of the translationally invariant ground state in quantum spin systems, Comm. Math. Phys. 126 (1990), 453-467.
  8. Datta N., Kennedy T., Expansions of one quasiparticle states in spin-1/2 systems, J. Statist. Phys. 108 (2002), 373-399, cond-mat/0104199.
  9. Goderis D., Maes C., Constructing quantum dissipations and their reversible states from classical interacting spin systems, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), 805-828.
  10. Dorlas T., Skrypnik W., On quantum XZ 1/2 spin models with Gibbsian ground states, J. Phys. A: Math. Gen. 37 (2004), 66623-6632.
  11. Skrypnik W., Order parameters in XXZ-type spin 1/2 quantum models with Gibbsian ground states, SIGMA 2 (2006), 011, 6 pages, math-ph/0601060.
  12. Skrypnik W., Quantum spin XXZ-type models with Gibbsian ground states and double long-range order, Phys. Lett. A 371 (2007), 363-373.
  13. Messager A., Nachtergaele B., A model with simultaneous first and second order phase transitions, J. Statist. Phys. 122 (2006), 1-14, cond-mat/0501229.
  14. van Enter A.C.D., Shlosman S.B., First-order transitions for n-vector models in two and more dimensions; rigorous proof, Phys. Rev. Lett. 89 (2002), 285702, 3 pages, cond-mat/0205455.
  15. van Enter A.C.D., Shlosman S.B., Provable first-order transitions for liquid crystal and lattice gauge models with continuous symmetries, Comm. Math. Phys. 255 (2005), 21-32, cond-mat/0306362.
  16. Thomas L.E., Quantum Heisenberg ferromagnets and stochastic exclusion processes, J. Math. Phys. 21 (1980), 1921-1924.
  17. Skrypnik W., Long-range order in nonequilibrium systems of interacting Brownian linear oscillators, J. Statist. Phys. 111 (2003), 291-321.
  18. Reed M., Simon B., Methods of modern mathematical physics, Vols. II, IV, Academic Press, 1975.
  19. Ginibre J., Reduced density matrices of quantum gases. I. Limit of infinite volume, J. Math. Phys. 6 (1964), 238-251.
  20. Kato T., Perturbation theory for linear operators, Springer-Verlag, 1966.
  21. Gantmacher F.R., Matrix theory, 4th ed., Nauka, Moscow, 1988.
  22. Gantmacher F.R., Applications of the theory of matrices, Interscience Publ., New York, 1959.
  23. Kelly D., Sherman S., Inequalities on correlations in Ising ferromagnets, J. Math. Phys. 9 (1968), 466-483.
  24. Bricmont J., Fontaine J.-R., Correlation inequalities and contour estimates, J. Statist. Phys. 26 (1981), 745-753.
  25. Fröhlich J., Lieb E., Phase transitions in anisotropic lattice spin systems, Comm. Math. Phys. 60 (1978), 233-267.
  26. Skrypnik W., Long-range order in linear ferromagnetic oscillator systems. Strong pair quadratic n-n potential, Ukrainian Math. J. 56 (2004), 964-972.
  27. Affleck I., Kennedy T., Lieb E., Tasaki H., Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys. 115 (1988), 477-528.
  28. Izenman M., Translation invariance and instability of phase coexistence in the teo dimensional Ising system, Comm. Math. Phys. 73 (1980), 84-92.
  29. Koma T., Tasaki H., Symmetry breaking and finite-size effects in quantum many-body systems, J. Statist. Phys. 76 (1994), 745-803, cond-mat/9708132.

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