Abstract
Quantum systems of particles obeying Bose statistics and moving in d-dimensional lattices are studied. The original Bose–Hubbard Hamiltonian is considered, together with model systems related to this Hamiltonian: the Bose–Hubbard model with exchange interaction of infinite radius and the Bose–Hubbard model with infinite interaction potential. Rigorous results concerning the proof of the existence of Bose condensation and a phase transition to the Mott insulator state in these systems are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Ruelle, Statistical Mechanics: Rigorous Results (W.A. Benjamin, New York, 1969).
B. Simon, The P(ϕ)2 Euclidean (Quantum) Field Theory (Princeton Univ. Press, Princeton, NJ, 1974).
D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005).
J. Hubbard, “Electron correlations in narrow energy bands,” Proc. R. Soc. London A 276, 238–257 (1963).
T. Matsubara and H. Matsuda, “A lattice model of liquid helium. I,” Prog. Theor. Phys. 16, 569–582 (1956).
H. A. Gersch and G. C. Knollman, “Quantum cell model for bosons,” Phys. Rev. 129, 959–967 (1963).
M. P. A. Fisher, P. B.Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid–insulator transition,” Phys. Rev. B 40, 546–570 (1989).
W. Zwerger, “Mott–Hubbard transition of cold atoms in optical lattices,” J. Optics B 5 (2), S9–S16 (2003).
J.-B. Bru and T. C. Dorlas, “Exact solution of the infinite-range-hopping Bose–Hubbard model,” J. Stat. Phys. 113, 177–196 (2003).
N. N. Bogolubov, Jr., A Method for Studying Model Hamiltonians: A Minimax Principle for Problems in Statistical Physics (Pergamon, Oxford, 1972).
J. Ginibre, “On the asymptotic exactness of the Bogoliubov approximation for many boson systems,” Commun. Math. Phys. 8, 26–51 (1968).
F. J. Dyson, E. H. Lieb, and B. Simon, “Phase transitions in quantum spin systems with isotropic and nonisotropic interactions,” J. Stat. Phys. 18, 335–383 (1978).
T. Kennedy, E. H. Lieb, and B. S. Shastry, “The XY model has long-range order for all spins and all dimensions greater than one,” Phys. Rev. Lett. 61, 2582–2584 (1988).
R. Fernández, J. Fröhlich, and D. Ueltschi, “Mott transition in lattice boson models,” Commun. Math. Phys. 266, 777–795 (2006).
M. Aizenman, E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, “Bose–Einstein quantum phase transition in an optical lattice model,” Phys. Rev. A 70, 023612 (2004).
R. Kubo, “Statistical-mechanical theory of irreversible processes. I: General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Japan 12, 570–586 (1957).
N. N. Bogoljubow, “Quasimittelwerte in Problemen der statistischen Mechanik. II,” Phys. Abh. Sowjetunion 6 (3), 229–284 (1962).
J. Naudts, A. Verbeure, and R. Weder, “Linear response theory and the KMS condition,” Commun. Math. Phys. 44, 87–99 (1975).
G. Roepstorff, “Bounds for a Bose condensate in dimensions v ≥ 3,” J. Stat. Phys. 18, 191–206 (1978).
R. Haag, N. M. Hugenholtz, and M. Winnink, “On the equilibrium states in quantum statistical mechanics,” Commun. Math. Phys. 5, 215–236 (1967).
D. Kastler, J. C. T. Pool, and E. T. Poulsen, “Quasi-unitary algebras attached to temperature states in statistical mechanics: A comment on the work of Haag, Hugenholtz and Winnink,” Commun. Math. Phys. 12, 175–192 (1969).
H. Falk and L. W. Bruch, “Susceptibility and fluctuation,” Phys. Rev. 180, 442–444 (1969).
D. P. Sankovich, “Local Gaussian domination and Bose condensation in lattice boson model,” Phys. Lett. A 377, 2905–2908 (2013).
D. P. Sankovich, “Gaussian dominance and phase transitions in systems with continuous symmetry,” Teor. Mat. Fiz. 79 (3), 460–472 (1989)
D. P. Sankovich, Theor. Math. Phys. 79, 656–665 (1989)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D.P. Sankovich, 2015, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2015, Vol. 290, pp. 335–343.
Rights and permissions
About this article
Cite this article
Sankovich, D.P. Rigorous results of phase transition theory in lattice boson models. Proc. Steklov Inst. Math. 290, 318–325 (2015). https://doi.org/10.1134/S0081543815060280
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543815060280