No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
Random loop representations for quantum spin systems
1. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Commun. Math. Phys. 115, 477–528 (1988).
2. 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).
4. C. Albert, L. Ferrari, J. Fröhlich, and B. Schlein, “Magnetism and the Weiss exchange field—A theoretical analysis motivated by recent experiments,” J. Stat. Phys. 125, 77–124 (2006).
5. G. Alon
and G. Kozma
, “The probability of long cycles in interchange processes
,” Duke Math. J.
(to be published); e-print arXiv:1009.3723
6. O. Angel, “Random infinite permutations and the cyclic time random walk,” Discrete Math. Theor. Comput. Sci. Proc. 9–16 (2003).
8. S. Bachmann
and B. Nachtergaele
, “On gapped phases with a continuous symmetry and boundary operators
,” e-print arXiv:1307.0716
10. N. Berestycki, “Emergence of giant cycles and slowdown transition in random transpositions and k-cycles,” Electron. J. Probab. 16, 152–173 (2011).
11. N. Berestycki
and G. Kozma
, “Cycle structure of the interchange process and representation theory
,” Bull. Soc. Math. Fr.
(to be published); e-print arXiv:1205.4753
12. J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics 102 (Cambridge University Press, 2006).
13. V. Betz and D. Ueltschi, “Spatial random permutations and Poisson-Dirichlet law of cycle lengths,” Electron. J. Probab. 16, 1173–1192 (2011).
14. M. Biskup, “Reflection positivity and phase transitions in lattice spin models,” in Methods of Contemporary Mathematical Statistical Physics, Lecture Notes in Mathematics Vol. 1970 (Springer, 2009), pp. 1–86.
16. J. Björnberg
, “Infrared bounds and mean-field behaviour in the quantum Ising model
,” Commun. Math. Phys.
(to be published); e-print arXiv:1205.3385
17. C. Borgs, R. Kotecký, and D. Ueltschi, “Low temperature phase diagrams for quantum perturbations of classical spin systems,” Commun. Math. Phys. 181, 409–446 (1996).
18. J. Conlon and J. P. Solovej, “Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet,” Lett. Math. Phys. 23, 223–231 (1991).
20. N. Datta, R. Fernández, and J. Fröhlich, “Low temperature phase diagrams of quantum lattice systems. I. Stability for perturbations of classical systems with finitely-many ground states,” J. Stat. Phys. 84, 455–534 (1996).
21. 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).
24. Y. A. Fridman, O. A. Kosmachev, and P. N. Klevets, “Spin nematic and orthogonal nematic states in S = 1 non-Heisenberg magnet,” J. Magn. Magn. Mater. 325, 125–129 (2013).
26. J. Fröhlich, R. B. Israel, E. H. Lieb, and B. Simon, “Phase transitions and reflection positivity. I. General theory and long range lattice models,” Commun. Math. Phys. 62, 1–34 (1978).
27. J. Fröhlich, R. B. Israel, E. H. Lieb, and B. Simon, “Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions,” J. Stat. Phys. 22, 297–347 (1980).
28. J. Fröhlich and C.-É. Pfister, “On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems,” Commun. Math. Phys. 81, 277–298 (1981).
30. J. Fröhlich, B. Simon, and T. Spencer, “Infrared bounds, phase transitions and continuous symmetry breaking,” Commun. Math. Phys. 50, 79–95 (1976).
33. C. Goldschmidt, D. Ueltschi, and P. Windridge, “Quantum Heisenberg models and their probabilistic representations,” in Entropy and the Quantum II, Contemp. Math. 552, 177–224 (2011);
34. G. R. Grimmett, “Space-time percolation,” in In and out of Equilibrium 2, Prog. Probab. 60, 305–320 (2008).
37. A. Hammond
, “Infinite cycles in the random stirring model on trees
,” Bull. Inst. Math. Acad. Sinica
(to be published); e-print arXiv:1202.1319
38. A. Hammond
, “Sharp phase transition in the random stirring model on trees
,” e-print arXiv:1202.1322
40. D. Ioffe, “Stochastic geometry of classical and quantum Ising models,” in Methods of Contemporary Mathematical Statistical Physics, Lecture Notes in Mathematics 1970 (Springer, 2009), pp. 87–127.
41. D. Ioffe, S. Shlosman, and Y. Velenik, “2D Models of statistical physics with continuous symmetry: The case of singular interactions,” Commun. Math. Phys. 226, 433–454 (2002).
42. S. Jansen
and N. Kurt
, “On the notion(s) of duality for Markov processes
,” e-print arXiv:1210.7193
44. T. Kennedy, E. H. Lieb, and B. S. Shastry, “Existence of Néel order in some spin- Heisenberg antiferromagnets,” J. Stat. Phys. 53, 1019–1030 (1988).
49. N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).
50. A. Messiah, Quantum Mechanics (Dover, 1999).
51. B. Nachtergaele
, “Quasi-state decompositions for quantum spin systems
,” in Probability Theory and Mathematical Statistics
, edited by B. Grigelionis et al.
), pp. 565
; e-print arXiv:cond-mat/9312012
52. B. Nachtergaele, “A stochastic geometric approach to quantum spin systems,” in Probability and Phase Transitions, Nato Science Series C 420, edited by G. Grimmett (Springer, 1994), pp. 237–246.
53. B. Nachtergaele, Lecture Notes (unpublished).
57. G. Schütz and S. Sandow, “Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered interacting particle systems,” Phys. Rev. E 49, 2726–2741 (1994).
58. B. Tóth, “Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet,” Lett. Math. Phys. 28, 75–84 (1993).
60. T. A. Tóth, A. M. Läuchli, F. Mila, and K. Penc, “Competition between two- and three-sublattice ordering for S = 1 spins on the square lattice,” Phys. Rev. B 85, 140403 (2012).
Article metrics loading...
We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.
Full text loading...
Most read this month