Journal of Statistical Physics

, Volume 135, Issue 3, pp 571–583 | Cite as

Critical Value of the Quantum Ising Model on Star-Like Graphs

  • Jakob E. Björnberg


We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the junction of several copies of ℤ at a single point. Our approach is to use the graphical, or fk-, representation of the model, and the probabilistic and geometric tools associated with it.


Ising model Random-cluster model Critical value 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994) CrossRefADSMathSciNetMATHGoogle Scholar
  2. 2.
    Aizenman, M., Barsky, D., Fernández, R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47, 343–374 (1987) CrossRefADSGoogle Scholar
  3. 3.
    Aizenman, M., Klein, A., Newman, C.M.: Percolation methods for dis-ordered quantum Ising models. In: Kotecký, R. (ed.) Phase Transitions: Mathematics, Physics, Biology. World Scientific, Singapore (1992) Google Scholar
  4. 4.
    Alexander, K.S.: Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32(1A), 441–487 (2004) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bezuidenhout, C., Grimmett, G.R.: Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19(3), 984–1009 (1991) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Björnberg, J.E.: Ph.D. thesis (2009). In preparation Google Scholar
  7. 7.
    Björnberg, J.E., Grimmett, G.R.: The phase transition of the quantum Ising model is sharp (2009). Submitted to JSP. arXiv:0901.0328
  8. 8.
    Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random-cluster models. Ann. Probab. 36(4), 1287–1321 (2008) CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Cerf, R., Messikh, R.: On the 2d Ising Wulff crystal near criticality. arXiv:math/0603178v3
  10. 10.
    Chamon, C., Oshikawa, M., Affleck, I.: Junctions of three quantum wires and the dissipative Hofstadter model. Phys. Rev. Lett. 91(20), 206403 (2003) CrossRefADSGoogle Scholar
  11. 11.
    Gandolfi, A., Keane, M., Russo, L.: On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16(3), 1147–1157 (1988) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Grimmett, G.R.: The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften, vol. 333. Springer, Berlin (2006) MATHGoogle Scholar
  13. 13.
    Grimmett, G.R.: Space-time percolation. In: Sidoravicius, V., Vares, M.E. (eds.) In and Out of Equilibrium 2. Progress in Probability, vol. 60, pp. 305–320. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
  14. 14.
    Higuchi, Y.: Coexistence of infinite (*)-clusters. II. Ising percolation in two dimensions. Probab. Theory Relat. Fields 97, 1–33 (1993) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Hou, C.Y., Chamon, C.: Junctions of three quantum wires for spin-(1/2) electrons. Phys. Rev. B. 77, 155422 (2008) CrossRefADSGoogle Scholar
  16. 16.
    Ioffe, D.: Stochastic Geometry of Classical and Quantum Ising Models. Lecture Notes in Mathematics. Springer, Berlin (2008) Google Scholar
  17. 17.
    Lal, S., Rao, S., Sen, D.: Junction of several weakly interacting quantum wires: a renormalization group study. Phys. Rev. B 66(16), 165327 (2002) CrossRefADSGoogle Scholar
  18. 18.
    Marchetti, R., Rasetti, M., Sodano, P., Trombettoni, A.: Critical behaviour at the junction of spin networks. Preprint June 12 (2007) Google Scholar
  19. 19.
    Martino, A.D., Moriconi, M., Mussardo, G.: Reflection scattering matrix of the Ising model in a random boundary magnetic field. arXiv:cond-mat/9707022v2
  20. 20.
    Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970) CrossRefADSGoogle Scholar
  21. 21.
    Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (1999) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University of CambridgeCambridgeUK
  2. 2.Royal Institute of TechnologyStockholmSweden

Personalised recommendations