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The Loop-Weight Changing Operator in the Completely Packed Loop Model

  • Bernard NienhuisEmail author
  • Kayed Al Qasimi
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 2)

Abstract

Loop models are statistical ensembles of closed paths on a lattice. The most well-known among them has a variety of names such as the dense O(n) loop model, the Temperley-Lieb (TL) model. This note concerns the model in which the weight of the loop n = 1, and a local operator which changes the weight of all the loops that surround the position of the operator to some other value. A conjecture of the expectation value of the one-point function of this operator was formulated 15 years ago. In this note we sketch the proof.

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Notes

Acknowledgements

We thank Christian Hagendorf for his important contribution to this result, and the MATRIx organization for the excellent opportunity for scientific exchange.

References

  1. 1.
    Baxter, R.J., Kelland, S.B., Wu, F.Y.: Equivalence of Potts model or Whitney polynomial with an ice-type model. J. Phys. A 9 397 (1976)CrossRefGoogle Scholar
  2. 2.
    Razumov, A.V., Stroganov, Y.G.: Spin chains and combinatorics. J. Phys. A 34, 3185 (2001). arXiv:cond-mat/0012141Google Scholar
  3. 3.
    Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. Math. 156, 835 (2002). arXiv:math/0008184Google Scholar
  4. 4.
    Batchelor, M.T., de Gier, J., Nienhuis, B.: The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions. J. Phys. A 34, L265 (2001). arXiv:cond-mat/0101385Google Scholar
  5. 5.
    Razumov, A.V., Stroganov, Y.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333 (2004); Teor.Mat.Fiz. 138 (2004) 395; arXiv:math/0104216Google Scholar
  6. 6.
    Cantini, L., Sportiello, A.: Proof of the Razumov-Stroganov conjecture. J. Combin. Theor. A 118, 1549 (2011). arXiv:1003.337Google Scholar
  7. 7.
    Mitra, S., Nienhuis, B.: Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders. J. Stat. Mech. P10006 (2004). arXiv:cond-mat/0407578Google Scholar
  8. 8.
    Hagendorf, C., Morin-Duchesne, A.: Symmetry classes of alternating sign matrices in the nineteen-vertex model. J. Stat. Mech. 053111 (2016). arXiv:1601.01859Google Scholar
  9. 9.
    Di Francesco, P., Zinn-Justin, P.: Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule. Electron. J. Combin. 12, R6 (2005). arXiv:math-ph/0410061Google Scholar
  10. 10.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current-algebra and Wess-Zumino model in 2 dimensions. Nucl. Phys. B 247, 83 (1984)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Di Francesco, P., Zinn-Justin, P., Zuber, J.-B.: Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. J. Stat. Mech. P08011 (2006). arXiv:math-ph/0603009Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of PhysicsAmsterdamThe Netherlands
  2. 2.Korteweg de Vries InstituteAmsterdamThe Netherlands

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