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Factors Generated by XY-Model with Competing Ising Interactions on the Cayley Tree

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Abstract

In the present paper, we consider a quantum Markov chain corresponding to the XY-model with competing Ising interactions on the Cayley tree of order two. Earlier, it was proved that this state does exist and is unique. Moreover, it has clustering property. This means that the von Neumann algebra generated by this state is a factor. In the present paper, we establish that the factor generated by this state may have type \(\mathop {\mathrm{III_{\lambda }}}\)\(\lambda \in (0,1)\) which is unusual for states associated with models with nontrivial interactions.

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Acknowledgements

The authors are grateful to an anonymous referee whose valuable comments and remarks improved the presentation of this paper.

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Correspondence to Farrukh Mukhamedov.

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Communicated by Claude-Alain Pillet.

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Mukhamedov, F., Gheteb, S.E. Factors Generated by XY-Model with Competing Ising Interactions on the Cayley Tree. Ann. Henri Poincaré 21, 241–253 (2020). https://doi.org/10.1007/s00023-019-00853-9

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Mathematics Subject Classification

  • 46L53
  • 46L60
  • 82B10
  • 81Q10