International Journal of Theoretical Physics

, Volume 50, Issue 4, pp 974–981 | Cite as

\(\mathcal{PT}\) Invariant Complex E8 Root Spaces



We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E8-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser-Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.


\(\mathcal{PI}\)-symmetry Non-Hermitian Hamiltonians Perturbed Ising model 


  1. 1.
    Zamolodchikov, A.B.: Integrals of motion and S matrix of the (scaled) T=T(c) Ising model with magnetic field. Int. J. Mod. Phys. A 4, 4235 (1989) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Coldea, R., Tennant, D.E., Wheeler, E., Wawrzynska, Prabhakaran, D., Telling, M., Habicht, K., Smeibidl, P., Kiefer, K.: Quantum criticality in an Ising chain: experimental evidence for emergent E(8) symmetry. Science 327, 177–180 (2010) ADSCrossRefGoogle Scholar
  3. 3.
    von Gehlen, G.: Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field. J. Phys. A 24, 5371–5399 (1991) ADSCrossRefGoogle Scholar
  4. 4.
    Castro-Alvaredo, O., Fring, A.: A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field. J. Phys. A 42, 465211 (2009) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Wigner, E.: Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960) MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT Symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998) MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Mostafazadeh, A.: Pseudo-Hermitian Quantum Mechanics. arXiv:0810.5643
  9. 9.
    Znojil, M., Tater, M.: Complex Calogero model with real energies. J. Phys. A 34, 1793–1803 (2001) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Znojil, M., Fring, A.: PT-symmetric deformations of Calogero models. J. Phys. A 41, 194010(17) (2008) Google Scholar
  11. 11.
    Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981) MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Wilson, G.: The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras. Ergod. Theory Dyn. Syst. 1, 361–380 (1981) MATHCrossRefGoogle Scholar
  13. 13.
    Olive, D.I., Turok, N.: The symmetries of Dynkin diagrams and the reduction of Toda field equations. Nucl. Phys. B 215, 470–494 (1983) MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Fring, A., Smith, M.: Antilinear deformations of Coxeter groups, an application to Calogero models. J. Phys. A 43, 325201(28) (2010) MathSciNetGoogle Scholar
  15. 15.
    Assis, P.E.G., Fring, A.: From real fields to complex Calogero particles. J. Phys. A 42, 425206(14) (2009) Google Scholar
  16. 16.
    Fring, A., Smith, M.: in preparation Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Centre for Mathematical ScienceCity University LondonLondonUK

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