International Journal of Theoretical Physics

, Volume 54, Issue 11, pp 4027–4033 | Cite as

Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type

  • Sanjib Dey
  • Andreas Fring
  • Thilagarajah Mathanaranjan
Article

Abstract

We propose a noncommutative version of the Euclidean Lie algebra E2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.

Keywords

Euclidean algebras PT-symmetry Pseudo-Hermitian Hamiltonians Non-commutative quantum systems 

Notes

Acknowledgments

SD is supported by a City University Research Fellowship. TM is funded by an Erasmus Mundus scholarship and thanks City University for kind hospitality.

References

  1. 1.
    Dey, S., Fring, A., Mathanaranjan, T.: Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra. Ann. Phys. 346, 28–41 (2014)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Turbiner, A.: Lie algebras and linear operators with invariant subspaces. In: Kamran, N., Olver, P.J. (eds.) Lie Algebras, Cohomologies and New Findings in Quantum Mechanics, Contemp. Math. AMS, vol. 160, pp. 263–310 (1994)Google Scholar
  3. 3.
    Assis, P.E.G., Fring, A.: Non-Hermitian Hamiltonians of Lie algebraic type. J. Phys. A 42, 015203 (23p) (2009)ADSGoogle Scholar
  4. 4.
    Assis, P.E.G.: Metric operators for non-Hermitian quadratic su(2) Hamiltonians. J. Phys. A 44, 265303 (2011)CrossRefADSGoogle Scholar
  5. 5.
    Bender, C.M., Kalveks, R.J.: Extending PT symmetry from Heisenberg Algebra to E2 Algebra. Int. J. Theor. Phys. 50, 955–962 (2011)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Jones-Smith, K., Kalveks, R.J.: Vector models in PT quantum mechanics. Int. J. Theor. Phys. 52, 2187–2195 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)CrossRefADSGoogle Scholar
  8. 8.
    Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT-symmetric optical lattices. Phys. Rev. A 81, 063807(10) (2010)CrossRefADSGoogle Scholar
  9. 9.
    Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902(4) (2009)ADSGoogle Scholar
  10. 10.
    Midya, B., Roy, B., Roychoudhury, R.: A note on the PT invariant potential 4c o s 2 x + 4i V 0 s i n2x. Phys. Lett. A 374, 2605–2607 (2010)MATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Jones, H.: Use of equivalent Hermitian Hamiltonian for PT-symmetric sinusoidal optical lattices. J. Phys. A 44, 345302 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Graefe, E., Jones, H.: PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold. Phys. Rev. A 84, 013818(8) (2011)CrossRefADSGoogle Scholar
  13. 13.
    Longhi, S., Della Valle, G.: Invisible defects in complex crystals. Ann. Phys. 334, 35–46 (2013)MATHCrossRefADSGoogle Scholar
  14. 14.
    Wigner, E.: Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960)MATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Bender, C.M., Boettcher, S.: Real spectra in Non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)MATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)CrossRefADSGoogle Scholar
  17. 17.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)MATHCrossRefGoogle Scholar
  18. 18.
    Heiss, W.D.: Repulsion of resonance states and exceptional points. Phys. Rev. E 61, 929–932 (2000)CrossRefADSGoogle Scholar
  19. 19.
    Rotter, I., Exceptional points and double poles of the S matrix. Phys. Rev. E 67, 026204 (2003)Google Scholar
  20. 20.
    Günther, U., Rotter, I., Samsonov, B.F.: Projective Hilbert space structures at exceptional points. J. Phys. A: Math. Theoret. 40(30), 8815 (2007)MATHCrossRefADSGoogle Scholar
  21. 21.
    Scholtz, F.G., Geyer, H.B., Hahne, F.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. 213, 74–101 (1992)MATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. 7, 1191–1306 (2010)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sanjib Dey
    • 1
  • Andreas Fring
    • 1
  • Thilagarajah Mathanaranjan
    • 1
  1. 1.Department of MathematicsCity University LondonLondonUK

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