Skip to main content
SpringerLink
Log in

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

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We propose a noncommutative version of the Euclidean Lie algebra E 2. 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

  3. Assis, P.E.G., Fring, A.: Non-Hermitian Hamiltonians of Lie algebraic type. J. Phys. A 42, 015203 (23p) (2009)

    ADS  Google Scholar 

  4. Assis, P.E.G.: Metric operators for non-Hermitian quadratic su(2) Hamiltonians. J. Phys. A 44, 265303 (2011)

    Article  ADS  Google Scholar 

  5. Bender, C.M., Kalveks, R.J.: Extending PT symmetry from Heisenberg Algebra to E2 Algebra. Int. J. Theor. Phys. 50, 955–962 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jones-Smith, K., Kalveks, R.J.: Vector models in PT quantum mechanics. Int. J. Theor. Phys. 52, 2187–2195 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  8. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT-symmetric optical lattices. Phys. Rev. A 81, 063807(10) (2010)

    Article  ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Jones, H.: Use of equivalent Hermitian Hamiltonian for PT-symmetric sinusoidal optical lattices. J. Phys. A 44, 345302 (2011)

    Article  MathSciNet  Google Scholar 

  12. Graefe, E., Jones, H.: PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold. Phys. Rev. A 84, 013818(8) (2011)

    Article  ADS  Google Scholar 

  13. Longhi, S., Della Valle, G.: Invisible defects in complex crystals. Ann. Phys. 334, 35–46 (2013)

    Article  MATH  ADS  Google Scholar 

  14. Wigner, E.: Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Bender, C.M., Boettcher, S.: Real spectra in Non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  ADS  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  18. Heiss, W.D.: Repulsion of resonance states and exceptional points. Phys. Rev. E 61, 929–932 (2000)

    Article  ADS  Google Scholar 

  19. Rotter, I., Exceptional points and double poles of the S matrix. Phys. Rev. E 67, 026204 (2003)

  20. Günther, U., Rotter, I., Samsonov, B.F.: Projective Hilbert space structures at exceptional points. J. Phys. A: Math. Theoret. 40(30), 8815 (2007)

    Article  MATH  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. 7, 1191–1306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, City University London, Northampton Square, London, EC1V 0HB, UK

    Sanjib Dey, Andreas Fring & Thilagarajah Mathanaranjan

Authors
  1. Sanjib Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Fring
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thilagarajah Mathanaranjan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Fring.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dey, S., Fring, A. & Mathanaranjan, T. Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type. Int J Theor Phys 54, 4027–4033 (2015). https://doi.org/10.1007/s10773-014-2447-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-014-2447-4

Keywords

Search

Navigation