Skip to main content

Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators

Abstract

We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice \(\mathbb {Z}\). These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in \(\mathbb {R}\). The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Aizenman, M., Warzel, S.: Random Operators: Disorder Effects on Quantum Spectra and Dynamics Graduate Studies in Mathematics, vol. 168. American Mathematical Society, Providence, Rhode Island (2015)

    Book  Google Scholar 

  2. 2.

    Bauer, H.: Measure and Integration Theory De Gruyter Studies in Mathematics, vol. 26. W. de Gruyter, Berlin (2001)

    Google Scholar 

  3. 3.

    Bhatia, R.: Matrix Analysis Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)

    Google Scholar 

  4. 4.

    Bourget, O., Moreno Flores, G.R., Taarabt, A.: One-dimensional discrete Dirac operators in a decaying random potential I: spectrum and dynamics. Math. Phys. Anal. Geom. 23, 20 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)

    Book  Google Scholar 

  6. 6.

    Carvalho, S.L., de Oliveira, C.R., Prado, R.A.: Sparse one-dimensional discrete Dirac operators II: spectral properties. J. Math. Phys. 52(073501), 1–21 (2011)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Cassano, B., Ibrogimov, O.O., Krejčiřík, D., Štampach, F.: Location of eigenvalues of non-self-adjoint discrete Dirac operators. Ann. Henri Poincaré, 21, 2193–2217 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. PMP, vol. 54. Basel, Birkhäuser (2008)

    Google Scholar 

  9. 9.

    de Oliveira, C.R., Prado, R.A.: Dynamical delocalization for the 1D Bernoulli discrete Dirac operator. J. Phys. A: Math. Gen. 38, L115–L119 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    de Oliveira, C.R., Prado, R.A.: Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator. J. Math. Phys. 46 (072105), 1–17 (2005)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Fukushima, M.: On the spectral distribution of a disordered system and the range of a random walk. Osaka J. Math. 11, 73–85 (1974)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Gebert, M., Müller, P.: Localization for random block operators. Mathematical Physics, Spectral Theory and Stochastic Analysis. Oper. Theory Adv. Appl. 232, 229–246 (2013). Birkhä,user, Basel

    MATH  Google Scholar 

  13. 13.

    Gebert, M., Rojas-Molina, C.: Lifshitz tails for the fractional Anderson model. J. Stat. Phys. 179, 341–353 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Golénia, S., Tristan, H.: On the a.c. spectrum of the 1D discrete Dirac operator. Methods Funct. Anal. Topol. 20, 252–273 (2014)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Kirsch, W.: An invitation to random Schrödinger operators. With an appendix by Frédéric Klopp, Panor. Synthèses, 25, Random Schrödinger Operators, 1–119, Soc. Math. France, Paris (2008)

  16. 16.

    Kirsch, W., Martinelli, F.: On the density of states of Schrödinger operators with a random potential. J. Phys. A: Math. Gen. 15, 2139–2156 (1982)

    ADS  Article  Google Scholar 

  17. 17.

    Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. Spectral Theory and Mathematical Physics. A festschrift in honor of Barry Simon’s 60th birthday, 649–696 (2007)

  18. 18.

    Lifshitz, I.M.: The energy spectrum of disordered systems. Adv. Phys. 13, 483–536 (1964)

    ADS  Article  Google Scholar 

  19. 19.

    Lifshitz, I.M.: Energy spectrum structure and quantum states of disordered condensed systems. Soviet Physics Uspekhi 7, 549–573 (1965)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Mezincescu, G.A.: Internal Lifshitz singularities of disordered finite-difference Schrödinger operators. Commun. Math. Phys. 103, 167–176 (1986)

    ADS  Article  Google Scholar 

  21. 21.

    Prado, R.A., de Oliveira, C.R.: Sparse 1D discrete Dirac operators I: quantum transport. J. Math. Anal. Appl. 385, 947–960 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Prado, R.A., de Oliveira, C.R., Carvalho, S.L.: Dynamical localization for discrete Anderson Dirac operators. J. Stat. Phys. 167, 260–296 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Rabinovich, V.: Exponential estimates of solutions of pseudodifferential equations on the lattice \((h\mathbb {Z})^{n}\): applications to the lattice Schrödinger and Dirac operators. J. Pseudo-Differ. Oper. Appl. 1(2), 233–253 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Romerio, M., Wreszinksi, W.: On the Lifshitz singularity and the tailings in the density os states for random lattice systems. J. Stat. Phys. 21, 169–179 (1979)

    ADS  Article  Google Scholar 

  25. 25.

    Simon, B.: Lifshitz tails for the Anderson model. J. Stat. Phys. 38, 65–76 (1985)

    ADS  Article  Google Scholar 

  26. 26.

    Simon, B.: Internal Lifshitz tails. J. Stat. Phys. 46, 911–918 (1987)

    ADS  Article  Google Scholar 

  27. 27.

    Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer, New York (1992)

    Google Scholar 

  28. 28.

    Yueh, W.C.: Eigenvalues of several tridiagonal matrices. Appl. Math. E-Notes 5, 66–74 (2005)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Zhu, S.-L., Zhang, D.-W., Wang, Z.D.: Delocalization of relativistic Dirac particles in disordered one-dimensional systems and its implementation with cold atoms. Phys. Rev. Lett. 102, 210403 (2009)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The first author RAP thanks support by PNPD-CAPES scholarship of the Post-Graduate Program in Applied Mathematics of the IMECC-UNICAMP (under contract 88887.466594/2019-00). CRdO thanks the partial support by CNPq (Brazilian agency, under contract 303503/2018-1).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Roberto A. Prado.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Roberto A. Prado is supported by CAPES (Brazil).

César R. de Oliveira is partially supported by CNPq (Brazil)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Prado, R.A., de Oliveira, C. & de Oliveira, E. Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators. Math Phys Anal Geom 24, 30 (2021). https://doi.org/10.1007/s11040-021-09403-4

Download citation

Keywords

  • Discrete Dirac operators
  • Random potentials
  • Density of states
  • Lifshitz tails

Mathematics Subject Classification (2010)

  • Primary 47B80
  • 47A75
  • 82B44. Secondary 81Q15