Relativistic quantum level-spacing statistics in chaotic graphene billiards

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 81, Issue 5, 2010, Pages

  Huang, Liang ^a,b   Lai, Ying Cheng ^a,c,d   Grebogi, Celso ^d  

^a Arizona State University   (United States)

^b LANZHOU UNIVERSITY   (China)

^c ARIZONA STATE UNIVERSITY   (United States)

^d UNIVERSITY OF ABERDEEN   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

DIRAC EQUATIONS; ELECTRONIC MOTIONS; ENERGY-LEVEL STATISTICS; GAUSSIAN ORTHOGONAL ENSEMBLES; GAUSSIAN UNITARY ENSEMBLE; LANDAU LEVELS; LOW ENERGIES; NON-LINEAR DYNAMICS; QUANTUM LEVELS; QUANTUM SYSTEM; RANDOM MATRICES; STRONG MAGNETIC FIELDS; WEAK MAGNETIC FIELDS;

EQUATIONS OF MOTION; GRAPHENE; GRAPHITE; LINEAR EQUATIONS; MAGNETIC FIELDS; QUANTUM ELECTRONICS; QUANTUM OPTICS;

CHAOTIC SYSTEMS;

EID: 77953336663     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.81.055203     Document Type: Article

References (33)

1. O. Bohigas and M.-J. Giannoni, in Mathematical and Computational Methods in Nuclear Physics, Lecture Notes in Physics, Vol. 209 (Springer, Berlin, 1984) 10.1007/3-540-13392-5-1
2. O. Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984) 10.1103/PhysRevLett.52.1
3. M. V. Berry, Proc. R. Soc. London, Ser. A 400, 229 (1985) 10.1098/rspa.1985.0078
4. M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990)
5. F. Haake, Quantum Signatures of Chaos, 2nd ed. (Springer, Berlin, 2001)
6. H. A. Weidenmüller and G. E. Mitchell, Rev. Mod. Phys. 81, 539 (2009). 10.1103/RevModPhys.81.539
7. A requirement for the applicability of random-matrix theory is that the system possess no geometric symmetry.
8. H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, England, 1999) 10.1017/CBO9780511524622
9. U. Stoffregen, J. Stein, H.-J. Stöckmann, M. Kuś, and F. Haake, Phys. Rev. Lett. 74, 2666 (1995) 10.1103/PhysRevLett.74.2666
10. P. So, S. M. Anlage, E. Ott, and R. N. Oerter, Phys. Rev. Lett. 74, 2662 (1995). 10.1103/PhysRevLett.74.2662
11. M. V. Berry and R. J. Mondragon, Proc. R. Soc. London, Ser. A 412, 53 (1987). 10.1098/rspa.1987.0080
12. Neutrinos have a minuscule, but nonzero mass. See G. Karagiorgi, Phys. Rev. D 75, 013011 (2007). 10.1103/PhysRevD.75.013011
13. K. S. Novoselov, Science 306, 666 (2004). 10.1126/science.1102896
14. F. Miao, Science 317, 1530 (2007). 10.1126/science.1144359
15. L. A. Ponomarenko, Science 320, 356 (2008). 10.1126/science.1154663
16. F. Libisch, C. Stampfer, and J. Burgdörfer, Phys. Rev. B 79, 115423 (2009). 10.1103/PhysRevB.79.115423
17. I. Amanatidis and S. N. Evangelou, Phys. Rev. B 79, 205420 (2009). 10.1103/PhysRevB.79.205420
18. H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002) 10.1103/PhysRevLett.89.266603
19. C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008). 10.1103/RevModPhys. 80.1337
20. ′, will modify the band structure. But the linear dispersion relation still holds near the Dirac points. Thus the same statistics persist for energy levels close to the Dirac points. This has been validated by direct numeric calculations.
21. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 10.1103/RevModPhys.81.109
22. The hopping energy for the boundary atoms could be 10% larger than inner atoms (see, for example, Z. F. Wang, Phys. Rev. B 75, 113406 (2007). 10.1103/PhysRevB.75.113406
23. R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 61, 2981 (2000) 10.1103/PhysRevB.61.2981
24. A. Rycerz, J. Tworzydło, and C. W. J. Beenakker, Nat. Phys. 3, 172 (2007) 10.1038/nphys547
25. V. V. Cheianov, V. Fal'ko, and B. L. Altshuler, Science 315, 1252 (2007) 10.1126/science.1138020
26. J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, Phys. Rev. Lett. 100, 236801 (2008). 10.1103/PhysRevLett.100.236801
27. Other orientations have been checked by rotating the graphene lattice to a certain angle then apply the confinement. The number of localized edge states could be different, but the level-spacing statistics are the same.
28. H. P. Baltes and E. R. Hilf, Spectra of Finite Systems (B.-I. Wissenschaftseverlag, Mannheim, 1976).
29. M. V. Berry and M. Tabor, Proc. R. Soc. London, Ser. A 356, 375 (1977). 10.1098/rspa.1977.0140
30. M. L. Mehta, Random Matrices (Academic, New York, 1967).
31. A. Bohr and B. R. Mottelson, Nuclear Structure (Benjamin, New York, 1969), Vol. 1, Appendix 2C, pp. 294-301.
32. F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963). 10.1063/1.1704008
33. Although an infinite graphene flake has the symplectic symmetry, the billiard system, with the sharp edge cuts, will couple the two valleys at the edge and thus break the symplectic symmetry, rendering the GSE statistics unobservable for such systems.