Abstract
In this paper, we have provided a matrix Hamiltonian model for honeycomb lattices and subsequently obtained the dispersion relation. Furthermore, we have constructed the C operator for the given non-Hermitian Hamiltonian model. The quadratic surfaces are sketched and the quantum Brachistochrone problem is discussed for the given honeycomb lattice model.
[1] K. S. Novoselov et al., Science 306, 666 (2004) http://dx.doi.org/10.1126/science.110289610.1126/science.1102896Search in Google Scholar PubMed
[2] A. K. Geim, K. S. Novoselov, Nature Mater. 6, 183 (2007) http://dx.doi.org/10.1038/nmat184910.1038/nmat1849Search in Google Scholar PubMed
[3] J. M. Cerveró, Int. J. Theor. Phys. 50, 2134 (2011) http://dx.doi.org/10.1007/s10773-010-0635-410.1007/s10773-010-0635-4Search in Google Scholar
[4] P. R. Wallace, Phys. Rev. 71, 622 (1947) http://dx.doi.org/10.1103/PhysRev.71.62210.1103/PhysRev.71.622Search in Google Scholar
[5] M. Mecklenburg, B. C. Regan, Phys. Rev. Lett. 106, 116803 (2011) http://dx.doi.org/10.1103/PhysRevLett.106.11680310.1103/PhysRevLett.106.116803Search in Google Scholar PubMed
[6] C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008) http://dx.doi.org/10.1103/RevModPhys.80.133710.1103/RevModPhys.80.1337Search in Google Scholar
[7] H. Ramezani et al., Phys. Rev. A 85, 013818 (2012) http://dx.doi.org/10.1103/PhysRevA.85.01381810.1103/PhysRevA.85.013818Search in Google Scholar
[8] C. E. Ruter et al., Nat. Phys. 6, 192 (2010) http://dx.doi.org/10.1038/nphys151510.1038/nphys1515Search in Google Scholar
[9] A. Guo et al., Phys. Rev. Lett. 103, 093902 (2009) http://dx.doi.org/10.1103/PhysRevLett.103.09390210.1103/PhysRevLett.103.093902Search in Google Scholar PubMed
[10] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, T. Kottos, Phys. Rev. A 84, 040101(R) (2011) http://dx.doi.org/10.1103/PhysRevA.84.04010110.1103/PhysRevA.84.040101Search in Google Scholar
[11] C. M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998) http://dx.doi.org/10.1103/PhysRevLett.80.524310.1103/PhysRevLett.80.5243Search in Google Scholar
[12] A. Szameit et al., Phys. Rev. A 84, 021806(R) (2011) http://dx.doi.org/10.1103/PhysRevA.84.02180610.1103/PhysRevA.84.021806Search in Google Scholar
[13] O. Panella, P. Roy, Phys. Lett. A 376, 2580 (2012) http://dx.doi.org/10.1016/j.physleta.2012.07.00210.1016/j.physleta.2012.07.002Search in Google Scholar
[14] M. Fagotti, C. Bonati, D. Logoteta, P. Marconcini, M. Macucci, Phys. Rev. B 83, 241406 (2011) http://dx.doi.org/10.1103/PhysRevB.83.24140610.1103/PhysRevB.83.241406Search in Google Scholar
[15] A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, M. Segev, Phys. Rev. A 84, 021806(R) (2011) http://dx.doi.org/10.1103/PhysRevA.84.02180610.1103/PhysRevA.84.021806Search in Google Scholar
[16] K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, Phys. Rev. B 84, 205128 (2011) http://dx.doi.org/10.1103/PhysRevB.84.20512810.1103/PhysRevB.84.205128Search in Google Scholar
[17] C. M. Bender et al, J. Phys. A: Math. Gen. 37, 10139 (2004) http://dx.doi.org/10.1088/0305-4470/37/43/00910.1088/0305-4470/37/43/009Search in Google Scholar
[18] N. Hatano, D. Nelson, Phys. Rev. Lett 77, 570 (1996) http://dx.doi.org/10.1103/PhysRevLett.77.57010.1103/PhysRevLett.77.570Search in Google Scholar PubMed
[19] Y. Nakamura, N. Hatano, J. Phys. Soc. Jpn. 75, 104001 (2006) http://dx.doi.org/10.1143/JPSJ.75.10400110.1143/JPSJ.75.104001Search in Google Scholar
[20] J. Feinberg, A. Zee, Phys. Rev. E 59, 6433 (1999) http://dx.doi.org/10.1103/PhysRevE.59.643310.1103/PhysRevE.59.6433Search in Google Scholar PubMed
[21] C.-S. Jia, A. de Souza Dutra, Ann. Phys. 323, 566 (2008) http://dx.doi.org/10.1016/j.aop.2007.04.00710.1016/j.aop.2007.04.007Search in Google Scholar
[22] C.-S. Jia, A. de Souza Dutra, J. Phys. A 39, 11877 (2006) http://dx.doi.org/10.1088/0305-4470/39/38/01310.1088/0305-4470/39/38/013Search in Google Scholar
[23] C.-S. Jia et al., Int. J. Theor. Phys. 47, 2513 (2008) http://dx.doi.org/10.1007/s10773-008-9685-210.1007/s10773-008-9685-2Search in Google Scholar
[24] S. Longhi, Phys. Rev. Lett. 105, 013903 (2010) http://dx.doi.org/10.1103/PhysRevLett.105.01390310.1103/PhysRevLett.105.013903Search in Google Scholar PubMed
[25] Ö. Yesiltas, J. Phys. A: Math. Theor. 46, 015302 (2013) http://dx.doi.org/10.1088/1751-8113/46/1/01530210.1088/1751-8113/46/1/015302Search in Google Scholar
[26] S. Murakami, N. Nagaosa, S.-C. Zhang, Science 301, 1348 (2003) http://dx.doi.org/10.1126/science.108712810.1126/science.1087128Search in Google Scholar PubMed
[27] V. M. Apalkov, T. Chakraborty, Eur. Phys. Lett. 100, 17002 (2012) http://dx.doi.org/10.1209/0295-5075/100/1700210.1209/0295-5075/100/17002Search in Google Scholar
[28] V. M. Apalkov, T. Chakraborty, Eur. Phys. Lett. 100, 67008 (2012) http://dx.doi.org/10.1209/0295-5075/100/6700810.1209/0295-5075/100/67008Search in Google Scholar
[29] P. K. Ghosh, J. Phys.: Condens. Matter 24, 145302 (2012) 10.1088/0953-8984/24/14/145302Search in Google Scholar PubMed
[30] K. Esaki et al., Phys. Rev. B 84, 205128 (2011) http://dx.doi.org/10.1103/PhysRevB.84.20512810.1103/PhysRevB.84.205128Search in Google Scholar
[31] M. J. Majid, S. S. Savinskii, Semiconductors 47, 141 (2013) http://dx.doi.org/10.1134/S106378261212011110.1134/S1063782612120111Search in Google Scholar
[32] A. Carlini, Phys. Rev. Lett. 96, 060503 (2006) http://dx.doi.org/10.1103/PhysRevLett.96.06050310.1103/PhysRevLett.96.060503Search in Google Scholar PubMed
[33] A. M. Frydryszak, V. M. Tkachuk, Phys. Rev. A 77(1), 014103 (2008) http://dx.doi.org/10.1103/PhysRevA.77.01410310.1103/PhysRevA.77.014103Search in Google Scholar
[34] C. M. Bender, D. C. Brody, H. F. Jones, B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007) http://dx.doi.org/10.1103/PhysRevLett.98.04040310.1103/PhysRevLett.98.040403Search in Google Scholar PubMed
[35] U. Günther, B. F. Samsonov, Phys. Rev. Lett. 101, 230404 (2008) http://dx.doi.org/10.1103/PhysRevLett.101.23040410.1103/PhysRevLett.101.230404Search in Google Scholar PubMed
[36] P. E. G. Assis, A. Fring, J. Phys. A: Math. Theor. 41, 244002 (2008) http://dx.doi.org/10.1088/1751-8113/41/24/24400210.1088/1751-8113/41/24/244002Search in Google Scholar
[37] A. Mostafazadeh, Phys. Rev. Lett. 99, 130502 (2007) http://dx.doi.org/10.1103/PhysRevLett.99.13050210.1103/PhysRevLett.99.130502Search in Google Scholar PubMed
[38] C. M. Bender et al., Phil. Trans. Roy. Soc. A-Math. Phys. Eng. Sci. 371, 20120160 (2013) http://dx.doi.org/10.1098/rsta.2012.016010.1098/rsta.2012.0160Search in Google Scholar PubMed
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