Wave Propagation and Localization in a Complex Random Potential with Power-law Correlations: A Numerical Study
Main Article Content
Abstract
In this paper, we investigate numerically wave propagation and localization in a complex random potential with power-law correlations. Using a discrete stationary Schrӧdinger equation with the simultaneous presence of the spatial correlation and the non-Hermiticity of the random potential in the diagonal on-site terms of the Hamiltonian, we calculate the disorder-averaged logarithmic transmittance and the localization length. From the numerical analysis, we find that the presence of power-law correlation in the imaginary part of the on-site disordered potential gives rise to the localization enhancement as compared with the case of absence of correlation. Depending on the disorder's strength, we show that there exist different behaviors of the dependence of the localization on the correlation strength.
Keywords:
Non-Hermitian Hamiltonian, complex disordered potential, spatial correlated disorder, Anderson localization.
References
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[31] A. Basiri, Y. Bromberg, A. Yamilov, H. Cao, T. Kottos, Light localization induced by a random imaginary refractive index, Phys. Rev. A 90 (2014) 043815. https://doi.org/10.1103/PhysRevA.90.043815.
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https://doi.org/10.1103/PhysRevB.92.094204.
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[36] J.M. Zuener, M.C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M.S. Rudner, M. Segev, A. Szameit, Observation of a topological transition in the bulk of a non-Hermitian system, Phys. Rev. Lett. 115 (2015) 040402. https://doi.org/10.1103/PhysRevLett.115.040402.
[37] B.P. Nguyen, D.K. Phung, K. Kim, Anomalous localization enhancement in one-dimensional non-Hermitian disordered lattices, J. Phys. A: Math. Theor. 53 (2020) 045003. https://doi.org/10.1088/1751-8121/ab5eb8.
[38] A.F. Tzortzakakis, K.G. Makris, E.N. Economou, Non-Hermitian disorder in two-dimensional optical lattices, Phys. Rev. B 101 (2020) 014202. https://doi.org/10.1103/PhysRevB.101.014202.
[39] Y. Huang, B.I. Shklovskii, Anderson transition in three-dimensional systems with non-Hermitian disorder, Phys. Rev. B 101 (2020) 014204. https://doi.org/10.1103/PhysRevB.101.014204.
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https://doi.org/10.1038/nphys971.
https://doi.org/10.1103/PhysRev.109.1492.
[2] E. Abrahams, 50 years of Anderson localization, World Scientific, Singapore, 2010.
[3] S.A. Gredeskul, Y.S. Kivshar, A.A. Asatryan, K.Y. Bliokh, Y.P. Bliokh, V.D. Freilikher, I.V. Shadrivov, Anderson localization in metamaterials and other complex media, Low. Temp. Phys. 38 (2012) 570-602.
https://doi.org/10.1063/1.4736617.
[4] M. Segev, Y. Silberberg, D.N. Christodoulides, Anderson localization of light, Nat. Photon. 197 (2013) 197-204. https://doi.org/10.1038/nphoton.2013.30.
[5] H.H. Sheinfux, Y. Lumer, G. Ankonina, A.Z. Genack, G. Bartal, M. Segev, Observation of Anderson localization in disordered nanophotonic structures, Science 356 (2017) 953-956.
https://science.sciencemag.org/content/356/6341/953
[6] S. Kim, K. Kim, Anderson localization and delocalization of massless two-dimensional Dirac electrons in random one-dimensional scalar and vector potentials, Phys. Rev B. 99 (2019) 014205.
https://doi.org/10.1103/PhysRevB.99.014205.
[7] D.H. Dunlap, H.L. Wu, P.W. Phillips, Absence of localization in a random-dimer model, Phys. Rev. Lett. 88 (1990) 88-91. https://doi.org/10.1103/PhysRevLett.65.88.
[8] V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G.B. Parravicini, F. Domínguez-Adame, R. Gómez-Alcalá, Experimental evidence of delocalized states in random dimer superlattices, Phys. Rev. Lett. 82 (1999) 2159-2162. https://doi.org/10.1103/PhysRevLett.82.2159.
[9] F.A.B. F de Moura, M.L. Lyra, Delocalization in the 1D Anderson model with long-range correlated disorder, Phys. Rev. Lett. 81 (1998) 3735-3738. https://doi.org/10.1103/PhysRevLett.81.3735.
[10] F.M. Izrailev, A.A. Krokhin, Localization and the mobility edge in one-dimensional potentials with correlated disorder, Phys. Rev. Lett. 82 (1999) 4062-4065. https://doi.org/10.1103/PhysRevLett.82.4062.
[11] J.M. Kantelhardt, S. Russ, A. Bunde, S. Havlin, I. Webman, Comment on "Delocalization in the 1D Anderson model with long-range correlated disorder'', Phys. Rev. Lett. 84 (2000) 198.
https://doi.org/10.1103/PhysRevLett.84.198.
[12] U. Kuhl, F.M. Izrailev, A.A. Krokhin, and H.J. Stӧckmann, Experimental observation of the mobility edge in a waveguide with correlated disorder, Appl. Phys. Lett. 77 (2000) 633-635. https://doi.org/10.1063/1.127068.
[13] H. Shima, T. Nomura, T. Nakayama, Localization-delocalization transition in one-dimensional electron systems with long-range correlated disorder, Phys. Rev. B 70 (2004) 075116.
https://doi.org/10.1103/PhysRevB.70.075116.
[14] T. Kaya, Localization-delocalization transition in chains with long-range correlated disorder, Eur. Phys. J. B 55 (2007) 49-56. https://doi.org/10.1140/epjb/e2007-00036-4.
[15] S. Nishino, K. Yakubo, H. Shima, Finite size effects in infinitely large electronic systems with correlated disorders, Phys. Rev. B 79 (2009) 033105. https://doi.org/10.1103/PhysRevB.79.033105.
[16] A.M. García-García, E. Cuevas, Differentiable potentials and metallic states in disordered one-dimensional systems, Phys. Rev. B 79 (2009) 073104. https://doi.org/10.1103/PhysRevB.79.073104.
[17] L.Y. Gong, P.Q. Tong, Z.C. Zhou, von Neumann entropy signatures of a transition in one-dimensional electron systems with long-range correlated disorder, Eur. Phys. J. B 77 (2010) 413-417.
https://doi.org/10.1140/epjb/e2010-00283-2.
[18] A. Croy, P. Cain, M. Schreiber, Anderson localization in 1D systems with correlated disorder, Eur. Phys. J. B 82 (2011) 107-112. https://doi.org/10.1140/epjb/e2011-20212-1.
[19] B.P. Nguyen, K. Kim, Influence of weak nonlinearity on the 1D Anderson model with long-range correlated disorder, Eur. Phys. J. B 84 (2011) 79-82. https://doi.org/10.1140/epjb/e2011-20608-9.
[20] F.M. Izrailev, A.A. Krokhin, N.M. Makarov, Anomalous localization in low-dimensional systems with correlated disorder, Phys. Rep. 512 (2012) 125-254. https://doi.org/10.1016/j.physrep.2011.11.002.
[21] G.M. Petersen, N. Sandler, Anticorrelations from power-law spectral disorder and conditions for an Anderson transition, Phys. Rev. B 87 (2013) 195443. https://doi.org/10.1103/PhysRevB.87.195443.
[22] W. Choi, C. Yin, I.R. Hooper, W.L. Barnes, J. Bertolotti, Absence of Anderson localization in certain random lattices, Phys. Rev. E 96 (2017) 022122. https://doi.org/10.1103/PhysRevE.96.022122.
[23] J.R.F. Lima, L.F.C. Pereira, A.L.R. Barbosa, Dirac wave transmission in Levy disordered systems, Phys. Rev. E 99 (2019) 032118. https://doi.org/10.1103/PhysRevE.99.032118.
[24] U. Kuhl, F.M. Izrailev, A.A. Krokhin, Enhancement of localization in one-dimensional random potentials with long-range correlations, Phys. Rev. Lett. 100 (2008) 126402. https://doi.org/10.1103/PhysRevLett.100.126402.
[25] W. Zhao, J.W. Ding, Enhanced localization and delocalization in surface disordered quantum waveguides with long-range correlation, EPL. 89 (2010) 57005. https://doi.org/10.1209/0295-5075/89/57005.
[26] C.S. Deng, H. Xu, Anomalous localization and dual role of correlation in one-dimensional electronic systems with long-range correlated disorder, Physica E 44 (2012) 1747-1751.
https://doi.org/10.1016/j.physe.2011.12.002.
[27] M.K. Nezhad, S.M. Mahdavi, A.R. Bahrampour, M. Golshani, Effect of long-range correlated disorder on the transverse localization of light in 1D array of optical waveguides, Opt. Commun. 307 (2013) 39-45.
https://doi.org/10.1016/j.optcom.2013.06.004.
[28] N. Hatano, D.R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77 (1996) 570-573. https://doi.org/10.1103/PhysRevLett.77.570.
[29] A. V. Kolesnikov, K.B. Efetov, Localization-delocalization transition in non-Hermitian disordered systems, Phys. Rev. Lett. 84 (2000) 5600. https://doi.org/10.1103/PhysRevLett.84.5600.
[30] T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D.N. Christodoulides, S. Nolte, A. Szameit, Mobility transition from ballistic to diffusive transport in non-Hermitian lattices, Nat. Commun. 4 (2013) 3533. https://doi.org/10.1038/ncomms3533.
[31] A. Basiri, Y. Bromberg, A. Yamilov, H. Cao, T. Kottos, Light localization induced by a random imaginary refractive index, Phys. Rev. A 90 (2014) 043815. https://doi.org/10.1103/PhysRevA.90.043815.
[32] B.P. Nguyen, K. Kim, Transport and localization of waves in ladder-shaped lattices with locally PT-symmetric potentials, Phys. Rev. A 94 (2016) 062122. https://doi.org/10.1103/PhysRevA.94.062122.
[33] S. Longhi, Bloch oscillations in complex crystals with PT symmetry, Phys. Rev. Lett. 103 (2009) 123601. https://doi.org/10.1103/PhysRevLett.103.123601.
[34] S. Longhi, D. Gatti, G. Della Valle, Non-Hermitian transparency and on-way transport in low-dimensional lattices by an imaginary gauge field, Phys. Rev. B 92 (2015) 094204.
https://doi.org/10.1103/PhysRevB.92.094204.
[35] M.S. Rudner, L.S. Levitov, Topological transition in a non-Hermitian quantum walk, Phys. Rev. Lett. 102 (2009) 065703. https://doi.org/10.1103/PhysRevLett.102.065703.
[36] J.M. Zuener, M.C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M.S. Rudner, M. Segev, A. Szameit, Observation of a topological transition in the bulk of a non-Hermitian system, Phys. Rev. Lett. 115 (2015) 040402. https://doi.org/10.1103/PhysRevLett.115.040402.
[37] B.P. Nguyen, D.K. Phung, K. Kim, Anomalous localization enhancement in one-dimensional non-Hermitian disordered lattices, J. Phys. A: Math. Theor. 53 (2020) 045003. https://doi.org/10.1088/1751-8121/ab5eb8.
[38] A.F. Tzortzakakis, K.G. Makris, E.N. Economou, Non-Hermitian disorder in two-dimensional optical lattices, Phys. Rev. B 101 (2020) 014202. https://doi.org/10.1103/PhysRevB.101.014202.
[39] Y. Huang, B.I. Shklovskii, Anderson transition in three-dimensional systems with non-Hermitian disorder, Phys. Rev. B 101 (2020) 014204. https://doi.org/10.1103/PhysRevB.101.014204.
[40] D.S. Wiersma, The physics and applications of random lasers, Nat. Phys. 4 (2008) 359-367.
https://doi.org/10.1038/nphys971.