One-dimensional discrete Dirac operators in a decaying random potential. I: Spectrum and dynamics. (English) Zbl 1441.82009
Summary: We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type \(n^{-\alpha}\) for \(\alpha > 0\). We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region \(\alpha >\frac{1}{2}\); a transition from pure point to singular continuous spectrum in the critical region \(\alpha =\frac{1}{2}\); and pure point spectrum in the sub-critical region \(\alpha <\frac{1}{2}\). From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 47B80 | Random linear operators |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 82C44 | Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
References:
| [1] | Andrei, EY; Du, X.; Duerr, F.; Lucian, A.; Skachko, I., Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene, Nature, 462, 192-5 (2009) |
| [2] | Aizenman, M.; Molchanov, S., Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phy., 157, 245-278 (1993) · Zbl 0782.60044 |
| [3] | Aizenman, M.; Sims, R.; Warzel, S., stability of the absolutely continuous spectrum of random schrödinger operators on tree graphs, Probab, Theory Rel. Fields, 136, 363-394 (2006) · Zbl 1169.82007 |
| [4] | Aizenman, M., Warzel, S.: Random Operators: Disordered effects on Quantum spectra and dynamics, Graduate Studies in Mathematics 168 AMS (2016) · Zbl 1333.82001 |
| [5] | Azuma, K., Weighted sums of certain dependent random variables, Tô,hoku Math. J., 19, 3, 357-367 (1967) · Zbl 0178.21103 |
| [6] | Barbaroux, J.-M., Cornean, H., Zalczer, S.: Localization for gapped Dirac Hamiltonians with random pertubations: Application to graphene antidot lattices, arXiv:1812.01868 · Zbl 1420.82028 |
| [7] | Bissbort, U.; Esslinger, T.; Greif, D.; Hofstetter, W.; Jotzu, G.; Messer, N.; Uehlinger, T., Artificial graphene with tunable interactions, Phys. Rev. Lett., 111, 185307 (2013) |
| [8] | Bourgain, J., On random schrödinger operators on \(\mathbb{Z}^2\) ℤ2, Discret Contin. Dyn. Syst., 8, 1-15 (2002) · Zbl 1006.47034 |
| [9] | Bourgain, J., Random Lattice Schrödinger Operators with Decaying Potential: Some Higher Dimensional Phenomena, Geometric Aspects of Functional Analysis, Lectures Notes in Math., vol. 1807, 70-98 (2003), Berlin-Heidelberg: Springer, Berlin-Heidelberg · Zbl 1071.47042 |
| [10] | Bourget, O., Moreno Flores, G.R., Taarabt, A.: Dynamical localization for the one-dimensional continuum Anderson model in a decaying random potential, preprint · Zbl 1453.82037 |
| [11] | Bucaj, V.: On the Kunz-Souillard approach to localization for the discrete one dimensional generalized Anderson model, preprint. · Zbl 1448.47044 |
| [12] | Bucaj, V., The Kunz-Souillard approach to localization for jacobi operators, Oper. Matrices., 12, 4, 1099-1127 (2018) · Zbl 1448.47044 |
| [13] | Bucaj, V.; Damanik, D.; Fillman, J.; Gerbuz, V.; VandenBoom, T.; Wang, F.; Zhang, Z., Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent, Trans. Amer. Math. Soc., 372, 3619-3667 (2019) · Zbl 1422.35022 |
| [14] | Bolotin, KI; Ghahari, F.; Kim, P.; Shulman, MD; Stormer, HL, Observation of the fractional quantum Hall effect in graphene, Nature, 462, 196-9 (2009) |
| [15] | Basu, C.; Macía, E.; Domínguez-Adame, F.; Roy, CL; Sánchez, A., Localization of relativistic electrons in a One-Dimensional disordered system, J. Phys. A, 27, 3285-3291 (1994) |
| [16] | Bolotin, KI; Jiang, Z.; Sikes, KJ, Ultrahigh electron mobility in suspended graphene, Solid State Commun., 146, 351-5 (2008) |
| [17] | Carmona, R., Exponential localization in one dimensional disordered systems, Duke, Math. J., 49, 191-213 (1982) · Zbl 0491.60058 |
| [18] | Carmona, R.; Klein, A.; Martinelli, F., Anderson localization for bernoulli and other singular potentials, commun, Math. Phys., 108, 41-66 (1987) · Zbl 0615.60098 |
| [19] | Carvalho, S.; de Oliveira, C.; Prado, R., Sparse one-dimensional discrete Dirac operators II: Spectral properties, J. Math. Phys, 073501, 52 (2011) · Zbl 1317.81096 |
| [20] | Carvalho, S.; de Oliveira, C.; Prado, R., Dynamical localization for discrete anderson dirac operators, J. Stat. Phys., 167, 2, 260-296 (2017) · Zbl 1372.35044 |
| [21] | Comets, F.; Yoshida, N., Branching random walks in Space-Time random environment: Survival probability, global and local growth rates, J. Theor. Prob., 24, 657-687 (2011) · Zbl 1235.60146 |
| [22] | Cycon, HL; Froese, RG; Kirsch, W.; Simon, B., Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer Study Edition (1987), Berlin: Springer-Verlag, Berlin · Zbl 0619.47005 |
| [23] | Castro Neto, AH; Guinea, F.; Peres, NMR; Novoselov, KS; Geim, AK, The electronic properties of graphene, Rev. Modern Phys., 81, 109-162 (2009) |
| [24] | Damanik, D., Gorodetski, A: An extension of the Kunz-Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators, Adv. Math (2016) · Zbl 1344.47021 |
| [25] | De Bièvre, S.; Germinet, F., dynamical Localization for the Random Dimer schrödinger Operator, J. Stat. Phys., 98, 5-6, 1134-1148 (2000) · Zbl 1005.82028 |
| [26] | Delyon, F., appearance of a purely singular continuous spectrum in a class of random schrödinger operators, J. Statist. Phys., 40, 621-630 (1985) · Zbl 0642.60042 |
| [27] | Delyon, F.; Simon, B.; Souillard, B., From power pure point to continuous spectrum in disordered systems, Ann. Henri Poincaré, 42, 6, 283-309 (1985) · Zbl 0579.60056 |
| [28] | Del Rio, R.; Jitomirskaya, S.; Last, Y.; Simon, B., What is localization?, Phys. Rev. Lett., 75, 117-119 (1995) |
| [29] | Del Rio, R.; Jitomirskaya, S.; Last, Y.; Simon, B., Operators with singular continuous spectrum IV: Hausdorff dimensions, rank one pertubations and localization, J. Anal. Math., 69, 153-200 (1996) · Zbl 0908.47002 |
| [30] | de Oliveira, C.; Prado, R., Dynamical delocalization for the 1D Bernoulli discrete Dirac operator, J. Phys. A, 38, 115-119 (2005) · Zbl 1067.81030 |
| [31] | de Oliveira, C.; Prado, R., Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator, J. Math. Phys., 072105, 46 (2005) · Zbl 1110.81079 |
| [32] | de Oliveira, C.; Prado, R., Dynamical lower bounds for 1D Dirac operators, Math. Z., 259, 1, 45-60 (2008) · Zbl 1138.81409 |
| [33] | de Oliveira, C.; Prado, R., Sparse 1D discrete Dirac operators I: Quantum transport, J. Math. Anal. Appl., 385, 947-960 (2012) · Zbl 1229.81077 |
| [34] | Durrett, R., Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics (2010), New York: Cambridge University Press, New York · Zbl 1202.60001 |
| [35] | Dean, CR; Wang, L.; Maher, P., Hofstadter’s butterfly and the fractal quantum Hall effect in moire superlattices, Nature, 497, 598-602 (2013) |
| [36] | Figotin, A.; Germinet, F.; Klein, A.; Müller, P., persistence of Anderson localization in schrödinger operators with decaying random potentials, Ark. Mat., 45, 15-30 (2007) · Zbl 1159.47059 |
| [37] | Froese, R.; Hasler, D.; Spitzer, W., Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem, Comm. Math. Phys., 269, 239-257 (2007) · Zbl 1117.82024 |
| [38] | Germinet, F.; Klein, A., Bootstrap multiscale analysis localization in random media, Commun. Math Phys., 222, 415-448 (2001) · Zbl 0982.82030 |
| [39] | Germinet, F.; Kiselev, A.; Tcheremchantsev, S., transfer matrices and transport for schrödinger operators, Ann. Inst. Fourier, 54, 787-830 (2004) · Zbl 1074.81019 |
| [40] | Germinet, F.; Taarabt, A., spectral properties of dynamical localization for schrödinger operators, Rev. Math. Phys., 25, 9 (2013) · Zbl 1278.81076 |
| [41] | Novoselov, KS; Geim, AK; Morozov, SV, Two-dimensional gas of massless Dirac fermions in graphene, Nature, 438, 197-200 (2005) |
| [42] | Goldsheid, I.; Molchanov, S.; Pastur, L., a pure point spectrum of the stochastic one-dimensional schrödinger equation, Funct. Anal. Appl., 11, 1-10 (1977) · Zbl 0368.34015 |
| [43] | Golénia, S.; Haugomat, T., On the a.c. spectrum of 1D discrete Dirac operator, Methods Funct. Anal. Topology, 20, 3, 252-273 (2014) · Zbl 1324.47066 |
| [44] | Hamza, E.; Stolz, G., Lyapunov exponents for unitary Anderson models, J. Math. Phys., 043301, 48 (2008) · Zbl 1137.82315 |
| [45] | Hunt, B.; Sanchez-Yamagishi, JD; Young, AF, Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure, Science, 340, 1427-30 (2013) |
| [46] | Jitomirskaya, S.; Zhu, X., Large deviations of the Lyapunov exponent and localization for the 1D Anderson model, Comm. Math. Phys., 370, 1, 311-324 (2019) · Zbl 07083976 |
| [47] | Klein, A., Extended states in the Anderson model on the Bethe lattice, Adv. Math., 133, 163-184 (1998) · Zbl 0899.60088 |
| [48] | Kiselev, A.; Last, Y.; Simon, B., modified prüfer and EFGP transforms and the spectral analysis of one-dimensional schrödinger operators, Comm. Math. Phys., 194, 1-45 (1998) · Zbl 0912.34074 |
| [49] | Katsnelson, MI; Novoselov, KS; Geim, AK, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys., 2, 9, 620-625 (2006) |
| [50] | Kiselev, A.; Remling, C.; Simon, B., effective perturbation methods for one-dimensional schrödinger operators, J. Diff. Equ., 151, 290-312 (1999) · Zbl 0921.34076 |
| [51] | Krishna, M., Anderson model with decaying randomness: existence of extended states, Proc. Indian Acad. Sci. (Math. Sci.), 100, 285-294 (1990) · Zbl 0731.47057 |
| [52] | Kunz, H.; Souillard, B., sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys., 78, 201-246 (1980) · Zbl 0449.60048 |
| [53] | Last, Y.; Simon, B., eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional schrödinger operators, Invent. Math., 135, 329 (1999) · Zbl 0931.34066 |
| [54] | Novoselov, KS; Geim, AK; Morozov, SV; Jiang, D.; Zhang, Y.; Dubonos, SV; Grigorieva, IV; Firsov, AA, Electric field effect in atomically thin carbon films, Science, 306, 5696, 666-669 (2004) |
| [55] | Rahu, S.; Haldane, FDM, Analogs of quantum-Hall-effect edge states in photonic crystals, Phys. Rev. A, 78, 033834 (2008) |
| [56] | Rahu, S.; Haldane, FDM, Possible realization of directional optical waveguides in photonic crystals with broken Time-Reversal symmetry, Phys. Rev. Lett., 100, 013904 (2008) |
| [57] | Roy, CL; Basu, C., Relativistic study of electrical conduction in disordered systems, Phys. Rev. B, 45, 14293-14301 (1992) |
| [58] | Sarma, SD; Adam, S.; Hwang, EH; Rossi, E., Electronic transport in two-dimensional graphene, Rev. Mod. Phys., 83, 407 (2011) |
| [59] | Simon, B., Some Jacobi matrices with decaying potential and dense point spectrum, Comm. Math. Phys., 87, 253-258 (1982) · Zbl 0546.35048 |
| [60] | Simon, B.: Spectral Analysis of rank one perturbations and applications, CRM Lectures Notes Vol. 8, Amer. Math. Soc, Providence, RI (1995) · Zbl 0824.47019 |
| [61] | Zhang, Y.; Tan, YW; Stormer, HL; Kim, P., Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature, 438, 201-4 (2005) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
