Abstract
We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.
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Altland, A., Simons, B.D., Zirnbauer, M.: Theories of low-energy quasi-particle states in disordered d-wave superconductors. Phys. Rep. 359, 283–354 (2002)
Altland, A., Zirnbauer, M.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997)
Anderson, P.W.: Theory of dirty superconductors. J. Phys. Chem. Solids 11, 26–30 (1959)
Balatsky, A.V., Vekhter, I., Zhu, J.X.: Impurity-induced states in conventional and unconventional superconductors. Rev. Mod. Phys. 78, 373–433 (2006)
Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)
de Gennes, P.G.: Superconductivity of Metals and Alloys. Benjamin, New York (1966)
Durst, A.C., Lee, P.A.: Impurity-induced quasiparticle transport and universal-limit Wiedemann-Franz violation in d-wave superconductors. Phys. Rev. B 62, 1270–1290 (2000)
Gebert, M.: Anderson localization for random block operators. Diploma thesis, LMU München (2011)
Hislop, P., Müller, P.: A lower bound for the density of states of the lattice Anderson model. Proc. Am. Math. Soc. 136, 2887–2893 (2008)
Jeske, F.: Über lokale Positivität der Zustandsdichte zufälliger Schrödinger-Operatoren. Ph.D. thesis, Ruhr-Universität Bochum (1992)
Kirsch, W.: Random Schrödinger operators: a course. In: Holden, H., Jensen, A. (eds.) Schrödinger Operators. Lect. Notes in Phys., vol. 345, pp. 264–370. Springer, Berlin (1989)
Kirsch, W.: An invitation to random Schrödinger operators. Panor. Synth. 25, 1–119 (2008)
Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Gesztesy, F., Deift, P., Galvez, C., Perry, P., Schlag, W. (eds.) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Part 2. Proc. Symp. Pure Math., vol. 76, pp. 649–696. Am. Math. Soc, Providence (2007)
Krüger, H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. Preprint arXiv:1006.5233 (2010)
Lifshitz, I.M.: The energy spectrum of disordered systems. Adv. Phys. 13, 483–536 (1964)
Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, San Diego (1978)
Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)
Veselić, I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré 11, 991–1005 (2010)
Vishveshwara, S., Senthil, T., Fisher, M.P.A.: Superconducting “metals” and “insulators”. Phys. Rev. B 61, 6966–6981 (2000)
Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44, 9–15 (1981)
Ziegler, K.: Quasiparticle states in disordered superfluids. Z. Phys. B 86, 33–38 (1992)
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Work supported by the German Research Foundation (DFG) through Sfb/Tr 12 “Symmetries and Universality in Mesoscopic Systems” and DFG Research Unit 718 “Analysis and Stochastics in Complex Physical Systems”.
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Kirsch, W., Metzger, B. & Müller, P. Random Block Operators. J Stat Phys 143, 1035–1054 (2011). https://doi.org/10.1007/s10955-011-0230-y
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DOI: https://doi.org/10.1007/s10955-011-0230-y