Abstract
In the tight-binding random Hamiltonian on Z d, we consider the charge transport induced by an electric potential which varies sufficiently slowly in time, and prove that it is almost surely equal to zero at high disorder. In order to compute the charge transport, we adopt the adiabatic approximation and prove a weak form of adiabatic theorem while there is no spectral gap at the Fermi energy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. Aizenman, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6:1163–1182 (1994).
M. Aizenman and G. M. Graf, Localization bounds for an electron gas, J. Phys. A: Math. Gen. 31:6783–6806 (1998).
M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Commun. Math. Phys. 157:245–278 (1993).
P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109:1492–1505 (1958).
J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Commun. Math. Phys. 203:445–463 (1999).
J. E. Avron, R. Seiler, and L. G. Yaffe, Adiabatic theorems and applications to the quantum Hall effect, Commun. Math. Phys. 110:33–49 (1987).
J. Bellissard, A. van Elst, and H. Schultz-Baldes, The non-commutative geometry of the quantum Hall effect, J. Math. Phys. 35(10):5373–5451 (1994).
J. M. Combes and L. E. Thomas, Asymptotic behaviour of eigenfunctions for multi-particle Schrödinger operators, Commun. Math. Phys. 34:251–270 (1973).
H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, Berlin, Heidelberg, New York, 1987).
R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69:153–200 (1996).
J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88:151–184 (1983).
T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Jpn. 5:435–439 (1950).
M. Klein and R. Seiler, Power-low corrections to the Kubo formula vanish in quantum Hall systems, Commun. Math. Phys. 128:141–160 (1990).
Q. Niu, Quantum adiabatic particle transport, Phys. Rev. B 34:5093 (1986).
Q. Niu, Towards a quantum pump of electric charge, Phys. Rev. Lett. 64:1812 (1990).
Q. Niu and D. J. Thouless, Quantized adiabatic charge transport in the presence of disorder and many-body interaction, J. Phys. A: Math. Gen. 17:2453–2462 (1984).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Functional Analysis (Academic Press, New York, 1975).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975).
H. Schultz-Baldes and J. Bellissard, A kinetic theory for quantum transport in aperiodic media, J. Stat. Phys. 91:991–1026 (1998).
B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random hamiltonians, Commun. Pure. Appl. Math. 39:75–90 (1986).
D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27:6083 (1983).
H. von Dreifus and A. Krein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys. 124:285–299 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nakano, F., Kaminaga, M. Absence of Transport Under a Slowly Varying Potential in Disordered Systems. Journal of Statistical Physics 97, 917–940 (1999). https://doi.org/10.1023/A:1004657913118
Issue Date:
DOI: https://doi.org/10.1023/A:1004657913118