Charge deficiency, charge transport and comparison of dimensions

Abstract

Abstract. We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.

Moment analysis for localization in random Schrödinger operators

“…The hallmark of localization is rapid (even exponential) decay of G E (x, y) at energies in the spectrum, though in this case it occurs with pre-factors which are not uniform in space and diverge at a dense countable set of eigenvalues. Rapid decay of the Green function is related to the non-spreading of wave packets supported in the corresponding energy regimes and various other manifestations of localization whose physical implications have been extensively studied in regards to the conductive properties of metals [8,55,73,1,54] and in particular to the quantum Hall effect [36,57,10,12,3].…”

Geometric Properties of Transport in Quantum Hall Systems

“…Simon (1994a), avoiding the language of non-commutative geometry completely. Note that the operator U de ned by m ultiplication with u := X 1 + iX 2 jX 1 + iX 2 j (14) is the gauge transformation related to unit ux tube piercing the IR 2 at the origin.…”

“…We consider now an application of the index approach t o a quantum Hall system, following J. E. Avron, R. Seiler, B. Simon (1994a). Our model describes non relativistic, non interacting fermions in the punctured plane C n f a g , a 2 C, with random impurities.…”

Advances in Quantum Mechanics

“…Later theoretical investigations showed that a topological phenomenon underlies this quantization: the integer n in the above formula was shown to be the first Chern number of a vector bundle, naturally associated to the quantum system [27,1,2]. The only role played by the magnetic field in quantum Hall systems is that of breaking time-reversal symmetry: if the system were time-reversal symmetric, then the Hall conductivity would vanish, and the system would remain in an insulating state.…”