The viscosity of quantum fluids with an energy gap at zero temperature is non-dissipative and is related to the adiabatic curvature on the space of flat background metrics (which plays the role of the parameter space). For a

It is shown that the integers found by Thouless

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing. * Electronic address: jansen@math.tu-berlin.de † Electronic address: seiler@math.tu-berlin.de ‡ Electronic address: Marybeth.Ruskai@tufts.edu

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.

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.

In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result

We describe a precise mathematical theory of the Laughlin argument for the quantization of the Hall conductance for general multiparticle Schrodinger operators with general background potentials. The quantization is a consequence of the geometric content of the conductance, namely, that it can be identified with an integral over the first Chern class. This generalizes ideas of Thouless et ai, for noninteracting Bloch Hamiltonians to general (interacting and nonperiodic) ones.PACS numbers: 03.65. Bz, 02.40. + m, 72.20.My The integer quantization of the Hall conductance has been explained by Laughlin 1 making clever use of a nontrivial geometry: a ring threaded by a flux tube, combined with a gauge argument. The impact of this work on the development of the subject cannot be overestimated.Our purpose here is to describe a precise mathematical theory of this argument. The two key issues are, first, Laughlin's identification of a physical quantity as the Hall conductance averaged over one unit of quantum flux (of the flux tube that threads the ring). Following Laughlin we shall slightly abuse the terminology and call it the Hall conductance.The second main theme will be the identification of the geometric content of the Hall conductance: Roughly speaking, there is a natural notion of curvature describing how the state of the system is parallel transported in the Hilbert space of states. The Hall conductance is a suitable integral of the corresponding curvature. More precisely, it is an integral over the first Chern class. 2 This was first recognized by Thouless et al in the special case of noninteracting Bloch Hamiltonians. 3 What is shown here is that this holds generally, with electron-electron interaction and general background potential. (No flux averaging is necessary for Bloch Hamiltonians.) Bellissard generalized the result of Thouless et al from rational to real magnetic flux. 4 We shall replace Laughlin's condition that the Fermi energy lies in the region of the localized states (which is not an appropriate condition for multiparticle Hamiltonians) by a condition of nondegeneracy of the multiparticle ground states (for all fluxes).Our work has been independent of, but is nevertheless closely related to, a recent published paper of Niu and Thouless. 5 The general framework is similar, although there are some differences in the details and in the approach. In both works one needs the ground state to be separated from the rest of the spectrum by a finite gap (the nondegeneracy condition). In both approaches one considers time-periodic Hamiltonians (in Niu-Thouless only up to unitary equivalence). In Niu and Thouless the time dependence resides in the substrate potential and it comes from a Galilean transformation that removes the electric field. In our case, the time dependence comes from generating the electromotive force by a flux tube, and so resides in the minimal coupling term in the Hamiltonian. In both, strict quantization is obtained only after a suitable averaging: In Niu-Thouless the averaging is o...

The amazingly precise quantization of Hall conductance in a two-dimensional electron gas can be understood in terms of a topological invariant known as the Chern number.