Chem integers describing quantized transport change (typically by~1 ) when the energy levels or bands with which they are associated become degenerate.We give a statistical treatment of these degeneracies and the consequent fluctuations of the Chem integers. The density of degeneracies is calculated exactly for a parametrization of the Gaussian unitary ensemble of matrices, and we present numerical results indicating that the "gas" of degeneracies has a "charge neutrality" property. The results apply to a broad class of complex systems after rescaling the parameter space.The Chem integer is a topological invariant which can characterize quantized transport without dissipation. It occurs in the analysis of the quantized Hall effect for electrons in a crystal lattice without disorder [1], and in models for transport with a time dependent periodic potential [2]. It also arises in more general models for quantized conductance [3], in which the potential need not be periodic. In all of these systems the Hamiltonian can be represented by a Hermitian operator which is a periodic function of two parameters, either Bloch wave vectors or magnetic fIuxes threaded through holes in the sample; we denote these parameters by X& and X2, and assume that they are scaled so that both periods are 2~. There is a Chem integer N, associated with each eigenvalue E"of the operator. In some systems the Chem integers may be large and impossible to calculate analytically.An example is illustrated in Fig. 1; the system is a chaotic quantum billiard pierced by three magnetic Iluxes, Fix;/e, i = 1, 2, 3. Chem integers are associated with each energy level, describing quantized transfer of charge around the second Aux in response to increasing the first Aux by one quantum. This Letter is the first of a pair of publications which characterize the Chem integers statistically, by considering the effect of varying a third parameter X3. It is not essential that the Hamiltonian be periodic in X3, this parameter could describe a change in the shape of the boundary or an externally applied electric field instead of a flux. Pairs of eigenvalues typically degenerate at isolated points in the space of the three parameters (Xt, X2 X3).The Chem integers generically change by~1 at values of X3 for which there is a degeneracy [4]; a sign can be attached to each degeneracy, positive if the Chem number of the lower degenerating level increases when X3 increases past the point of degeneracy, negative otherwise. Reference [5] lists some earlier papers which have investigated the changes of Chem numbers at degeneracies. If the positions and signs of degeneracies were randomly distributed in both parameter space and over energy level labels, the Chem integers would perform a random walk Here 23 is the density of degeneracies in parameter space between a given level and one of its neighbors, A. = 4' is the area of the (Xt t X2) torus, and g(AX3) is a function which contains information about the correlations between positions of degeneracies: If they were randomly d...

We investigate the addition spectrum of disordered quantum dots containing spinless interacting fermions using the self-consistent Hartree-Fock approximation. We concentrate on the regime r s > ∼ 1, with finite dimensionless conductance g. We find that in this approximation the peak spacing fluctuations do not scale with the mean single particle level spacing for either Coulomb or nearest neighbour interactions when r s > ∼ 1. We also show that Koopmans' approximation to the addition spectrum can lead to errors that are of order the mean level spacing or larger, both in the mean addition spectrum peak spacings, and in the peak spacing fluctuations.