Universal Scrambling Properties of Spectra and Wave functions in Disordered Interacting Systems

Recent experiments on quantum dots in the Coulomb Blockade regime have shown how adding successive electrons into a dot modifies the energy spectrum and the wave functions of the electrons already present in the dot. Using a microscopic model, we study the importance of electron-electron interaction on these ``scrambling'' effects. We compute the Hartree-Fock single particle properties as function of the number $p$ of added electrons. We define parametric correlation functions that characterize the scrambling properties of the Hartree-Fock wave functions and energy spectra. We find that each of these correlation functions exhibit a universal behavior in terms of the ratio $p/p^{\ast}$ where $p^{\ast}$ is a characteristic number that decreases with increasing either the interaction strength, the disorder strength or the system sizeComment: proceedings Recontres de Moriond, les Arcs 2001, 2 figure

Berry Curvature and Quantum Metric in NN-band systems -- an Eigenprojector Approach

The eigenvalues of a parameter-dependent Hamiltonian matrix form a band structure in parameter space. In such $N$-band systems, the quantum geometric tensor (QGT), consisting of the Berry curvature and quantum metric tensors, is usually computed from numerically obtained energy eigenstates. Here, an alternative approach to the QGT based on eigenprojectors and (generalized) Bloch vectors is exposed. It offers more analytical insight than the eigenstate approach; in particular, it is shown that the full QGT of each band can be computed from the Hamiltonian and the respective band energy alone. Most saliently, the well-known two-band formula for the Berry curvature in terms of the Hamiltonian vector is generalized to arbitrary $N$. The formalism is illustrated using three- and four-band multifold fermion models that have very different geometrical and topological properties despite an identical band structure. From a broader perspective, the methodology used in this work can be applied to compute any physical quantity or to study the quantum dynamics of any observable without the explicit construction of energy eigenstates.Comment: 11+12 pages, 3+1 figures. Revised version for a more compact and direct presentation of the main results on the quantum geometric tensor. An extended discussion of the multifold fermion models of v1 (Section 6) will appear elsewher

Winding vector: how to annihilate two Dirac points with the same charge

The merging or emergence of a pair of Dirac points may be classified according to whether the winding numbers which characterize them are opposite ($+-$ scenario) or identical ($++$ scenario). From the touching point between two parabolic bands (one of them can be flat), two Dirac points with the {\it same} winding number emerge under appropriate distortion (interaction, etc), following the $++$ scenario. Under further distortion, these Dirac points merge following the $+-$ scenario, that is corresponding to {\it opposite} winding numbers. This apparent contradiction is solved by the fact that the winding number is actually defined around a unit vector on the Bloch sphere and that this vector rotates during the motion of the Dirac points. This is shown here within the simplest two-band lattice model (Mielke) exhibiting a flat band. We argue on several examples that the evolution between the two scenarios is general.Comment: 5 pages, 6 figure

A new magnetic field dependence of Landau levels on a graphene like structure

We consider a tight-binding model on the honeycomb lattice in a magnetic field. For special values of the hopping integrals, the dispersion relation is linear in one direction and quadratic in the other. We find that, in this case, the energy of the Landau levels varies with the field B as E_n(B) ~ [(n+\gamma)B]^{2/3}. This result is obtained from the low-field study of the tight-binding spectrum on the honeycomb lattice in a magnetic field (Hofstadter spectrum) as well as from a calculation in the continuum approximation at low field. The latter links the new spectrum to the one of a modified quartic oscillator. The obtained value $\gamma=1/2$ is found to result from the cancellation of a Berry phase.Comment: 4 pages, 4 figure