Luca G. Molinari


THESIS PROPOSALS

  • Functional integral and effective potential for electronic systems
    • (Tesi magistrale)
    Construction of the functional integral for the generator of density and current correlators for interacting electrons at null or finite temperature. Evaluation of the effective potential via Legendre transform and saddle point expansion. Comparison with effective potential of density functional theory.
    (For the first part, see textbooks by Negele & Orland, Popov).

  • Type II superconductors in Ginzburg-Landau theory
    • (Tesi triennale)
    Derive the Ginzburg Landau equations for superconductivity. Study 1D problem with external H field in the linearized case and obtain Abrikosov's lattice of vortices. Numerically study the non-linearized 1D problem.
    (see: Schmidt, Introduction to superconuctivity; De Gennes, Superconductivity of metals and alloys).

  • Spectral density of random matrices and the electrostatic analogy
    • (Tesi triennale)
    The eigenvalues of a matrix may be viewed as unit point charges in the plane. The electrostatic picture is useful in the derivation of the spectral density of random matrices.
    (see books by M.L.Mehta on random matrices, or by Forrester on log-gases).

  • The Hartree-Fock description of the electron gas
    • (Tesi triennale)
    Obtain the total energy in HF approximation for the uniform case, and discuss the choice of ground state (read existing literature about this) Study the problem of semi-infinite electron gas in HF.

  • Gutzwiller's trace formula
    • (Tesi triennale)
    Feynman's path integral gives the quantum amplitude of motion from x to x' in time t in terms of trajectories. The levels of a quantum system are the poles of the trace of the resolvent, which is related by Fourier transform to the Feynman amplitude for closed trajectories. This gives an evaluation of the spectrum in terms of the periodic orbits of the system (Gutzwiller).
    Refs: Schulman (for Feynman path integral), Gutwiller's book (for the trace formula).

  • Anderson localization in quasi-1D devices
    • (Tesi triennale/magistrale)
    Electrons in a lattice with random impurities are subject to Anderson localization. The mathematical model is a Hamiltonian matrix H=T+V where T is the Laplacian matrix of the lattice, and V is a diagonal matrix with independent random entries. Localization of eigenstates is described by the spectrum of a transfer matrix, for which a spectral duality holds. The proposal is to study the Lyapunov spectrum for narrow strips. The thesis work requires some computational skill.

  • Ising model on random graphs
    • (tesi triennale)
    Cubic and quartic matrix models of two coupled matrices are equivalent, in the large N limit, to the Ising model on random planar lattices with coordination 3 and 4 (Kazakov and Boulatov). They were analytically solved by means of orthogonal polynomials, and the critical exponents were obtained in the continuum limit.

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