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Consider a many-body system, like a molecule or solid, with Hamiltonian = + , where is the interaction, = Σ is the kinetic energy with ground state |Φ〉 and eigenvalue . I shall write to mean that the product is over occupied spin-orbitals, and produces a Fermi sphere in the thermodynamic limit. How can we find the ground state energy of ? Standard perturbation theory fails unless is small compared to the unperturbed energy difference; in practice, it fails almost always, since in most interesting problems, the spectrum is continuous.

For Jellium, due to momentum conservation, we re-label the diagram 11.10 d) involving the bubble as in Figure 12.1 and (11.73) becomes and (11.76) becomes, restoring , with So, the self-energy (11.86) becomes and since , this diverges at small . The electron gains self-energy by exciting the medium and then re-adsorbing the excitations, but the process runs out of control for the long-wavelength ones.What is going wrong at long distances? It is the Coulomb interaction , which is causing the divergence by its long range, but should actually be replaced by a shorter ranged screened interaction .

Phenomena like electron transport, electronic transitions in chemisorption, desorption, molecule-surface collision processes, and sputtering are naturally described by time-dependent electronic Hamiltonians. What happens in the Fano-Anderson model (Chapter 5.1.2) if the and parameters depend on time? Such a time-dependence occurs because the Hamiltonian of electrons depends on the position of the nuclei, and, through this, from the time. Such problems are completely outside the scope of the diagram method discussed so far; the equation of motion method (EOM) can work but depends on some approximate truncation procedure. This Chapter is devoted to the quantum theory of such non-equilibrium processes. A powerful generalized perturbation method enables us to deal with time dependent problems and processes that are far from equilibrium, generalizing the Kubo approach [29] to all orders. It is a general and in principle exact technique; the Feynman method is recovered as a particular case. There is no need to assume that the perturbation is small or that the system deviates little from equilibrium; one does not need to make any assumption that the system evolves in a reversible way. As one could expect, the study of the excited states and out-of-equilibrium situations involves special difficulties, however this generalized theoretical framework is also powerful for exploring the equilibrium nonlinear response to strong perturbations, that can be treated to all orders.

Any stationary quantum problem, with any number of degrees of freedom, can be mapped into a solvable linear chain one-body tight-binding model . From an abstract mathematical viewpoint, the method for putting a symmetric matrix in three-diagonal form which is readily diagonalized was invented by Lanczos[58] in 1950. This is enough to calculate the Green’s function of systems with a finite number of states, and the algorithm is powerful and convenient even for large matrices provided that they are sparse, that is, most elements vanish. Haydock and coworkers[59] extended the method and developed applications to physical systems with many degrees of freedom. Let , = 0, 1, 2 ... be the site orbitals of a semi-infinite chain, with local levels . The Hamiltonian is defined by where ’s are hopping integrals, and its matrix is three-diagonal: For this model, we want the local retarded Green’s function is which is the 00 element of

For strong electromagnetic fields, like those produced by a laser, the linear response theory fails. Interesting processes leading to color changes become important, and this has implications and applications. In this Chapter we shall study the Raman effect, Second Harmonic Generation, the diffusion of Coherent light and the Dynamical Stark effect.

Without the gauge invariance, any theory is untenable. In classical theory, the Hamiltonian of a charged particle is where p is the kinetic momentum and A the vector potential. Both are unobservable; one could have started with new potentials ′, ′ such that where (, ) is a completely arbitrary function. Only fields appear in the classical equations of motion; quite the other way, the gauge transformed Schrödinger equation in terms of the old potentials reads

Long ago, Kohn and Luttinger [174] proposed that a superconducting instability should occur in Jellium. They focussed on pairs of parallel spin electrons of large relative angular momentum L. The large L and the triplet state keep the electrons apart, so the Coulomb repulsion is particularly mild in such pairs. At long distances, the screened interaction undergoes Friedel oscillations due to the singularity of the dielectric function at 2 (See Section 12.1.1). This means a slight over-screening of the repulsion in some distance ranges: one gets an effective attraction from the repulsion via a quantum mechanical correlation effect. Attraction in some distance ranges does not necessarily imply binding, but Kohn and Luttinger suggested that this effect could indeed produce pairing. While no such superconducting instability appears to be relevant to ordinary metals, the paradoxical theoretical idea that attraction could result from repulsion is fascinating and the discovery of high- superconductivity [175] has stimulated a discussion on the possibility that something similar is realized in the Cuprates (although the pairs are singlets with L=2, so some important modification is needed). The fact that can be above liquid N (rather than liquid He) temperatures, and the evidence that this occurs in strongly correlated materials suggests to part of the community that the mechanism must be different from the conventional phonon-assisted BCS one[181].

Consider a repulsive Hubbard model, defined on a d-dimensional lattice or graph of || sites, with nearest-neighbor hopping. No symmetry is assumed; however we can use as an example a simple square Hubbard Model with periodic boundary conditions and hopping between nearest neighbors (n.n.):

We used EOM several times (see Sections 4.3,4.4,5.1.2); now we extend the approach used in Equation (4.39) for the free propagator to interacting problems. Using the many-body Hamiltonian (1.63) one readily obtains with all the operators in the Heisenberg representation ( is a first-quantized one-body operator), with the notation = (, ). We multiply on the left by () and perform an interacting ground state average:

Consider a repulsive Hubbard model, defined on a d-dimensional lattice or graph of || sites, with nearest-neighbor hopping. No symmetry is assumed; however we can use as an example a simple square Hubbard Model with periodic boundary conditions and hopping between nearest neighbors (n.n.):