Nonlinear Sigma Model for Normal and Superconducting Systems: A Pedestrian Approach

The nonlinear sigma model (NLSM) epitomises a field-theoretical approach to (interacting) electrons in disordered media. These lectures are aimed at the audience who might have vaguely heard about its existence but know very little of what is that, even less so of why it should be used and next to nothing of how it can be applied. These what, why and mainly how are the subject of the present lectures. In the first part, after a short description of why to be bothered, the NLSM is derived from scratch in a relatively simple (but still rather mathematical) way for non-interacting electrons in the presence of disorder, and some illustration of its perturbative usage is given. In the second part it is generalised, not without some leap of faith, to include the Coulomb repulsion and superconducting pairing.Comment: 21 pages, 5 eps figures. To appear in "Proceedings of the International School of Physics "Enrico Fermi" Course CLI, edited by B. Altshuler and V. Tognetti, IOS Press, Amsterdam 200

One-Dimensional Transport of Ultracold Bosons

Advances in cooling and trapping of atoms have enabled unprecedented experimental control of many-body quantum systems. This led to the observation of numerous quantum phenomena, important for fundamental science, indispensable for high-precision simulations of condensed-matter systems and promising for technological applications. However, transport measurements in neutral quantum gases are still in their infancy in contrast to the central role they play in electronics. In these lectures, after reviewing nascent experiments on quantum fermionic transport, I will focus on our theoretical prediction sand the possibility of experimental observations of qualitatively new phenomena in transport of ultracold bosons which do not have a direct counterpart in quantum electronic transport in condensed matter systems. The description of this transport is based on the Luttinger liquid (LL) theory. So in the first part of the lectures I will introduce main concepts of the LL based on the functional bosonisation approach.Comment: Lecture notes for 13th International School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter", Sep 2018, Rzesz\'ow, Polan

Impurity in the Tomonaga-Luttinger model: a Functional Integral Approach

In this tutorial notes we review a functional bosonization approach in the Keldysh technique to one-dimensional Luttinger liquid in the presence of an impurity.Comment: 15 pages, 1 figure, Proceedings of LXXXI Les Houches School on "Nanoscopic quantum transport", Les Houches, France, June 28-July 30, 200

Many-body effects in Landau levels: Non-commutative geometry and squeezed correlated states

We discuss symmetry-driven squeezing and coherent states of few-particle systems in magnetic fields. An operator approach using canonical transformations and the SU(1,1) algebras is developed for considering Coulomb correlations in the lowest Landau levels.Comment: 3 pages, 2 figures; to be reported at 17th Int. Conf. on High Magnetic Fields in Semiconductor Physics, Wuerzburg, Germany, July 30 - Aug 4, 200

Low temperature decoherence and relaxation in charge Josephson-junction qubits

In this lectures, we have described some essential features of loss of coherence by a qubit coupled to the environment. We have first presented well known semiclassical arguments that relate both decoherence and relaxation to the environmental noise. Then we have shown that models with pure decoherence (but no relaxation in qubit states) can be exactly solvable. As an example, we have treated in detail the model of fluctuating background charges which is believed to describe one of the most important channels for decoherence for the charge Josephson junction qubit. We have shown that the decoherence rate saturates at `high' temperatures while becoming linear in T at low temperatures and showing in all regimes a non-monotonic behaviour as a function of the coupling of the qubit to the fluctuating background charges. We have also considered, albeit only perturbatively, the qubit relaxation by the background charges and demonstrated that a quasi-linear behaviour of the spectral density of noise deduced from the measurements of the relaxation rate can be qualitatively explained.Comment: Lecture notes for International workshop on mesoscopic and nanoscopic systems, Kolkata, India, February 2006, to be published by Springe

The Level Spacing Distribution Near the Anderson Transition

For a disordered system near the Anderson transition we show that the nearest-level-spacing distribution has the asymptotics $P(s)\propto \exp(-A s^{2-\gamma })$ for s\gg \av{s}\equiv 1 which is universal and intermediate between the Gaussian asymptotics in a metal and the Poisson in an insulator. (Here the critical exponent $0<\gamma<1$ and the numerical coefficient $A$ depend only on the dimensionality $d>2$). It is obtained by mapping the energy level distribution to the Gibbs distribution for a classical one-dimensional gas with a pairwise interaction. The interaction, consistent with the universal asymptotics of the two-level correlation function found previously, is proved to be the power-law repulsion with the exponent $-\gamma$.Comment: REVTeX, 8 pages, no figure