Courses

Course I - Excitons, Cooper pairs and condensed phases of composite bosons 

Vitor Pereira - U. Porto/CF-UM-UP 

States of matter displaying macroscopic quantum phase coherence are of much interest, both fundamentally and for the materialization of quantum technologies. Bose-Einstein condensates (BEC) and superconducting states are two prominent and actively explored classes of quantum matter with that feature. Excitons and Cooper pairs are composite bosons that arise from many-body electronic interactions among fermions, and provide building blocks for such condensed states. Being able to stabilize and probe them in solid-state systems opens the opportunity to integrate and manipulate macroscopically quantum coherence within optoelectronic devices, with immense advantages for storing and processing quantum information in a scalable manner.

Though fundamentally different in their origin and in their associated phenomenology, excitons and Cooper pairs share, besides the composite-boson nature, a number of similar analytical techniques used to describe them in different scenarios. We will explore this relationship and introduce some of the basic theoretical approaches to study them, with particular emphasis on their condensed phases: neutral exciton condensates in one case, charged BEC and superconductors in the other.

For illustration, we will also see how two-dimensional materials provide a rich and evolving platform to probe and reach new regimes of these quantum condensates that were previously much harder, sometimes impossible, to achieve.

Course II - Artificial Lattices: A Means to Simulate Topological Quantum Matter

Ricardo Dias & Anselmo Marques, U. Aveiro/i3N

The goal of this course is twofold. First, it aims to provide a basic understanding of topological insulators in one and two dimensions, and in particular, topologically protected edge states, topological invariants, and bulk-edge correspondence. Secondly, the experimental realization of these topological insulators using artificial lattices, in particular acoustic lattices, is discussed. Artificial lattices replicate the atomic-level characteristics of quantum materials and offer the advantage of selectively modifying individual parameters of the quantum Hamiltonians.

In the hands-on session, we address simple cases of topological insulators and 2D artificial lattices.

Needs: Mathematica, Jupyter notebooks

Course III -  Linear Response Signatures of Collective Excitations

Bruno Amorim - U. Minho/CF-UM-UP 

In electronic systems, electron-electron interactions affect not only the ground-state of the system, but also its excitations. While in non-interacting systems, excitations are due to a single-particle, interacting systems become collective modes. Examples of these collective excitations include plasmons in metals, excitons in insulators and even the Higgs amplitude mode in superconductors.

In this course, we will see how such collective excitations can be theoretically described. In particular we will focus on the use of time-dependent Hartree-Fock (TDHF) theory. We will see how linear response within the TDHF approximation leads to an eigenvalue problem, the Bethe-Salpeter equation (BSE), with eigenvalues corresponding to the energies of the collective excitations. We will then establish how the linear response TDHF is equivalent to the diagrammatic summation of ladder diagrams in many-body perturbation theory.

In the hands-on session, we will solve the BSE for some simple cases.

Needs: Python with Numpy and Matplotlib.