List of Projects

Monte Carlo Methods, Stochastic Dynamics and Critical Phenomena

José Guilherme Boura de Matos (Univ. do Porto)

One of the most interesting phenomena in physics is the way, the same breed of interactions in similar substances, can give rise to very distinct states of matter. The only difference between ice and water vapour is temperature and its effect on the symmetry of the molecular arrangement. In this tutorial, we shall employ stochastic methods (dubbed Monte Carlo Methods) in simple models, analysing this essential competition between energy minimization and entropy maximization. This is the chief mechanism driving critical phenomena in physical systems.

Anderson Localization and Matter Wave Propagation in Disordered Media

João Pedro dos Santos Pires (Univ. do Porto)

Since the seminal work of P.W. Anderson, in 1958, it is clear that conduction electrons in solid state systems show their underlying quantum nature when temperatures are low and propagate through a crystalline matrix hosting impurities. When seen as matter-waves, electrons can have their space propagation exponentially suppressed in a random medium, giving rise to a disorder-induced insulating phase called an Anderson dielectric. In this tutorial, we will use numerical integration of a 1D and 2D disordered Schrödinger’s Equation to depict these effects and see the transition between a ballistic (clean) propagation and a localized one.

Spectral Methods for Simulating Quantum Matter

Simão Meneses João (Univ. do Porto)

Differential equations appear in most physical contexts and, over time, thousands of methods have been developed to solve them, such as Euler's method, Runge-Kutta methods, etc... Spectral methods are one such class of methods that is especially useful to solve boundary value problems such as Schrodinger's equation. Recently, they have been gaining increasing popularity thanks to the Kernel Polynomial Method (KPM) approach to quantum transport, and they offer one of the best compromises available between numerical efficiency and accuracy.

In this tutorial, we will learn how to use spectral methods to solve Schrodinger's equation numerically, comparing it with a real-space discretization. Then, to show the power of spectral methods, we will compute the Density of States (DoS) of a quantum well.

Artificial atoms: quantum dots in graphene

Tatiana Rappoport (Univ. Federal Rio de Janeiro & Instituto de Telecomunicações IST)

We will model a quantum dot in a graphene sheet, which is an atom-thick layer made of carbon atoms. A quantum dot is modeled as a potential well that confines the electrons to a small region of graphene. To characterize our system, we will analyze its energy spectrum in function of the size and depth of the potential well. We also plan to explore its wave-functions and compare the energy spectrum and eigenstates to the ones we learnt in a quantum mechanics course. 
Techniques: graphene is modeled in real-space (tight-binding approach). For the modeling and analysis, the student will use pybinding, a tight-binding package for python.  
No previous knowledge of solid-states physics or python is needed, although elementary knowledge of python is a plus.

Decoherence in Open Quantum Systems

Pedro Ribeiro (IST, Univ. de Lisboa)

The unavoidable leaking of information from a quantum system to its environment leads to relaxation and decoherence and shall be ultimately responsible for the emergence of the classical world from quantum mechanics laws. 

Although not all is understood, some definite insights can be learned from the theory of open quantum systems. This tutorial is meant as a smooth hands-on introduction to these topics.

For simplicity, we shall address decoherence and relaxation in Markovian environments where the non-unitary evolution can be described in terms of a master equation of the Lindblad type, and explore its link with measurement-induced quantum jumps that model quantum evolution in the presence of continuous measurements. 

Applications of these concepts are of direct interest for modelling quantum systems in interaction with their environment, such as qubits in quantum computers or nuclear spins in NMR experiments. 

Topological Quantum Matter

Miguel Gonçalves (IST, Univ. de Lisboa)

The topic of topological phases of matter is highly attractive, mostly due to the robustness of topological properties against perturbations. This tutorial will be a simple hands-on introduction. We will cover some needed solid-state background, namely, tight-binding models. Then, we will play with a simple tight-binding model in one-dimension associated with a quantum phase transition between a conventional and a topological insulating phase. This example will allow us to understand important physical consequences of non-trivial topology and to introduce important concepts such as the bulk-boundary correspondence and topological invariants. If time allows, we will briefly comment on generalizations to higher dimensions, in particular, quantum Hall effects. 
Required software: Mathematica

Two Dimensional Materials

Eduardo Castro (Univ. do Porto)

One atom thick materials were condemned to exist only as building blocks of 3D matter, as they were thought to be unstable at finite temperatures. Graphene was the first truly 2D material to be found, and since then many more were brought to light. Some are metals, others insulators, some others semimetals and even magnetic 2D materials were found. They are highly tunable, and when combined show unexpected properties. 

In this tutorial we intend to work out some of the fascinating properties of 2D matter: we will show that charge carriers in graphene are massless Dirac fermions, while in bilayer graphene they become chiral massive fermions; we will understand why graphene, a semimetal, becomes an insulator by breaking a simple symmetry; we will see that strained graphene behaves as in a huge magnetic field; and  will learn that hexagonal Boron Nitride (h-BN) is the best 2D insulator, which combined with twisted layers of graphene gave rise to one of the hottest research topics in quantum matter since 2018.

Required software: Mathematica