# 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.

### Introduction to quantum chaos

**Lucas Sá **(IST, Univ. de Lisboa)

Abstract: Solving quantum systems more complicated than the standard textbook examples (particle in a box, harmonic oscillator, hydrogen atom, ...) seems an impossible task. However, if the system is complicated enough, universal behavior emerges and we can treat the Hamiltonian of the system as a large random matrix. Powerful tools from random matrix theory then allow us to develop a statistical theory of quantum chaos. In this tutorial, we will discuss several different physical systems where quantum chaotic behavior arises (e.g., nuclear matter, quantum billiards, and black holes) and what the main random-matrix signatures are, which we will study numerically. Finally, we will perform a simple calculation to find the repulsion between levels of a random matrix, perhaps the most striking feature of quantum chaotic systems.

### Density matrix renormalization group: formalism and numerical implementation

**Gonçalo Catarina** (INL)

A quantum system is often expressed in the form of a matrix, termed Hamiltonian. The properties of a quantum system can be found by diagonalizing its Hamiltonian: the eigenvalues give the energy levels and the eigenvectors represent eigenstates that allow the computation of other observables (e.g., magnetization).

For quantum many-body systems, the dimension of the Hamiltonian matrix scales exponentially with the number of degrees of freedom. Therefore, only very small systems can be treated. In order to go beyond this exponential wall [1], several methods have been developed, such as the density functional theory, the quantum Monte Carlo and the density matrix renormalization group (DMRG). In this tutorial, I will give a pedagogical introduction to DMRG, invented by Steven R. White in 1992 [2], which has become the numerical method of choice to obtain the low-energy properties of one-dimensional interacting quantum systems [3].

In the theoretical class, I will cover the basics of the DMRG formalism. In the practical part, we will write together a DMRG code to obtain the ground state energy of the spin-1 Heisenberg model. Time permitting, I will also show how to use a public library [4] that efficiently implements DMRG using matrix product states [5].

[1] Kohn W, 1999, Rev. Mod. Phys. 71 1253

[2] White S R, 1992, Phys. Rev. Lett. 69 2863

[3] Schollwöck U, 2005, Rev. Mod. Phys. 77 259

[4] Fishman M, White S R, Stoudenmire E M, 2007, arXiv:2007.14822

[5] Schollwöck U, 2011, Annals of Physics 326 96

Needs

- Python programming language (https://www.python.org/)

Optional but recommended:

- Julia programming language (https://julialang.org/)

- ITensor Julia library (https://itensor.github.io/ITensors.jl/stable/)

### 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 e Hugo Loio** (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.

Recommended 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