Courses
Monte Carlo Methods
João Viana Lopes, João Pedro Pires, David Capitão Lima e Júlio Oliveira - U. Porto/CF-UM-UP
Monte Carlo methods are amongst the most powerful techniques to investigate thermodynamic equilibrium properties in both quantum and classical interacting systems. Their success can be traced to the great efficiency in the stochastic evaluation of any integral in a high-dimensional space, such as the phase space of large classical systems or the Hilbert spaces of interacting quantum systems.
In this course, we give a general overview of Monte-Carlo methods (MCM), starting from the basic notions of Markovian stochastic dynamics as a tool to simulate equilibrium microstates of large classical systems. We begin by introducing the Markov Matrix as a way to specify a stochastic dynamics and observe convergence properties of the sampled chain of states, namely considering its auto-correlation and error analysis. These concepts will be used to analyse thermodynamic properties of the Ising spin model. Finally, we will further apply the MCM to some Quantum problems where a formal mapping to a classical field is possible. Potential drawbacks from this approach will also be indicated (e.g. the sign problem).
Quantum Chaos, Random Matrices and Localization
Eduardo V. Castro, U. Porto/CF-UM-UP
Lucas Sá, IST/CeFEMA
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 course, 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, in particular, level statistics. Then, we will see how the same tools can be applied to the seemingly unrelated problem of electrons hopping in a disordered potential. If the disorder is strong enough, electrons localize and quantum chaos is destroyed.
Numerical Methods for Quantum Many-Body Physics
Gonçalo Catarina - INL
Miguel Gonçalves - IST/CeFEMA
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 with exact diagonalization (ED). 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, we will start by explaining the basics of ED. Then, we 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]. Time permitting, we will also give a brief introduction to matrix product states, used for efficient implementations of DMRG codes [4].
[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] Schollwöck U, 2011, Annals of Physics 326 96
Out-of-Equilibrium Quantum Transport
Bruno Amorim, Univ. do Minho/CF-UM-UP
In usual bulk systems, their electric transport properties can be characterized in terms of an intensive property: the conductivity. As an intensive property, the conductivity does not depend on the particular geometric details of an electronic device. For sufficiently small devices, with size smaller than the phase coherence length, this picture breaks down. In this mesoscopic regime, the transport properties become dependent on the particular geometry of the device: the notion of conductivity breaks down, and one must consider the conductance instead.
In this course, we will see how we can describe transport of non-interacting electrons in a fully quantum mechanical way. We will consider two possible frameworks: the scattering states approach, which leads to the Landauer-Buttiker equation, and the non-equilibrium Green’s function approach, which leads to the Caroli formula. Despite their different appearance, we will see that the two approaches actually coincide. We will then see how these methods can be implemented.