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
The notion of thermodynamic equilibrium is fundamental in physics. However, one is rarely interested in systems in equilibrium. Typically, one wants to know how a system reacts to an external drive. If the drive is weak enough, one can assume that the response of the system is proportional to the perturbation. In this linear response regime, the “proportionality constant”, generally referred to as susceptibility, is a property of the system in equilibrium. However, in many cases the external drive is not weak. In these cases, the system is strongly driven out-of-equilibrium and the drive cannot be treated in perturbation theory.
In this course, we will introduce the Non-Equilibrium Green’s Function (NEGF) or Keldysh formalism, a powerful framework suitable to study systems strongly driven away from equilibrium. We will start by presenting the basis of the formalism: the idea of contour ordering, the Schwinger-Keldysh double-time contour and the contour-ordered Green’s function. As an application, we will use the NEGF formalism to derive the Caroli formula for the current in a two-terminal device (a particular formulation of the Landauer formula). In the hands-on session of the course, the Caroli formula will be used to evaluate current through a single-level system coupled to 1D chains. The role of electron-electron interactions will be studied at a mean-field level.
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.
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).