Classical and Quantum Monte Carlo Methods

Course I

João Viana Lopes - 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).

Periodically Driven Quantum Systems

Course II

Pedro Ribeiro, IST/CeFEMA

Periodically driving a quantum system profoundly alters its long-time dynamics and can cause non-trivial phenomena such as drive-induced topological order and time crystallinity. However, in most cases, driving also leads to heat production in the system, thus yielding a featureless infinite temperature state in long time scales.

In this course, we give an introduction to drive-induced physical phenomena to help comprehend and navigate the many recent developments in the field. We begin by introducing basic concepts on periodically-driven systems and the Floquet formalism. The formulation of Floquet Green’s functions will be then explained and applied to driven quadratic models. Then, the concepts of infinite temperature state and energy absorption are explored in generic models. Finally, we will explore the means of escaping infinite temperature, by means of dissipation and many-body localization. Some drive-induced physical phenomenology (crystallinity and drive-induced topological phases) will also be demonstrated.

Needs: Mathematica

Spectral Properties using Kernel Polynomial Methods

Course III

Simão M. João, Univ. do Porto/CF-UM-UP

The exotic properties of quantum matter emerge from an interplay between the quantum behaviour of its fundamental constituents (e.g. electrons, phonons, spins) and the sheer number of such constituents. In principle, studying observables in macroscopic quantum mechanical systems is a straightforward task but, in practice, can present a formidable technical challenge that cannot be tackled using standard methods of linear algebra.

In this course, we demonstrate the use of spectral methods to numerically analyse single-particle and transport properties of quantum systems with algorithms that scale linearly with the system’s Hilbert space dimension. In particular, we explore a class of particularly stable methods, known as Kernel Polynomial Methods (KPM), which are based upon expanding the observable of interest as an operator-valued series of orthogonal Chebyshev polynomials. These methods can be used for real-space calculations in systems that are not required to have translation symmetry. This makes them particularly relevant to study more realistic systems hosting disorder or defective lattices. As an application, we apply KPM to evaluate the density of states and electrical conductivity in non-interacting electronic systems at low temperatures, with or without the disorder. The density of states of a small interacting transverse-field quantum Ising chain will also be done.

Needs: Python with Numpy and Matplotlib. Ideally, the participants should also have Jupyter Notebook.

Out-of-Equilibrium Quantum Transport

Course IV

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.

Needs: Julia + Jupyter notebooks. Setup instructions and material for the session can be found here.