2.1. Introduction#
In this tutorial, the most important features of Kwant are explained using simple, but still physically meaningful examples. Each of the examples is commented extensively. In addition, you will find notes about more subtle, technical details at the end of each example. At first reading, these notes may be safely skipped.
A scientific article about Kwant is available as well, see Citing Kwant.
The article introduces Kwant with a somewhat different focus than the tutorial and it is the authors’ intention that both texts complement each other. While the tutorial is more “hands-on”, the article presents Kwant in a more conceptual way, as well as discussing questions of design and performance.
Quantum transport#
This introduction to the software Kwant is written for people that already have some experience with the theory of quantum transport. Several introductions to the field are available, the most widely known is probably the book “Electronic transport in mesoscopic systems” by Supriyo Datta.
The Python programming language#
Kwant is a library for Python. Care was taken to fit well with the spirit of the language and to take advantage of its expressive power. If you do not know Python yet, do not fear: Python is widely regarded as one of the most accessible programming languages. For an introduction we recommend the official Python Tutorial. The Beginner’s Guide to Python contains a wealth of links to other tutorials, guides and books including some for absolute beginners.
Kwant#
There are two steps in obtaining a numerical solution to a problem: The first is defining the problem in a computer-accessible way, the second solving it. The aim of a software package like Kwant is to make both steps easier.
In Kwant, the definition of the problem amounts to the creation of a tight binding system. The solution of the problem, i.e. the calculation of the values of physical observables, is achieved by passing the system to a solver.
The definition of a tight binding system can be seen as nothing else than the creation of a huge sparse matrix (the Hamiltonian). Equivalently, the sparse Hamiltonian matrix can be seen as an annotated graph: the nodes of the graph are the sites of the tight binding system, the edges are the hoppings. Sites are annotated with the corresponding on-site Hamiltonian matrix, hoppings are annotated with the corresponding hopping integral matrix.
One of the central goals of Kwant is to allow easy creation of such annotated graphs that represent tight binding system. Kwant can be made to know about the general structure of a particular system, the involved lattices and symmetries. For example, a system with a 1D translational symmetry may be used as a lead and attached to a another system. If both systems have sites which belong to the same lattices, the attaching can be done automatically, even if the shapes of the systems are irregular.
Once a tight binding system has been created, solvers provided by Kwant can be used to compute physical observables. Solvers expect the system to be in a different format than the one used for construction – the system has to be finalized. In a finalized system the tight binding graph is fixed but the matrix elements of the Hamiltonian may still change. The finalized format is both more efficient and simpler – the solvers don’t have to deal with the various details which were facilitating the construction of the system.
The typical workflow with Kwant is as follows:
Create an “empty” tight binding system.
Set its matrix elements and hoppings.
Attach leads (tight binding systems with translational symmetry).
Pass the finalized system to a solver.
Please note that even though this tutorial mostly shows 2-d systems, Kwant is completely general with respect to the number of dimensions. Kwant does not care in the least whether systems live in one, two, three, or any other number of dimensions. The only exception is plotting, which out-of-the-box only works for up to three dimensions. (But custom projections can be specified!)