This repository contains a tutorial on tangent space methods for uniform matrix product states, originally written to be taught at the 2020 school on Tensor Network based approaches to Quantum Many-Body Systems held in Bad Honnef, Germany. The tutorials are based on the lecture notes 'Tangent-space methods for uniform matrix product states' by Laurens Vanderstraeten, Jutho Haegeman and Frank Verstraete.

The tutorial is given in the form of IPython Notebooks written in either Julia or Python, which provide a detailed guide through the relevant concepts and algorithms, interspersed with checks and demonstrations. Reference implementations are also given in MATLAB.

For those wondering where to start, we highly recommend using the Julia version of the tutorials. Aside from demonstrating the relevant MPS algorithms, the Julia notebooks also serve as a gentle introduction to the TensorOperations.jl and TensorKit.jl packages for tensor computation. This not only allows for more concise and intuitive implementations compared to ones based on plain multidimensional arrays, but also provides a gateway to more involved tensor network applications using TensorKit.jl and other packages such as for example MPSKit.jl.

To run the notebooks, you should have a working Jupyter installation (see below) as well as a functioning IJulia Jupyter kernel. After this, simply instantiate the environment specified in the 'Project.toml' file in the Julia notebook folder and you are good to go.

The Python version of the notebooks is slightly more barebones, and implements everything using plain NumPy arrays. This way of working requires less intuition into using tensors as multilinear maps as is done in TensorKit.jl, but comes with some extra overhead in the form of additional reshapes, permutes, casting to and from vectors and the like.

The easiest way to get all of the tools needed to open and run the notebooks is to install Python via the anaconda distribution. This automatically comes with all packages needed for this tutorial.

If you already have a python version installed, you can install Jupyter seperately via the Python Package Index by running

pip install notebook

Once jupyter notebook is installed, clone this repository to a local folder of your choosing, open a terminal and navigate to this folder, and simply run

jupyter notebook

For performing contractions of tensor networks, we have opted to use the Python implementation of the ncon contractor, which can be found here. There are undoubtedly many contraction tools that work equally well and any of them may of course be used, but this one has a particularly intuituve and handy syntax. To install ncon, you may run

pip install ncon

The tutorial consists of three parts:

This part is concerned with the basics of MPS in the thermodynamic limit: normalization, fixed points, algorithms for gauging an MPS, and computing expectation values.

This part describes how to perform a variational search in the space of MPS to find the ground state of a local Hamiltonian. We start by considering a simple gradient based approach, and then introduce the VUMPS algorithm. We also briefly touch upon how excited states may be constructed using the result of the VUMPS ground state search. The algorithms are demonstrated for the case of the one-dimensional quantum spin-1 Heisenberg model.

In this part the VUMPS algorithm is extended to transfer matrices in the form of matrix product operators (MPOs), which can then be used to contract infinite two-dimensional tensor networks. The algorithm is demonstrated for the case of the classical two-dimensional Ising model, and is for example used to compute its partition function and evaluate the expectation value of the magnetization.

Each chapter provides a notebook (.ipynb) file written in Julia or Python with guided exercices on implementing the algorithms, as well as a solution notebook. Similar files are available for MATLAB. The approach itself is very basic, where all algorithms are broken down and illustrated in digestible steps. The implementations are very simple: there are no black boxes or fancy tricks involved, everything is built from the ground up. While these tutorials were originally intended to be taught at a school on tensor networks, they now serve as a general reference on uniform MPS and how one would go about implementing these concepts.