├── .gitignore ├── matlab ├── doPath.m └── AuxiliaryFunctions │ ├── ArrayNorm.m │ ├── randcomplex.m │ ├── ArrayIsEqual.m │ ├── lq.m │ ├── qrpos.m │ ├── Heisenberg.m │ └── poldec.m ├── julia ├── Project.toml └── img │ ├── traceNorm.svg │ ├── leftOrth.svg │ ├── expVal3.svg │ ├── rightOrth.svg │ ├── expVal2.svg │ ├── rightTerms.svg │ ├── leftGauge.svg │ ├── gaugeTransform.svg │ ├── leftTerms.svg │ ├── VlX.svg │ └── q.svg ├── README.md └── python └── img ├── traceNorm.svg ├── leftOrth.svg ├── expVal3.svg ├── rightOrth.svg ├── expVal2.svg ├── rightTerms.svg ├── leftGauge.svg ├── gaugeTransform.svg ├── leftTerms.svg ├── VlX.svg └── q.svg /.gitignore: -------------------------------------------------------------------------------- 1 | .ipynb_checkpoints 2 | *~ 3 | *.pyc 4 | -------------------------------------------------------------------------------- /matlab/doPath.m: -------------------------------------------------------------------------------- 1 | function doPath() 2 | addpath(genpath(pwd)); 3 | end 4 | 5 | -------------------------------------------------------------------------------- /matlab/AuxiliaryFunctions/ArrayNorm.m: -------------------------------------------------------------------------------- 1 | function y=ArrayNorm(Y) 2 | 3 | y=sqrt(Y(:)'*Y(:)); 4 | 5 | end -------------------------------------------------------------------------------- /matlab/AuxiliaryFunctions/randcomplex.m: -------------------------------------------------------------------------------- 1 | function x=randcomplex(varargin) 2 | 3 | x=rand(cell2mat(varargin))+1i*rand(cell2mat(varargin)); 4 | 5 | end -------------------------------------------------------------------------------- /matlab/AuxiliaryFunctions/ArrayIsEqual.m: -------------------------------------------------------------------------------- 1 | function y=ArrayIsEqual(x1,x2,tol) 2 | 3 | if nargin==2 || isempty(tol) 4 | tol=eps; 5 | end 6 | 7 | y=ArrayNorm(x1-x2)= n, and orthonormal rows if m <= n. 7 | % U and H are computed via an SVD of A. 8 | % U is a nearest unitary matrix to A in both the 2-norm and the 9 | % Frobenius norm. 10 | 11 | % Reference: 12 | % N. J. Higham, Computing the polar decomposition---with applications, 13 | % SIAM J. Sci. Stat. Comput., 7(4):1160--1174, 1986. 14 | % 15 | % (The name `polar' is reserved for a graphics routine.) 16 | 17 | [m, n] = size(A); 18 | 19 | try 20 | [P, S, Q] = svd(A, 0); % Economy size. 21 | catch 22 | [P, S, Q] = svd(A+1e-10*randn(size(A)), 0); % Economy size. 23 | end 24 | if m < n % Ditto for the mncon contractor](https://arxiv.org/abs/1402.0939), which can be found [here](https://github.com/mhauru/ncon). 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 32 | ```console 33 | pip install ncon 34 | ``` 35 | 36 | 37 | ## Contents 38 | 39 | The tutorial consists of three parts: 40 | 41 | #### 1. Matrix product states in the thermodynamic limit 42 | 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. 43 | 44 | #### 2. Finding ground states of local Hamiltonians 45 | 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](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.045145). 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. 46 | 47 | #### 3. Transfer matrices and fixed points 48 | 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. 49 | 50 | 51 | ## Use of this tutorial 52 | 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. 53 | -------------------------------------------------------------------------------- /julia/img/traceNorm.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | -------------------------------------------------------------------------------- /python/img/traceNorm.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | -------------------------------------------------------------------------------- /julia/img/leftOrth.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | -------------------------------------------------------------------------------- /python/img/leftOrth.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | -------------------------------------------------------------------------------- /julia/img/expVal3.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | -------------------------------------------------------------------------------- /python/img/expVal3.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | -------------------------------------------------------------------------------- /julia/img/rightOrth.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | -------------------------------------------------------------------------------- /python/img/rightOrth.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | -------------------------------------------------------------------------------- /julia/img/expVal2.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | -------------------------------------------------------------------------------- /python/img/expVal2.svg: 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