mptikz - graphical tensor notation for LuaTeX

The mptikz package provides convenient functions for drawing tensor networks in graphical notation. Right now, it manly deals with the 1D tensor networks, i.e. matrix-product states and operators, but it's readily extensible. The package needs LuaLatex or LuaTex to work, I would reccomend to make the diagrams in a standalone file and import the pdf file with \includegraphics.

Tip: \includegraphics also works in math environments. The valign option to \includegraphics allows the user to set the vertical alignmnet. To enable it, you need to import \usepackage[export]{adjustbox}.

Installation

Just copy the mptikz.sty file in your working directory or any directory in the search path of LuaTeX.

Drawing single nodes

Drawing a single tensor is as easy as loading the mptikz package via \usepackage{mptikz} and running the following command inside a tikzpicture.

\tensor{{N=2, S=3, E=1}}

Note the double {{}}. This draws a tensor with 1, 2, and 3 legs on the left (EAST), top (NORTH), and bottom (SOUTH) position, respectively.

<p align='center'> <img height='150' src='img/example_1.svg'> </p>

See example_1.tex for the full source code. Note that all examples rely on lualatex.

mptikz automatically names tensors and legs, which then can be used to add annoations to the graph (see example_2.tex for the full source code). Also, the example below shows how to customize the look of the tensors using the tensor_style properties passed to the \tensor command. It accepts any valid TikZ style including (as shown below) the name of predefined styles.

\tikzstyle{tensornode}=[draw,minimum size=1, fill=orange, rounded corners=0.1cm]
\tensor{{N=1, tensor_name='A', tensor_style='tensornode', len_vertical_legs=1, leg_style='line width= .2mm', leg_color_NS='black'}}
\node at (A) {$A$};
\node [anchor=west] at (A_N1) {$i$};

<p align='center'> <img height='150' src='img/example_2.svg'> </p>

For each legs, three different TikZ coordinates are defined: A_N1 stands for the middle first northern leg of the tensor A, A_N1e stands for the end of that leg and A_N1b for the base of the leg.

Drawing more complex MPAs

For drawing chain of tensors (so called Matrix Product Arrays), we provide the \mpa functions shown below. We also show how to manipulate the default values for styling using the \tensorstyle command. See example_3.tex for the full code

\tensorstyle{{len_vertical_legs=0.25, tensor_style='draw, fill=orange, rounded corners=0.1cm', leg_style='line width= .2mm', leg_color_NS='black', leg_color_EW='black'}}
% Draw MPA manually
\tensor{{S=1, W=1, E=1, x=0}}
\tensor{{S=1, W=1, E=1, x=1.25}}
\tensor{{S=1, W=1, E=1, x=2.5}}

% Draw MAP using appropriate function
\tensorstyle{{tensor_style='draw, fill=green, rounded corners=0.1cm'}}
\mpa{3}{{N=1, E=1, W=1, y=-1.25, tensor_name='A'}}
\node at (A_1_2) {$A$};

<p align='center'> <img height='150' src='img/example_3.svg'> </p>

Block structure

More generally, the first input to \mpa is expected to be an array of integers, and the function will draw blocks of tensors separated by ellipses according to the input. As shown in example_4.tex for the input {3,2,1} we get 3 blocks: the first with 3 tensors, the second with 2 and the last with 1. The function \tlabel also allows to name every tensor in the array at once (to use a \ in the label, to put for instance a greek letter, one has to use \\ instead, more on that below).

\tensorstyle{{len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85 }}
\mpa{{3,2,1}}{{N=1,E=1,W=1, y=0.65, tensor_name='A'}}

\tlabel{'A'}{{ label='$A$' }}

<p align='center'> <img width='500' src='img/example_4.svg'> </p>

For naming the tensors, we simply append _i_j to the given name, where i is the number of the block and j is the number of the tensor within the block (both starting with 1). Therefore, if we want to address the northern leg of the 2nd tensor in the 1st block, we can use the keys A_1_2_N1, A_1_2_N1e and A_1_2_N1b.

Traces and removing external legs

To draw structures akin to those of Matrix Product States one often has to trace out elements in the diagram by connecting opposite lines. The \mpa command can do so automatically in two different ways. The \mpa command has an optional argument that can take the values -1, 0, 1 and 2.

• 0 is the default value and does nothing and yields results similar to example_4.tex.

• -1 removes the most external (East-West) legs, as shown in example_5.tex:

\tensorstyle{{len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85 }}
\mpa[-1]{{2,1,2}}{{N=1,E=1,W=1, y=0.65, tensor_name='A'}}

\tlabel{'A'}{{ label='$A$' }}

<p align='center'> <img width='500' src='img/example_5.svg'> </p>

• 1 adds red dots on the most external (East-West) legs, as shown in example_6.tex:

\tensorstyle{{len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85 }}
\mpa[1]{{2,1,2}}{{N=1,E=1,W=1, y=0.65, tensor_name='A'}}

\tlabel{'A'}{{ label='$A$' }}

<p align='center'> <img width='500' src='img/example_6.svg'> </p>

This is a more "tidy" version of 2.

• 2 makes most external (East-West) legs loop around as shown in example_7.tex, so they are easier to link as shown in example_8.tex.

Example 7:

\tensorstyle{{len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85 }}
\mpa[2]{{2,1,2}}{{N=1,E=1,W=1, y=0.65, tensor_name='A'}}

\tlabel{'A'}{{ label='$A$' }}

<p align='center'> <img width='500' src='img/example_7.svg'> </p>

Example 8:

\tensorstyle{{trace_offsetEW=-0.3, len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85}}
\mpa[2]{{2,1,2}}{{N=1,E=1,W=1, y=0.65, tensor_name='A1'}}
	
\draw[line width=1mm, leg_color_EW] (A1_1_1_W1e) -- (A1_3_2_E1e);
	
\tensorstyle{{trace_extensionEW=3}}
	
\mpa[2]{{2,1,2}}{{S=1,E=1,W=1, y=-0.65, tensor_name='A2', trace_inverterE=-1, trace_inverterW=-1}}
	
\tlabel{'A1'}{{ label='$A$' }}
\tlabel{'A2'}{{ label='$\\bar{A}$' }}

<p align='center'> <img width='500' src='img/example_8.svg'> </p>

This examples features two ways to link the lines that were looped by the option:

1. By using the \draw command of tikz with the tikz coordinates for the ends of the lines. The color names leg_color_EW and leg_color_NS are loaded with the package to the default color values for horizontal and vertical legs respectively.
2. By the trace_extensionEW option. This styling option only works with the 2 optional argument of \mpa. It extends the legth of the line by the given argument, therefore changing the position of the tikz coordinate corresponding to the end of the line. It also accepts negative inputs to reduce the length of the line.

The tikz coordinate A1_1_1_W1 is halfway on the "external"/"end" line, with this option there is the new coordinate A1_1_1_W1m which is half way on "turn".

The vertical displacement of the looped line changed between example_7.tex and example_8.tex. This is thanks to the trace_offsetEW option which allows to control for the displacement of the line looping back.

The options trace_inverterE=-1 and trace_inverterW=-1 allow us to change the side on which the line is looped around in example 8.

The same effect can be obtained for the vertical legs of the chain with the style option trace_NS. It takes as argument an array of an array of integers. Each sub-array corresponds to the block of the same index and contains the indices of the tensors within the block that we want to loop around. As is shown in example_9.tex:

\tensorstyle{{trace_offsetNS=-0.2, len_horizontal_legs=0.22, len_vertical_legs = 0.3, x=1.4, tensor_width=0.85, tensor_height=0.85 }}
	
\mpa[-1]{{2,1,2}}{{N=1,E=1,W=1, y=0.65, tensor_name='A1', traceNS = {{1,2},{},{1}} }}
\mpa[-1]{{2,1,2}}{{S=1,E=1,W=1, y=-0.65, tensor_name='A2', traceNS = {{1},{},{1}} }}
	
\tlabel{'A1'}{{ label='$A$' }}
\tlabel{'A2'}{{ label='$\\bar{A}$' }}

<p align='center'> <img width='500' src='img/example_9.svg'> </p>

Here we used traceNS = {{1,2},{},{1}} for the first line, so for the first block the 1st and 2nd tensor had the vertical leg looped around, for the second block no leg is looped around since the corresponding array is empty and for the last block we have the array {1} therefore only the first tensor has the vertical leg looped around. Whereas for the second line we used traceNS = {{1},{},{1}} therefore for the first block only the first tensor has the looped around leg.

With the option legs_order, which takes as input a string of the form 'NSWE' or 'EWNS', one can decide in which order the legs of the tensor are drawn so to decide which is on top of the other at the intersections.

The options trace_inverterN, trace_inverterS, trace_offsetNS and trace_extensionNS also exists and have the analogous effect of their horizontal counterparts.

There are many more options that come into play for \mpa[2] and traceNS, one can use them both in the second input of \mpa or in \tensorstyle. All these options are listed here:

• trace_extensionNS and trace_extensionEW correspond to how much the "long trace" is extended (can take negative values).

  • e.g. trace_extensionNS = e in the visual help below.
  • default value: 0.

• trace_widthNS and trace_widthEW give control to the value of w in the visual help below. In the example given w and w' increase proportionally to `t