# Physics (SiteType) System Examples

## Make a Custom Local Hilbert Space / Physical Degree of Freedom

ITensor provides support for a range of common local Hilbert space types, or physical degrees of freedom, such as S=1/2 and S=1 spins; spinless and spinful fermions; and more.

However, there can be many cases where you need to make custom degrees of freedom. You might be working with an exotic system, such as $Z_N$ parafermions for example, or need to customize other defaults provided by ITensor.

In ITensor, such a customization is done by overloading functions on specially designated Index tags. Below we give an brief introduction by example of how to make such custom Index site types in ITensor. Other code formulas following this one explain how to build on this example to expand the capabilities of your custom site type such as adding support for quantum number (QN) conservation and defining custom mappings of strings to states.

Throughout we will focus on the example of $S=3/2$ spins. These are spins taking the $S^z$ values of $+3/2,+1/2,-1/2,-3/2$. So as tensor indices, they are indices of dimension 4.

The key operators we will make for this example are $S^z$, $S^+$, and $S^-$, which are defined as:

\begin{aligned} S^z &= \begin{bmatrix} 3/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 &-1/2 & 0 \\ 0 & 0 & 0 &-3/2\\ \end{bmatrix} \\ S^+ & = \begin{bmatrix} 0 & \sqrt{3} & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & \sqrt{3} \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \\ S^- & = \begin{bmatrix} 0 & 0 & 0 & 0 \\ \sqrt{3} & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \sqrt{3} & 0 \\ \end{bmatrix} \\ \end{aligned}

Code Preview

First let's see the minimal code needed to define and use this new $S=3/2$ site type, then we will discuss what each part of the code is doing.

using ITensors

ITensors.space(::SiteType"S=3/2") = 4

ITensors.op(::OpName"Sz",::SiteType"S=3/2") =
[+3/2   0    0    0
0  +1/2   0    0
0    0  -1/2   0
0    0    0  -3/2]

ITensors.op(::OpName"S+",::SiteType"S=3/2") =
[0  √3  0  0
0   0  2  0
0   0  0 √3
0   0  0  0]

ITensors.op(::OpName"S-",::SiteType"S=3/2") =
[0   0  0   0
√3  0  0   0
0   2  0   0
0   0  √3  0]


Now let's look at each part of the code above.

The SiteType

The most important aspect of this code is a special type, known as a SiteType, which is a type made from a string. The string of interest here will be an Index tag. In the code above, the SiteType we are using is

SiteType"S=3/2"

What is the purpose of a SiteType? The answer is that we would like to be able to select different functions to call on an ITensor Index based on what tags it has, but that is not directly possible in Julia or indeed most languages. However, if we can map a tag to a type in the Julia type system, we can create function overloads for that type. ITensor does this for certain functions for you, and we will discuss a few of these functions below. So if the code encounters an Index such as Index(4,"S=3/2") it can call these functions which are specialized for indices carrying the "S=3/2" tag.

The space Function

One of the overloadable SiteType functions is space, whose job is to describe the vector space corresponding to that site type. For our SiteType"S=3/2" overload of space, which gets called for any Index carrying the "S=3/2" tag, the definition is

ITensors.space(::SiteType"S=3/2") = 4

Note that the function name is prepended with ITensors. before space. This prefix makes sure the function is overloading other versions of the space inside the ITensors module.

The only information needed about the vector space of a "S=3/2" Index in this example is that it is of dimension four. So the space function returns the integer 4. We will see in more advanced examples that the returned value can instead be an array which specifies not only the dimension of a "S=3/2" Index, but also additional subspace structure it has corresponding to quantum numbers.

After defining this space function, you can just write code like:

s = siteind("S=3/2")

to obtain a single "S=3/2" Index, or write code like

N = 100
sites = siteinds("S=3/2",N)

to obtain an array of N "S=3/2" indices. The custom space function will be used to determine the dimension of these indices, and the siteind or siteinds functions provided by ITensor will help with extra things like putting other Index tags that are conventional for site indices.

The op Function

The op function lets you define custom local operators associated to the physical degrees of freedom of your SiteType. Then for example you can use indices carrying your custom tag with OpSum and the OpSum system will know how to automatically convert names of operators such as "Sz" or "S+" into ITensors so that it can make an actual MPO.

In our example above, we defined this function for the case of the "Sz" operator as:

ITensors.op(::OpName"Sz",::SiteType"S=3/2") =
[+3/2   0    0    0
0  +1/2   0    0
0    0  -1/2   0
0    0    0  -3/2]

As you can see, the function is passed two objects: an OpName and a SiteType. The strings "Sz" and "S=3/2" are also part of the type of these objects, and have the meaning of which operator name we are defining and which site type these operators are defined for.

The body of this overload of ITensors.op constructs and returns a Julia matrix which gives the matrix elements of the operator we are defining.

Once this function is defined, and if you have an Index such as

s = Index(4,"S=3/2")

then, for example, you can get the "Sz" operator for this Index and print it out by doing:

Sz = op("Sz",s)
println(Sz)
ITensor ord=2
Dim 1: (dim=4|id=315|"S=3/2")'
Dim 2: (dim=4|id=315|"S=3/2")
NDTensors.Dense{Float64, Vector{Float64}}
4×4
1.5  0.0   0.0   0.0
0.0  0.5   0.0   0.0
0.0  0.0  -0.5   0.0
0.0  0.0   0.0  -1.5

Again, through the magic of the SiteType system, the ITensor library takes your Index, reads off its tags, notices that one of them is "S=3/2", and converts this into the type SiteType"S=3/2" in order to call the specialized function ITensors.op defined above.

You can use the op function yourself with a set of site indices created from the siteinds function like this:

N = 100
sites = siteinds("S=3/2",N)
Sz1 = op("Sz",sites[1])
Sp3 = op("S+",sites[3])

Alternatively, you can write the lines of code above in the style of Sz1 = op("Sz",sites,1).

This same op function is used inside of OpSum (formerly called AutoMPO) when it converts its input into an actual MPO. So by defining custom operator names you can pass any of these operator names into OpSum and it will know how to use these operators.

Further Steps

See how the built-in site types are defined inside the ITensor library:

• S=1/2 sites - Dimension 2 local Hilbert space. Similar to the "Qubit" site type, shares many of the same operator definitions.
• Qubit sites - Dimension 2 local Hilbert space. Similar to the "S=1/2" site type, shares many of the same operator definitions.
• S=1 sites - Dimension 3 local Hilbert space.
• Fermion sites - Dimension 2 local Hilbert space. Spinless fermion site type.
• Electron sites - Dimension 4 local Hilbert space. Spinfull fermion site type.
• tJ sites - Dimension 3 local Hilbert space. Spinfull fermion site type but without a doubly occupied state in the Hilbert space.
• Boson sites - General d-dimensional local Hilbert space. Shares the same operator definitions as the "Qudit" site type.
• Qudit sites - General d-dimensional local Hilbert space. Generalization of the "Qubit" site type, shares the same operator definitions as the $Boson$ site type.

## Make a Custom Local Hilbert Space with QNs

In the previous example above, we discussed the basic, minimal code needed to define a custom local Hilbert space, using the example of a $S=3/2$ spin Hilbert space. In those examples, the space function defining the vector space of a $S=3/2$ spin only provides the dimension of the space. But the Hilbert space of a $S=3/2$ spin has additional structure, which is that each of its four subspaces (each of dimension 1) can be labeled by a different $S^z$ quantum number.

In this code formula we will include this extra quantum information in the definition of the space of a $S=3/2$ spin.

Code Preview

First let's see the minimal code needed to add the option for including quantum numbers of our $S=3/2$ site type, then we will discuss what each part of the code is doing.

using ITensors

function ITensors.space(::SiteType"S=3/2";
conserve_qns=false)
if conserve_qns
return [QN("Sz",3)=>1,QN("Sz",1)=>1,
QN("Sz",-1)=>1,QN("Sz",-3)=>1]
end
return 4
end

ITensors.op(::OpName"Sz",::SiteType"S=3/2") =
[+3/2   0    0    0
0  +1/2   0    0
0    0  -1/2   0
0    0    0  -3/2]

ITensors.op(::OpName"S+",::SiteType"S=3/2") =
[0  √3  0  0
0   0  2  0
0   0  0 √3
0   0  0  0]

ITensors.op(::OpName"S-",::SiteType"S=3/2") =
[0   0  0   0
√3  0  0   0
0   2  0   0
0   0  √3  0]



Now let's look at each part of the code above.

The space function

In the previous code example above, we discussed that the function space tells the ITensor library the basic information about how to construct an Index associated with a special Index tag, in this case the tag "S=3/2". As in that code formula, if the user does not request that quantum numbers be included (the case conserve_qns=false) then all that the space function returns is the number 4, indicating that a "S=3/2" Index should be of dimension 4.

But if the conserve_qns keyword argument gets set to true, the space function we defined above returns an array of QN=>Int pairs. (The notation a=>b in Julia constructs a Pair object.) Each pair in the array denotes a subspace. The QN part of each pair says what quantum number the subspace has, and the integer following it indicates the dimension of the subspace.

After defining the space function this way, you can write code like:

s = siteind("S=3/2"; conserve_qns=true)

to obtain a single "S=3/2" Index which carries quantum number information. The siteind function built into ITensor relies on your custom space function to ask how to construct a "S=3/2" Index but also includes some other Index tags which are conventional for all site indices.

You can now also call code like:

N = 100
sites = siteinds("S=3/2",N; conserve_qns=true)

to obtain an array of N "S=3/2" indices which carry quantum numbers.

The op Function in the Quantum Number Case

Note that the op function overloads are exactly the same as for the more basic case of defining an "S=3/2" Index type that does not carry quantum numbers. There is no need to upgrade any of the op functions for the QN-conserving case. The reason is that all QN, block-sparse information about an ITensor is deduced from the indices of the tensor, and setting elements of such tensors does not require any other special code.

However, only operators which have a well-defined QN flux–-meaning they always change the quantum number of a state they act on by a well-defined amount–-can be used in practice in the case of QN conservation. Attempting to build an operator, or any ITensor, without a well-defined QN flux out of QN-conserving indices will result in a run time error. An example of an operator that would lead to such an error would be the "Sx" spin operator since it alternately increases $S^z$ or decreases $S^z$ depending on the state it acts on, thus it does not have a well-defined QN flux. But it is perfectly fine to define an op overload for the "Sx" operator and to make this operator when working with dense, non-QN-conserving ITensors or when $S^z$ is not conserved.

## Extending an Existing Local Hilbert Space

In the two previous examples above, we discussed the basics of how custom local Hilbert spaces a.k.a. site types can be defined from scratch in ITensor. However, there are cases where a custom site type is already designed for you, such as the site types "S=1/2", "S=1", "Fermion", "Electron" and others included with ITensor.

A nice feature of the ITensor SiteType system is that you can arbitrarily extend operator and other definitions, even of existing SiteTypes created in other code or by someone else.

Extending op Function Definitions

Perhaps the most common part of the SiteType system one wishes to extend are the various op or op! function overloads which allow code like

s = siteind("S=1/2")
Sz = op("Sz",s)

to automatically create the $S^z$ operator for an Index s based on the "S=1/2" tag it carries. A major reason to define such op overloads is to allow the OpSum (formerly AutoMPO) system to recognize new operator names, as discussed more below.

Let's see how to introduce a new operator name into the ITensor SiteType system for this existing site type of "S=1/2". The operator we will introduce is the projector onto the up spin state $P_\uparrow$ which we will denote with the string "Pup".

As a matrix acting on the space $\{ |\!\uparrow\rangle, |\!\downarrow\rangle \}$, the $P_\uparrow$ operator is given by

\begin{aligned} P_\uparrow &= \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \end{aligned}

To add this operator to the ITensor op system, we just need to introduce the following code

using ITensors

ITensors.op(::OpName"Pup",::SiteType"S=1/2") =
[1 0
0 0]

This code can be defined anywhere, such as in your own personal application code and does not have to be put into the ITensor library source code.

Note that we have to name the function ITensors.op! and not just op! so that it overloads other functions of the name op! inside the ITensors module.

Having defined the above code, we can now do things like

s = siteind("S=1/2")
Pup = op("Pup",s)

to obtain the "Pup" operator for our "S=1/2" Index s. Or we can do a similar thing for an array of site indices:

N = 40
s = siteinds("S=1/2",N)
Pup1 = op("Pup",s[1])
Pup3 = op("Pup",s[3])

Using Custom Operators in OpSum (AutoMPO)

A key use of these op system extensions is allowing additional operator names to be recognized by the OpSum (formerly AutoMPO) system for constructing matrix product operator (MPO) tensor networks. With the code above defining the "Pup" operator, we are now allowed to use this operator name in any OpSum code involving "S=1/2" site indices.

For example, we could now make an OpSum involving our custom operator such as:

N = 100
sites = siteinds("S=1/2",N)
ampo = OpSum()
for n=1:N
ampo += "Pup",n
end
P = MPO(ampo,sites)

This code makes an MPO P which is just the sum of a spin-up projection operator acting on every site.