## Learn to Use ITensor

main / formulas / sitetype_basic Julia | C++

# 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{align} 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{align}

## 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

function ITensors.op!(Op::ITensor,
::OpName"Sz",
::SiteType"S=3/2",
s::Index)
Op[s'=>1,s=>1] = +3/2
Op[s'=>2,s=>2] = +1/2
Op[s'=>3,s=>3] = -1/2
Op[s'=>4,s=>4] = -3/2
end

function ITensors.op!(Op::ITensor,
::OpName"S+",
::SiteType"S=3/2",
s::Index)
Op[s'=>1,s=>2] = sqrt(3)
Op[s'=>2,s=>3] = 2
Op[s'=>3,s=>4] = sqrt(3)
end

function ITensors.op!(Op::ITensor,
::OpName"S-",
::SiteType"S=3/2",
s::Index)
Op[s'=>2,s=>1] = sqrt(3)
Op[s'=>3,s=>2] = 2
Op[s'=>4,s=>3] = sqrt(3)
end


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 is really the heart of the SiteType system. This is the function that 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 AutoMPO and the AutoMPO 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:

function ITensors.op!(Op::ITensor,
::OpName"Sz",
::SiteType"S=3/2",
s::Index)
Op[s'=>1,s=>1] = +3/2
Op[s'=>2,s=>2] = +1/2
Op[s'=>3,s=>3] = -1/2
Op[s'=>4,s=>4] = -3/2
end


As you can see, the function is passed an ITensor Op and an Index s. The other arguments are there to select which of the various functions named op! get called. It is guaranteed by the op system that the ITensor Op will have indices s and s'.

The body of this overload of ITensors.op! is just setting the elements of the Op ITensor to the correct values that define the "Sz" operator for an $S=3/2$ spin.

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)
@show Sz


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 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 AutoMPO and it will know how to use these operators.

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