Learn to Use ITensor
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:
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 most important aspect of this code is a special type, known as a
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
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
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
"S=3/2" tag, the definition is
ITensors.space(::SiteType"S=3/2") = 4
Note that the function name is prepended with
This prefix makes sure the function is overloading other versions of the
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
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
Index, but also additional subspace structure it has corresponding to quantum
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
will be used to determine the dimension of these indices, and the
siteinds functions provided by ITensor will help with extra things like
putting other Index tags that are conventional for site indices.
The op Function
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
"S+" into ITensors so that it can make an actual MPO.
In our example above, we defined this function for the case of the
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
The body of this overload of
ITensors.op! is just setting the elements of the
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
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
siteinds function like this:
N = 100 sites = siteinds("S=3/2",N) Sz1 = op("Sz",sites) Sp3 = op("S+",sites)
Alternatively, you can write the lines of code above in the style
Sz1 = op("Sz",sites,1).
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.
- Add QN conservation to a custom local Hilbert space
- Extending an existing local Hilbert space
- See how the built-in site types are defined inside the ITensor library:
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