Learn to Use ITensor
Make a Custom Local Hilbert Space with QNs
In a previous code formula 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
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.
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 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 space function
In the code formula for defining a basic site type 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
As in that code formula, if the user does not request that quantum numbers be included
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
space function we
defined above returns an array of
QN=>Int pairs. (The notation
a=>b in Julia constructs
Pair object.) Each pair in the array denotes a subspace.
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.
siteind function built into ITensor relies on your custom
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
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.
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