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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 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";
  if conserve_qns
    return [QN("Sz",3)=>1,QN("Sz",1)=>1,
  return 4

function ITensors.op!(Op::ITensor,
  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

function ITensors.op!(Op::ITensor,
  Op[s'=>1,s=>2] = sqrt(3)
  Op[s'=>2,s=>3] = 2
  Op[s'=>3,s=>4] = sqrt(3)

function ITensors.op!(Op::ITensor,
  Op[s'=>2,s=>1] = sqrt(3)
  Op[s'=>3,s=>2] = 2
  Op[s'=>4,s=>3] = sqrt(3)

 Download this example code

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 "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.

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