Observable

We use a similar tree structure for calculating observables. For instance, we first generate a random product state.

using FiniteMPS

# construct a random MPS
L = 4
Ψ = randMPS(L, ℂ^2, ℂ^1)
# display
for i in 1:L
println(Ψ[i])
end

MPSTensor{3}(TensorMap((ℂ^1 ⊗ ℂ^2) ← ℂ^1):
[:, :, 1] =
 -0.5548825211008388  0.831928715562083
)
MPSTensor{3}(TensorMap((ℂ^1 ⊗ ℂ^2) ← ℂ^1):
[:, :, 1] =
 -0.9993260886819206  0.03670651821807647
)
MPSTensor{3}(TensorMap((ℂ^1 ⊗ ℂ^2) ← ℂ^1):
[:, :, 1] =
 -0.7718808685210725  0.6357671938777234
)
MPSTensor{3}(TensorMap((ℂ^1 ⊗ ℂ^2) ← ℂ^1):
[:, :, 1] =
 -0.10724699565836393  -0.9942324084047224
)

Here we use randMPS to construct a random MPS whose physical spaces are all $\mathbb{C}^2$ and bond spaces are all $\mathbb{C}^1$ thus represents a random produce state.

Next, we calculate the on-site values and spin correlations.

# define the S^z operator following the syntax of TensorKit.jl
Sz = TensorMap([0.5 0.0; 0.0 -0.5], ℂ^2, ℂ^2)

# construct the observable tree
Tree = ObservableTree(L)
for i in 1:L
     addObs!(Tree, (Sz,), (i,), (false,); name = (:Sz,))
end
for i in 1:L, j in 1:L
     addObs!(Tree, (Sz, Sz), (i, j), (false, false); name = (:Sz, :Sz))
end
calObs!(Tree, Ψ)
Obs = convert(Dict, Tree)

Dict{String, Dict} with 2 entries:
  "SzSz" => Dict{Tuple{Int64, Int64}, Number}((1, 2)=>-0.0957939, (3, 1)=>-0.01…
  "Sz"   => Dict{Tuple{Int64}, Number}((4,)=>-0.488498, (2,)=>0.498653, (3,)=>0…

Here Tree is an ObservableTree object that contains all observables to be calculated, and addObs! is the standard interface to add terms to it, analog to InteractionTree and addIntr!. Then, we call calObs! to trigger the in-place calculation in the tree, with the given MPS Ψ. Finally, we use the convert method to extract the data from the tree to a dictionary Obs. For example,

Obs["SzSz"][(1, 2)]

-0.09579385714411875

is the correlation $\langle S_1^z S_2^z\rangle$. One can perform a simple quantum mechanics calculation to check this result.

Here is just a simple example to show the basic usage, more complex examples that contain fermion correlations and multi-site correlations (e.g. pairing correlations) can be found in the concrete example for Hubbard model.