How to compute inner product of MPS with another MPS translated by 1 right sites

DMRG and Numerical Methods
1

Hello everyone,

I would like to ask if there is a way to compute the inner product ⟨ψ | T₁ | ψ⟩, where T₁ is a cyclic translation operator that shifts site n-to-n+1, by successively applying ITensor’s movesite functions, without the need to explicitly build an MPO for T₁.

Could this be considered a correct interpretation?

I would appreciate any help or insights you can provide.

Best regards,

Juan.

2

Hi @JuanD_H,
yes movesite is a way to implement it and you should actually do a single call (movesite(mps, i => j) does not swap site i with site j but it performs internal swaps so that, taking j>i, in i you have i+1, in i+1 you have i+2 and so on, which is directly what I think you wanted to do), see this example:

julia> using ITensorMPS

julia> sites = siteinds("S=1/2", 4);

julia> mps = MPS(sites, _ -> rand(["Up", "Dn"]))
MPS
[1] ((dim=2|id=514|"S=1/2,Site,n=1"), (dim=1|id=994|"Link,l=1"))
[2] ((dim=1|id=994|"Link,l=1"), (dim=2|id=934|"S=1/2,Site,n=2"), (dim=1|id=247|"Link,l=2"))
[3] ((dim=1|id=247|"Link,l=2"), (dim=2|id=916|"S=1/2,Site,n=3"), (dim=1|id=359|"Link,l=3"))
[4] ((dim=1|id=359|"Link,l=3"), (dim=2|id=24|"S=1/2,Site,n=4"))


julia> expect(mps, "Sz")
4-element Vector{Float64}:
  0.5
 -0.5
  0.5
  0.5

julia> mps = movesite(mps, 1 => 4)
MPS
[1] ((dim=2|id=934|"S=1/2,Site,n=2"), (dim=2|id=481|"Link,l=1"))
[2] ((dim=2|id=481|"Link,l=1"), (dim=2|id=916|"S=1/2,Site,n=3"), (dim=2|id=343|"Link,l=2"))
[3] ((dim=2|id=343|"Link,l=2"), (dim=2|id=24|"S=1/2,Site,n=4"), (dim=2|id=110|"Link,l=3"))
[4] ((dim=2|id=110|"Link,l=3"), (dim=2|id=514|"S=1/2,Site,n=1"))


julia> expect(mps, "Sz")
4-element Vector{Float64}:
 -0.5
  0.5
  0.5
  0.5
3

Just to clarify, internally the way it is implemented it is equivalent to doing a series of pairwise swaps.

4

Thank you, I have edited the answer to make this clearer