Create a symmetric copy of a MPS: How to transpose QN?

Question

Hi folks,
I am trying to copy the first 1...L÷2 matrices of an MPS to the last L÷2+1...L matrices (in order to get the symmetric copy of the MPS). the layout is as following:

A - B - C - U λ - V - D - E - F 
|   |   |    |    |   |   |   |     

 => A - B - C - U λ - U - C - B - A
    |   |   |    |    |   |   |   |

The problem I am having right now is that I am not able to link Uλ with U. However, since λ is diagonal, this should be possible in principle. In particular, this works without quantum numbers, but with quantum numbers somehow the order of the blocks is reversed. Do you have an idea how I could go on in that situation? Here some code:

L      = 8
maxdim = 20
sites  = siteinds("S=1/2",L,conserve_szparity=true)
ψ      = randomMPS(sites,[rand(Bool) ? "↑" : "↓" for i in 1:L]);
truncate!(ψ,maxdim=maxdim)
orthogonalize!(ψ,L÷2);
U, λ, V = svd(ψ[L÷2]*ψ[L÷2+1], uniqueinds(ψ[L÷2],ψ[L÷2+1]))
ψ_new   = copy(ψ)
s_ind1  = siteindex(ψ,L)
s_ind2  = siteindex(ψ,1)
bondL   = commonind(ψ[1],ψ[2])
ψ_new[L] = prime(replaceind(ψ[1],s_ind2,s_ind1),bondL)
for i in 2:L÷2-1
    s_ind1 = siteindex(ψ,L-i+1)
    s_ind2 = siteindex(ψ,i)
    bondL  = commonind(ψ[i],ψ[i+1])
    bondR  = commonind(ψ[i],ψ[i-1])
    ψ_new[L-i+1] = replaceind(ψ[i],s_ind2,s_ind1)
    prime!(ψ_new[L-i+1],bondL)
    prime!(ψ_new[L-i+1],bondR)
end
ψ_new[L÷2]   = U*λ
linku = commonind(U,λ)
linkv = commonind(V,λ)
bondL = commonind(ψ[L÷2],ψ[L÷2-1])
s_ind1 = siteindex(ψ,L÷2+1)
s_ind2 = siteindex(ψ,L÷2)
ψ_new[L÷2+1] = replaceind(U,s_ind2,s_ind1)
prime!(ψ_new[L÷2+1],bondL)
replaceind!(ψ_new[L÷2+1],linku,linkv)

The error I get in the last line here is

Indices must have the same spaces to be replaced

If we look at linku and linkv, it becomes apparent that they do not coincide (in dimensionality and quantum numbers), even though λ is diagonal. This is something I do not quite understand, does this diagonality somehow change when using quantum numbers?

Best,

v.

Comments

Hi,

It looks like you've already fixed your problem. To answer the question about the singular value matrix output by ITensors.jl, as you noticed it is diagonal but the indices don't necessarily have the same space if QNs are conserved. This is because the singular value matrix doesn't always have a QN flux of zero, so in general the indices won't have the same QNs. They can be made to have the same QN if the tensor you are SVDing has a QN flux of 0, but that is not guaranteed.

I was going to recommend using a decomposition like the polar decomposition, which would make a non-diagonal center-site matrix that has indices with the same space. Out of curiosity, and in case someone else has a similar issue, what did you do to solve your problem?

-Matt

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Hi Matt, I solved it now as following


    function symmetrize(ψ::MPS)
        L = length(ψ)
        @assert iseven(L)
        sites = siteinds(ψ)
        orthogonalize!(ψ,L÷2)
        U, λ, V = svd(ψ[L÷2]*ψ[L÷2+1], uniqueinds(ψ[L÷2],ψ[L÷2+1]))
        ψ_new   = copy(ψ)
        s_ind1  = siteindex(ψ,L)
        s_ind2  = siteindex(ψ,1)
        bondL   = commonind(ψ[1],ψ[2])
        ψ_new[L] = prime(replaceind(ψ[1],s_ind2,s_ind1),bondL)
        for i in 2:L÷2-1
            s_ind1 = siteindex(ψ,L-i+1)
            s_ind2 = siteindex(ψ,i)
            bondL  = commonind(ψ[i],ψ[i+1])
            bondR  = commonind(ψ[i],ψ[i-1])
            ψ_new[L-i+1] = replaceind(ψ[i],s_ind2,s_ind1)
            prime!(ψ_new[L-i+1],bondL)
            prime!(ψ_new[L-i+1],bondR)
        end
        ψ_new[L÷2]   = U*λ
        u1 = commonind(ψ_new[L÷2],ψ_new[L÷2-1])
        u2 = commonind(U,λ)
        v1 = commonind(V,λ)
        s_ind2 = siteind(ψ_new, L÷2)
        s_ind1 = siteindex(ψ,L÷2+1)
        @assert itensor(store(tensor(U)),(s_ind2,u1,u2)) ≈ U
        ψ_new[L÷2+1] = itensor(store(conj(tensor(U))),(s_ind1,u1',v1))
        orthogonalize!(ψ_new,L÷2)
        return ψ_new
    end

Following from your answer, this should not always work? However, this worked for some tests.
Best,
v.

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It would take me some time to read through your code to see if it is what I had in mind, but if it works for you, that's all that matters.