Unreasonable results occurred when using TDVP to calculate the long-range interaction Hamiltonian

ITensor Julia Questions
1

I recently encountered a puzzling issue while using TDVP, and it took me a long time to find the root cause. Therefore, I would greatly appreciate any assistance.

L = 2
tₜₒₜₐₗ = 10
τ = 0.01
tₛₜₑₚₛ = convert(Int64, tₜₒₜₐₗ /τ)
sites = siteinds("Boson",2L)
states = ["1","1","0","0"]
ψ₀ = MPS(sites,states)
os = OpSum()
os += "a", 2, "a†", 3
os += "a", 3, "a†", 2
ℋ = MPO(os,sites)

ψₜ = ψ₀
N̂ₛₜₑₚₛ = []
for i in 1:tₛₜₑₚₛ
   N̂ = expect(ψₜ ,"N")
   push!(N̂ₛₜₑₚₛ ,N̂')
   ψₜ = tdvp(ℋ,ψₜ ,-im*τ,ishermitian=true)
end

In the above code, I created a boson model with an initial state of “1100” and applied the Hamiltonian of a_{2}^{\dagger} a_{3} + a_{3}^{\dagger} a_{2}. After drawing the graph of the number of particles at each site over time, as we expected, the particles oscillated back and forth between the second and third sites.

image
image1547×526 51.6 KB

But when I modify the Hamiltonian to a_{1}^{\dagger} a_{3} + a_{3}^{\dagger} a_{1}

os = OpSum()
os += "a", 1, "a†", 3
os += "a", 3, "a†", 1
ℋ = MPO(os,sites)

Strange things happened, as the particles seemed to be fixed in place and did not oscillate between the first and third sites as expected.

image
image1549×516 7.11 KB

At the same time, I still used TEBD for the same time evolution, and luckily, I got the image I wanted to see.

L = 2
tₜₒₜₐₗ = 10
τ = 0.1
tₛₜₑₚₛ = convert(Int64, tₜₒₜₐₗ /τ)
sites = siteinds("Boson",2L)
states = ["1","1","0","0"]
ψ₀ = MPS(sites,states)

gates=ITensor[]
tg=op("Adag",sites[1])*op("A",sites[3])
tg+=op("Adag",sites[3])*op("A",sites[1])
tevol=exp(-im*τ/2*tg)
push!(gates,tevol)
append!(gates,reverse(gates))

ψₜ = ψ₀
for i in 1:tₛₜₑₚₛ
     N̂=expect(ψₜ,"N") 
     push!( N̂ₛₜₑₚₛ , N̂')
     ψₜ=apply(gates,ψₜ)
end

image
image1550×520 49.3 KB

Therefore, I would like to know where I made mistakes or made some thought-provoking mistakes when using TDVP to calculate Hamiltonians with long-range interactions. Please point them out for me. Thank you very much！

2

Try with nsite=2 argument in tdvp function, to invoke 2-site tdvp optimization.

3

Yes, I have modified the function TDVP to

ψₜ = tdvp(ℋ,ψₜ ,-im*τ,ishermitian=true;nsite=2)

based on your suggestion. No modifications have been made to the other parts. But the result is still incorrect.

image
image1552×520 7.26 KB

Thank you very much for your suggestion.

4

You could try manually increasing the bond dimensions of the initial state MPS before starting the evolution. I tried it, and a bond dimension equal to 2 is already enough to see the right result. I guess even the TDVP2 algorithm can’t “see” long range correlations/interactions if the MPS being evolved isn’t “big” enough, and non-nearest-neighbour terms get lost when the Hamiltonian gets projected.

As for how you can increase the bond dimensions, I don’t know if there is an MPS constructor which allows you to do it directly; I usually multiply each bond of the MPS with a delta ITensor of the desired new dimension.

5

May I ask if it is convenient for you to provide a simple example code?

6

This is what I used to enlarge the MPS manually:

for bond in 1:(length(ψ₀)-1)
        bond_index = commonind(ψ₀[bond], ψ₀[bond + 1])
        aux = Index(new_dim; tags=tags(bond_index))
        ψ₀[bond] = ψ₀[bond] * delta(bond_index, aux)
        ψ₀[bond + 1] = ψ₀[bond + 1] * delta(bond_index, aux)
end

You could plug it in your code just after you create the MPS.

2 Likes
7

Oh yes, I have modified the program according to your suggestions and indeed obtained the correct results. I will continue to follow this approach to modify other issues. Thank you for your help！

1 Like