# VUMPS in a two leg ladder model

**URL:** <https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669>\
**Category:** ITensor Julia Questions\
**Tags:** julia\
**Created:** [February 27, 2023, 7:41pm UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669 "2023-02-27T19:41:53Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![ntausend](https://avatars.discourse-cdn.com/v4/letter/n/f07891/32.png) [@ntausend](https://itensor.discourse.group/u/ntausend)\
**Post date:** [February 27, 2023, 7:41pm UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669/1 "2023-02-27T19:41:53Z")

</div>

Dear all,

I am currently trying to use the implementation of the _vumps_ algorithm from the `ITensorInfiniteMPS.jl`  
package.

The model I want to study consists of two wires of Josephson Junctions described by  
bosonic modes b\_{\alpha,j}, \alpha = a,b. The modes are defined w.r.t. the mean filling of the  
superconducting islands, i.e. n = diag(\dots, -2, -1, 0 , 1 ,2, \dots).  
The Hamiltonian looks similar to:

H = \sum\_{j,\alpha} b\_{j,\alpha}^\dagger b\_{j,\alpha} + \text{h.c.} + E\_c\, n\_{j,\alpha}^2 + \sum\_j + \Omega b\_{j,a}^\dagger b\_{j,b} + \text{h.c.} + V n\_{j, a} n\_{j, b} 

In previous finite size DMRG studies, I “enrolled” the unitcell to only have a local hilbertspace dimension of 2n\_b + 1 with a chain of length 2L. I.e. I defined new operators:

\tilde{b}\_j = \begin{cases} b\_{(j + 1)/2, a}\ ,\ \text{for } j\ \text{odd}\\ b\_{j/2, b}\ ,\ \text{for } j\ \text{even}\\ \end{cases}

The above Hamiltonian gets then transformed into a model with next nearest neighbor hoppings,  
and interactions + hoppings **only** between even and odd sites 2j - 1 and 2j, but not between  
2j and 2j+1.

I tried to define a new model with unit\_cell terms as:

```auto
function ITensorInfiniteMPS.unit_cell_terms(::Model"JosphsonLadder"; Ej = 1, Ec = 1, V = 0, Omega = 0)
    opsum = OpSum()
    opsum += -Ej/2, "Cr", 1, "Anh", 3
    opsum += -Ej/2, "Cr", 3, "Anh", 1
    opsum += -Ej/2, "Cr", 2, "Anh", 4
    opsum += -Ej/2, "Cr", 4, "Anh", 2
    opsum += Ec, "N * N", 1

    opsum += V, "N", 1, "N", 2
    
    opsum += Omega, "Cr", 1, "Anh", 2
    opsum += Omega, "Cr", 2, "Anh", 1
    
    return opsum
end

```

But this is too naive, since it will not recognize that the \Omega term needs to be repeated only every  
second site. What would be the correct way to define such a Hamiltonian?  
Moreover, for \Omega = 0, the subspace expansion will fail with the error message:

`Impossible to do a subspace expansion, probably due to conservation constraints`

It seems, that there needs to be a nearest neighbor term for the subspace expansion.

Thanks for any help in advance!

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<div class="post-metadata">

**Author:** ![ntausend](https://avatars.discourse-cdn.com/v4/letter/n/f07891/32.png) [@ntausend](https://itensor.discourse.group/u/ntausend)\
**Post date:** [March 2, 2023, 9:35am UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669/2 "2023-03-02T09:35:27Z")

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Ok, I think I defined the model in a correct way now:

```auto
function ITensorInfiniteMPS.unit_cell_terms(::Model"JosphsonLadder"; Ej = 1, Ec = 1, V = 0, Omega = 0)
    opsum = OpSum()
    opsum += -Ej/2, "Cr", 1, "Anh", 3
    opsum += -Ej/2, "Cr", 3, "Anh", 1
    opsum += -Ej/2, "Cr", 2, "Anh", 4
    opsum += -Ej/2, "Cr", 4, "Anh", 2
    
    opsum += Ec, "N * N", 1   
    opsum += Ec, "N * N", 2

    opsum += V, "N", 1, "N", 2
    
    opsum += Omega, "Cr", 1, "Anh", 2
    opsum += Omega, "Cr", 2, "Anh", 1
    
    return opsum
end

```

At least this definition gives the same results in a finite system by using `MPO(Model("JosephsonLadder"), ...)`. I think I also found a bug. Namely if there is only one term in the  
`OpSum` acting on one position in the unit cell, the creation will fail. This is because in Line 136 of  
`models.jl` the line

```auto
opsum_cell_dict = groupreduce(minimum ∘ ITensors.sites, +, opsum)

```

will not convert such a single term into a `OpSum` object, but it will simply stay a `Scaled{ComplexF64, Prod{Op}}`. I think this will be solved by using

```auto
opsum_cell_dict = groupreduce(minimum ∘ ITensors.sites, identity, +, opsum; init = OpSum())

```

instead.

And is there already an implementation for correlation functions?

Thanks!

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<div class="post-metadata">

**Author:** ![ntausend](https://avatars.discourse-cdn.com/v4/letter/n/f07891/32.png) [@ntausend](https://itensor.discourse.group/u/ntausend)\
**Post date:** [March 2, 2023, 9:37am UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669/3 "2023-03-02T09:37:31Z")

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Ah and I notice that the energy reported by vumps is not equal for both positions in the unit-cell, is this bad or something which can happen?

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**Author:** ![mtfishman](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/mtfishman/32/11_2.png) [@mtfishman](https://itensor.discourse.group/u/mtfishman)\
**Post date:** [March 2, 2023, 11:47am UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669/4 "2023-03-02T11:47:15Z")

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Could you make a PR for that fix for single terms in the OpSum? [Pull requests · ITensor/ITensorInfiniteMPS.jl · GitHub](https://github.com/ITensor/ITensorInfiniteMPS.jl/pulls)

We’re adding an example for computing correlation matrices here: [Added example file with Infinite MPS to finite MPS function by nbaldelli · Pull Request #72 · ITensor/ITensorInfiniteMPS.jl · GitHub](https://github.com/ITensor/ITensorInfiniteMPS.jl/pull/72) (the goal is to just use finite MPS tools to do that, to minimize the amount of code I have to support in that library, since I don’t have much time to develop it and it will be superseded by next-generation infinite tensor network tools in [GitHub - mtfishman/ITensorNetworks.jl: A package with general tools for working with higher-dimensional tensor networks based on ITensor.](https://github.com/mtfishman/ITensorNetworks.jl) once those become available).

Unequal energies would depend on the physics, for example if there is spontaneous symmetry breaking (or the Hamiltonian already breaks the symmetry). Otherwise it could just be due to convergence, it of course depends on details of the calculation and the magnitude of the difference (and how the difference changes as you change the bond dimension, i.e. does it trend towards zero in the infinite bond dimension limit?).

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<div class="post-metadata">

**Author:** ![ntausend](https://avatars.discourse-cdn.com/v4/letter/n/f07891/32.png) [@ntausend](https://itensor.discourse.group/u/ntausend)\
**Post date:** [March 2, 2023, 1:04pm UTC](https://itensor.discourse.group/t/vumps-in-a-two-leg-ladder-model/669/5 "2023-03-02T13:04:35Z")

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Hey thanks!  
yes I can do that later.

Ok that approach sounds good, will look into that. Are these tools already on Github in some branch?  
If yes I would rather switch to that since it will be supported in the future.

Mhm ok, I guess its because of the additional interaction term between site 1 and 2 not present between  
site 2 and 3. On the other hand, all local observables are homogeneous on the unitcell.

Thanks!