# TEBD on 2D fermionic system

**URL:** <https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865>\
**Category:** ITensor Julia Questions\
**Created:** [July 11, 2024, 6:23am UTC](https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865 "2024-07-11T06:23:10Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![MaxSilvester](https://avatars.discourse-cdn.com/v4/letter/m/aeb1de/32.png) [@MaxSilvester](https://itensor.discourse.group/u/MaxSilvester)\
**Post date:** [July 11, 2024, 6:23am UTC](https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865/1 "2024-07-11T06:23:10Z")

</div>

Hello everyone,

My code below is meant to perform Trotterized imaginary-time evolution on an initial Neel state, for a 2 x 2 Hubbard square lattice at half filling (open boundary conditions). There is nothing wrong with the construction of the Hamiltonian MPO, etc. as confirmed by the DMRG run at the end that yield the correct energy.

```julia
using ITensors

function hubbard_TEBD_gates(NN,sites,t,U,τ)
    # NN: list of nearest-neighbor sites
    # sites: siteinds of MPS
    # t, U: Hubbard parameters
    # τ: imaginary-time step
    
    gates = ITensor[]
    
    for edge in NN
        edge[2] < edge[1] && reverse!(edge)
        
        s1, s2 = sites[edge[1]], sites[edge[2]]
        hj = -t*op("Cdagup",s1)*op("Cup",s2)
            - t*op("Cdagup",s2)*op("Cup",s1)
            - t*op("Cdagdn",s1)*op("Cdn",s2)
            - t*op("Cdagdn",s2)*op("Cdn",s1)
        Gj = exp(-1.0 * τ/2 * hj)
        push!(gates,Gj)
    end

    for i in 1:length(sites)
        si = sites[i]
        hi = U*op("Nupdn",si)
        Gi = exp(-1.0 * τ/2 * hi)
        push!(gates,Gi)
    end
    
    append!(gates,reverse(gates))
    
    return gates
end

function lattice_hubbard_MPO(NN,sites,nx,ny,t,U)
    N = nx*ny
    ampo = OpSum()
    for edge in NN
        ampo += (-t,"Cdagup",edge[1],"Cup",edge[2])
        ampo += (-t,"Cdagup",edge[2],"Cup",edge[1])
        ampo += (-t,"Cdagdn",edge[1],"Cdn",edge[2])
        ampo += (-t,"Cdagdn",edge[2],"Cdn",edge[1])
    end
    for i in 1:N
        ampo += U,"Nupdn",i
    end
    H = MPO(ampo,sites)
    return H
end

let
    nx = 2 #number of rows in lattice
    ny = 2 #number of columns
    t, U = 1, 6 #parameters in Hubbard model
    τ = 0.1 #imaginary-time evolution step-size
    NN = [[1,2],[1,4],[2,3],[3,4]] #nearest-neighbor sites
    
    sites = siteinds("Electron",nx*ny;conserve_qns=true)
    gates = hubbard_TEBD_gates(NN,sites,t,U,τ)
    
    H = lattice_hubbard_MPO(NN,sites,nx,ny,t,U)
    
    ψTEBD = productMPS(sites,n -> isodd(n) ? "Up" : "Dn")
    
    for i in 1:10
        ψTEBD = apply(gates, ψTEBD)
        normalize!(ψTEBD)
        E = inner(ψTEBD',H,ψTEBD)
        
        println()
        @show i
        @show E
    end
    
    ψDMRG = productMPS(sites,n -> isodd(n) ? "Up" : "Dn")
    dims = [4,8,12,16]
    sweeps = Sweeps(length(dims))
    setmaxdim!(sweeps,dims...)
    EDMRG, ψDMRG = dmrg(H,ψDMRG,sweeps)
end

```

Instead, the problem is coming from construction of my gates in the function `hubbard_TEBD_gates( )`. I have not performed swaps to make sites adjacent to each other before applying gates, because my understanding is that the `apply( )` function should handle all of this for me in ITensor. I apologize if I am asking a version of a question you have already answered, but I genuinely unsure of the best approach for applying these gates for cases with long-range interactions.

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<div class="post-metadata">

**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\
**Post date:** [July 14, 2024, 2:02pm UTC](https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865/2 "2024-07-14T14:02:14Z")

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Hi Max, to make sure I understand your question, are you finding that when you use your approach you are getting incorrect results? Or are you just in the planning stage and asking if you need to perform any extra steps when making fermionic evolution gates that are not nearest-neighbor?

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**Author:** ![ryanlevy](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/ryanlevy/32/150_2.png) [@ryanlevy](https://itensor.discourse.group/u/ryanlevy)\
**Post date:** [July 14, 2024, 2:21pm UTC](https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865/3 "2024-07-14T14:21:37Z")

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Just wanted to quickly add that if you are doing imaginary time evolution you should use

```julia
Gj = exp(-1.0im * τ/2 * hj)

```

so the operators are properly complex

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**Author:** ![wangfh5](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/wangfh5/32/619_2.png) [@wangfh5](https://itensor.discourse.group/u/wangfh5)\
**Post date:** [September 12, 2024, 2:19pm UTC](https://itensor.discourse.group/t/tebd-on-2d-fermionic-system/1865/4 "2024-09-12T14:19:52Z")

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Hi Miles, actually I am really curious about the latter question you proposed. When the interactions are beyond nearest-neighbor (NN), a usual workaround is to use SWAP gate to make the interacting sites being NN ([Minimally entangled typical thermal state algorithms - IOPscience](https://iopscience.iop.org/article/10.1088/1367-2630/12/5/055026)). For fermions, SWAP gates will cause additional signs. I would like to ask: (1) if the `apply` function in `Itensor.jl` utilizes the SWAP gates method to deal with long-range interaction? (2) should I worry about the fermionic cases?

Thank you!  
Fo-Hong