Partial Filling in VUMPS

DMRG and Numerical Methods
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1

I am interested in making an electronic initial MPS state at various filling fractions for VUMPS. The simplest state for half filling that conserves Sz=0 for example is

initstate(n) = isodd(n) ? "↑" : "↓"
s = infsiteinds("Electron", N; initstate, conserve_qns)
ψ = InfMPS(s, initstate)

If I consider 1/4 filling, I would naively write

initstate(n) = isodd(n) ? "↑" : "0"
s = infsiteinds("Electron", N; initstate, conserve_qns)
ψ = InfMPS(s, initstate)

but this now does not conserve Sz=0. Do I need a 4 site unit cell then ? If I wanted filling 1/8 would I then also need 8 sites in order to conserve Sz=0 ?

Thank you for your time.

2

Yes, you would need to use larger unit cells for those fillings.

1 Like
3

@mtfishman As I understand, the unit cell roughly resembles the width of cylinders in iDMRG. Does this mean that computationally I’m limited to fillings around 1/8th or so?

4

If you are using a 1D model (a model with local interactions in 1D) the cost should scale linearly with the unit cell size.

5

Sorry, If I am trying say a 2-leg ladder for example, I’m not clear how to construct a 1/4th filling unit cell as it seems the unit cell size can be used to extend either along the infinitely repeated line or along the new dimension?

6

For a 2-leg ladder, it would make sense to me to make a 4-site unit cell based on 2 rungs of the ladder.

7

Hi Matt, our solution for this has been to use the following initial state function.

function InitState(; N::Int64=4, filling::Int64=2, SzTot::Float64=0.0)::Vector{String}
    ##### Filling the state with up and down electrons #####
    Nup = Int((filling + (2 * SzTot)) / 2)
    Ndn = Int((filling - (2 * SzTot)) / 2)
    ##### Choosing random sites to fill the up-electrons and down-electrons #####
    stateUp = state_fill(N, Nup, "Up")
    stateDn = state_fill(N, Ndn, "Dn"; starting = (filling/N) == 1.0 ? 2 : 3)
    state =  stateUp .* stateDn
    state = [x == "" ? "Emp" : x for x in state]
    return state
end

This seems to work for half-filling (N=4, filling = 4 for example), but for large interaction strengths (J = 5 for a 1D t-J model) for partial filling ( (N=4, filling = 2) , we’re getting the error

┌ Warning: Right precision error 1.052545211436737 is too high!
└ @ ITensorInfiniteMPS ~/.julia/packages/ITensorInfiniteMPS/v5Lx4/src/vumps_nonlocalham.jl:404
┌ Warning: Left precision error 1.397765999575384 is too high!
└ @ ITensorInfiniteMPS ~/.julia/packages/ITensorInfiniteMPS/v5Lx4/src/vumps_nonlocalham.jl:402
VUMPS iteration 4 (out of maximum 200). Bond dimension = 32, energy = ([-5.313809909935317, -6.956322928281491, -8.465293657965256, -1.723316220609907], [-3.390841907800686, -5.141353622968244, -0.3790025062582366, -6.223079887932087]), ϵᵖʳᵉˢ = 1.397765999575384, tol = 1.0e-6, iteration time = 1.663362496 seconds

and the VUMPs does not converge. Do you have a suggestion for going about debugging this? Since it only occurs for large interaction strengths (J = 4.5 does not do it for example).