Create a hump-like density profile in Fermi-Hubbard model

How do I use this algorithm? What does that parameter do?
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ykharkov
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Joined: 28 Jan 2021, 01:42

Create a hump-like density profile in Fermi-Hubbard model

Post by ykharkov »

Hi,

I'm trying to create a hump-like density profile in a spinless 1D Fermi-Hubbard model using DMRG engine.
To do so I'm using a standard FermiHubbardModel class in TenPy and adding an onsite potential V_i*n_i at each site i=1,...,N, where V_i is the external potential with the Gaussian shape, V(x) = -A*exp(-(x-x0)^2/sigma^2).

Then I run DMRG engine and find the ground state of the model. To be concrete, I am considering Fermi-Hubbard model near filling nu=N_fermions/N_sites=0.1.
Although this approach sometimes works I found somewhat strange behaviour (which is also related to the choice of conserved quantum numbers).

(1) If I set the DMRG engine parameter to conserve=None and try to adjust chemical potential mu, then below certain critical value of mu=mu_c the DMRG converges to a state with density=0. While I decrease the chemical potential continuously in order to reach the desired background filling factor=0.1, the density in the DMRG solution at large distances jumps from a finite value=0.3 at some mu=mu_c, to exact zero at mu<mu_c at all lattice sites. So the DMRG solution contains zero fermions.
I suspect there is some sort of leakage from the sector with a given total number of fermions to a sector with zero fermions.

(2) In order to fight the 'leakage' problem I set conserve='N' (or conserve='best') to enforce particle conservation.
Then I initialize my initial wave function as a bitstring [0..0011..11], so that the total number of fermions corresponds to the desired fermion density=0.1. Next I run DMRG engine to find the ground state.
The density profile that I obtain with this method has some unphysical 'ripples' at large distances, see Fig. attached, The ripples persist even if the I set interaction term in Fermi Hubbard mode to zero, U=0.
I tried to change parameters of DMRG engine and change number of fermions in the initial bitstring, but the problem still persists.
Image


I use infinite MPS boundary condition when initialize FermiHubbardModel.
The values of other parameters of DMRG engine are listed below:

Code: Select all

dmrg_params = {
    'mixer': True, 
    'max_E_err': 1.e-10,
    'trunc_params': {
        'chi_max': 50,
        'svd_min': 1e-16
    },
    'verbose': False,
    'combine': True,
    'max_sweeps': 20
}
Would really appreciate any advice on how to deal with the problem!

Thanks!
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Johannes
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Joined: 21 Jul 2018, 12:52
Location: TU Munich

Re: Create a hump-like density profile in Fermi-Hubbard model

Post by Johannes »

First of all, let me clarify: Do you want some uniform "background density" and then some more particles in the center?
Or do you only want particles in the center?

For 1): If you don't enforce charge conservation, DMRG is "allowed" to switch to another charge sector, even if you initial state has zero overlap with it.
It might happen because (a) there is some numerical noise eventually giving you small overlap with another charge sector, or (b) because TeNPy's DMRG code actually does some exact diagonalization instead of lanczos at small bond dimensions, which explores other sectors as well.
Don't rely on the switching of charge sectors, though!
I would guess that for the mu< mu_c, the density-0 filling actually has a lower ground state energy than if you keep the density, right?

For 2):
Indeed, if you want to target a specific particle density, you should enforce it by setting conserve='N'.
Let me mention that it is better to chose the initial state roughly with the density profile you expect to have in the end,
e.g. in your case something like [0, 1, 0, 1, ..., 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, ...., 0, 1] with an appropriate number of 0s and 1s to fix the desired filling.
For the "unphysical ripples": are you really sure that they are unphysical?
Did you try to solve the free fermion hamiltonian and look at the density profile of it?
The density profile of a pure particle in a box has also "ripples", right? So I wouldn't be too surprised to also see that here...
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