The question of calculating the zero-frequency Green's function for a one-dimensional interacting fermion model with open and periodic boundary conditions.

DMRG and Numerical Methods
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Hi,

Recently, I’ve been calculating the Green’s function

G(k, \omega) = \langle 0| c_{k}^{\dagger} \frac{1}{\omega+\mu+E_{N}^{0}-H} c_{k}|0\rangle +\langle 0| c_{k} \frac{1}{\omega+\mu-E_{N}^{0}+H} c_{k}^{\dagger}|0\rangle

for a one-dimensional model

H=-t \sum_{i}\left(c_{i+1}^{\dagger} c_{i}+\text { H.c. }\right)-t^{\prime} \sum_{i}\left(c_{i+2}^{\dagger} c_{i}+\text { H.c. }\right)+V \sum_{i} n_{i} n_{i+1}

with periodic boundary conditions for small sizes, and then scaling to the thermodynamic limit. I used linsolve in the computation, and I found that it gives me the desired results. Now, I want to apply this method to larger system sizes, but it seems that periodic boundary conditions are difficult to implement, and while open boundary conditions can be used for systems with up to around 210 sites, the results are not reliable. My idea is to use large systems with open boundary conditions to approximate periodic boundary conditions, but I’m wondering if this approach is too simplistic.

I have several questions:

  1. Is it possible to implement periodic boundary conditions for large system sizes? If so, it likely involves more than just adding interaction terms between the first and last sites.
  2. The results with open boundary conditions deviate significantly from the periodic case. Is this due to boundary effects? How can we mitigate these boundary effects? It seems that simply increasing the system size doesn’t eliminate the problem. What might be the reason for this? Also, could calculating the entanglement entropy or other quantities provide a way to estimate the impact of boundary effects on the system?
  3. As stated on the website DMRG FAQs · ITensors.jl, my model may be more suitable for infinite MPS calculations. However, I’m unclear whether I can use linsolve for iteration after obtaining the ground state. I know that IDMRG is still under development.

I apologize for the many questions, but this has been quite confusing for me. I would greatly appreciate any guidance or answers you can provide. Thank you in advance!

Best regards.