# How To Extrapolate Ground State Energy in Thermodynamic Limit via DMRG [closed]

I am implementing a system of fermions in a trap potential with tensor networks and want to find the ground state energy via DMRG (I'm using SyTen). As usual in quantum chemistry the Hamiltonian has the following form:

$$H=\sum_{i,j=0}^{d-1}h_{ij}c^\dagger_i c_j + \sum_{i_1,i_2,j_1,j_2}^{d-1}V_{i_1,i_2,j_1,j_2}c^\dagger_{i_1}c^\dagger_{i_2}c_{j_1}c_{j_2},$$

where $$d$$ denotes the maximum number of orbitals - of course one needs infinitely many orbitals in principle, which requires extrapolation of the ground state energies. I got into tensor networks from the condensed matter side and there it is usual that one regards e.g. some spin lattice system, fixes the filling fraction (e.g. half filling) and then one computes the ground state energies for different system sizes $$L$$ - then plotting the energies v.s. $$1/L$$ (for 1D systems) and finds the "true" ground state energy for $$1/L=0$$ for the fixed filling fraction.

However, now I'm not interested in the energy for some fixed filling fraction, instead I want the energy of a fixed particle number. I can fix the number of fermions in the system by initializing a start state in the correct symmetry sector (we have U(1) symmetry). My question is the following: I want to find out the "true" ground state energy for the "thermodynamic" limit of $$d=\infty$$ orbitals and $$N$$ fermions in the system. Is it valid to fix $$N$$, compute the ground state energies for different $$d$$ and then plot and fit the energy v.s. $$1/d$$ and taking the energy at $$1/d=0$$ as the true ground state energy for particle number $$N$$? It seems reasonable for me but I'd like to make sure that this is indeed the correct thing to do.

I was looking through the literature about this but could not find an explicit answer which explains the extrapolation process for such cases. I hope that I could explain my issue clearly enough - an explanation would be great but also some references which address exactly such questions would be very helpful!

• Have you tried just calculating the energies for a few different $d$ values and plotting them? Usually gives a decent idea of what to expect. Apr 12, 2023 at 2:35
• @Anyon yes, I did try the procedure in question, but it does not give me the result I'd expect (comparing the analytical solution of this problem), which is why I am questioning it. Maybe there is also another fault in my implementation but it would be good to know whether I can exclude the extrapolation process as error source or not. Apr 12, 2023 at 6:37
• I gave my +1 long ago, but I wonder if there's any update you can give us after all these months? Also, are you still actively in need of an answer to this question? Please update us! Oct 12, 2023 at 15:52
• @Nike Dattani Sorry for the lack of responses!! It turned out that I had a bug in my implementation and when I fixed it, I could get very accurate results without extrapolating at all such that I did not work on this question anymore. As a side note: There are extrapolation schemes out there for more sophisticated implementation schemes, such as e.g. proposed in arXiv:2209.14190. Oct 20, 2023 at 10:51
• That paper cited two of mine! In fact the top portion of Table III was all made by me, and the bottom portion shows their results. However I don't see anything about the "thermodynamic limit" in that paper. Do you? Oct 20, 2023 at 16:54