We know that DMRG and MPS are essentially 1D algorithm. And when one wants to use DMRG to solve 2D system, it is possible to connect each finite 1D part to make a quasi 1D system.
My question is, how does ITensors treat this? There are two methods come into my mind, like the plot1, combining each column together into a big local Hamiltonian, and sweep each column respectively.
Another method is shown in plot2. Just like 1D update. First construct the total Hilbert space as order 1,2,3,4,1,2,3,4… from left to right. Then sweep each site in order, but with non-local Hamiltonian.
I am not sure how does ITensors operate this and abovementioned ways are just my guess.
Since the limited access of my account, I put the second schematic plot here.
The approach to using DMRG for quasi-2D systems, of the type you drew, is the second idea you mentioned, that of ordering the sites in 2D in a one-dimensional ordering and then just letting the Hamiltonian be longer-ranged (or “non-local” though that term can have different meanings).
This is the idea mentioned around Fig. 2 of this review article about DMRG for 2D systems:
https://arxiv.org/abs/1105.1374
I wouldn’t exactly say that this is how “ITensor” does 2D: rather it’s just the standard approach to doing DMRG in 2D so it is the one we advertise and show users how to do. But you can actually use the Hamiltonian and DMRG tools provided by ITensor to do DMRG on any system or Hamiltonian that you like (provided you can represent it as an MPO or a sum of MPOs). So you could in principle also use ITensor to try the first idea you mentioned. But the drawback of that idea is that the site dimensions would grow exponentially with the transverse size of the system, while the bond dimensions of the MPS would still be just as large, so the second approach you mentioned is probably superior.
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Thanks for your response and explanations.
Blockquote Hamiltonian be longer-ranged (or “non-local” though that term can have different meanings).
Would this type of long-range interaction affect the run time and sweeping time till convergence?
Yes, working with 2D Hamiltonians encoded as 1D MPOs has strong effects on the run time and resources needed to converge. Sometimes the number of sweeps does not have to be too much but it depends a lot on the particular system and the quality of the initial state.
I would encourage you to read the review article linked above which is precisely about such things, and to experiment with some 2D systems yourself (we have code examples provided with ITensor of 2D DMRG) to get a feel for what the calculations are like and how quickly the energy converges compared to the 1D case.
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