I want to ask what are the limits of the DMRG to investigate the GS of long range interactions Hamiltonian. In particular, can the DMRG be extended to Hamiltonian with interactions between 3rd neighbors sites? I have an Hamiltonian that is translational invariant, so I expect to observe expectation values independent on the site on which I’m doing the measure. Indeed, this what I observe putting in the Hamiltonian nearest neighbors and next to nerarest neighbors interactions. When I put 3rd neighbors interactions, the situation changes, the expectation values are different from site to site.
So, my question is: as you know, can the Itensors.jl DMRG have problems with interactions that are more than next to nearest neighbors?
Hi Francesco,
Basically there is no restriction on the kind of Hamiltonian you can input to DMRG. So yes you can certainly input Hamiltonians with third-neighbor interactions and it should work quite well as long as your system is 1D or quasi-1D.
I guess you’re saying you expected to get translationally invariant results but didn’t? There could be two reasons:
you didn’t succeed in converging your calculation, so you need to do more sweeps
you are using a finite-size system with open boundary conditions (which is good, and is what you should do) and are finding effects coming from the open edges. So you need to measure properties sufficiently far from the edges and/or use a larger system size or extrapolate results over a number of finite sizes.
It seems very sensible that the longer-range the interactions are, that the open boundary and finite-size effects could become larger. So that’s my guess as to what you are seeing.
Thank you for the answer. I corrected some imperfections of the code and what I observed Is that It works well (homogenous values of the observables on the sites) if I truncate the long range interactions until 3rd neighbors. However, if I put the full Matrix of long range interactions, I have the problem, there is no more homogeneity in the observables; at the same time the convergence of the ground state is good.
It seems that the Energy of the GS Is the correct one, but the observables that i compute (like magnetization and correlation functions) does not work well. Is It possible to improve the computation of that observables, i.e. do something like managing the bond link.
Also, maybe I defined in a wrong way the Matrix that contains long range interactions, but I checked many times and It seems correct.
Thanks for providing more details. When you say “closed boundaries” do you mean periodic? Or do you mean that the interactions terminate on each end? If the second, it’s equivalent to what I call “open boundaries”.
As best I can understand, what’s happening here is that you are simulating a system which is not translationally invariant, but expecting the results to be translationally invariant.
The only way to get strictly translation-invariant results are (1) simulating an infinite system (such as with infinite DMRG or VUMPS) or (2) simulating a periodic system. I’m not recommending you do that, just setting up a contrast.
For a finite-size, open boundary system such as what you’re simulating, there can be a kind of “emergent translation invariance” where on a large enough system, properties in the center of the system, sufficiently far away from the edges, obey a kind of local translation invariance. However, this will only be true if the correlations decay quickly from the edges and you study a system size which is significantly bigger than this decay length. (Note that these statements are not specific to DMRG and are just how finite-size systems behave.)
So when you say the results do not work well, you might actually be observing totally correct results but just not accounting for finite-size or open-boundary effects.