Question of calculate degenerate states

DMRG and Numerical Methods
1

Hi itensor group，

I seem to need to consider a question about excited states. I’m simulating a kagome lattice of Rydberg atoms on itensor. When I take a set of parameters, the calculated ground state result is that one of the three sites in each unit cell is in the exicted state.

The literature tells me that the true ground state is triply degenerate within this phase. But I tried repeatedly and did not find a triple degenerate case, the excited state is always in the same site of the unit cell. This makes me a little suspicious.

I’ve tried changing some parameters, but the result doesn’t change. I also tried the method of calculating the first excited state to find the degeneracy, but after I tried many weight parameters, the result still did not change.

I sincerely hope to get some understanding and views on such issues. At the same time, I also want to know if there are any requirements for the selection of the weight parameter for calculating the excited state.Is it a purely empirical parameter?

Thanks to the developers and maintainers of itensor, thank you for your efforts.

2

Hi Kevinh,
Sorry to hear your calculations aren’t going as you were expecting so far.

To help you with this, we will need to know some more technical details.

  • What size systems are you simulating? Especially, what is the transverse, y-direction, size?
  • What DMRG parameters are you choosing? How many sweeps and what maximum bond dimension and cutoff?
  • What kind of output are you seeing? Does the energy converge or keep changing from sweep to sweep?

In terms of what may be happening, I can only guess right now, but some possibilities may include:

  • your Hamiltonian or lattice has a bug in how it is programmed
  • your DMRG calculations aren’t converging, which could be for a variety of reasons from choice of initial state, to settings of DMRG parameters
  • possibly, the results actually are correct but care is needed in interpreting them when comparing to the literature. For example, degenerate ground states are often discussed in one basis or linear combination in theoretical papers, whereas DMRG will find a different linear combination with lower entanglement. This behavior of DMRG can be confusing to newcomers.

Finally, regarding the weight parameter for obtaining excited states: basically yes it is empirical. The only guideline I am confident about is that it needs to be set larger than the gap from the ground state(s) to the first excited state. But apart from that it’s less obvious and some experimentation is needed. We discuss that a little bit in this part of the documentation.

Looking forward to hearing more details about your calculations.

3

Thank you very much for your kind suggestion. Based on your answer above, I think my problem may be that the ground state used in the calculation is not fully converged. In fact, due to inexperience, this may be the first time I have encountered such a difficult system to converge.

According to my tests over the past few days, I think I should first use a small bond dimension to make the ground state energy close to a stable value, and then gradually expand the bond dimension to a higher level. Instead of using a larger bond dimension while the energy is still at a high level like I did before. Obviously, this is a waste of computing resources.

Thank you for your kindness and help.

4

Glad to hear about the progress. I would still be curious to know what system sizes you are trying.

5

  1. Thank you very much for your kindness.What I am testing is "Nematic" in this paper This is a kagome lattice with 8*4 unit cells. In each unit cell There are 3 sites. To avoid spurious effects due to sharp edges, unit cell at the right boundary of the cylinder contains only two sites. Therefore, there are 104 sites in total. And, s=1/2 on each site. The picture of “Nematic” as follows，
    Nematic

  2. This paper mentions: The nematic order spontaneously breaks the threefold rotational (C3) symmetry of the underlying kagome lattice, so, for aninfinite system, the true ground state is
    triply degenerate within this phase.I think this can be find out by calculating the excited states.

  3. My sweeps parameters are also written below. Maybe there is something wrong with it, but I still hope for your suggestion.In particular ,I think the ground state used to calculate the excited state is not converged enough .Will this have a destructive effect on the computation of excited states?

sweeps = Sweeps(50) 
maxdim!(sweeps,10,10,10,20,20,20,100,100,100,200,200,400,800,1000,2000) 
cutoff!(sweeps,1E-16)

weight=4.0
println("weight=",weight)
energy1,psi1 = dmrg(H,[psi],psi1_init,sweeps; weight)
setnoise!(sweeps, 1e-6,1e-6,1e-7,1e-7,1e-8,1e-8,1e-8,1e-10,1e-10,1e-12,1e-12,0)
@show inner(psi1,psi)
6

Those are helpful details. Here are my main thoughts about your calculation and what you might expect in terms of convergence and results you’ll be able to get. Note that these are just speculative and each DMRG calculation is different and you never know ahead of time exactly how easy or hard things will be.

  • First of all, your system size is quite large in the y direction. It depends on the Hamiltonian, but transverse sizes that large can often take bond dimensions of many thousands to converge well. If you can start by studying transverse sizes more like Ny=4 or so you may find it much easier to reach convergence.
  • The total number of sites is much less important for DMRG than the transverse size. DMRG scales linearly in Nx and exponentially in Ny.
  • I think the reason you are only seeing one of the ground states is that your lattice explicitly breaks rotation symmety itself, so it automatically prefers one of the infinite-system ground states over the other. There isn’t much you can do about this: I think it’s just something you have to accept about working on finite-size systems for this case. If you are lucky, you might be able to see the other ground states, which will here be excited states, getting closer to the finite-size ground state as you reach larger sizes. But you may not be able to reach very large sizes.
  • your pattern of sweeps looks good to me. A good guideline for challenging systems is letting the energy converge or stop changing much for one maxdim value before going to the next maxdim value. So it might not hurt to do, say, 20 sweeps of maxdim=10 then 20 more of maxdim=20 then four or even eight at 100 and then the rest like you have now. The maxdim=10 or 20 sweeps will only take at most a few minutes each (and probably less) so it doesn’t hurt to do a lot of them.
7

Thank you for your sincerely reply, my understanding of the problem in such a large Ny system has improved, and it is indeed difficult to calculate the triply degenerate in the case of such a large Ny. After communicating with a friend today, he suggested that if I only need to draw the excited state of the first position in each unit cell, I only need to put an operator of -\delta ni that only works in the position of site=1. Although this method modifies the Hamiltonian

In your reply, you suggested keep bond dimesion=10 to sweeps 10 or 20 times, which I also find very effective. But if I use something like maxdim!(sweeps,10,10,10,10,10........10,100,200,...) Repeatedly inputing 10 doesn’t seem very convenient. It seems that only Int can be input in maxdim! function. I don’t know if it can be written separately as follows

sweeps = Sweeps(20)
maxdim!(sweeps,10)
energy, psi = @time dmrg( H,psi0,sweeps)

sweeps = Sweeps(20)
maxdim!(sweeps,100,200,200,400,800,1000)
energy, psi = @time dmrg(H,psi,sweeps)

Or the itensor provides a better way to write the parameters need to repeat input

8

Your question about how to program a large number of sweeps with the same maxdim more conveniently is a good one. Would you please start a new topic (new forum post) asking that question? I bet others have the same question and it deserves its own separate topic.

About your system, I wanted to reiterate that not finding the triply degenerate ground states is probably not to do with the size of Ny, mainly. I mentioned the Ny size to talk about whether you were reaching large enough bond dimensions (maxdim). But I would guess the main reason you aren’t seeing the triply degenerate ground states is that you are working on a finite-size system, and so the rotational symmetry present in the infinite system simply isn’t present in your finite system. Unfortunately I can’t think of any way to make that symmetry be present using DMRG which can only treat finite sizes in the y direction. So I think you’ll just have to know that your single ground state turns into three in the infinite 2D limit.

9

Thank you for your help in correcting the misunderstanding of the above reply. And I have already start a new topic of sweeps.

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