Possible Local Order Parameter

[JerryChen97]

1. A table for all the possible ops
2. The overlap between wavefunctions in some phase (remember to avoid the cat states)

[JerryChen97]

I designed a function to generate all the possible candidates for (local) order parameters according to the following rules:

• Commuting with the local symmetries Dn
• Anti-commuting with at least one of the global symmetries Sigma_x or Sigma_y

And then with its help, the correlation functions of candidates within a given range can be calculated.

[JerryChen97]

[JerryChen97]

The capital letters in the fig above represent local operators: I := Id, X:= Sigma_x, Y := Sigma_y, Z := Sigma_z.

[JerryChen97]

Though quite inefficient, it can also generate any arbitrary range (w.r.t. how many unicells on the chain the local operator involves). I also tested all range-3 operators and all vanished.

[JerryChen97]

P.S. The MPS I used was generated by periodic DMRG with 80 physical spin-1/2 sites.

[JerryChen97]

@aaronszasz from these results I just don't believe the existence of any local order parameter.

However, I am still unable to exclude this possibility in theory. I am wondering if anyone else has tried showing that in a 1D local spin system, the range of any possible local order parameter operator will have an upper bound scaling with the range of the system's interaction (e.g. I will say the range in our Kitaev ladder to be 3); if this is true, then we don't have to show all the vanishing results till infinity, which is obviously impossible.

[JerryChen97]

Right now I am thinking about the other possible explanation: such a Kitaev ladder may be an SPT. But I have no clue how to prove this either numerically or theoretically.

[aaronszasz]

I'm not aware of such a result, but that doesn't mean there isn't one. I haven't really put any time into thinking about this question, but I can do so this week to hopefully have something useful to say.

[aaronszasz]

-> By "this question" I mean the existence of a local order parameter in this model

[JerryChen97]

I'm not aware of such a result, but that doesn't mean there isn't one. I haven't really put any time into thinking about this question, but I can do so this week to hopefully have something useful to say.

Yeah definitely I didn't show that there is absolutely no local order parameter in this model. Right now I can only say that this is my own guess; btw the property of being an SPT and that of being an SSB should not be contradicting right?

By running over all the possible range-5 candidates (involving up-to-10 spin-1/2 sites) satisfying the two rules above, I found the total number is 7168; this is not a huge number in principle but it takes some time to check all of them. In a word, it is almost impossible to confidently claim that there is no such stuff if it does not exist.

[JerryChen97]

Here is a map of the XY local order parameter generated from the ~20k points I've got recently.

[JerryChen97]

[JerryChen97]

[JerryChen97]

@aaronszasz I hope you enjoy the 3 figures above!

[aaronszasz]

Looks great!

[aaronszasz]

This is indeed exactly what I was asking to see, thanks!

[aaronszasz]

I'm actually a bit surprised that it's so large even at Jx=Jy=2. Very neat

[JerryChen97]

This is indeed exactly what I was asking to see, thanks!

Your idea is excellent and also matches perfectly with my iDMRG simulation! I am also working on the large-X limit derivation; will update it on Overleaf and email you and Yin-Chen once it's done~

[aaronszasz]

Great, I look forward to seeing it!

[aaronszasz]

How is the large-X perturbation theory coming?

[JerryChen97]

I think I've got the reason why the OBC system is two-fold degenerate instead of 4: the cluster model our Kitaev ladder corresponds to is slightly different from the model mentioned in Xie's textbook. A ring of the cluster model with N spins is composed of N local cluster Hamiltonian ZXZ, but when the ring is broken into an open chain, Xie's treatment is getting rid of two of those ZXZ, corresponding to the start and the end of the chain; therefore, in their OBC cluster model, there will be a four-fold degeneracy. However, in our case, the cluster model is induced from the original ladder, where only one of the effective ZXZ is lost when the PBC is changed to OBC; so we will only get a two-fold degeneracy. I am still creating some figures to illustrate this point, and I will attach them to the Overleaf doc and email you and Yin-Chen.

Still, I couldn't well explain another problem: why the PBC Kitaev ladder still seems to be 2-fold degenerate in the large-X limit? According to the effective Hamiltonian, there should be no more degeneracy. I am still checking if something in my code went wrong.

[aaronszasz]

Did you try Yin-Chen's suggestion to do the PBC case in finite-system DMRG and see if you still get the degeneracy?

[JerryChen97]

Did you try Yin-Chen's suggestion to do the PBC case in finite-system DMRG and see if you still get the degeneracy?

Yes, I tried and there is no more degeneracy. Also, I did the matching for smaller sizes (8, 12 and 16 spins) between fDMRG of PBC with Exact Diagonalization on the energy and it seems perfect.

[aaronszasz]

So it sounds like your first excited state energy from the analytical method is wrong?

[JerryChen97]

I also did the benchmark between the numerical integral over the spectra and ED, and an interesting fact is: for GS energy, the numerical integral works perfectly for any finite PBC, but not always good for the first excited energy; it doesn't work when the system is around the large-X limit.

[JerryChen97]

So it sounds like your first excited state energy from the analytical method is wrong?

Exactly. I am still wondering about the essential reason, but I couldn't get a very good answer to this. I will share and discuss with you and Yin-Chen hours later.

[aaronszasz]

Ok, I'll look forward to hearing about it later then.