On the phase diagram of Spin-1 cases

[JerryChen97]

I spent some days improving the phase diagram of spin-1 cases but there has been no significant change. Here is a diagram just as usual, where we calculate the correlation length (logarithm) for each phase point at two different bond dimensions 64 and 128.

The main part is still composed of gapped points that have no change in the correlation length, which is fine; but for certain points gathered in the center of two diagonal directions, they seemed gapless but also sometimes calculated inaccurately, which seems quite creepy.

Then, I look closer into the line Jx=Jy, do the DMRG in a series (iteratively using the previous psi as the input of the next DMRG engine), and here is an interesting figure for the correlation length (the original value, not log)

The two peak points correspond to Jx=Jy=.84 and Jx=Jy=1.40 respectively.

[JerryChen97]

Here is a fig for the finite-scaling of entanglement entropy w.r.t. the log of correlation length at Jx=Jy=1.4, Jz=1

[JerryChen97]

Here is a fig for the finite-scaling of entanglement entropy w.r.t. the log of correlation length at Jx=Jy=1.4, Jz=1

A fitting gives the central charge to be 0.5 here.

[JerryChen97]

Here is a fig for the finite-scaling of entanglement entropy w.r.t. the log of correlation length at Jx=Jy=1.4, Jz=1

A fitting gives the central charge to be 0.5 here.

chi_list = [16, 32, 64, 128, 256]

[JerryChen97]

The .84 point seems a little bit not so typical. I will try finding a 'more gapless' point.

[JerryChen97]

Here is a rough phase diagram given by directly calculating the correlation length of each point.

It seems that there is a rocket-shaped gapless circle lying in the center.

[JerryChen97]

Here is a much more precise phase diagram given by log xi (all the log below zero were trimmed to be just zero). @aaronszasz How do you like it?

[JerryChen97]

Here is the original version:

[JerryChen97]

I got a feeling that maybe the real phase diagram is much more complicated and elegant than I've thought

[aaronszasz]

This looks really interesting! So it looks like there are really three phases, maybe?

[JerryChen97]

This looks really interesting! So it looks like there are really three phases, maybe?

Yeah that's my current guess.

[JerryChen97]

And the shape of the left-below region seems really unusual; also I am curious about what happened to the border between the left-below region and the central region.

[JerryChen97]

Some other figs along the line Jx=Jy, where the resolution is 0.01

[JerryChen97]

Next: fDMRG for the critical points

[JerryChen97]

Here is a map for XY local order parameter using the same dataset of spin-1 case

[aaronszasz]

Excellent, so that's one phase identified!

[aaronszasz]

I suspect that one of the two remaining phases will correspond to the other spin-1/2 phase, and the last one will be new, but we'll see

[JerryChen97]

I just got the diagrams for the X-string order parameter (numerically calculated by <X_1 ... X_51>) and the Z-string order parameter (numerically calculated by <Z_0 ... Z_50>) using the same dataset.

[JerryChen97]

PS: only the right-below triangular part of the diagram was really computed; the other half was produced by simply flipping the same values along the diagonal line Jx=Jy. Therefore, in the left-upper part the real op should be the Y-string order parameter

[JerryChen97]

@aaronszasz I got a feeling that something novel happened here when the system was changed from spin-1/2 to spin-1: in spin-1/2 case, both XY and Z-string can indicate the Z-phase, and X-string indicates the X-phase; however, in spin-1 case, if my iDMRG data are correct enough, the XY order parameter and Z-string order parameter are 'taken apart' from each other.

[JerryChen97]

In theory, do we have any technique to map the physical spin from discrete values to continuous values? I heard from someone that there is a way to transform spin models to another set of models, which implements such stuff, but right now I couldn't recall the details....

[aaronszasz]

These string order parameters for spin 1 look to me like they are not providing any information. Rather, there is a tendency when Jz is large to orient along Z, likewise for Jx and X. This is why these order parameters seem to decay to 0 at random places rather than the phase boundary.

Can you show the equivalent of these figures for spin-1/2?

[JerryChen97]

Spin 1/2 case:

[JerryChen97]

Spin 1/2 case:

These diagrams were created by multiply only 10 ops together; I didn't include more because TeNPy didn't provide 'Sigma' operators for the SpinSite model and this made me multiply an extra 2**n where n is the number of spin operators, and if too many operators are considered I think the final result will be kinda spoiled. This choice made the results not to perfectly reflect the precise theoretical prediction.

In theory, as the number of ops approaches infinity, X-string parameter immediately vanishes in the Z region and vice versa for Z-string.

[JerryChen97]

There are also some diagrams I created before that might be useful, in another issue

[aaronszasz]

This looks very different from the spin 1 case, to me. The "boundary" of the string order parameter is exactly at the phase boundary here.

[JerryChen97]

This looks very different from the spin 1 case, to me. The "boundary" of the string order parameter is exactly at the phase boundary here.

I agree. The boundary lines can still be observed somehow but in quite an unclear way, especially for Z-string which only provides a diff in values instead of whether or not to vanish. I am not sure whether this is caused by the finite length of string operators I computed or it's something more fundamental.

[aaronszasz]

It appears from the figure you posted in the other issue that even if the string expectation value does not vanish at the phase transition, it would have infinite slope at that point in the limit where the string length goes to infinity, ie there is a singularity there. You can possibly check this by extrapolation up to ~20 spins? I'm not sure of the physical significance of that, precisely, but it is again distinct from the spin-1 case.