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JerryChen97 / KitaevLadder

Using TeNPy to simulate Kitaev Ladders. This will be part of my PSI essay work.
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Difference in the translational invariance of Z-phase and X-phase #39

Open JerryChen97 opened 4 years ago

JerryChen97 commented 4 years ago

By adding a shift=2 to the transfer matrix, I tested the 2-translational invariance of the whole diagram. Surprisingly, they are quite different in two phases: image image

JerryChen97 commented 4 years ago

This implies that in the X-phase, the ground state is 2-translational invariance, while in the Z-phase it's not the case. In the other phase it's 4-translational invariant

JerryChen97 commented 4 years ago

@aaronszasz Do you think this result is reasonable?

aaronszasz commented 4 years ago

I think so. From the effective site in each phase, you can see what the translation invariance should be. In the X phase, it's from one X bond to the next X bond, which I believe is a two-site translation in your setup, and in the Z phase it's from one Z bond to the next Z bond, which is a translation from sites 0 and 1 to sites 3 and 2 rather than 2 and 3. If you want to properly test this, you could check a shift of two again but with either the bra or ket having sites 2 and 3 swapped. I think in that case you would see the two-site translation reappear.

JerryChen97 commented 4 years ago

Ah that makes sense! I will try this

JerryChen97 commented 4 years ago

@aaronszasz I used the swap_sites function provided in the MPS class of TeNPy, and 'swapped' the site 2 and the site 3 of the iMPS of 4 physical spins, but the transfer matrix produced afterwards has the largest eigenvalue smaller than 1...

JerryChen97 commented 4 years ago

I am gonna rebuild the model in the order that works naturally for the large-Z limit so that the whole numerical process keeps the same regardless of anything we don't know about the usage or hidden bugs of swap_sites.

aaronszasz commented 4 years ago

That's reasonable. Norm being less than 1 is probably not a huge deal after swapping though as long as it's not too much less, since there's some extra truncation involved in the swap. Was it much less than 1?

JerryChen97 commented 4 years ago

That's reasonable. Norm being less than 1 is probably not a huge deal after swapping though as long as it's not too much less, since there's some extra truncation involved in the swap. Was it much less than 1?

~0.8

aaronszasz commented 4 years ago

That sounds large enough that you can try the transfer matrix with shift and you might get useful results already.

JerryChen97 commented 4 years ago

I just reproduce the simulation at Jx=Jy=Jz=1 with another setup which should be able to naturally show the correct translational invariance, and the transfer matrix with shift=2 indeed has the largest eigenvalue equal to 1. I think the actual translational invariance of large-z ground states can be verified; when I got some time I will think about what's wrong with the previous direct swap or permutation.

aaronszasz commented 4 years ago

Ok, sounds good! At least we do understand correctly what's happening.