Entanglement Entropy near phase transition in 1D

Question

Hi,

I am trying to evaluate entanglement entropy for 1D system with OBC by keeping fix J and variable t2. There is a phase transition from 2-Majorana phase to 4-Majorana phase. For J=0, entanglement entropy is non-vanishing and there is a peak in entanglement entropy at phase transition but as soon as I turn on J, entanglement entropy turns out to be zero (near 10^(-7)). I used DMRG (ITensor Julia) to calculate ground state wavefunction and it seems like it is choosing a state with least entanglement for non-zero J interaction. Is there any way to force dmrg in ITensor Julia to get correct entangled ground state? I would really appreciate any help or suggestions.

Thank you so much!

I am using following Hamiltonian:

sites = siteinds("S=1/2",N1)

for j=1:N1-1
ampo += -1/2,"Sx",j,"Sx",j+1
ampo += J/4,"Sz",j,"Sz",j+1
end
for j=1:N1-2
ampo += t2,"Sx",j,"Sz",j+1,"Sx",j+2
end

Answer 1

HI, thanks for the question.

I don't immediately know the answer, but I would like to know if the energy of the unentangled state you are getting is matching the expected ground state energy for the J > 0 case? In other words, is the product state (unentangled state) you are finding a valid ground state?

I ask because you mentioned Majorana physics but the Hamiltonian here is a spin model. I know of some cases, like the Kitaev chain, where the Majorana model is equivalent to a spin model but the ground state in the fermion language is more entangled, while in the spin language is has less entanglement. In fact, it is an equal superposition of the two spin ground states. Is something like that expected here?

Thanks,
Miles

Comments

Hi Miles,

I am sorry, I didn't mention it earlier; Hamiltonian has been transformed from Majorana basis to spin model using Jordan-Wigner Transformation. I am getting a good phase boundary (t2, J values) between the 2MF phase ( double ground state degeneracy) and 4-MF ( quartic ground state degeneracy) by probing the behavior of different energy eigenvalues E0, E1, E2, and E3 ( ground state, 1st, 2nd, and 3rd excited state). At the phase boundary E2-E0 ~1/N where N is the system size. Though phase boundary is good, its not that precise which will give correct values for finite-size effects and other critical coefficients for the CFT at the phase boundary. Entanglement entropy can provide further precise values for phase boundary (t2,J). I should mention one more thing that earlier for small system size I got finite entanglement entropy for J=0 but I kept increasing system size from N=20 to N=400, I am getting very very small values of entanglement entropy (10^{-6} or 10^{-7}) for J=0 as well as J non-zero. I have learned about another idea (using Binder cumulants) that might help in locating the very precise location of the phase boundary. But at the moment I don't know much about it.
For finding the entanglement, I used the usual SVD decomposition method and using square of Schmidt values.   
Thank you,

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Thank you for the information. My guess (just a guess, since I haven't studied this exact model before) is that like the Kitaev chain, when you do a Jordan-Wigner transformation to the spin basis then the topological phase in the Majorana basis corresponds to a symmetry-breaking state in the spin basis. And symmetry breaking states can have very low or even zero entanglement. So I bet that DMRG is actually giving the right answer here but in the spin variables.

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Thank you so much, Miles. I think you are right though I am not aware of many papers related to it. I found one paper that talked about a similar issue in the Kitaev chain and Ising model: arXiv:1402.5262. Is there any way to deal with it in ITensor?

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Hi, so coincidentally I actually recently had a long discussion about this very topic with another user. Here is that thread: http://itensor.org/support/3729/symmetry-breaking-transverse-field-ising-model-kitaev-chain?show=3729#q3729

I think the key thing is that the symmetry breaking behavior you are seeing is first and foremost just how DMRG behaves when you work in the spin variables for this system. It's definitely not an ITensor-specific thing, but how any DMRG code would work, and indeed most numerical techniques, even exact diagonalization.

The simplest solution to see the entangled superposition ground states you are seeking is to work in the fermionic variables. Was there a reason you needed to work in the spin language versus fermion language for this system?

Miles

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Hi Miles,
There is no specific reason at the moment for using spin variable. To be honest, I am not familiar with using Fermion variable in ITensor dmrg. I would really appreciate if you can suggest some example code in ITensor which work with Fermion variable.  Also, If I am using Fermion variables, how do we specify whether they are Majorana fermion and how do we take in to account density-density interactions while building the Hamiltonian?

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I forgot it would have to be Majorana fermions if you want to simulate the model as written. Here are your two best options:

- I had forgotten, but a trick to get the entangled 'cat' state in the topological phase is to conserve Z2 parity of the spin variables. You'd have to work out the details, but basically you define conserve_szparity=true and conserve_sz=false when making the siteinds and then you may also need to work in a different operator basis (like rotated by 90 degrees) when defining the Hamiltonian terms. See the discussion linked above for more about this.

- to do Majorana fermions, you'd have to use the definitions a_{2j-1} = (c_j + c^\dagger_j)/2 and a_{2j} = i (c_j - c^\dagger_j)/2  which I might have slightly off by a minus sign to map the Majorana operators into regular c and c^\dagger operators, resulting in a more complicated Hamiltonian. Then you could input this fermion Hamiltonian into ITensor just like in our code examples for the extended Hubbard model but with a different pattern of hoppings and interactions.

Miles