# Local parity operator expansion for electron site parity operator **URL:** \ **Category:** ITensor Julia Questions\ **Created:** [October 3, 2023, 5:53pm UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181 "2023-10-03T17:53:12Z")\ **Posts on this page:** 6\ **Page:** 1
**Author:** ![oroszl](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/oroszl/32/68_2.png) [@oroszl](https://itensor.discourse.group/u/oroszl)\ **Post date:** [October 3, 2023, 5:53pm UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/1 "2023-10-03T17:53:12Z")
Continuing the discussion from [Issues with Nf parity conservation](https://itensor.discourse.group/t/issues-with-nf-parity-conservation/1176): Dear Miles, I have a comment and a question regarding @vernek 's question on parity conservation. Am I correct that in Edson’s case the expectation value of the total parity operator e.g. \displaystyle (-1)^{N}=\prod\_{m,\sigma}(1-2n\_{m,\sigma}) should capture the appropriate parity eigenvalue ? If so what is the simplest way of implementing it? I spent a couple of hours with a sanity check regarding parties in a related problem e.g. I also have superconductivity so parity is the only conserved quantity. In the long run, I want to calculate parity content associated with a certain part of my system, so a nice operator built up of local parts would be nice. I have came up, based on the documentation, the following code snippet: ```julia s = siteinds(PSI) PSI0=PSI for i in 1:Length # this is (1-2n_up)(1-2n_dn)=(1-2n_tot-4Nupnd) O=op("Id", s[i])-2*op("Ntot",s[i])+4*op("Nupdn",s[i]) psi=O*PSI0[i] noprime!(psi) PSI0[i]=psi end inner(PSI,PSI0) ``` it seems however that the last inner product is always 1. independent of the actual parity sector my original wavefunction PSI0 is located. ---
**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\ **Post date:** [October 3, 2023, 6:16pm UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/2 "2023-10-03T18:16:17Z")
Thanks for the question. Some simple ways of implementing a product operator of the form you wrote, so some operator of the form W = \prod\_j O\_j, would be: 1. making a copy of your original MPS, then looping over each site of the copied MPS and applying each O\_j onto the j’th MPS tensor, then using `inner` to overlap this modified MPS with your original MPS 2. putting each O\_j into an MPO, i.e. making each MPO tensor equal O\_j, then using the `inner(psi,W,psi)` to compute its expected value ---
**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\ **Post date:** [October 3, 2023, 6:18pm UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/3 "2023-10-03T18:18:49Z")
Also, for what it’s worth, you can be sure that if you are using conserved quantum numbers which either explicitly conserve parity, or something that includes parity like total particle number, then if you call `flux(psi)` on your MPS `psi`, then it will accurately return the total quantum number flux (total QN) of your MPS. If that flux is in the parity sector you want, that the overall MPS is also guaranteed to be in that parity sector. ---
**Author:** ![oroszl](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/oroszl/32/68_2.png) [@oroszl](https://itensor.discourse.group/u/oroszl)\ **Post date:** [October 3, 2023, 7:55pm UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/4 "2023-10-03T19:55:46Z")
Thanks, Miles for the prompt answer! I am aware that I can check the total parity with `flux(psi)`, but my goal is to get the parity of two disjoint subsystems. This might not be well defined in general, but I have good reason to believe that for my special case, it is going to deliver a proper parity eigenvalue, e.g. \pm 1. Regarding your suggestion in your first answer, I believe that my code included in the original post does just that, yet it does not give me the appropriate answer. I prepared a more complete example below using a random state with a fixed parity ```auto using ITensors N=10 # make this even just for the sake of concreteness @time sites = siteinds("Electron", N; conserve_nfparity = true, conserve_sz = false, conserve_nf = false); state = ["Up" for i in 1:N] state[1]="Emp" # odd state if this line is commented even state can be checked psi0 = randomMPS(sites, state, 5) @show flux(psi0) # this shows that the flux is odd PSI=psi0 s = siteinds(PSI) for i in 1:N # this is (1-2n_up)(1-2n_dn)=(1-2n_tot-4Nupnd) O=op("Id", s[i])-2*op("Ntot",s[i])+4*op("Nupdn",s[i]) psi=O*PSI[i] noprime!(psi) PSI[i]=psi end inner(PSI,psi0) # this should be -1 for odd and +1 for even ``` ---
**Author:** ![oroszl](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/oroszl/32/68_2.png) [@oroszl](https://itensor.discourse.group/u/oroszl)\ **Post date:** [October 4, 2023, 7:51am UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/5 "2023-10-04T07:51:38Z")
I found my bug. Instead of `PSI=psi0` I needed `PSI=copy(psi0)`. I guess this solves my Issue! Thanks for the suggestions Miles! ---
**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\ **Post date:** [October 4, 2023, 11:24am UTC](https://itensor.discourse.group/t/local-parity-operator-expansion-for-electron-site-parity-operator/1181/6 "2023-10-04T11:24:25Z")
Glad you found the bug and the code is working now.