## Mapping of Spinful Fermion to Spinless fermion

General discussions on the ideas behind the algorithm.
Ricky_Pang
Posts: 3
Joined: 16 May 2021, 17:15

### Mapping of Spinful Fermion to Spinless fermion

I am writing to ask the physics behind the mapping between spinful fermion and spineless fermion in https://tenpy.readthedocs.io/en/latest/ ... igner.html. I want to know what is the meaning of " spin-1/2 fermions on a chain as spinless fermions living on a ladder." In this algorithm, we still need to use the spinful operators ["Cu", "Cd", "Cdu", "Cdd"] to construct the MPO. If the MPO is constructed by spinful operators, why it can map spinful fermions system to spineless fermion system? I would appreciate for any response.
Ricky_Pang
Posts: 3
Joined: 16 May 2021, 17:15

### Re: Mapping of Spinful Fermion to Spinless fermion

Hi all. I have solved my problem. To understand this algorithm, we need to look back the Hilbert space on both chain and spin ladder. The general idea is that the local Hilbert space of a site on spin chain is H_{i} = V_{i up} \otimes V_{i down}, where the V_{i up} is the Hilbert space of spin up fermion and V_{i down} for spin down. When we map this spin chain to spin ladder, the local site Hamiltonian will become H_{i', i'+1} = V_{i' up} \otimes V_{i'+1, down}. Therefore, when we impose an ordering of the sites on ladder, we implicitly include the information of spin of each site. For instance, spin up particle correspond to even sites on spin ladder and spin down particles correspond to odd sites. Now, in each site on the ladder, they are just spinless fermions. This is the essence of the ladder mapping.
Johannes
Posts: 316
Joined: 21 Jul 2018, 12:52
Location: UC Berkeley

### Re: Mapping of Spinful Fermion to Spinless fermion

Yes, that's right.
I tried to explain this in the section "spinful fermions" of the Jordan-Wigner user guide you referenced.
If you have a suggestion how to formulate this a bit clearer, feel free to submit a corresponding pull request on github!
Ricky_Pang
Posts: 3
Joined: 16 May 2021, 17:15

### Re: Mapping of Spinful Fermion to Spinless fermion

Thank you for your comment. I have a detailed explanation regarding the spin ladder mapping on stackecxahnge
( https://physics.stackexchange.com/a/637545/246547 ). If this explanation is correct, I will submit in to the corresponding pull request.