# How to deal with fermionic operators in density matrix renormalization group (DMRG)?

Let we have 1D Hubbard model with spinless fermions $$H = -t\sum_i^{L-1} \big(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\big) +V\sum_i^{L-1} n_i n_{i+1}$$ Though this model can be mapped onto XXZ Heisenberg model using Jordan-Wigner transformation, I want to solve it in this creation and annihilation operators formalism using DMRG because I will be using this concept for more complicated calculations in future.

I have one thing to ask:

How to write fermionic operators ($$c_i^\dagger, c_i)$$ in matrix form for Hubbard model, especially, spinless Hubbard model?

Note: I want to do DMRG in traditional formalism instead of MPS formalism.

• The Jordan-Wigner transformation is the matrix representation of fermionic operators. So it's not clear what kind of answer you want here. Of course one can probably devise alternative matrix representations, but they will either be unitarily equivalent to the standard Jordan-Wigner or will have a higher local dimension which is undesirable for DMRG. Jan 29, 2019 at 11:02
• @MarkMitchison actually I got the answer of this question from your answer on my other question. I am trying to learn DMRG on my own. Unfortunately, I do not have supervision of any professor or specialist. I was able to solve spin systems using DMRG but when it comes to fermionic systems, my naive understanding does not help. Moreover, I could not find any introductory reference to DMRG for fermions. Do you happen to know any introductory book for fermionic DMRG? Thank you for reading. Jan 29, 2019 at 13:59
• Well, to numerically simulate fermions using DMRG, you first need to represent fermionic operators in a tensor product vector space. To do this, you use Jordan-Wigner to map onto a spin system. So DMRG for spin systems also applies to fermions. The only caveat is that local interactions in the fermion picture do not necessarily map onto local interactions for the Jordan-Wigner strings. But for fermions in 1D with nearest-neighbour interactions there should be no problem. Jan 29, 2019 at 22:57
• @MarkMitchison Very useful comment. And it worked, well at least, it worked for non-interacting part. I am wondering why $V\neq 0$ is wrong. Can you please say few lines about how to handle filling density in DMRG i.e. let's we want to study half-filling case. We add 2 sites in every iteration, how do we set # of particles in each iteration step? Feb 1, 2019 at 11:38
• DMRG is basically a minimisation problem. So to minimise energy at finite density you just minimise $H-\mu N$, where $\mu$ is a Lagrange multiplier (the chemical potential) and $N= \sum_k c_k^\dagger c_k$ is the number operator (which also has a nice Jordan-Wigner representation). Feb 1, 2019 at 15:33