1
$\begingroup$

Can you solve analytically the following model spinless Fermi-Hubbard model:

$\hat H = -t \displaystyle\sum_i (c^+_ic_{i+1} + h.c.) +V\sum_{\langle i,j\rangle} n_in_j$

If yes, how do you proceed? and is there also an exact solution in higher dimensions than 1D? Are there any references?

$\endgroup$

1 Answer 1

3
$\begingroup$

In 1D, using Jordan-Wigner transformation it is shown that the above model can be mapped to Heisenberg XXZ model: $$H=\sum_{j=1}^{N}J_{x}\left(\sigma_{j}^{x} \sigma_{j+1}^{x}+\sigma_{j}^{y} \sigma_{j+1}^{y}\right)+J_{z} \sigma_{j}^{z} \sigma_{j+1}^{z}-B \sigma_{j}^{z}$$ where the $\sigma^\alpha$ are Pauli matrices and $J_{x}=-t / 2, J_{z}=V / 4, B=-V/2$. This has a Bethe ansatz solution, see for example arXiv:hep-th/9605187.

Of course, both JW transformation and Bethe-ansatz work only (except in some tricky cases) for 1D. To the best of my knowledge, I don't know any higher than 1D solution.

$\endgroup$
2
  • $\begingroup$ thank you for your answer! does one also have to use the jordan wigner transform in the non-interacting case? (V=0) $\endgroup$
    – relaxon
    Nov 19, 2021 at 17:25
  • 2
    $\begingroup$ @relaxon no. This is a quadratic form, i.e., free fermion on the lattice. It has a simple close-form solution in momentum space. $\endgroup$ Nov 20, 2021 at 6:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.