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?


1 Answer 1


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.

  • $\begingroup$ 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

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