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angelariva authored May 2, 2024
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Expand Up @@ -29,7 +29,7 @@ where the $J_{i,n}$ coefficients are the couplings between the chain sites and t

## Double chain mapping

![image](doublechain.pdf)
![image](doublechain.png)

In the double chain geometry of Fig. \ref{subfig:double_ferm}, the MPOs bond dimension is: $\chi = 4$. The MPO has the structure defined in Eq. \ref{eq:mpo} and therefore can be seen as the product of the following matrices:
```math
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## Interleaved chain mapping

![image](foldedchain.png)

The drawback of the double chain representation is that the particle-hole pairs are spatially separated in the MPS, creating correlations and therefore leading to a dramatic increase in the bond dimensions. This is why Kohn and Santoro propose an interleaved geometry, the advantages of which are thoroughly explained in [^khon_eff_2022]. Exploiting the interleaved representation, the interaction comes to be between next-nearest neighbors: a string operator appears in the Jordan-Wigner transformation from bosons to fermions:
```math
\hat a_{i}^\dagger \hat a_{i+2} + \hat a_{i+2}^\dagger \hat a_{i} = \hat b_{i}^\dagger \hat F_{i+1} \hat b_{i+2} + \hat b_{i} \hat F_{i+1} \hat b_{i+2}^\dagger,
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