# Pertubation Theory

For long-range barrier tunneling, consider three qubits governed by the Hamiltonian $$H$$. $$H = -J(\sigma_1^+\sigma_2^-+\sigma_1^-\sigma_2^++\sigma_3^+\sigma_2^-+\sigma_3^-\sigma_2^+)+\frac{U}2\sigma^z_2, \ \text{for} \ J \ll U$$ How do we modify it so that there are $$L$$ high-energy sites between two endpoints such as ($$L+2L$$ sites total)? How can we show that the effective Hamiltonian between the two endpoints is an exchange term with leading order coefficient such as $$-J^{L+1}/U^L$$?

• $L=2L$ sites total What does $L=2L$ mean? Oct 24 '20 at 4:49
• Split the Hamiltonian into two pieces, $\hat{H}_{0}$ and $\hat{H}'$ with the former containing a term proportional to $U$, then apply a Schrieffer-Wolf transformation and identify the first and second-order perturbative terms which will be proportional to $J/U$ and $J^{2}/U$ respectively. Neglect the first-order term since they take the fermions to a higher Hubbard sector. Thus, the effective Hamiltonian for the lowest Hubbard sector will have $\hat{H}_{0}$ and $\hat{H}_{eff}^{2}\propto J^{2}/U,\ \ldots\ \hat{H}_{eff}^{n}\propto J^{n+1}/U^{n}$ that are the terms which allow virtual excursions. Oct 24 '20 at 10:28
• About the L +2L ? Oct 24 '20 at 21:12