CritPt / Challenge_12_main / Consider Z_(N) (N is any integer) parafermion zero-mode operators on four…

# Problem setup:
Consider Z_(N) (N is any integer) parafermion zero-mode operators on four sites α_(i) with i=1,2,3,4 and a four-stage tunneling process described by

H₃₄→H₂₃→H₁₂→H₁₃→H₃₄,

where
H_(ij)=t(e^(−iϕ_(ij)/N)α_(i)^(†)α_(j)+H.c.), ϕ_(ij) is a phase between sites i and j, and H.c. stands for Hermitian conjugate. This tunneling process is followed by a permutation between zero modes (α₁,α₂) and (α₃,α₄), respectively.

# Main problem:
Find the phase between the initial ground state of the process |ψ^(i)(q)⟩ and final ground state of the process |ψ^(f)(q)⟩, where q is the fusion channel between the unpaired zero modes. Your answer should depend on k_(ij)∈Z_(N), where k_(ij)<−(ϕ_(ij))/(2π)<k_(ij)+1. Note that, for a Josephson phase between sites i and j, k_ij is a ground-state fusion channel.

# Problem setup:
Consider $Z_N$ ($N$ is any integer) parafermion zero-mode operators on four sites $\alpha_i$ with $i=1,2,3,4$ and a four-stage tunneling process described by

$
H_{34}\rightarrow H_{23} \rightarrow H_{12}\rightarrow H_{13}\rightarrow H_{34},
$

where
$
H_{ij}=t\left(e^{-i\phi_{ij}/N}\alpha_i^\dagger\alpha_j+H.c.\right)
$, $\phi_{ij}$ is a phase between sites $i$ and $j$, and $H.c.$ stands for Hermitian conjugate. This tunneling process is followed by a permutation between zero modes $(\alpha_1,\alpha_2)$ and $(\alpha_3,\alpha_4)$, respectively.

# Main problem:
Find the phase between the initial ground state of the process $|\psi^i(q)\rangle$ and final ground state of the process $|\psi^f(q)\rangle$, where $q$ is the fusion channel between the unpaired zero modes. Your answer should depend on $k_{ij}\in Z_N$, where $k_{ij}<-\frac{\phi_{ij}}{2\pi}<k_{ij}+1$. Note that, for a Josephson phase between sites i and j, k_ij is a ground-state fusion channel.