phase between the neibouring site

The formula (128) can be used to calculate the current of the link $ij$ , $J_L(t)_{ij}$,

$$J_L(t) = \sum_{i i} J_L(t)_{ij}$$ (267)

$$J_L(t)_{ij} = 2e \; {\rm Re} \left[ t_{i j} G_{j i}^<(t,t) \right]$$ (268)

The wavefunction, $\Psi$, is solved in the time-independent non-equilibrium state. The charge at $(r,E)$ is calculated as

$$n(r,E) = \Psi^*(r,E) \Psi(r,E)$$ (269)

while

$$n(r,E) = \frac{1}{2\pi i} G^<(r,E)$$ (270)

The total charge is $n(r) = \int dE\; n(r,E)$. These reads that $\Psi(r,E)^*\Psi(r,E)$ is associated with $1/(2\pi i)\; G^<(r,E)$.[9]

Then $J_{L,ij}$ is [8]

$$J_{L,ij} = 2e \;{\rm Re} \; \frac{1}{2\pi} \int\! dE G^<(r,E)$$ (271)

$$= 2e \;{\rm Re} \; \frac{1}{2\pi} \int\! dE \left[ t_{ij} (2\pi i) \langle j\vert E\rangle \langle E \vert i\rangle \right]$$ (272)

$$= 2e \; {\rm Im} \; \int\! dE \left[ t_{i j} \langle j \vert E \rangle \langle E\vert i\rangle \right]$$ (273)

$$= 2e \; {\rm Im} \; \int\! dE \left[ \langle E\vert i\rangle t_{i j} \langle j\vert E \rangle \right]$$ (274)

When $\Psi(E) = \sum_i a_{Ei} \phi_i$, $J_{L,ij} = 2e \int\!dE t_{i j} \vert a_{Ei}\vert\vert a_{Ej}\vert \sin(\theta_{Ei}-\theta_{Ej})$, with the phase factor $\theta_{Ei}$ of $a_{Ei}$. This means that no current flows if the phase difference between the site $i,j$ is 0 or $\pi$.

Compare it with another definition of the current density

$$j(r) = (1/m^*) \; {\rm Im} \; \left[\psi^+(r) \nabla \psi(r) \right]$$ (275)

$\langle E\vert i\rangle$ in eq.(276) is weight at site $i$ of the wavefunction $\psi(E)$, while $\psi(r)$ is weight at $r$ of the wavefunction. So eq.(276) is similar to eq.(277). (Of course!) if $\psi(r)=\exp(i k r)$, $j(r)=k/m^*$. In this case, $\psi(r+dr)=\exp(i k r)\exp(ik\; dr)$. Then $\theta=k\;dr$.