content_image/363371.jpg content_image/363370.jpg content_image/363369.jpg content_image/363368.jpg
Contrary to the non-interacting wire where both the charge and spin currents are con- served, the constraints in the wire allow only for the conservation of the electric current ˆ I$_{e}$ . Due to the fact that the commutation equations and the conservation laws for the currents in the leads and in the wire are not the same, we have at the interface between the wire and the leads incoherent scattering. (Electrons with a well defined spin (spin up or spin down), are injected from the reservoirs. The leads-wire interface randomizes the spin and causes the electrons to behave as spinless particles at finite temperatures.) 

## 6. The current at finite temperatures for a finite wire L >> d and L > L$_{T}$ 

In this section we consider explicitly the limit of U → ∞ taken at finite temperature T . In this limit, spin exchange processes are suppressed and the only allowed processes are the particles-holes (non-zero modes ) and particles excitations (zero modes). Due to the fact that the system is finite, we can separate the Hamiltonian, and the Bosonic fields into the periodic non-zero modes Bosons ϑ ( n ̸ =0) R,σ ( x ) ≡ ˆ ϑ$_{R,σ}$ ( x ), ϑ ( n ̸ =0) L,σ ( x ) ≡ ˆ ϑ$_{L,σ}$ ( x ) and the zero mode Bosons ϑ ( n =0) R,σ ( x ) ≡ x √ π L N$_{R,σ}$ , ϑ ( n =0) L,σ ( x ) ≡ x √ π L N$_{L,σ}$ where N$_{R,σ}$ N$_{L,σ}$ measure the added charges with respect the Fermi energy [32, 33]. 

Using the Boson decomposition we find the representation: 

<img src='content_image/363371.jpg'>

Similarly we decompose the Hamiltonian into two parts: H ( n ̸ =0) wire + H ( n ̸ =0) leads = ∑ σ = ↑ , ↓ ∫ L 2 − L 2 dx ¯ hv 2 [( ∂$_{x}$ ˆ ϕ$_{σ}$ ( x )) 2 + ( ∂$_{x}$ ˆ ϑ$_{σ}$ ( x )) $^{2}$] ( n =0) ( n =0) 

and the zero mode part H wire + H leads : 

<img src='content_image/363370.jpg'>

For a constant external voltage, V = µ$_{R}$ ( x ) − µ$_{L}$ ( x ) e we have only the zero mode reservoir Hamiltonian: 

<img src='content_image/363369.jpg'>

Since the commutator is space dependent, we will introduce a space dependent electrostatic potential δv ( x ) = µ$_{R}$ ( x )+ µ$_{L}$ ( x ) − e which will couple to the non-zero mode densities. The effect of the electrostatic potential δv ( x ) is introduced using the non-zero mode reservoir : 

<img src='content_image/363368.jpg'>

