constrained partition function Z = T$_{r}$ [ $_{e}$− H n =0) K B T ] computed with the unconstrained Hamil- tonian given in equation (22). As a result, the thermal occupation function << F | ( N$_{σ,L}$ − N$_{σ,R}$ ) | F >>$_{T}$ ≈ 1 $_{Z}$T$_{r}$ [( N$_{σ,L}$ − N$_{σ,R}$ ) $_{e}$− H ( n =0) K B T ] is given in terms of the thermal Fermi-Dirac occupation functions f$_{F.D.}$ ( ϵ$_{n}$ − µ$_{L}$ ) = 1 1+ e$^{(}$$^{ϵ}$$^{n}$$^{−}$$^{µ}$$^{L}$$^{)}$ K B T , f$_{F.D.}$ ( ϵ$_{n}$ − µ$_{R}$ ) = 1 1+ e$^{(}$$^{ϵ}$$^{n}$$^{−}$$^{µ}$$^{R}$$^{)}$ K B T expressed in terms of the single particle spectrum ϵ$_{n}$ = hv L ( n − 1 $_{2}$), n=1,2,3... . [32, 33]: In order to compute the static current caused by the static potential V , we have to perform a space average which singles out the zero mode current. This can be achieved if we perform a space average over the length L$_{T,L}$ . After performing the space average, we see that on a length scale L$_{T}$ that particle-hole current ˆ I ( n =0) e vanishes. This result is seen from the the result V (+) eff. + V ( − ) eff. = 0 given in equation (38). Therefore, we find ( see equation (38)) that the thermal expectation value of the current is given by zero mode current operator << F | ˆ I ( n =0) e | F >>$_{T}$ . Using the length L$_{T,L}$ = smallest [ L, L$_{T}$ ], we perform a space and thermodynamic average ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ << F | ˆ I$_{e}$ | F >>$_{T}$ and find: 16 ∑ σ = ↑ , ↓ << F | N$_{σ,L}$ − N$_{σ,R}$ | F >>$_{T}$ ≈ 1 Z T$_{r}$ [( N$_{σ,L}$ − N$_{σ,R}$ ) $_{e}$− H ( n =0) K B T ] ≈ 2$^{n}$$^{=}$ ∞ ∑ n =1 [ f$_{F.D.}$ ( ϵ$_{n}$ − µ$_{L}$ )] − f$_{F.D.}$ ( ϵ$_{n}$ − µ$_{R}$ )]) ≈ L L T > 1 eV h ( 2 L v ) (37) I$_{e}$ = ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ << F | ˆ I$_{e}$ | F >>$_{T}$ = ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ << F | ˆ I ( n =0) e | F >>$_{T}$ + ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ << F | ˆ I ( n =0) e | F >>$_{T}$ = ( v v$_{eff.}$ )( e $^{2}$n$_{e}$ h )( 2 πe κ ) ∫ L T 2 0 dx (1 − h(x,d) 2 ) 2 ∫ L 2 − L 2 dy 1 2 ∂$_{x}$ [ h(y,d) √ ( x − y ) 2 + a 2 − h(y,d) √ ( x − y ) 2 + ξ $_{2}$] +( v v$_{eff.}$ )( e $^{2}$n$_{e}$ h )( 2 πe κ ) ∫ 0 − L T,L 2 dx (1 − h(x,d) 2 ) 2 ∫ L 2 − L 2 dy 1 2 ∂$_{x}$ [ h(y,d) √ ( x − y ) 2 + a 2 − h(y,d) √ ( x − y ) 2 + ξ $_{2}$] + ev L ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ [1 − h ( x, d ) 2 ] ∑ σ = ↑ , ↓ << F | N$_{σ,L}$ − N$_{σ,R}$ | F >>$_{T}$ = 2 e 2 h ( V (+) eff. + V ( − ) $_{eff.}$) + [1 − ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ h ( x, d ) 2 ](2 e 2 h V ) = [1 − ∫ L T,L 2 − L T,L 2 dx L$_{T,L}$ h ( x, d ) 2 ](2 e 2 h V ) = (1 − 0 . 5 d L$_{T,L}$ )(2 e 2 h ) V (38)