The persistent current
(I) of the spinless-electron Luttinger liquid is calculated by numerical microscopic methods in the continuous one-dimensional ring containing a weak link with transmission amplitude ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}⪡1$. The electrons interact via a screened interaction of range
d. If $d\ensuremath{\lesssim}1/2{k}_{F}$, the interaction modifies the transmission as ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{N}^{\ensuremath{-}\ensuremath{\alpha}}$ and the current as $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where
N is the electron number,
L the ring length,
N/L a fixed density, and $\ensuremath{\alpha}$ depends on the interaction in accord with the renormalization-group (RG) theory. Unlike our results, the RG theory predicts for a finite
d the power laws ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ and $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$, both depending on two interaction parameters,
d and $\ensuremath{\alpha}$. To explain why our results depend solely on $\ensuremath{\alpha}$, we show analytically that the interaction-matrix elements depend only on $\ensuremath{\alpha}$ (not on
d) when $d\ensuremath{\lesssim}1/2{k}_{F}$. As
d exceeds
1/2kF, our result for
LI/evF starts to depend on
d as well and likely approaches the result $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ in the limit
L⪢d⪢1/2kF, not feasible by our numerical methods.
We study the Josephson effect through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. The interactions are treated in the framework of the Luttinger model. At zero temperature, the Josephson critical current decays algebraically with increasing distance between the junctions. In a Luttinger liquid of finite size, the Josephson current depends on the total number of electrons modulo 4. These parity effects are studied for a ring, coupled capacitively to a gate voltage and threaded by a magnetic flux. Interactions introduce qualitatively new features compared to the noninteracting case.
The quantized conductance was produced due to dissipation within the Luttinger liquid from the parity inversion and time reversal symmetric interaction. For unitary time with Hermitian Luttinger liquid Hamiltonian and produced nonequilibrium conductivity and was renormalized due to interaction. We have studied the nonequilibrium dynamics and transport of parity inversion symmetric Luttinger liquid and the model was quenched from domain wall. The study was extended to initial ground state of a Hermitian interacting Luttinger liquid. We have used the model using the time evolving block decimation and have found the conductance inside the light core and depended on post-quench Hamiltonian and was quantized when the system was quenched to the non Hermitian Hamiltonian. The conductance was found when the system was quenched from an interacting initial state and showed that the initial state had no effect on conductance. The strength of the initial or final interactions in the case of Hermitian dynamics showed the quenched displayed a conductance that remained universal.