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.
The concept of a Luttinger liquid in 1D is extended to include twisted boundary conditions on a ring and mesoscopic parity effects due to evenness and oddness of the particle number
N0. Using Haldane's notion of topological excitations, a proof of Leggett's conjecture is presented, stating that the ground state of interacting 1D quantum systems is diamagnetic or paramagnetic depending on the parity of
N0. The persistent currents produced by an Aharonov-Bohm flux are calculated and shown to have period and amplitude that are in agreement with recent experiments.
We consider a ring of strongly interacting electrons connected to two external leads by tunnel junctions. By studying the positions of conductance resonances as a function of gate voltage and magnetic flux, the interaction parameter
g can be determined experimentally. For a finite ring the minimum conductance is strongly influenced by device geometry and electron-electron interactions. In particular, if the tunnel junctions are close to one another, the interaction-related orthogonality catastrophe is suppressed and the valley current is unexpectedly large.
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.
We consider spinless electrons in two dimensions with the bare spectrum ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{p}}$=
vF(\ensuremath{\Vert}
px\ensuremath{\Vert}+\ensuremath{\Vert}
py\ensuremath{\Vert}). In momentum space, the interactions among electrons have a finite range
q0, which is small compared to the Fermi momentum. A golden-rule calculation of the electron lifetime indictes a breakdown of Landau's Fermi-liquid theory in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.