The nature of the order as $T\ensuremath{\rightarrow}0$ in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer (
g2 and
g4) and large-momentum-transfer (
g1 and
g3) electron-electron interactions, focusing upon the particular value of
g1 for which Luther and Emery have recently found an exact solution. For this value of ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, we calculate the low-frequency behavior of the zero-temperature response functions explicitly. For the singlet superconducting and charge-density-wave responses, our results are consistent with those of Luther and Emery, implying that these response functions are divergent as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for certain values of
g2. For the triplet superconducting and spin-density-wave responses, however, our results differ from those of Luther and Emery, as we show explicitly that these responses have a gap for low frequencies, as was predicted by Lee. Furthermore, they do not diverge at the gap edge for any value of
g2. We also consider the effect of interactions between electrons on the same side of the Fermi "surface," and find that the low-momentum-transfer process of that type (
g4) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" (
g3), we may solve the problem exactly for
g2=0 and ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, and find that the ground state of the system exhibits only long-range charge-density-wave order. By a mapping onto the classical two-dimensional Coulomb gas problem, we may extend those results for ${g}_{3}\ensuremath{\ne}0$ to the region
g1<0 and ${g}_{1}\ensuremath{-}2{g}_{2}<|{g}_{3}|$.
A model of one-dimensional fermions characterized by a large momentum-transfer interaction
g1 and a small momentum-transfer interaction
g2 is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of
g1 dominates the behavior of the system. For
g1>0, ${g}_{1}\ensuremath{-}2{g}_{2}>0$ the triplet Cooper response diverges as $T\ensuremath{\rightarrow}0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for
g1>0 and diverges to zero for
g1<0. The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.
An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.
One-dimensional (1D) interacting Fermi gases underpin a rich variety of quantum liquids, ranging from the Tomonaga-Luttinger liquid and the Fulde-Ferrell-Larkin-Ovchinnikov paired state to the Luther-Emery liquid (LEL). Notably, the LEL-characterized by gapless excitations in charge or spin degree of freedom and gapped excitations in the other-has long been theoretically predicted but has eluded definitive experimental observation to date. By exactly solving the attractive Yang-Gaudin model via the Bethe ansatz, we fully determine the low-energy excitation spectra for both spin and charge channels. For fully paired fermions, the spectrum features gapped spin excitation coexisting with gapless charge excitation. We further employ bosonization to derive analytical expressions for the dynamical structure factor (DSF) of the gapped spin excitation, which is a long-standing challenge in this study. In the trapped gas, we find that the spin Bragg spectroscopy peak occurs at a much higher frequency than the charge-mode peak, where the splitting frequency between these two peaks scales approximately linearly with the interaction strength. Crucially, the spin gap manifests as a sharp threshold in the spin DSF, below which spectral weight is completely absent, proving a direct and unambiguous spectroscopic fingerprint of the LEL. Our work establishes quasi-1D trapped attractive Fermi gases as a pristine experimental platform for the long-sought direct observation of the LEL.
A slightly corrected version of the 'boson representation' of fermions in the Luttinger model (that avoids the alpha to 0 limit) is used to obtain explicitly the lowest-order interactions between charge- and spin-density waves in the generalised model that includes backscattering. This is in contrast to the decoupling of these degrees of freedom reported by Luther and Emery. Instead, the results obtained by these authors are identified as applying to a model with long-range coupling between spin densities at different points.
The energy spectrum of the one-dimensional electron gas, solved by Luther and Emery for a particular attractive interaction strength
U of the backward scattering across the Fermi surface, has been calculated for small deviations from this particular limit. The energy spectrum shows a gap at the Fermi energy, whose magnitude is very sensitive to the interaction
U. The density of states shows a gap at zero temperature, which is smeared out by the quasiparticle lifetime for $T\ensuremath{\ne}0$.