We discuss the effective interactions between two localized perturbations in one-dimensional quantum liquids. For noninteracting fermions, the interactions exhibit Friedel oscillations, giving rise to a Ruderman-Kittel-Kasuya-Yosida-type interaction familiar from impurity spins in metals. In the interacting case, at low energies, a Luttinger-liquid description applies. In the case of repulsive fermions, the Friedel oscillations of the interacting system are replaced, at long distances, by a universal Casimir-type interaction which depends only on the sound velocity and decays inversely with the separation. The Casimir-type interaction between localized perturbations embedded in a fermionic environment gives rise to a long-range coupling between quantum dots in ultracold Fermi gases, opening an alternative to couple qubits with neutral atoms. We also briefly discuss the case of bosonic quantum liquids in which the interaction between weak impurities turns out to be short ranged, decaying exponentially on the scale of the healing length.
Virtual phonons of a quantum liquid scatter off impurities and mediate a long-range interaction, analogous to the Casimir effect. In one dimension the effect is universal and the induced interaction decays as 1/r3, much slower than the van der Waals interaction ∼1/r6, where r is the impurity separation. The sign of the effect is characterized by the product of impurity-phonon scattering amplitudes, which take a universal form and have been seen to vanish for several integrable impurity models. Thus, if the impurity parameters can be independently tuned to lie on opposite sides of such integrable points, one can observe an attractive interaction turned into a repulsive one.
We compute an analog Casimir effect in a one-dimensional spinless Luttinger liquid confined to a segment in the presence of a nearly impenetrable partition dividing the segment into two compartments. The Casimir interaction is found to be a bounded piecewise-continuous oscillatory function whose maxima are points of force discontinuity and correspond to resonant tunneling across the partition. The well-known regularization-based results are reproduced by the lower envelope of this function, which corresponds to an approximation that ignores the rather large oscillations due to particle discreteness. These macroscopic conclusions are tested and confirmed via a rigorous analysis of the Casimir effect in an exactly-solvable model of a one-dimensional nonrelativistic spinless gas of free fermions. Additionally we confirm the result of a recent calculation which employed an effective low-energy theory with a cutoff to find the Casimir interaction between two strong well-separated impurities placed in a Luttinger liquid.
The Casimir effect for two parallel slabs immersed in an ideal Fermi sea is investigated at both zero and nonzero temperatures. It is found that the Casimir effect in a Fermi gas is distinctly different from that in an electromagnetic field or a massive Bose gas. In contrast to the familiar result that the Casimir force decreases monotonically with the increase of the separation L between two slabs in an electromagnetic field and a massive Bose gas, the Casimir force in a Fermi gas oscillates as a function of L. The Casimir force can be either attractive or repulsive, depending sensitively on the magnitude of L. In addition, it is found that the amplitude of the Casimir force in a Fermi gas decreases with the increase of the temperature, which also is contrary to the case in a Bose gas, since the bosonic Casimir force increases linearly with the increase of the temperature in the region T < T c , where T c is the critical temperature of the Bose—Einstein condensation.
We investigate the indirect interaction between two isolated impurities in a Luttinger liquid described by a microscopic lattice model. To treat the electron-electron interaction
U, the functional renormalization group method is used. For comparison, we also study the
U=0 case. We find that for a wide range of impurity parameters the impurity interaction
V12 as a function of their separation
r oscillates with decaying amplitude between being attractive and repulsive. For half-filling of the band and in a crossover regime between weak and strong impurities, the interaction becomes purely attractive. For
U=0 and independent of the impurity strength, the amplitude of the interaction energy falls off as
1∕r. For
U>0, the decay for small separations and weak to intermediate impurities is governed by a
U dependent exponent larger than $\ensuremath{-}1$, which crosses over to $\ensuremath{-}1$ for large
r. The crossover scale depends on the impurity strength and
U. We present simple pictures which explain our results in the limits of weak and strong impurities. We finally also consider attractive interactions
U<0.