Phonon-Mediated Casimir Interaction between Mobile Impurities in One-Dimensional Quantum Liquids

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 $1/r^3$, much slower than the van der Waals interaction $\sim 1/r^6$, 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.

Figure 1

Casimir interaction between mobile impurities in a quantum liquid. (a) Schematic depiction of two impurities experiencing an induced Casimir attraction mediated by phonon fluctuations of a one-dimensional quantum liquid. (b) Single-phonon exchange (wavy line) leading to a spatially local inter-impurity interaction (impurities $i$, $j$ are denoted by straight lines with coordinates $X_{i,j}$). (c) Two-phonon exchange responsible for the long-range Casimir interaction between impurities (see Eqs. (3) and (4) for the corresponding Lagrangian).