Dispersion relation and spectral function of an impurity in a one-dimensional quantum liquid

We consider the motion of an impurity particle in a general one-dimensional quantum fluid at zero temperature. The dispersion relation $\Omega \left(P\right)$ of the impurity is strongly affected by interactions with the fluid as the momentum approaches $\pm \pi \hslash n,\pm 3\pi \hslash n,\dots$, where $n$ is the density. This behavior is caused by singular $\pm 2\pi \hslash n$ scattering processes and can be understood by analogy to the Kondo effect, both at strong and weak couplings, with the possibility of a quantum phase transition where ${\Omega }^{\prime }\left(\pm \pi n\right)$ jumps to zero with increasing coupling. The low energy singularities in the impurity spectral function can be understood on the same footing.

Figure 1

Schematic dispersion relation of an impurity moving in $K>1$ Luttinger liquid. At weak coupling a cusp persists at $P=\pi n$ but vanishes discontinuously at some critical coupling. In the strong-coupling limit the dispersion takes the simple form $\Omega \left(P\right)\propto {\text{sin}}^{2}P/2n$. $\Omega \left(P\right)$ is the ground-state energy of the system at given $P$: the singular form of the impurity spectral function near this threshold is indicated in the inset. The dashed curve illustrates the ground state in the absence of an impurity for reference.

Figure 2

Dispersion relation (top) and spectral function exponent (bottom) for the exactly soluble limit of an impurity of mass $M=m$ moving in a Fermi gas, corresponding to Luttinger parameter $K=1$. Results are given for $\gamma \equiv mu/{\hslash }^{2}n=0.01(○)$, 0.2 (◻), 0.5 (△), 1 (◇), and $2(\star )$. The dashed curve gives the exponent $\alpha \left(P\right)=\frac{1}{2}\left[1+{\left(\frac{P}{\pi n}\right)}^2\right]$ valid in the $\gamma \to \infty$ limit.