Fractional Angular Momentum and Anyon Statistics of Impurities in Laughlin Liquids

The elementary excitations of a fractional quantum Hall liquid are quasiparticles or quasiholes that are neither bosons nor fermions, but are so-called anyons. Here we study impurity particles immersed in a quantum Hall liquid that bind to the quasiholes via repulsive interactions with the liquid. We show that the angular momentum of an impurity is given by the multiple of a fractional “quantum” of angular momentum, and can directly be observed from the impurity density. In a system with several impurities bound to quasiholes, their total angular momentum interpolates between the values for free fermions and for free bosons. This interpolation is characterized by the fractional statistical parameter of the anyons, which is typically defined via their braiding behavior.

Figure 1

We plot the angular momentum $⟨{\widehat{L}}_b⟩$ of an impurity in a Laughlin liquid at (a) $\nu =1/3$ ($N_a=8$ particles at $L=L_3+N_a+4=96$), and at (b) $\nu =1/5$ ($N_a=6$ particles at $L=L_5+N_a+4=85$). The 12 lowest states (on the left to the red-dotted line) are states of zero interaction energy. On average, the impurity takes fractionally quantized values $L_b=[(m+\nu )/(1-\nu )]$ (indicated through the blue-dashed lines).

Figure 2

(a) We plot the radial density ${\rho }_b^{m,q}(|w|)$ of an impurity, which is excited to the $m$th level ($m=0$, 1, 4), and which is immersed in a FQH liquid at $\nu =1/q$ (for $q=3$ and $q=5$). For concreteness, we have assumed a liquid of $N_a=8$ particles at $L=L_3+N_a+4=96$ for $\nu =1/3$, and a liquid of $N_a=6$ particles at $L=L_5+N_a+4=85$ at $\nu =1/5$, and different $m$ levels correspond to different edge modes. We also plot the density ${\rho }_m(|w|)=|{\phi }_m(w){|}^2$ of a single impurity in the absence of a liquid ($\nu =0$). (b) For different levels $m$, we plot the mean square $⟨r^2{⟩}_m$ of the radial position of the impurity in the presence of a liquid at $\nu =1/3$, $\nu =1/5$, and in the absence of the liquid. The slope of the linear relation between $m$ and $⟨r^2{⟩}_m$ characterizes the quantization of angular momentum.

Figure 3

The impurity angular momentum $⟨{\widehat{L}}_b⟩$ is plotted as a function of impurity number $N_b$, for different filling factors $\nu$ of the majority liquid in the Laughlin state. The numerical results are obtained from Monte Carlo sampling in the wave function Eq. (4) for $N_a=20$ majority particles. We also plot $L_{b,\text{Bose}}(N_b,\nu )$ and $L_{b,\text{Fermi}}(N_b,\nu )$, the values expected if free bosons or fermions would fill the effective single-particle levels for impurities bound to quasiholes, as well as the anyonic interpolation between both curves, $L_{b,\text{Any}}(N_b,\nu )$, defined in Eq. (5). The numerical data is found to match very well the anyonic prediction.