Strongly interacting mesoscopic systems of anyons in one dimension

Using the fractional statistical properties of so-called anyonic particles, we present solutions of the Schrödinger equation for up to six strongly interacting particles in one-dimensional confinement that interpolate the usual bosonic and fermionic limits. These solutions are exact to linear order in the inverse coupling strength of the zero-range interaction of our model. Specifically, we consider two-component mixtures of anyons and use these to eludicate the mixing-demixing properties of both balanced and imbalanced systems. Importantly, we demonstrate that the degree of demixing depends sensitively on the external trap in which the particles are confined. We also show how one may in principle probe the statistical parameter of an anyonic system by injection a strongly interacting impurity and doing spectral or tunneling measurements.

Figure 1

Four-body system of two $A$ and two $B$ particles in a hard-wall box with statistical parameters ${\theta }_A$ and ${\theta }_B$ respectively. The exact solution for large short-range interaction strength, $g$, have linear spectral slopes as shown schematically for (a) Fermi-Fermi (FF), (b) Bose-Fermi (BF), and (c) Bose-Bose (BB) mixtures. (d) The probabilities of different particle configurations for the ground state is shown as a function of first ${\theta }_B$ (keeping ${\theta }_A=0$) and then ${\theta }_A$ (keeping ${\theta }_B=\pi$). The particle configurations of each line is shown in the left-hand panel. The system is dominantly antiferromagnetic in the FF limit (${\theta }_B={\theta }_A=0$), then becomes demixed in the BF case (${\theta }_B=\pi$ and ${\theta }_A=0$), and finally becomes completely mixed in the BB limit (${\theta }_B={\theta }_A=\pi$) where all configurations are equally likely.

Figure 2

Same as in Fig. 1 but for a balanced six-body case where the $A$ particles are fermions (${\theta }_A=\pi$) assuming an external confinement that is a box (solid lines) or a harmonic trap (dashed lines). The configurations as indicated above each set of lines. For simplicity we show only the configurations with the largest probabilities.

Figure 3

Imbalanced five- (upper row) and six-body (low row) systems. All $A$ particles are fermions (${\theta }_A=0$). The configurations follow the lines in the ${\theta }_B=0$ limit (left-hand side) from top to bottom. Solid lines are for box and dashed lines for harmonic trapping. As in Fig. 2 we show only the dominant configurations.

Figure 4

Probabilities to find an impurity on the side (left column) or in the middle (right column) of an anyonic system with up to five anyons as function of statistical parameter, ${\theta }_B$. The upper row is for a box and the lower row for a harmonic trap. The middle is defined as either the single (for even $N_B$) or the two equivalent central positions (for odd $N_B$). For ${\theta }_B=\pi$ all configurations have equal probability (BB regime).

Figure 5

Schematic of the coordinate space for the three-body problem where the $A$ particles are anyons with statistical parameter $\theta$ while single the $B$ particle can be considered an impurity.