Topological phases in small quantum Hall samples

Topological order has proven a useful concept to describe quantum phase transitions which are not captured by the Ginzburg-Landau type of symmetry-breaking order. However, lacking a local order parameter, topological order is hard to detect. One way to detect it is via direct observation of anyonic properties of excitations which are usually discussed in the thermodynamic limit, but so far has not been realized in macroscopic quantum Hall samples. Here we consider a system of few interacting bosons subjected to the lowest Landau level by a gauge potential, and theoretically investigate vortex excitations in order to identify topological properties of different ground states. Our investigation demonstrates that even in surprisingly small systems anyonic properties are able to characterize the topological order. In addition, focusing on a system in the Laughlin state, we study the robustness of its anyonic behavior in the presence of tunable finite-range interactions acting as a perturbation. A clear signal of a transition to a different state is reflected by the system's anyonic properties.

Figure 1

Statistical phase angle and effective charge of vortices in the Laughlin states at $L=N(N-1)$ [(a),(c)] and $L=2N(N-1)$ [(b),(d)]. $R$ is the distance between the hole(s) and the center of the cloud (in units $\lambda$).

Figure 2

Statistical phase angle and effective charge of vortices in the $L=N(N-2)$ sector. The ones on the exact ground state are depicted in panels (a) and (c), while the corresponding ones for the Laughlin quasiparticle state with $\xi =0$ are depicted in (b) and (d). $R$ is the distance between the hole(s) and the center of the cloud (in units $\lambda$).

Figure 3

Statistical phase angle (a) and effective charge of vortices (b) in the $\left(221\right)$-Halperin state. $R$ is the distance between the hole(s) and the center of the cloud (in units $\lambda$).

Figure 4

Effective charge (a) and statistical phase angle (b) of vortices averaged over different vortex positions. The represented value is the average $\pm$ the standard deviation, in a $\nu =\frac{1}{2}$ Laughlin system ($N=6$, $L=30$) perturbed by finite-range interactions characterized by a Haldane pseudopotential with relative strength $V_2/V_0$.

Figure 5

Density profiles (in units ${\lambda }^{-2}$) of the ground state of the system ($N=6$, $L=30$) for different Haldane pseudopotential strengths $v=V_2/V_0$. The curves denoted by HP and PH show the density profiles of a trial wave function obtained from the Laughlin by adding a quasiparticle and subsequently a quasihole (PH) or a quasihole and subsequently a quasiparticle (HP) in the origin.

Figure 6

Statistical phase ($N=6$, $L=30$) as a function of radial vortex position $R$ (in units $\lambda$) for different Haldane pseudopotential strengths. The labels correspond to the values of $V_2/V_0$.

Figure 7

(a) Gaps (in units $V_0$) above ground state at $N=6$ and $L=30$ in the presence of a Haldane pseudopotential with relative strength $V_2/V_0$. (b) Interaction energy (in units $V_0$) of lowest states at $N=6$ and $L=30$ and $L=31$.

Figure 8

Overlaps of the exact ground state at $N=6$ and $L=30$ with different trial wave functions in the presence of a Haldane pseudopotential with relative strength $V_2/V_0$.

Figure 9

Ground state correlation functions for $N=6$ at different values of $V_2/V_0$ with one particle fixed to $(x,y)=(2.5,0)$ in units $\lambda$. Bright colors denote relatively large values of the correlation function, that is, a high probability of finding a particle at the respective position. From (a) to (f), $V_2/V_0$ takes the values 0, 0.3, 0.5, 0.8, 1.5, and 4.0.

Figure 10

Pair correlation functions for $N=6$ for ${\Psi }_{\mathrm{Hf}/\mathrm{Hf}}$ given in Eq. (21) with one particle fixed to $(x,y)=(2.5,0)$ in units $\lambda$.