Few-particle systems: An analysis of some strongly correlated states

The analysis of the quantum Hall response of a small system of interacting ultracold bosonic atoms through the variation of its Hall resistivity against the applied gauge magnetic field provides a powerful method to unmask its strongly correlated states in a quite exhaustive way. Within a fixed range of values of the magnetic field in the lowest Landau-level regime, where the resistivity displays two successive plateaux, we identify the implied states as the Pfaffian and the state with filling factor $\nu =2/3$ in the thermodynamic limit. We fix the conditions to have good observability.

Figure 1

Variation of the expectation value of the angular momentum as a function of the rotation frequency $\mathrm{\Omega }$ in units of ${\omega }_{\perp }/2$. We consider $N=4, g_2=1, \gamma =0.1$, and $\mathit{a}=(1,0)$ in our unit of length (see text). The stepwise dashed line refers to the symmetric case ($\gamma =0$).

Figure 2

The same as Fig. 1 for $N=5$.

Figure 3

Hall resistivity as a function of $\mathrm{\Omega }$. We considered $N=4, a=(1.0,0), \gamma =0.1$, and $g_2=1$. The small plateau is approximately at $\mathrm{\Omega }=1.963$. We considered our units of length, energy, and frequency (see text).

Figure 4

The same as Fig. 3 for $N=5, a=(0.6,0), \gamma =0.1$, and $g_2=1$. The small plateaux are localized at approximately $\mathrm{\Omega }=1.915$ and $\mathrm{\Omega }=1.947$; no three-body interaction is considered.

Figure 5

Spectrum around the Laughlin state ($L_{GS}=12$) for $N=4$. The gap at $L=12$ is equal to $0.1585$ in units of $\hslash {\omega }_{\perp }/2$. The bottom states at $L=13,...,18$ are degenerated. $\mathrm{\Omega }=1.97$ has been considered.

Figure 6

Spectrum around the $\nu =2/3$ state ($L_{GS}=12$) at the second plateau of Fig. 4 for $N=5$. The gap at $L=12$ is equal to $0.1091$ in our units. $\mathrm{\Omega }=1.947$ has been considered.

Figure 7

Absolute value of the overlap between the exact solution and the analytical Laughlin expression as a function of $\mathrm{\Omega }$. $N=4, g_2=1, \gamma =0.1$, and $\mathit{a}=(0.6,0)$ are considered. Vertical lines mark the frontiers of the plateau shown in Fig. 3, from $\mathrm{\Omega }=1.963$ to $1.9663$.

Figure 8

The same as Fig. 7 for $N=5$. The analytical function considered on the overlap is the Pfaffian state. The significant values appear over the step $\langle L\rangle =8$. The full line corresponds to $g_3=0$, the plus symbols to $g_3=1$, and the crosses to $g_3=5$. For all of them, $g_2=1$. We consider $\mathit{a}=(0.6,0)$ and $\gamma =0.1$. The vertical lines mark the frontiers of the first (form 1.913 to 1.918) and the second plateau (from 1.945 to 1.95), respectively. Notice that in general, as the interaction grows, there is a whole shift to the left of the critical values of $\mathrm{\Omega }$ where $L$ jumps.

Figure 9

Hall resistivity as a function of $\mathrm{\Omega }$. The short full line contains only two-body interaction ($g_2=1$), while the long dotted one contains also three-body interaction ($g_3=2$). We consider $N=5$ and $[\mathit{a}=(0.8,0\left)\right]$. The marked points are $A$: $(1.91,58.94)$ and $B$: $(1.9172,62.61)$ on the full line and $C$: $(1.911,56.21)$ on the long dotted line.

Figure 10

Three-dimensional image of the GS density for $N=5$ at $\mathrm{\Omega }=1.911$ (in units of ${\omega }_{\perp }/2$) with an impurity at $\mathit{a}=(0.8,0)$ (in units of ${\lambda }_{\perp }$) and $\gamma =0.1$. We consider $g_2=1$ and $g_3=2$.

Figure 11

Three-dimensional image of the GS density for $N=5$ at $\mathrm{\Omega }=1.911$ (in units of ${\omega }_{\perp }/2$) with an impurity at $\mathit{a}=(0.8,0)$ (in units of ${\lambda }_{\perp }$) and $\gamma =0.1$. We consider $g_2=1$ and $g_3=0$.