Ring-shaped fractional quantum Hall liquids with hard-wall potentials

We study the physics of $\nu =1/2$ bosonic fractional quantum Hall droplets confined in a ring-shaped region delimited by two concentric cylindrically symmetric hard-wall potentials. Trial wave functions based on an extension of the Jack polynomial formalism including two different chiral edges are proposed and validated for a wide range of confinement potentials in terms of their excellent overlap with the eigenstates numerically found by exact diagonalization. In the presence of a single repulsive potential centered in the origin, a recursive structure in the many-body spectra and a massively degenerate ground-state manifold are found. The addition of a second hard-wall potential confining the fractional quantum Hall droplet from the outside leads to a nondegenerate ground state containing a well-defined number of quasiholes at the center and, for suitable potential parameters, to a clear organization of the excitations on the two edges. The utility of this ring-shaped configuration in view of theoretical and experimental studies of subtle aspects of fractional quantum Hall physics is outlined.

Figure 1

Typical behavior of the ${\mathcal{U}}_m$ confinement potentials as a function of the single-particle angular momentum $m$ for two different sets of confining parameters summarized in the legend and corresponding to FQH liquids in, respectively, the ring (black pluses) and pierced (red circles) geometries. The inset provides a magnified view of the main panel. As expected, the extremely steep condition discussed in the text is not fulfilled for the experimentally realistic potential parameters considered here.

Figure 2

(a) Radial density profiles of a 3QH state and of Jacks with root partitions $\lambda =\mathrm{\Omega }+{\kappa }^3-[1,1]=[13,11,9,7,4,2]$ and $\lambda =\mathrm{\Omega }+{\kappa }^3-\left[2\right]=[13,11,9,7,5,1]$ for an $\mathcal{N}=6$ particle system. All density profiles show the same behavior around the outer edge. On the other hand, the density profiles are characterized by very different shapes on the inner edge, as expected from states presenting IEEs. (b) Rescaled density difference $\mathcal{D}\left(r\right)$ of different Jacks compared to an ($n=3$)-QH state. The comparison of the $\lambda =\mathrm{\Omega }+\rho$ and $\lambda =\mathrm{\Omega }+{\kappa }^3+\rho$ cases allows us to identify the latter with OEEs of the 3QH state: The rescaled density differences only differ by a global shift which reflects the presence of three empty orbitals in the 3QH state and a weak attenuation of the peaks, which is consequence of the slightly different length of the edges. At the same time, since the behavior of the $\mathcal{D}\left(r\right)$ associated with Jacks of partitions $\lambda =\mathrm{\Omega }+{\kappa }^3-\rho$ is similar to those characterizing OEEs of the 3QH state, with the difference that the peaks occurs in proximity to the inner edge of the 3QH state, we can state that Jacks with partitions of that form well describe the IEEs of the 3QH state. Again the difference in the peaks high can be imputed to a difference in the length of the edges. (c) Radial density profiles characterizing the 3QH, 4QH, and 5QH states and Jacks with root partitions $\lambda =\mathrm{\Omega }+[4,4,4,3,3,3]$ and $\lambda =\mathrm{\Omega }+[5,5,5,4,4,4]$. While at small distances the density profile of the former (latter) Jack polynomial is very similar to the one of the 3QH state (4QH state), at large distances it follows the behavior of the 4QH state (5QH state) density profile.

Figure 3

Many-body spectra for (a) and (b) $\mathcal{N}=5$ and (c) $\mathcal{N}=6$ particle systems experiencing a piercing potential of parameters $V_{\mathrm{in}}=2V_0$ and $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$. Such spectra clearly show an energetic behavior characterized by the presence of branches and subbranches. By comparing (a) and (b), a recursive structure taking place at different energy scales can also be appreciated. On the other hand, the comparison between (a) and (c) proves that the structure of the spectra does not depend much on the number of particles, as expected from the arguments based on the eigenstate description in terms of Jack polynomials.

Figure 4

(a) Radial density profiles and (b) occupation number expectation values associated with the 3QH state with the $\rho =\left[1\right]$ IEE, the 4QH state with the $\rho =\left[2\right]$ IEE, and the 5QH state with the $\rho =\left[3\right]$ IEE. Despite that for all these states the lowest $m$ occupied orbital is the $m=2$ one, the occupation of this orbital is different for the considered states; in particular, the 5QH state with the $\rho =\left[3\right]$ IEE is the one characterized by the highest occupation. In terms of densities, this is reflected in the higher density of the 5QH state with the $\rho =\left[3\right]$ IEE in the inner region $r<1.5\sqrt{2}{l}_B$.

Figure 5

(a) Values of the ${\mathcal{U}}_m$ potentials for confinement potentials with $V_{\mathrm{in}}=2V_0$, $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$, $V_{\mathrm{ext}}=10V_0$, and $R_{\mathrm{ext}}=5.4\sqrt{2}{l}_B$ (blue pluses) or $R_{\mathrm{ext}}=5.7\sqrt{2}{l}_B$ (red circles). (b) Many-body energy spectrum of an $\mathcal{N}=5$ particle system subject to the potential illustrated with blue pluses in (a): The ground state is nondegenerate and corresponds to the 3QH state, while it is hard to point out any structure in the excited states. (c) Energy spectrum of an $\mathcal{N}=6$ particle system subject to the potential illustrated by red circles in (a). The similarity to the one in (b) is a consequence of the similarity of the ${\mathcal{U}}_m$ in the two cases. In particular, for large $m$ the $U_m$ potentials have a very similar shape except for a outward shift by two orbitals. As a result, the energy shifts coming from the outermost and innermost occupied orbitals are roughly the same for Jacks with the same EP describing $\mathcal{N}=5,6$ particle states. The inset shows the three lowest-energy states in the $L=47$ angular momentum sector.

Figure 6

(a) The $m$ dependence of the ${\mathcal{U}}_m$ values for $R_{\mathrm{in}}=0.3\sqrt{2}{l}_B$ (blue bars) or $R_{\mathrm{in}}=0.7\sqrt{2}{l}_B$ (bars with red stripes). Blue circles and red squares summarize the root configuration of the 1QH and 2QH states, respectively. (b) Energy of the different $n\mathrm{QH}$ states as a function of $R_{\mathrm{in}}$. As expected, by increasing $R_{\mathrm{in}}$, states containing a higher number $n$ of QHs become energetically favorable. Moreover, by comparing (a) and (b), we can note that the lowest-energy QH state is the one whose root configuration minimizes the energy contribution coming from the different ${\mathcal{U}}_m$. (c) Overlap between numerical states $|\mathrm{\Psi }\rangle$ and Jacks describing the associated QH states, which proves that for any value of $R_{\mathrm{in}}$ the lowest-energy states is extremely well described by an $n\mathrm{QH}$ state. For all plots $V_{\mathrm{in}}=2V_0$, $V_{\mathrm{ext}}=100V_0$, and $R_{\mathrm{in}}$ and $R_{\mathrm{ext}}$ satisfying Eq. (24) with $\mathcal{A}=\pi {\left(5.25\sqrt{2}{l}_B\right)}^2$ have been considered.

Figure 7

(a) Plot of the ${\mathcal{U}}_m$ values for a confinement potential of parameters $V_{\mathrm{in}}=20V_0$, $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$, $V_{\mathrm{ext}}=75V_0$, and $R_{\mathrm{ext}}=5.5\sqrt{2}{l}_B$. (b) Associated $\mathcal{N}=6$ particles spectrum. Blue circles in (a) indicate the occupied orbitals in the root configuration of the 3QH state. By recalling that root configurations for the IEEs (OEEs) can be obtained by moving inward (outward) particles occupying the innermost (outermost) single-particle orbitals, this representation of the ground-state root configuration pictorially explains which orbitals are associated with a given inner or outer edge excitation. Since the ${\mathcal{U}}_2$ and ${\mathcal{U}}_{14}$ values (red dashed lines) are comparable, the Hamiltonian eigenstates in the spectrum (or at least those with $n_i=\langle \psi |a_i^{†}{a}_i|\mathrm{\Psi }\rangle =0\forall i<2$ or $i>14$) are organized in an almost symmetric way. Eigenstates of the lowest branch between the 2QH and the 3QH state correspond to Jacks with EPs of the form $\eta =[3,\cdots ,2]$ and are characterized by a nonvanishing occupation of the $m=2$ orbital, while those between the 3QH and the 4QH state have nonvanishing occupation of the $m=14$ orbital and are well described by Jacks with $\eta =[4,\cdots ,3]$. Finally, states in the upper branches can be associated with Jacks of EPs like $\eta =[4,\cdots ,2]$ and therefore describing states presenting both inner and outer EEs.

Figure 8

In the left column, the top plot shows the ${\mathcal{U}}_m$ potentials associated with the confining parameters $V_{\mathrm{in}}=2V_0$, $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$, $V_{\mathrm{ext}}=10V_0$, and $R_{\mathrm{ext}}=5.7\sqrt{2}{l}_B$ and the three lower plots display mock ${\mathcal{U}}_m$ potentials obtained by shifting the previous ones by one, two, and three orbitals, respectively. In each panel, the blue circles indicate again the occupied orbitals in the root configurations of Jacks describing the ground state. The right column shows the corresponding many-body energies. Their structure is almost the same for all cases considered and no periodic behavior can be observed as a function of the number of QHs. The slow decrease of the excited-state energies as a function of the number of QHs pinned in the origin can be imputed to the bigger length of the ring-shaped system edges.

Figure 9

Rescaled density differences $\mathcal{D}\left(r\right)$ associated with the same (a) OEE or (b) IEE lying on the edges of QH states containing a different number of QHs. To be precise, the OEE (IEE) with OEP (IEP) of the form [1,1] has been considered. As it is clearly visible in both panels, having additional QHs in the origin simply results in an outward translation of the curve describing $\mathcal{D}\left(r\right)$, which reflects the presence of additional empty orbitals, and in an attenuation of the oscillations characterizing $\mathcal{D}\left(r\right)$, which is the consequence of having a longer edge.

Figure 10

Many-body spectra for (a) and (b) $\mathcal{N}=5$ and (c) $\mathcal{N}=6$ particle systems subject to a piercing potential of parameters $V_{\mathrm{in}}=2V_0$ and $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$ and a harmonic potential characterized by $\upsilon =3\times {10}^{-4}{V}_0/\hslash$. As we can see, the addition of a harmonic confinement preserves the organization of many-body energy levels into branches and subbranches as in the pierced geometry, but at the same time leads to a nondegenerate ground state corresponding to a QH state.

Figure 11

Energy spectra for an $\mathcal{N}=5$ particle system subject to a confinement potential of parameters $V_{\mathrm{in}}=2V_0$, $R_{\mathrm{in}}=0.5\sqrt{2}{l}_B$, $V_{\mathrm{ext}}=10V_0$, and $R_{\mathrm{ext}}=5.4\sqrt{2}{l}_B$. Red circles are obtained by diagonalizing the system Hamiltonian on the full Hilbert space ${\mathcal{H}}_{\text{full}}$ and black pluses by restricting the system Hamiltonian to the squeezed Hilbert spaces ${\mathcal{H}}_{\text{sq}}\left(\overline{\lambda }\right)$. To be precise, for ${\mathcal{H}}_{\text{full}}$ a single-particle angular momentum cutoff $m=15$ has been chosen, while for the different ${\mathcal{H}}_{\text{sq}}\left(\overline{\lambda }\right)$ partitions of the form $\overline{\lambda }=\mathrm{\Omega }+{\kappa }^2+\rho$, where $\rho$ is the dominant configuration among those of the integer $L-L_L-2\mathcal{N}$ which satisfy ${\rho }_1\le 5$, have been considered. As we can see, the agreement between the different spectra is excellent and small deviations can only be observed for the highest-energy states.