Fractional quantum Hall states of bosons on cones

Motivated by a recent experiment, which synthesizes Landau levels for photons on cones [Schine et al., Nature (London) 534, 671 (2016)], and more generally the interest in understanding gravitational responses of quantum Hall states, we study fractional quantum Hall states of bosons on cones. A variety of trial wave functions for conical systems are constructed and compared with exact diagonalization results. The tip of a cone is a localized geometrical defect with singular curvature, which can modify the density profiles of quantum Hall states. The density profiles on cones can be used to extract some universal information about quantum Hall states. The values of certain quantities are computed numerically using the density profiles of some quantum Hall states and they agree with analytical predictions.

Figure 1

(a) A cone can be built from a disk by removing a certain part and gluing the resulting two edges together. (b) The Landau levels on a cone contain type-I states Eq. (2) (red solid lines) and type-II states Eq. (3) (blue dashed lines). The quantum numbers for the states are displayed as $(s,m)$ above the lines.

Figure 2

The density profile $\rho \left(r\right)$ of the Laughlin state with $N_b=10$. (a) The ground state; (b) the state with one quasihole at the tip. The inset of (a) shows that the radius of the droplet scales with $\sqrt{\beta }$ as we expect for a system with average density $1/\left(4\pi \right)$. $\ell$ is the magnetic length.

Figure 3

The excessive charge $\mathrm{\Delta }n$ and the first moment ${\chi }_1$ computed from Fig. 2. (a) $\mathrm{\Delta }n$ of the ground state; (b) $\mathrm{\Delta }n$ of the state with one quasihole at the tip; (c) ${\chi }_1$ of the ground state; (d) ${\chi }_1$ of the state with one quasihole at the tip. The lines in (a) and (b) are linear fits using the data points at $\beta \ge 4$. The lines in (c) and (d) are theoretical predictions (not fitting results using the data points).

Figure 4

The same quantities as in Fig. 3 but for the $N_b=40$ system. The data points at $\beta =2,3$ in (a) and (b) are also used in the linear fits and the agreement for ${\chi }_1$ is much better than Fig. 3.

Figure 5

The left parts show energy spectra of bosons on cones at $L_z=80$ and 81 with $L_{\max }=25$ in units of $P_0$. The magenta lines are exact eigenstates and the black dots are composite fermion states. For $\beta =1.0$ at $L_z=81$, two states have very close energies so we use a yellow square in addition to a black dot. The right parts show the composite fermion configurations for the states. The red solid lines and blue dashed lines are defined as in Fig. 1. The presence of a green dot means that this state is occupied. For $\beta =2.0$ at $L_z=80$, there are multiple configurations with the same effective cyclotron energy and we only give one representative.

Figure 6

The left parts show energy spectra of bosons on cones at $L_z=72$ and 74 with $L_{\max }=25$ in units of $P_0$. The magenta lines are exact eigenstates and the black dots are composite fermion states. For $\beta =1.0$ and $\beta =1.2$ at $L_z=74$, several states have very close energies so we use a yellow square, a red star, and a blue cross in addition to a black dot. The right parts show the composite fermion configurations for the states. The red solid lines and blue dashed lines are defined as in Fig. 1. The presence of a green dot means that this state is occupied. For $\beta =2.0$ at $L_z=72$ and $\beta =1.0$ and 1.2 at $L_z=74$, there are multiple configurations with the same effective cyclotron energy and we only give one representative.

Figure 7

The density profile $\rho \left(r\right)$ and the excessive charge $\mathrm{\Delta }n$ of the Moore-Read state. (a) $\rho \left(r\right)$ of the $N_b=10$ system; (b) $\rho \left(r\right)$ of the $N_b=40$ system; (c) $\mathrm{\Delta }n$ computed from (a); (d) $\mathrm{\Delta }n$ computed from (b). The lines in (c) and (d) are linear fits where (c) uses the data points at $\beta \ge 4$ and (d) uses all the data points.

Figure 8

The density profile $\rho \left(r\right)$ and the excessive charge $\mathrm{\Delta }n$ of the Halperin 221 state with $N_b=16$. The line in (b) is a linear fit of the data points at $\beta \ge 4$.