Particle-hole-symmetric model for a paired fractional quantum Hall state in a half-filled Landau level

The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.

Figure 1

Energy spectrum of the two-body model, $H_2$ (black dashes), and three-body model that generates the Moore-Read Pfaffian, $H_3$ (red circles), for $N=14$ and $Q=12.5$ on the sphere.

Figure 2

Energy gap between the lowest energy excited state and the uniform ground state of $H_2$ (denoted by ${\mathrm{\Delta }}_{{\mathrm{\Psi }}_2}$) and $H_3$ (denoted ${\mathrm{\Delta }}_{{\mathrm{\Psi }}_{\text{Pf}}}$), respectively, as a function of inverse particle number. Linear extrapolations find the gaps in the thermodynamic limit to be ${\mathrm{\Delta }}_{{\mathrm{\Psi }}_2}=0.267\left(24\right)$ and ${\mathrm{\Delta }}_{{\mathrm{\Psi }}_{\mathrm{Pf}}}=0.277\left(47\right)$. The numbers in parenthesis are the standard deviation in the linear extrapolation.

Figure 3

The top panel shows the spectrum of $H_2$ for $N=15$ at $2Q=27$ and identifies the neutral fermion mode. The bottom panels show the neutral fermion modes for systems up to $N=13$ for $H_2$ (left) and $H_3$ (right). These figures can be compared to those for $H_3$ and the second LL Coulomb Hamiltonian in Ref. [48]. The wave vector is $k=L/\sqrt{Q}$.

Figure 4

Energy of $H\left(\alpha \right)$ relative to the ground state $E_0$. The middle panel shows the low-energy spectrum (lowest approximately 15 states) for $H\left(\alpha \right)$ as a function of $\alpha$ for $N=14$ and $Q=12.5$. The left and right panels plot relative energy as a function of angular momentum $L$ for $\alpha =0$ and 1, respectively. The angular momentum of each state is indicated by color. The gap stays open and relatively constant indicating adiabatic continuity.

Figure 5

The thermodynamic limit of the energy gap of $H\left(\alpha \right)$ is shown versus $\alpha$. Similar to the finite-size system results of Fig. 4 the gap remains finite and largely flat, adiabatically connecting ${\mathrm{\Psi }}_2$ with ${\mathrm{\Psi }}_{\mathrm{Pf}}$. The error bars indicate the standard deviation in the linear extrapolation.

Figure 6

Relative energy of $H\left(\alpha \right)$ in the torus geometry for $N=8$ (upper-left panel) and $N=12$ (upper-right panel) for an aspect ratio $\tau =0.95$ as a function of $\alpha$. The threefold ground-state degeneracy remains quasidegenerate, and well below the gap, as $\alpha$ is tuned from $H_3$ to $H_2$. The bottom panels show the corresponding overlap $\langle \mathrm{\Psi }|\overline{\mathrm{\Psi }}\rangle$, indicating that ${\mathrm{\Psi }}_2$ is PH-symmetric at $\alpha =1$ since $\langle {\mathrm{\Psi }}_2|\overline{{\mathrm{\Psi }}_2}\rangle =1,{\mathrm{\Psi }}_{\mathrm{Pf}}$ breaks PH-symmetry since $\langle {\mathrm{\Psi }}_{\mathrm{Pf}}|\overline{{\mathrm{\Psi }}_{\mathrm{Pf}}}\rangle \ne 1$, and $\mathrm{\Psi }$, the ground state of $H\left(\alpha \right)$, remains largely PH-symmetric for finite $\alpha$ less than unity.

Figure 7

Energy of $H\left(\alpha \right)$ relative to the ground state (see Fig. 4) for the system containing two quasihole excitations ($N=14$ and $Q=13$) (the blue sections on the left and middle panels indicate there are higher energy states in the continuum we did not calculate). Nonzero $\alpha$ causes the zero-energy degenerate non-Abelian quasihole manifold to be broken; however, the spread of states stays below the continuum of generic excitations all the way to $\alpha =1$.

Figure 8

The thermodynamic limit of the quasihole average energy, ${\delta }_{\mathrm{therm}}$ (red), and the energy gap to the generic continuum of states, ${\mathrm{\Delta }}_{\mathrm{therm}}$ (green), as a function of $\alpha$. At $\alpha \sim 0.7$ the spread of the quasidegenerate quasiholes bleeds into the continuum and appears to indicate the adiabatic continuity between the low-energy states in the quasihole sector of $H_2$ and $H_3$ is lost.

Figure 9

The thermodynamic limit of the gaps between the lowest energy state in the continuum and the smallest angular momentum states of the quasidegenerate manifold ${\mathrm{\Delta }}_{\mathrm{therm}}^1$, i.e., $L=0$, 1, 0, and 1 for $N=8$, 10, 12, and 14, respectively. This gap (black open circles) is well behaved and linear in $1/N$ as indicated by the small error bars in the extrapolation. Finite-size effects have been greatly reduced in comparison to Fig. 8 and the spread in the (quasi)degenerate quasihole energy remains below the gap to generic states in the continuum.

Figure 10

Entanglement entropy $S_{l_A}\left(N\right)$ versus $1/N$ for various $l_A$ for ${\mathrm{\Psi }}_2$ at $2Q=2N-3$ (left panel). The right panel shows the thermodynamic limit $S_{l_A}\left(\infty \right)$ versus $\sqrt{l_A}$ to determine the topological entanglement entropy ${\gamma }_{{\mathrm{\Psi }}_2}$. The error bars are uncertainty of the extrapolation (explained in the text) and the shaded region depicts the upper and lower bounds of the extrapolation.

Figure 11

Orbital entanglement spectrum for ${\mathrm{\Psi }}_2$ and Moore-Read Pfaffian for $N=14$. The inset shows the ES of the “root” configuration [70] of both states. The inset shows a zoom in where the states are barely distinguishable. The partition is along the spherical equator, i.e., $P\left[1\right|1]$ in the notation of Ref. [70].

Figure 12

The finite-sized Haldane pseudopotentials $V_1^{2Q},V_3^{2Q}$, and the ratio $V_1^{2Q}/V_3^{2Q}$ of $H_2$ are shown versus the inverse system size ($1/Q$). The values of $V_1,V_3$, and $V_1/V_3$ in the thermodynamic limit ($1/Q\to 0$) are shown on the $y$ axis and found via a fourth-order polynomial extrapolation. The value $V_1/V_3\to 3$ is used in the first line of Eq. (2).

Figure 13

The lowest 50 states as of $H\left(\alpha \right)$ a function of “index” for $N=8$, 10, 12, and 14 for $\alpha =0$, 0.5, and 1.

Figure 14

The wave function overlap of the exactly degenerate quasihole states, labeled by their angular momentum $L$, of $H_3$ with the quasidegenerate quasihole states of $H\left(\alpha \right)$ as a function of $\alpha$ for $N=8$, 10, 12, and 14. Additionally, the open symbols are the overlaps between the Moore-Read Pfaffian and the ground state of $H\left(\alpha \right)$ for the ground-state sector. All overlaps are trivially one for $\alpha =0$ and remain large (typically above 0.9 for states with small angular momentum $L$).