Fractional quantum Hall effect at $\nu =5∕2$: Ground states, non-Abelian quasiholes, and edge modes in a microscopic model

We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\nu =5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\pm e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\mu \mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).

Figure 1

(Color online) Illustration of the edge locations of the $\nu =1∕2$ Moore-Read state and the $\nu =2$ integer quantum Hall state. Although the positive background charge and total electron charge are five-times that of the electrons forming the Moore-Read state, the edge is hiding behind the integer quantum Hall edge in the lowest Landau level. The Moore-Read edge may thus be protected since it is farther away from the electrostatic edge.

Figure 2

(Color online) Total angular momentum of the global ground state as a function of the mixing parameter $\lambda$ and background charge distance $d$ for 12 electrons in 22 orbitals with the mixed Hamiltonian [Eq. (1)]. The ground state in region (ii) has the same $M_{gs}=126$ as the Moore-Read (or Pfaffian) wave function. The ground state in regions (i) and (iii) are believed to be stripe phases. In region (iv), the ground state is a candidate for the so-called anti-Pfaffian state (see Sec. 7 for detail).

Figure 3

(Color online) The densities of the $+e∕4$ Moore-Read quasihole state [Eq. (12)] and the Moore-Read condensate [Eq. (3)] for a 30-electron system.

Figure 4

(Color online) Ground state angular momentum as a function of trapping potential and three-body interaction strengthen. We have 12 electrons in 22 orbitals, with the mixed Hamiltonian [Eq. (1)] and the Gaussian tip potential [Eq. (14)]. $\lambda$ and $W$ characterize the three-body interaction and tip potential strength, respectively. The background charge is fixed at $d=0.7l_B$ above the electron layer (the ground state has $M_{gs}=126$, same as the Moore-Read state in the absence of the Gaussian potential). For large enough $\lambda$, as the tip potential strength $W$ increases, states with $M_{gs}=132$ or $M_{gs}=138$, believed to contain a $+e∕4$ quasihole or a $+e∕2$ quasihole, become the global ground state. For small $\lambda$, another ground state with $M_{gs}=126$ (which is a stripe state with occupation pattern ∣0000011111111111100000⟩) separates these two quasihole states.

Figure 5

(Color online) Low-energy excitations $\Delta E\left(\Delta M\right)$ from exact diagonalization for $N=12$ electrons in 26 orbitals in the 1LL (corresponding to $\nu =1∕2$) for the mixed Hamiltonian in Eq. (1) with $\lambda =0.5$. The neutralizing background charge for the Coulomb part is deposited at $d=0.6l_B$ above the electron plane. The red solid bars, the black dashed bars, and the blue dotted bars mark fermionic, bosonic, and mixed edge excitations, respectively (see Appendix for detail). Bulk excitations are represented by thin dotted bars (green).

Figure 6

(Color online) (a) Dispersion curves of bosonic $\left({ϵ}_b\right)$ and fermionic modes $\left({ϵ}_f\right)$ for $N=12$ electrons at half filling (in 26 orbitals) in the first Landau level for $\lambda =0.5$. These energies can be used to construct the complete edge spectrum for the 12-electron system up to $\Delta M=4$ (Ref. 26). (b) Dispersion curves for the same system with $\lambda =0.1$, or less three-body interaction. (c) Bosonic $\left(v_c\right)$ and fermionic velocities $\left(v_n\right)$ extrapolated to the pure Coulomb case. We obtain the $\lambda =0.5$ and 0.1 points from fitting the fermionic modes to a straight line in (a) and (b), respectively. At $\lambda =1$ (pure three body interaction), $v_n=0$ as all edge states have zero energies. We thus obtain $v_c=0.046$ and $v_n=0.0036$, in units of $\left(R{e}^2\right)∕\left(ϵl_B\hslash \right)$, for the pure Coulomb case.

Figure 7

(Color online) Low-energy excitations $\Delta E\left(\Delta M\right)$ from exact diagonalization (solid lines) for $N=12$ electrons in 26 orbitals in the 1LL (corresponding to $\nu =1∕2$) for the mixed Hamiltonian in Eq. (1) with $\lambda =0.1$. The neutralizing background charge for the Coulomb part is deposited at $d=0.6l_B$ above the electron plane. The red solid bars, the black dashed bars, and the blue dotted bars mark fermionic, bosonic, and mixed edge excitations, respectively. Bulk excitations are represented by thin dotted bars (green). While the bosonic edge excitations mix significantly with the bulk excitations, the fermionic edge excitations are still well separated from the rest (see Appendix for detail).

Figure 8

(Color online). Low-energy excitations $\Delta E\left(\Delta M\right)$ from exact diagonalization (solid lines) for $N=12$ electrons in 26 orbitals in the 1LL (corresponding to $\nu =1∕2$) for pure Coulomb interaction $(\lambda =0.0)$. The neutralizing background charge for the Coulomb part is deposited at $d=0.6l_B$ above the electron plane. The red solid bars, the black dashed bars, and the blue dotted bars mark fermionic, bosonic, and mixed edge excitations, respectively. Bulk excitations are represented by thin dotted bars (green). In this case, fermionic edge excitations also mix with the bulk excitations (see Appendix for detail).

Figure 9

(Color online) Low-energy spectra in a system of 12 electrons in 24 orbitals for the mixed Hamiltonian with $\lambda =0.1$ and $d=0.7l_B$. (a) In the absence of the external tip potential, the ground state $(M=126)$ is Moore-Read-like. There are 0, 1, 1, and 2 low-energy (fermionic) excitations for $\Delta M=1–4$. (b) In the presence of a Gaussian tip potential ($W=0.1$ and $\sigma =2.0$), the $+e∕4$ quasihole state emerge at $M=132$. There are 1, 1, 2, and 2 low-energy (fermionic) excitations for $\Delta M=1–4$. This suggests that a single $+e∕4$ quasihole changes the fermionic mode spectrum. (The states of interest are marked by red solid bars.)