Ground state and edge excitations of a quantum Hall liquid at filling factor 2/3

We present a numerical study of fractional quantum Hall liquid at Landau-level filling factor $\nu =2/3$ in a microscopic model including long-range Coulomb interaction and edge confining potential based on the disk geometry. We find that the ground state is accurately described by the particle-hole conjugate of a $\nu =1/3$ Laughlin state. We also find that there are two counterpropagating edge modes, and the velocity of the forward-propagating mode is larger than the backward-propagating mode. The velocities have opposite responses to the change in the background confinement potential. On the other hand changing the two-body Coulomb potential has qualitatively the same effect on the velocities; for example, we find that increasing layer thickness (which softens of the Coulomb interaction) reduces both the forward mode and the backward mode velocities.

Figure 1

(Color online) Schematic picture of electron-density profile along radial direction of a $\nu =2/3$ FQH droplet. We assume there is a hole Laughlin state embedded in the $\nu =1$ electron background. Therefore, there are two interfaces: one is between the 1/3 hole Laughlin state and the integral filling background and the other is between the $\nu =1$ integer quantum Hall droplet and the vacuum.

Figure 2

(Color online) Phase diagram for systems with 20 electrons at filling factor $\nu =2/3$, as a function of number of single-electron orbitals $N_{\text{orb}}$ and background charge distance $d$. The ground-state total angular momentum $M$ changes as $d$ increases. Angular momenta $M=270$, 280, 288, and 294 correspond to five-, six-, seven-, and eight-hole Laughlin ground states, respectively. However the $M=294$ state with $N_{\text{orb}}=27$ is expected to be a stripe state which can be described by $|{\Psi }_{\text{SP}}⟩=|011111100000011111111111111⟩$.

Figure 3

(Color online) Overlap between the 20-electron ground state in 26 orbitals and the particle-hole conjugate of the six-electron Laughlin state as a function of $d$. The decrease in the overlap indicates that the particle-hole conjugate Laughlin state favors smaller $d$, i.e., stronger confinement.

Figure 4

(Color online) Density profiles for the six-electron Laughlin state, the 20-electron ground state in 30 orbitals with $M=280$ and $d=0.7l_B$, the sum of them, and the 26-electron IQH state. The sum is almost the same as the density of IQH.

Figure 5

(Color online) Low-energy spectrum for 20 electrons in 28 orbitals with $d=0.5$. The inner- and outer-edge states are labeled by red (dark gray) solid bars and the simplest combination of the two is labeled by a black solid bar (the fourth lowest level at $M=280$). Its energy (0.1049) is roughly the sum of the two lowest excitation energies for the two modes: $0.02807\left(\text{inner}\right)+0.07687\left(\text{outer}\right)$. This state has a moderately large overlap (0.3959) with the particle-hole conjugate of a six-electron Laughlin state with $\Delta M=1$ edge excitation and embedded in the IQH edge state with $\Delta M=1$: $|{\overline{L}}_{17}^6\left(\Delta M=1\right)\dots 1101⟩$.

Figure 6

(Color online) Comparison of the low-energy excitation spectrum of the $\nu =1/3$, (a) six-electron Laughlin edge states (six electrons in 22 orbitals with Coulomb interaction, $d=0.5l_B$) and (b) $\nu =2/3$, 6-hole Laughlin edge states (20 electrons in 28 orbitals, $d=0.5l_B$). The edge states are labeled by red (dark gray) solid bars. The overlaps between these edge states are shown in Table .

Figure 7

(Color online) (a) The dispersion relation of the inner edge mode for 20 electrons in 28 orbitals at $\nu =2/3$ and (b) the edge mode of a 6-electron Laughlin droplet in 20 orbitals at $\nu =1/3$ with different background confinement potentials. The evolutions of the edge velocities as a function of $d$ are plotted in (c). It can be seen that the velocity of the counterpropagating edge mode in the hole Laughlin state crosses that of the electron Laughlin state at around $d=0.82l_B$, and the velocity of the electron Laughlin state has an opposite response to the change in the background confinement to the hole Laughlin state.

Figure 8

(Color online) The energy gap (red circle point line) $\Delta E=E_2-E_1$ and the overlap between the integral edge state $|{\overline{L}}_{16}^6\dots 11010⟩$ and the first two lowest states (square and triangular point line) in $M=281$ subspace as a function of $d$. The energy gap reaches its minima and the overlap has a crossover between the first state and the second state at around $d_c=0.84l_B$, indicating that the first state becomes the outer-edge state after $d>d_c$.

Figure 9

(Color online) The dispersion curves of a $\nu =2/3$ FQH droplet with 20 electrons in 28 orbitals (a) for both the inner and outer-edge modes at different $d$’s. As in the six-electron Laughlin state, the velocities of the outer-edge mode decrease linearly as a function of $d$ and have larger velocities than the inner edge (b) in the whole parameter range where the global ground state resides in the $M=280$ subspace.

Figure 10

(Color online) The energy spectrum of the $P\text{−}H$ conjugated Hamiltonian with hard-core interaction [Eq. (14)] for 20 electrons in the 28 orbitals. $M$ is the total angular momentum for the 20 electrons. The first 50 energy levels are plotted for each $M$. The lowest-energy states (they should be the exact zero-energy state if we include the constant term) at $M=294,\dots ,289$ are degenerate with degeneracy 1,1,2,3,5,7.

Figure 11

(Color online) (a) The energy spectrum for the pure Coulomb Hamiltonian with $\lambda =0$ and (b) a mixed Hamiltonian with $\lambda =0.5$. When $\lambda =0$ the overlaps between the outer-edge states (and ground state) and their corresponding conjugated states are: ${\left|{⟨{\Psi }_{M={280}_1}|{\overline{L}}_{16}^6⟩}_{26}^{20}\right|}^2=0.684$, ${\left|⟨{\Psi }_{M={281}_1}|{\overline{L}}_{16}^6\dots 11010⟩\right|}^2=0.385$, ${\left|⟨{\Psi }_{M={282}_3}|{\overline{L}}_{16}^6\dots 10110⟩\right|}^2=0.133$, and ${\left|⟨{\Psi }_{M={282}_{15}}|{\overline{L}}_{16}^6\dots 11001⟩\right|}^2=0.158$; the overlap for the simplest linear combination state is ${\left|⟨{\Psi }_{M={280}_3}|{\overline{L}}_{17}^6\left(\Delta M=1\right)\dots 11010⟩\right|}^2=0.194$. In the case of a mixed Hamiltonian with $\lambda =0.5$, they are ${\left|{⟨{\Psi }_{M={280}_1}|{\overline{L}}_{16}^6⟩}_{26}^{20}\right|}^2=0.66$, ${\left|⟨{\Psi }_{M={281}_1}|{\overline{L}}_{16}^6\dots 11010⟩\right|}^2=0.539$, ${\left|⟨{\Psi }_{M={282}_2}|{\overline{L}}_{16}^6\dots 10110⟩\right|}^2=0.218$, ${\left|⟨{\Psi }_{M={282}_{10}}|{\overline{L}}_{16}^6\dots 11001⟩\right|}^2=0.107$, and ${\left|⟨{\Psi }_{M={280}_2}|{\overline{L}}_{17}^6\left(\Delta M=1\right)\dots 11010⟩\right|}^2=0.296$.

Figure 12

(Color online) The dispersion curves for both the inner-edge and the outer-edge modes for 20 electrons in 28 orbitals with $d=0.9l_B$ and different layer thicknesses. The layer thickness softens the interaction between the electrons and reduces the edge mode velocities.

Figure 13

(Color online) The density profile of the 2/3 state (1/3 hole Laughlin state) and its quasihole and quasiparticle excitations. The system contains 20 electrons in 30 orbitals with $d=0.5l_B$ and the Gaussian tip potential $H_W=W_g{\sum }_m\text{exp}\left(-m^2/2s^2\right)c_m^{+}{c}_m$ has a width $s=2l_B$. (a) and (b) are the density profiles for the ground state with angular momentum $M=280$, (c) and (d) are the density profiles for one quasihole state with $M=286$ when $W_g=0.2$, and (e) and (f) are for one quasiparticle state with $M=274$ and $W_g=-0.2$.