Neutral collective modes in spin-polarized fractional quantum Hall states at filling factors $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{7}$, and $\frac{4}{9}$

We determine the lowest and higher order collective modes in both spin-conserving and spin-reversed sectors by calculating the energy differences of appropriate linear combinations of different levels of the excitons of the composite fermions and the fully spin-polarized ground states at filling factors $\nu =1/3$, 2/5, 3/7, and 4/9. Apart from providing a detailed study of previously reported modes that have also been observed in the experiments, we predict additional higher energy modes at different filling factors. The lowest and the next higher spin-conserving modes have equal number of “magnetorotons” and the number is the same as the number of filled effective Landau-like levels of composite fermions. The higher energy modes at $\nu =1/3$ merge with the lowest mode at long wavelengths. The spin-conserving modes do not merge at other filling factors. Apart from showing a zero-energy spin-wave mode at zero momentum, thanks to Larmor's theorem, the lowest spin-reversed modes at the ferromagnetic ground states of $\nu =2/5$, 3/7, and 4/9 display one or more “spin rotons” at negative energies, signaling unstable fully polarized ground states at sufficiently small Zeeman energies. The high-energy spin-reversed modes also have spin rotons but at positive energies. The energies of these excitations depend on the finite width of the quantum well as the Coulomb interaction gets screened. We perform a finite thickness correction to the Coulomb interaction by the standard method of local density approximation and use it to calculate the critical energies such as rotons and long- and short-wavelength modes, which are detectable in inelastic light scattering experiments.

Figure 1

A schematic diagram of the fully polarized FQHE ground state and some of the excited states with a spin-zero exciton of CFs. Solid(dashed) lines representing up(down)-spin $\Lambda \mathrm{Ls}$ are labeled on the left. Filled circles with two arrows represent CFs with two vortices attached to each of them; an open circle symbolizes the absence of a CF, i.e., a CF hole. (a) The spin-polarized ground state of a FQHE filling factor $\nu =n/(2n+1)$ corresponds to completely filled $0,1,\cdots ,n-2,n-1$ up-spin $\Lambda \mathrm{Ls}$ and empty down-spin $\Lambda \mathrm{Ls}$. (b) The $\text{level}-1^{+}$ CF exciton in which a CF particle is in the $(n,\uparrow )$ level and a CF hole is in the $(n-1,\uparrow )$ $\Lambda \mathrm{L}$. (c) An excitonic state with a $\text{level}-2^{+}$ CF exciton in which a CF particle is in the $(n+1,\uparrow )$ level and a CF hole is in the $(n-1,\uparrow )$ level. (d) The second $\text{level}-2^{+}$ CF exciton in which a CF particle is in the $(n,\uparrow )$ level and a CF hole is in the $(n-2,\uparrow )$ level.

Figure 2

A schematic diagram of some of the spin-one excitons of CFs in the fully polarized ground state of the filling factor $\nu =n/(2n+1)$. Filled circles with two arrows represent CFs with two vortices attached to each of them; an open circle symbolizes the absence of a CF, i.e., a CF hole. (a) A level-0 exciton with a CF particle in the $(n-1,\downarrow )$ level and a CF hole in the $(n-1,\uparrow )$ level. (b) Another level-0 exciton with a CF particle in the $(n-2,\downarrow )$ level and a CF hole in the $(n-2,\uparrow )$ level. (c) A level-${(n-2)}^{-}$ exciton with a CF particle in the $(1,\downarrow )$ level and CF hole in the $(n-1,\uparrow )$ level. (d) A level-${(n-1)}^{-}$ exciton with a CF particle in the $(0,\downarrow )$ level and a CF hole in the $(n-1,\uparrow )$ level.

Figure 3

(a) Dispersion of the excitations due to $\text{level}-1^{+}$, $\text{level}-2^{+}$, and $\text{level}-3^{+}$ spin-zero excitons in the fully polarized ground state of filling factor $1/3$ for $N=200$. (b) The lowest three spin-zero modes determined by considering the mixing of excitons up to level 5. All the modes are extrapolated (dotted lines) up to $q=0$, and they merge at $q=0$ corresponding to the excitation energy ${\Delta }_0^0$. A typical Monte Carlo uncertainty for obtaining these modes is shown at the end of these modes. Each of these modes exhibit one magneto-roton denoted as ${\Delta }_{R1,1}^0$, ${\Delta }_{R1,2}^0$, and ${\Delta }_{R1,3}^0$. Their energies at the high-momentum limit are denoted as ${\Delta }_{\infty ,1}^0$, ${\Delta }_{\infty ,2}^0$, and ${\Delta }_{\infty ,3}^0$. (c) The three lowest spin-one excitonic modes, viz., $(0,\uparrow )\to (0,\downarrow )$, $(0,\uparrow )\to (1,\downarrow )$, and $(0,\uparrow )\to (2,\downarrow )$ for $N=100$. (d) The three lowest spin-one modes arising from the mixing of spin-one excitonic modes up to $\text{level}-4^{+}$. The lowest mode is the spin-wave mode. Each of the other two spin-flip modes exhibit one spin roton denoted as ${\Delta }_{R1,2}^1$ and ${\Delta }_{R1,3}^1$. The extrapolated energies of these two spin-flip modes up to $q=0$ are denoted as ${\Delta }_{0,2}^1$ and ${\Delta }_{0,3}^1$. The high-momentum limit of the energies of these three modes are denoted as ${\Delta }_{\infty ,1}^1$, ${\Delta }_{\infty ,2}^1$, and ${\Delta }_{\infty ,3}^1$. The total energy for these spin-one modes is $\omega =E_z+{\Delta }^{\left(1\right)}$.

Figure 4

Spin-zero and spin-one modes at the fully polarized filling factor $2/5$. (a) The $\text{level}-1^{+}$ and $\text{level}-2^{+}$ spin-zero excitonic modes for $N=200$. (b) Two lowest spin-zero modes arising from the mixing up to $\text{level}-3^{+}$ excitons. Both modes exhibit two rotons and they are denoted as ${\Delta }_{R1,1}^0$, ${\Delta }_{R2,1}^0$, ${\Delta }_{R2,1}^0$, and ${\Delta }_{R2,2}^0$. The dotted lines denote the extrapolation of these modes up to $q=0$ where their energies are denoted as ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$. The energies of these modes at high momentum limit are denoted as ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$. (c) Two level-0 and one $\text{level}-1^{-}$ spin-one excitonic modes for $N=100$. The former two behave as spin waves at long wavelength, and the latter one displays a spin roton. (d) Two lowest spin-one modes obtained by mixing up to $\text{level}-1^{+}$ spin-one excitonic modes. The higher mode has one spin roton denoted as ${\Delta }_{R1,2}^1$. The energies of these modes at $q\to 0$ are zero and ${\Delta }_{0,2}^1$, respectively; the high-momentum limits of these modes are denoted as ${\Delta }_{\infty ,1}^1$ and ${\Delta }_{\infty ,2}^1$. The total energy for spin-one excitations is given by $\omega =E_z+{\Delta }^{\left(1\right)}$.

Figure 5

Spin-zero and spin-one modes at the fully polarized filling factor $3/7$. (a) The $\text{level}-1^{+}$ and $\text{level}-2^{+}$ spin-zero excitonic modes for $N=136$. (b) The two lowest spin-zero modes arising from the mixing up to $\text{level}-3^{+}$ excitons. Both modes exhibit three rotons and they are denoted as ${\Delta }_{R1,1}^0$, ${\Delta }_{R2,1}^0$, ${\Delta }_{R3,1}^0$, ${\Delta }_{R2,1}^0$, ${\Delta }_{R3,1}^0$, and ${\Delta }_{R3,2}^0$. The dotted lines denote the extrapolation of these modes up to $q=0$ where their energies are denoted as ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$. The energies of these modes at high momentum limit are denoted as ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$. (c) One level-0, one $\text{level}-1^{-}$, and one $\text{level}-2^{-}$ spin-one excitonic modes for $N=136$. (d) The two lowest spin-one modes obtained by mixing up to $\text{level}-1^{+}$ spin-one excitonic modes. Both modes have one spin roton denoted as ${\Delta }_{R1,1}^1{\Delta }_{R1,2}^1$. The energies of these modes at $q\to 0$ are zero and ${\Delta }_{0,2}^1$, respectively; the high-momentum limits of these modes are denoted as ${\Delta }_{\infty ,1}^1$ and ${\Delta }_{\infty ,2}^1$. The total energy for spin-one excitations is given by $\omega =E_z+{\Delta }^{\left(1\right)}$.

Figure 6

Spin-zero and spin-one modes at the fully polarized filling factor $4/9$. (a) The $\text{level}-1^{+}$ and $\text{level}-2^{+}$ spin-zero excitonic modes for $N=160$. (b) Two lowest spin-zero modes arising from the mixing up to $\text{level}-3^{+}$ excitons. Both modes exhibit three rotons and they are denoted as ${\Delta }_{R1,1}^0$, ${\Delta }_{R2,1}^0$, ${\Delta }_{R3,1}^0$, ${\Delta }_{R4,1}^0$, ${\Delta }_{R1,2}^0$, ${\Delta }_{R2,2}^0$, ${\Delta }_{R3,2}^0$, and ${\Delta }_{R4,2}^0$. The dotted lines denote the extrapolation of these modes up to $q=0$ where their energies are denoted as ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$. The energies of these modes at high momentum limit are denoted as ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$. (c) One level-0, one $\text{level}-2^{-}$, and one $\text{level}-3^{-}$ spin-one excitonic modes for $N=160$. (d) Two lowest spin-one modes obtained by mixing up to $\text{level}-1^{+}$ spin-one excitonic modes. The lowest mode has two spin rotons denoted as ${\Delta }_{R1,1}^1$ and ${\Delta }_{R2,1}^1$, and the next higher mode has one spin roton denoted as ${\Delta }_{R1,2}^1$. The lowest mode has a maxon denoted by ${\Delta }_{M1,1}^1$ as well. The energies of these modes at $q\to 0$ are zero and ${\Delta }_{0,2}^1$, respectively; the high-momentum limit of these modes is denoted as ${\Delta }_{\infty ,1}^1$ and ${\Delta }_{\infty ,2}^1$. The total energy for spin-one excitations is given by $\omega =E_z+{\Delta }^{\left(1\right)}$.

Figure 7

Effective two-dimensional Coulomb potential $V_{\mathrm{eff}}\left(r\right)$ obtained by the local density approximation in quantum wells of finite transverse width. Dotted lines for bare Coulomb potential. $V_{\mathrm{eff}}\left(r\right)$ (a) for various electron densities in the unit of ${10}^{10}{\mathrm{cm}}^{-2}$ at a fixed width $d=25$ nm, and (b) for various widths of quantum wells at a fixed electron density $n_e=2\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 8

The variation of critical energies corresponding to spin-zero modes [Fig. 3] with electron densities and width of the quantum wells at filling factor 1/3. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^0$, ${\Delta }_{R1,2}^0$, and ${\Delta }_{R1,3}^0$; (b) high-momentum modes: ${\Delta }_{\infty ,1}^0$, ${\Delta }_{\infty ,2}^0$, and ${\Delta }_{\infty ,3}^0$; (c) long-wavelength mode: ${\Delta }_{0,1}^0$.

Figure 9

The variation of critical energies corresponding to spin-one modes [Fig. 3] with electron densities and width of the quantum wells at filling factor 1/3. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,2}^1$, and ${\Delta }_{R1,3}^1$; (b) high-momentum modes: ${\Delta }_{\infty ,1}^1$, ${\Delta }_{\infty ,2}^1$, and ${\Delta }_{\infty ,3}^1$; (c) long-wavelength mode: ${\Delta }_{0,2}^1$.

Figure 10

The variation of critical energies corresponding to spin-zero modes [Fig. 4] with electron densities and width of the quantum wells at filling factor 2/5. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^0$, ${\Delta }_{R2,1}^0$; (b) rotons: ${\Delta }_{R1,2}^0$, ${\Delta }_{R2,2}^0$; (c) high-momentum modes: ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$; (d) long-wavelength modes: ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$.

Figure 11

The variation of critical energies corresponding to spin-one modes [Fig. 4] with electron densities and width of the quantum wells at filling factor 2/5. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^1$ and ${\Delta }_{R1,2}^1$; (b) high-momentum modes: ${\Delta }_{\infty ,1}^1$ and ${\Delta }_{\infty ,2}^1$; (c) long-wavelength mode: ${\Delta }_{0,2}^1$.

Figure 12

The variation of critical energies corresponding to spin-zero modes [Fig. 5] with electron densities and width of the quantum wells at filling factor 3/7. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^0$, ${\Delta }_{R1,2}^0$; (b) rotons: ${\Delta }_{R2,1}^0$, ${\Delta }_{R2,2}^0$; (c) rotons: ${\Delta }_{R3,1}^0$, ${\Delta }_{R3,3}^0$; (d) high-momentum modes: ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$; (e) long-wavelength modes: ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$.

Figure 13

The variation of critical energies corresponding to spin-one modes [Fig. 5] with the electron densities and width of the quantum wells at filling factor 3/7. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^1$ and ${\Delta }_{R1,2}^1$; (b) high-momentum modes: ${\Delta }_{\infty ,1}^1$ and ${\Delta }_{\infty ,2}^1$; (c) long-wavelength mode: ${\Delta }_{0,2}^1$.

Figure 14

The variation of critical energies corresponding to spin-zero modes [Fig. 6] with electron densities and width of the quantum wells at filling factor 4/9. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^0$, ${\Delta }_{R1,2}^0$; (b) rotons: ${\Delta }_{R2,1}^0$, ${\Delta }_{R2,2}^0$; (c) rotons: ${\Delta }_{R3,1}^0$, ${\Delta }_{R3,2}^0$; (d) rotons: ${\Delta }_{R4,1}^0$, ${\Delta }_{R4,2}^0$; (e) high-momentum modes: ${\Delta }_{\infty ,1}^0$ and ${\Delta }_{\infty ,2}^0$; (e) long-wavelength modes: ${\Delta }_{0,1}^0$ and ${\Delta }_{0,2}^0$.

Figure 15

The variation of critical energies corresponding to spin-one modes [Fig. 6] with electron densities and width of the quantum wells at filling factor 4/9. Critical energies in the unit of $e^2/\left(\epsilon \ell \right)$ for (a) rotons: ${\Delta }_{R1,1}^1$, ${\Delta }_{R1,2}^1$ and ${\Delta }_{R3,1}^2$; (b) maxon: ${\Delta }_{M1,1}^1$; (c) high-momentum mode: ${\Delta }_{\infty ,1}^1$; (d) long-wavelength mode: ${\Delta }_{0,2}^1$.