$\mathrm{SU}\left(4\right)$ skyrmions and activation energy anomaly in bilayer quantum Hall systems

The bilayer quantum Hall (QH) system has four energy levels in the lowest Landau level, corresponding to the layer and spin degrees of freedom. We investigate the system in the regime where all four levels are nearly degenerate and equally active. The underlying group structure is $\mathrm{SU}\left(4\right)$. At $\nu =1$ the QH state is a charge-transferable state between the two layers and the $\mathrm{SU}\left(4\right)$ isospin coherence develops spontaneously. Quasiparticles are isospin textures to be identified with $\mathrm{SU}\left(4\right)$ skyrmions. The skyrmion energy consists of the Coulomb energy, the Zeeman energy and the pseudo-Zeeman energy. The Coulomb energy consists of the self-energy, the capacitance energy and the exchange energy. At the balanced point only pseudospins are excited unless the tunneling gap is too large. Then, the $\mathrm{SU}\left(4\right)$ skyrmion evolves continuously from the pseudospin-skyrmion limit into the spin-skyrmion limit as the system is transformed from the balanced point to the monolayer point by controlling the bias voltage. Our theoretical result explains quite well the experimental data due to Murphyet al. [S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 72, 728 (1994)] and Sawada et al. [A. Sawada, D. Terasawa, N. Kumada, M. Morino, K. Tagashira, Z. F. Ezawa, K. Muraki, T. Saku, and Y. Hirayama, Physica E 18, 118 (2003); D. Terasawa, M. Morino, K. Nakada, S. Kozumi, A. Sawada, Z. F. Ezawa, N. Kumada, K. Muraki, T. Saku, and Y. Hirayama, Physica E 22, 52 (2004)] on the activation energy anomaly induced by applying parallel magnetic field.

Figure 1

The LLL contains four energy levels corresponding to the two layers and the two spin states. (We call them the $g$ level, $s$ level, $p$ level and $r$ level.) At $\nu =1$ the ground state is the up-spin bonding state. An electron may be moved to any one of the other three levels to form an electron-hole pair excitation.

Figure 2

The energy of an electron-hole pair state is depicted as a function of the imbalance parameter ${\sigma }_0$ based on the formulas (5.5, 5.10) with typical sample parameters as indicated. Three curves correspond to the state $\mid {\psi }_{{s}}⟩$, $\mid {\psi }_{\mathrm{sr}}⟩$ and $\mid {\psi }_p⟩$. We have normalized the activation energy of the pseudospin excitation to zero at ${\sigma }_0=0$.

Figure 3

The skyrmion energy is estimated as a function of the imbalanced parameter ${\sigma }_0$ with sample parameters ${\Delta }_{\mathrm{SAS}}=1\mathrm{K}$, and ${\rho }_0=1.2\times {10}^{11}∕{\mathrm{cm}}^2$. The experimental data are taken from Terasawa et al. (see Ref. 15). Solid lines show the energies of a ppin skyrmion and a spin skyrmion, normalized to the data at the balanced point and the monolayer point, respectively. The experimental data would be explained by the excitation of a genuine $\mathrm{SU}\left(4\right)$ skyrmion.

Figure 4

The activation energy at $\nu =1$ is plotted as a function of the tilting angle $\Theta$ in several samples with different tunneling gaps ${\Delta }_{\mathrm{SAS}}$. It shows a rapid decrease towards the critical angle ${\Theta }^{*}$, and then becomes almost flat. The data are taken from Murphy et al. (see Ref. 11). They are well fitted by the theoretical formula (9.11), where the activation energy decreases in the commensurate phase and becomes flat in the incommensurate phase. To fit the data, we have adjusted the activation energy at $\Theta =0$ with the experimental value, and assumed that the number of flipped pseudospins $N_{\mathrm{ppin}}$ is constant for all values of the tilting angle.

Figure 5

The activation energy at $\nu =1$ is plotted as a function of $\Theta$ in one sample (${\Delta }_{\mathrm{SAS}}=11\mathrm{K}$, ${\rho }_0=0.6\times {10}^{11}∕{\mathrm{cm}}^2$) with different imbalance parameter ${\sigma }_0$. It shows a decrease towards the critical angle ${\Theta }^{*}$, and then begins to increase for ${\sigma }_0=0.3$ and ${\sigma }_0=0.7$. The data are taken from Sawada et al. (see Ref. 14). To fit the data by the theoretical formula (9.15), we have adjusted the activation energy at $\Theta =0$ with the experimental value, and assumed that the flipped spin number $N_{\mathrm{ppin}}$ and the flipped pseudospin number $N_{\mathrm{ppin}}$ are constant for all values of the tilting angle.

Figure 6

The activation energy at $\nu =1$ is plotted as a function of $\Theta$ in a sample (${\Delta }_{\mathrm{SAS}}=33\mathrm{K}$, ${\rho }_0=1.0\times {10}^{11}∕{\mathrm{cm}}^2$). The data are taken from Terasawa et al. (see Ref. 15). To fit the data by the theoretical formula, we have adjusted the activation energy at $\Theta =0$ with the experimental value. We have also assumed that $N_{\mathrm{spin}}$ and $N_{\mathrm{ppin}}$ are constant for all values of the tilting angle. A better fitting is obtained as indicated by a solid line when both spins and pseudospins are excited.