SU(4) Skyrmions in the $\nu =\pm 1$ Quantum Hall State of Graphene

We explore different Skyrmion types in the lowest Landau level of graphene at a filling factor $\nu =\pm 1$. In addition to the formation of spin and valley pseudospin Skyrmions, we show that another type of spin-valley entangled Skyrmions can be stabilized in graphene due to an approximate SU(4) spin-valley symmetry that is affected by sublattice symmetry-breaking terms. These Skyrmions have a clear signature in spin-resolved density measurements on the lattice scale, and we discuss the expected patterns for the different Skyrmion types.

Figure 1

Pseudospin Skyrmions in an easy-axis [(a)] and an easy-plane pseudospin [(b)] FM background. The two subfigures sketch the lattice-resolved total electronic density ${\rho }_{\mathrm{tot}}$ in the $n=0$ LL. The insets show the pseudospin Bloch sphere, which is entirely covered. In each Bloch sphere, the orange and gray arrows show the polarization of the FM background at $r\to \infty$ and in the center, respectively. At intermediate values of $r$, all other parts of the Bloch sphere are explored. The spin (not shown) is homogeneously polarized. For illustration, we have used much smaller Skyrmion sizes (on the order of some lattice spacings) than encountered in an experimental situation (see text). Panels (a) and (b) correspond to points 3 and 2 in Fig. 2, respectively.

Figure 2

Phase diagram of $\mathcal{Q}=1$ Skyrmions (labeled in red, blue, and yellow) and the pseudospin FM background state (sketched in black), as a function of $u_z$ and $u_{\perp }$ (in units of the Zeeman energy ${\mathrm{\Delta }}_Z$). The parameters used correspond to a field $B=10\mathrm{T}$.

Figure 3

Entanglement Skyrmions in an unentangled (a) and an entangled (b) FM background. The two subfigures sketch the lattice-resolved profiles of the $z$ component of the spin magnetization $S_z$ in a color plot (blue for spin-up and red for spin-down densities). Insets show the entanglement Bloch sphere spanned by $\alpha$ and $\beta$ [see Eq. (10)]. In each Bloch sphere, the orange and gray arrows show the directions corresponding to the FM background and to the Skyrmion center, respectively. Panels (a) and (b) correspond to points 6 and 7 in Fig. 2, respectively.