Charge density wave with meronlike spin texture induced by a lateral superlattice in a two-dimensional electron gas

The combined effect of a lateral square superlattice potential and the Coulomb interaction on the ground state of a two-dimensional electron gas in a perpendicular magnetic field is studied for different rational values of $\mathrm{\Gamma }$, the inverse of the number of flux quanta per unit cell of the external potential, at filling factor $\nu =1$ in Landau-level $N=0.$ When Landau-level mixing and disorder effects are neglected, increasing the strength $W_0$ of the potential induces a transition at a critical strength of $W_0^{\left(c\right)}$ from a uniform and fully spin-polarized state to a two-dimensional charge density wave (CDW) with a meronlike spin texture at each maximum and minimum of the CDW. The collective excitations of this “vortex CDW” are similar to those of the Skyrme crystal that is expected to be the ground-state near filling factor $\nu =1$. In particular, a broken $U\left(1\right)$ symmetry in the vortex CDW results in an extra gapless phase mode that could provide a fast channel for the relaxation of nuclear spins. The average spin-polarization $S_z$ changes in a continuous or discontinuous manner as $W_0$ is increased depending on whether $\mathrm{\Gamma }\in [1/2,1]$ or $\mathrm{\Gamma }\in [0,1/2]$. The phase mode and the meronlike spin texture disappear at a large value of $W_0$ leaving as the ground state a partially spin-polarized CDW if $\mathrm{\Gamma }\ne 1/2$ or a spin-unpolarized CDW if $\mathrm{\Gamma }=1/2.$

Figure 1

(a) Hartree-Fock energy per electron and (b) average spin per electron $S_z/\hslash$ as a function of the applied external field $W_0$ for different values of $\mathrm{\Gamma }$ at filling factor $\nu =1$ in Landau-level $N=0$ and for the Zeeman coupling ${\mathrm{\Delta }}_Z/(e^2/\kappa \ell )=0.015$.

Figure 2

Density of states for the uniform fully spin-polarized phase at $\nu =1$ in Landau-level $N=0$ for Zeeman coupling ${\mathrm{\Delta }}_Z=0.015$. (a)–(c) $\mathrm{\Gamma }=1$ and $W_0=0,0.05,0.1$, respectively; (d) and (e) $\mathrm{\Gamma }=2/3$ and $W_0=0.05,0.1$, respectively; (f) $\mathrm{\Gamma }=4/5$ and $W_0=0.1$. The rapid oscillations in some of the graphs are a numerical artifact. All energies are in units of $e^2/\kappa \ell$.

Figure 3

Dispersion relation of the spin-wave mode in the uniform fully spin-polarized state at $\nu =1,N=0$ for $\mathrm{\Gamma }=1,{\mathrm{\Delta }}_Z=0.015$ and different values of the external potential $W_0$. The wave-vector $\mathbf{k}$ follows the path $\mathrm{\Gamma }-X^{\prime }-M^{\prime }-\mathrm{\Gamma }$ along the irreducible (density) Brillouin zone of the square lattice. All energies are in units of $e^2/\kappa \ell$.

Figure 4

Vortex-CDW phase of the 2DEG at $\nu =1$ in Landau-level $N=0$ for $W_0=0.12$ and Zeeman coupling ${\mathrm{\Delta }}_Z=0.015$. The electronic density $n\left(\mathbf{r}\right)$ is in units of ${\left(2\pi {\ell }^2\right)}^{-1}$. The vector field shows the vortex structure on the $xy$ plane for the parallel component of the spin vector. All energies are in units of $e^2/\kappa \ell$.

Figure 5

Density of states in the vortex CDW at $\nu =1$ in Landau-level $N=0$ for $\mathrm{\Gamma }=2/3$, Zeeman coupling ${\mathrm{\Delta }}_Z=0.015$, and different values of $W_0$ (all energies are in units of $e^2/\kappa \ell$).

Figure 6

Behavior with $W_0$ of different contributions to the total energy of the uniform and vortex-CDW states for $\mathrm{\Gamma }=1$ and the Zeeman coupling ${\mathrm{\Delta }}_Z=0.015$.

Figure 7

Dispersion relations of the collective modes at $\nu =1$ in Landau-level $N=0$ for $\mathrm{\Gamma }=2/3$ and Zeeman coupling ${\mathrm{\Delta }}_Z=0.015$. (a) $W_0=0.1$; (b) $W=0.115$; and (c) $W=0.13$. All energies are in units of $e^2/\kappa \ell$.

Figure 8

Energy (left) and corresponding average spin $S_z/\hslash$ per electron (right) of the vortex CDW (squares) and antivortex CDW (triangles) for $\mathrm{\Gamma }=1/3$ and Zeeman couplings: (a) and (b) ${\mathrm{\Delta }}_Z=0.015$; (c) and (d) ${\mathrm{\Delta }}_Z=0.006$; (e) and (f) ${\mathrm{\Delta }}_Z=0.002$ in units of $e^2/\kappa \ell$. The vertical dashed lines indicate the value of the potential $W_0$ at which the transition between the two vortex CDWs takes place.

Figure 9

Spin-polarization $S_z/\hslash$ as a function of the applied external potential $W_0$ for several values of $\mathrm{\Gamma }\le 1/2$ and Zeeman coupling ${\mathrm{\Delta }}_Z=0.001e^2/\kappa \ell$. The vertical dashed lines indicate the potential strength $W_0$ for each value of $\mathrm{\Gamma }$ where the transition from the vortex (filled symbols) to the antivortex CDW (open symbols) takes place. For $\mathrm{\Gamma }=1/2$, the transition is from the uniform (with $S_z/\hslash =1/2M)$ to the normal CDW with $S_z=0$.