Skyrme and Wigner crystals in graphene

At low energy, the band structure of graphene can be approximated by two degenerate valleys $\left(K,K^{\prime }\right)$ about which the electronic spectra of the valence and conduction bands have linear dispersion relations. An electronic state in this band spectrum is a linear superposition of states from the $A$ and $B$ sublattices of the honeycomb lattice of graphene. In a quantizing magnetic field, the band spectrum is split into Landau levels with level $N=0$ having zero weight on the $B\left(A\right)$ sublattice for the $K\left(K^{\prime }\right)$ valley. Treating the valley index as a pseudospin and assuming the real spins to be fully polarized, we compute the energy of Wigner and Skyrme crystals in the Hartree-Fock approximation. We show that Skyrme crystals have lower energy than Wigner crystals (WCs), i.e., crystals with no pseudospin texture in some range of filling factor $\nu$ around integer fillings. The collective mode spectrum of the valley-skyrmion crystal has three linearly dispersing Goldstone modes in addition to the usual phonon mode, while a WC has only one extra Goldstone mode with a quadratic dispersion. We comment on how these modes should be affected by disorder and how, in principle, a microwave absorption experiment could distinguish between Wigner and Skyrme crystals.

Figure 1

(Color online) Pseudospin texture in a MC at filling factor ${\nu }_0=0.8$ in Landau level $n=0$. The crystal has four merons per unit cell. In each unit cell, two merons with the same vorticities have opposite phases as explained in the text. Contours (ranging from $-0.5$ to 0.5) indicate the $z$ component of the pseudospin with dark regions corresponding to positive values.

Figure 2

(Color online) Pseudospin texture in a MPC at filling factor ${\nu }_0=0.8$ in Landau level $n=0$. The lattice is triangular and there are four merons per unit cell. Merons are bound in pairs with same value of $P_z$ and vorticities but opposite phases at each lattice site. Contours (ranging from $-0.5$ to 0.5) indicate the $z$ component of the pseudospin with dark regions corresponding to positive values.

Figure 3

(Color online) Hartree-Fock energy per electron as a function of filling factor for various crystal phases in Landau level $n=0$.

Figure 4

(Color online) Dispersion relation of the two Goldstone modes of the eBC1 state at ${\nu }_0=0.2$ in Landau level $n=0$. The dispersion is plotted along the irreducible Brillouin zone of the triangular lattice. The left (right) $y$ axis gives the phonon (pseudospin) frequency. The phonon mode has its biggest weight in ${\chi }_{{\rho }_n,{\rho }_n}$ and ${\chi }_{S_x,S_x}$ while the pseudospin wave mode has its biggest weight in ${\chi }_{S_y,S_y}$ and ${\chi }_{S_z,S_z}$.

Figure 5

(Color online) Dispersion relation of the two Goldstone modes of the hBC1 state at ${\nu }_1=0.55$ in Landau level $n=1$. The dispersion is plotted along the irreducible Brillouin zone of the triangular lattice. The phonon mode has its biggest weight in ${\chi }_{{\rho }_n,{\rho }_n}$ and ${\chi }_{S_x,S_x}$ while the pseudospin wave mode has its biggest weight in ${\chi }_{S_y,S_y}$ and ${\chi }_{S_z,S_z}$.

Figure 6

(Color online) Dispersion relation of the Goldstone modes of the MPC state at ${\nu }_0=0.8$ in Landau level $n=0$. The dispersion is plotted along the irreducible Brillouin zone of the triangular lattice. The legend indicates in what response function each collective mode has its biggest weight.

Figure 7

(Color online) Dispersion relation of the Goldstone modes of the MC state at ${\nu }_0=0.625$ in Landau level $n=0$. The dispersion is plotted along the irreducible Brillouin zone of the square lattice. The legend indicates in what response function each collective mode has its biggest weight.

Figure 8

(Color online) Hartree-Fock energy per electron as a function of filling factor for various crystal phases in Landau level $n=1$.

Figure 9

(Color online) Hartree-Fock energy per electron as a function of filling factor for various crystal phases in Landau level $n=2$.

Figure 10

(Color online) Dispersion relation of the phonon and pseudospin modes of the WC at ${\nu }_0=0.4$ in Landau level $n=0$ in an external potential with $W_K=-0.005$ and $W_{K^{\prime }}=0$. The dispersion is plotted along the irreducible Brillouin zone of the triangular lattice (see Fig. 5). Both the phonon and pseudospin wave mode are gapped by the external field.

Figure 11

Imaginary part of the current response function. From left to right: $k_x=k_y=0.001,0.01,0.05,0.1,0.3$ in units of $2\pi /a_0$. These peaks come from the phonon mode. There is no peak coming from the pseudospin mode.