Dynamical matrix of two-dimensional electron crystals

In a quantizing magnetic field, the two-dimensional electron gas has a rich phase diagram with broken translational symmetry phases such as Wigner, bubble, and stripe crystals. In this paper, we derive a method to obtain the dynamical matrix of these crystals from a calculation of the density response function performed in the generalized random-phase approximation (GRPA). We discuss the validity of our method by comparing the dynamical matrix calculated from the GRPA with that obtained from standard elasticity theory with the elastic coefficients obtained from a calculation of the deformation energy of the crystal.

Figure 1

Deformation energy as a function of $u_0$ for a shear deformation, $u\left(\mathbf{r}\right)=u_{0}y\widehat{\mathbf{x}}$, in a triangular Wigner crystal in Landau level $N=0$ with filling factor $\nu =0.15$. The square symbols are the HFA result, while the solid line is a polynomial fit of order 2 (the linear term is negligible).

Figure 2

Deformation energy as a function of $u_0$ for a one-dimensional dilatation of a triangular Wigner crystal in Landau level $N=0$ with filling factor $\nu =0.15$. The square symbols are the HFA result (left axis), while the solid line is a polynomial fit. The dashed line (right axis) is the deformation energy with the linear term removed.

Figure 3

First Brillouin zone of the triangular lattice with the irreducible Brillouin zone shown as the dark area. The arrows represent the reciprocal lattice vectors on the second shell, while (1) corresponds to the vector $\mathbf{G}=0$.

Figure 4

(Color online) Component $D_{xx}\left(\mathbf{k}\right)$ of the dynamical matrix along $k_x$ computed for different couples of vectors $(\mathbf{G},{\mathbf{G}}^{\prime })$. The numbers in the legend refer to the numeration of the reciprocal lattice vectors shown in Fig. 3.

Figure 5

(Color online) Elastic coefficients (a) $c_{66}$ and (b) $c_{12}$ of the triangular Wigner crystal for Landau level $N=2$ computed using the different approximations listed in the legend. For the GRPA, the coefficients are computed using three different couples of reciprocal lattice vectors. The numbers in the legend correspond to the numeration of the vectors given in Fig. 3. The empty circles give an average of the three GRPA results. For $c_{66}$, the different symbols are superimposed.

Figure 6

(Color online) Elastic coefficients (a) $c_{66}$ and (b) $c_{12}$ of the triangular Wigner crystal in Landau level $N=0$ computed in the different approximations indicated in the legend. For the GRPA, the coefficients are computed using three different couples of reciprocal lattice vectors. The numbers in the legend correspond to the numeration of the vectors given in Fig. 3. The empty circles give the average of the three GRPA results. For $c_{66}$, the different symbols are superimposed.

Figure 7

The first four shells of reciprocal lattice vectors of the anisotropic stripe crystal.

Figure 8

(Color online) Component $D_{xx}\left(\mathbf{k}\right)$ of the GRPA and HFA dynamical matrices of the stripe crystal computed along the direction $k_y=0$ for partial filling factor $\nu =0.42$ in Landau level $N=2$, computed using two different couples of reciprocal lattice vectors. The numbers in the legend refer to numbering of the reciprocal lattice vectors in Fig. 7.

Figure 9

(Color online) Component $D_{xy}\left(\mathbf{k}\right)$ of the GRPA and HFA dynamical matrices of the stripe crystal computed along the direction $k_y=k_x$ for partial filling factor $\nu =0.42$ in Landau level $N=2$, computed using two different couples of reciprocal lattice vectors. The numbers in the legend refer to numbering of the reciprocal lattice vectors in Fig. 7.

Figure 10

(Color online) Component $D_{yy}\left(\mathbf{k}\right)$ of the GRPA and HFA dynamical matrices of the stripe crystal computed along the direction $k_x=0$ for partial filling factor $\nu =0.42$ in Landau level $N=2$, computed using two different couples of reciprocal lattice vectors. The numbers in the legend refer to numbering of the reciprocal lattice vectors in Fig. 7.

Figure 11

(Color online) Component $D_{xy}\left(k\right)$ of the GRPA and HFA dynamical matrices of the stripe crystal computed along the direction $k_x=k_y$ for different filling factors in Landau level $N=2$.