Analytical theory of strongly correlated Wigner crystals in the lowest Landau level

In this work, we present an analytical theory of strongly correlated Wigner crystals (WCs) in the lowest Landau level (LLL) by constructing an approximate, but accurate effective two-body interaction for composite fermions (CFs) participating in the WCs. This requires integrating out the degrees of freedom of all surrounding CFs, which we accomplish analytically by approximating their wave functions by delta functions. This method produces energies of various strongly correlated WCs that are in excellent agreement with those obtained from the Monte Carlo simulation of the full CF crystal wave functions. We compute the compressibility of the strongly correlated WCs in the LLL and predict discontinuous changes at the phase boundaries separating different crystal phases.

Figure 1

Schematic diagram showing (a) the isolated and (b) the renormalized two-body formalism, accompanied by the probability density of the two-body CF wave function in each formalism, (c) ${\rho }_{\mathrm{isol}}\left(\mathbf{r}\right)=\int d^2{\mathbf{r}}^{\prime }{\left|{\psi }_{\mathrm{isol}}^{{}^{2p}\mathrm{CFC}}(\mathbf{r},{\mathbf{r}}^{\prime })\right|}^2$, and (d) ${\rho }_{\mathrm{renorm}}\left(\mathbf{r}\right)=\int d^2{\mathbf{r}}^{\prime }{\left|{\psi }_{\mathrm{renorm}}^{{}^{2p}\mathrm{CFC}}(\mathbf{r},{\mathbf{r}}^{\prime })\right|}^2$, respectively. Here, we set $2p=2$ and the nominal distance between crystal centers to be $6l_{\mathrm{B}}$. Note that probability densities are plotted in the natural logarithmic scale.

Figure 2

Energy per particle of various CFC states, $E^{{}^{2p}\mathrm{CFC}}$, in reference to that of the MZ WC, $E^{\mathrm{MZ}}$, as a function of filling factor, $\nu$, obtained in (a) the isolated and (b) the renormalized two-body formalism. Symbols and dashed lines are obtained from the MC simulation of the full CFC wave functions. Note that each ${}^{2p}\mathrm{CFC}$ is the lowest energy state (at least, among various CFC states) within the range of $1/(2p+3)<\nu \lesssim 1/(2p+1)$ for $2p\ge 2$. The MZ WC, or ${}^0\mathrm{CFC}$ is not energetically favorable at any filling factor ranges.

Figure 3

Shear modulus, $C^t$, of various CFC states as a function of filling factor, $\nu$, obtained in (a) the isolated and (b) the renormalized two-body formalism. For convenience, $C^t$ is expressed as its relative ratio with respect to that for the classical WC, $C_{\mathrm{classical}}^t/(e^2/\epsilon l_B)=0.0978\sqrt{\nu }$. Note that $C^t$ of each ${}^{2p}\mathrm{CFC}$ is valid only within the range of $1/(2p+3)<\nu \lesssim 1/(2p+1)$, where its curve is plotted in a solid line, denoting that the ${}^{2p}\mathrm{CFC}$ is the lowest energy state here.

Figure 4

Inverse of the compressibility, ${\kappa }^{-1}$, of various CFC states as a function of filling factor, $\nu$, obtained in (a) the isolated and (b) the renormalized two-body formalism. Similar to $C^t, {\kappa }^{-1}$ of each ${}^{2p}\mathrm{CFC}$ is valid only within the range of $1/(2p+3)<\nu \lesssim 1/(2p+1)$, where its curve is plotted in a solid line.