Fractionally Quantized Hall Effect: Liquid-Crystal Ground-State and it Excitations

It is shown that the ground-state and the lowest excited-states of two-dimensional electron system (2DES), with ion jellium background, correspond to partial crystal-like (with the period $L_{x}^{\square}=\sqrt{2 m \pi} \ell_{0}$) correlation order among $N$ electrons of the main region (MR; $L_{x} \times L_{y}$). Many-body variational ground-state wave function of 2DES is presented at the fractional and the integral filling factors $\nu=1/m$; $m=2\ell+1$ and $\ell=0, 1, 2,...$. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the $y-$ direction, with the period $L_{x}^{\square}/m$, and independent of $x$. At $m=3, 5$, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state; the same holds at $m=1$. At $m \geq 3$, the compound form of the many-body ground-state wave function leads to the compound structure of each electron within the main strip (MS; $L_{x}^{\square} \times L_{y}$). Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from $m$ strongly correlated quasielectrons (quasiholes) of the total charge $e/m$ ($-e/m$) each. The activation gap is obtained: it is given by the excitation gap of relevant compound exciton, at $m \geq 3$, and by the gap of pertinent compound spin-exciton, at $m=1$. Quantized Hall conductance $\sigma_{H}=e^{2}/(2 m \pi \hbar)$ is obtained. The theory is in good agreement with experiments.