At filling factor $ν=1/m$, $m$ odd integer, I present variational ground-state and excited-state wave functions, of two-dimensional electron system with homogeneous ion background, that show the condensation into a liquid-crystal state. For $m=1, 3, 5$, the ground-state energy per electron is substantially lower than the Laughlin one, for uniform liquid state.
Liquid-Crystal State of ν=1/m Quantum Hall Effects
At filling factor ν=1/m, m odd integer, I present variational ground-state and excited-state wave functions, of two-dimensional electron system with homogeneous ion background, that show the condensation into a liquid-crystal state. For m=1,3,5, the ground-state energy per electron is substantially lower than the Laughlin one, for uniform liquid state.
Ground-state of fractional and integral quantum Hall systems at ν≤1 and it excitations
Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)Lx□×Ly of the finite width Lx□=2πmℓ0 (and the periodic boundary condition (PBC) imposed along x−direction), is presented at the fractional and the integral filling factors ν=1/m for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); ℓ0 is the magnetic length, m=2ℓ+1 and ℓ=0,1,2,.... It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among N electrons of the main region (MR) Lx×Ly; then the study of 2DES of N electrons within MR is exactly reduced to the treatment of 2DES of N~=NLx□/Lx electrons localized within MS, with PBC along x. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y− direction, with the period Lx□/m, and independent of x. For IJB, at m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at m=1. 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 charge e/m (−e/m. Quantized Hall conductance σH=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
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 Lx□=2mπℓ0) correlation order among N electrons of the main region (MR; Lx×Ly). Many-body variational ground-state wave function of 2DES is presented at the fractional and the integral filling factors ν=1/m; m=2ℓ+1 and ℓ=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 Lx□/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≥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; Lx□×Ly). 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≥3, and by the gap of pertinent compound spin-exciton, at m=1. Quantized Hall conductance σH=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
A scaling procedure on the particle number is developed to construct the wave function of the uniform liquid ground state of the two-dimensional electron system in a strong magnetic field. When the filling factor is equal to the inverse of an odd number, this wave function is the correlated basis function introduced by Laughlin. In more general cases, when the filling factor is equal to a rational number, this wave function also can be regarded as a Laughlin wave function on the scale of a certain cluster. By this procedure, we can prove that an energy gap is always present between such a uniform liquid ground state and the excited states when the filling factor is equal to a rational number.
I present variational solutions of the many-body Schrödinger equation for a two-dimensional electron system in strong magnetic fields, which have, at ν=1, the energy about 46% lower than the energy of the Laughlin liquid. At ν=2/3, I obtain the energy, about 29% lower than was reported so far.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.