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.
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.
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.
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor ν=1) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at ν=1. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.
A simple one-dimensional model is proposed, in which N spinless interacting fermions occupy M>N degenerate states on a circle. 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 νN/M=1/(2m+1). Within this model the complimentary wavefunction for ν=l−I/(2m+1) is found explicitly.
This is a mathematical study of certain aspects of the interacting electron system in very high perpendicular magnetic field.\nWe analyse restrictions imposed upon the density correlation functions of this system and propose a set of sum rules which they must obey.\nWe study the possibility of building a bosonisation scheme for the projected density operators in the lowest Landau level. We suggest a second order bosonisation, along with an approximation scheme, which may be useful for carrying out calculations in the lowest Landau level.\nWe analyse the possible ground states of the system. We suggest a set of variational wavefunctions which can have lower energy than the Laughlin state for sufficiently soft interaction potentials.\nWe study the collective excitations of the system, paying particular attention to its symmetries. We suggest a set of variational excited states and discuss their applicability to finite as well as infinite systems.