Anisotropic electronic states in the fractional quantum Hall regime Isaac Berry and Orion Ciftja Department of Physics, Prairie View A&M University, Prairie View, Texas 77446, USA Abstract
Phases in the Lowest Landau Level
Configuration of N=25 electrons
This study is concerned with the quantum Hall state at filling factor 1/6 of the lowest Landau level. This state is very close to the critical filling factor where the liquid-solid transition takes place. In this work we investigate whether an anisotropic Coulomb interaction potential is able to stabilize an anisotropic electronic liquid state at this filling factor. Monte Carlo calculations for small systems of electrons in disk geometry are implemented to obtain various quantities of interest . The results are consistent with the existence of an anisotropic liquid state of electrons in the lowest Landau level.
2D electronic systems in a perpendicular magnetic field form sets of discrete energy states called the Landau levels (LLs). An important characteristic parameter of such a system is the filling factor defined as the ratio of the total number of electrons relative to the capacity of each LL. The most robust fractional QHE state occurs in the lowest Landau level (LLL) at filling factors, υ=1/3 and 1/5. Such states are very well described by a wave function that is called Laughlin’s wave function. Wigner crystallization happens at filling factor υ=1/7 , thus the liquid-solid phase transition happens in the range of filling factors 1/7-1/5.
Fractional Quantum Hall Regime
Anisotropic Coulomb Potential
Energy Difference Per Electron
Anisotropic Wave Function (ν=1/6)
Conclusions
Ψ=∏j>k (zj-zk)4 (zj-zk+a)(zj-zk-a) exp[-(1/4)∑j|zj|2] Det[Plane waves]
We describe a possible anisotropic quantum Hall liquid state at filling factor 1/6 of the lowest Landau level stabilized by the presence of an anisotropic Coulomb interaction potential. The state is described by a wave function that contains a built-in anisotropy parameter that breaks rotational symmetry. Monte Carlo results for small systems of electrons indicate stability of liquid crystalline order in presence of an anisotropic interaction potential.
We consider a two-dimensional (2D) system of fully spinpolarized electrons subject to a strong perpendicular magnetic field in the z direction. The magnetic field leads to the creation of discrete energy levels (Landau levels). Degeneracy of Landau levels is proportional to the value of the magnetic field. For large magnetic fields, electrons fill a fraction of the states in the lowest Landau level. This is the fractional quantum Hall regime. Under these conditions, the system manifests novel behavior such as the phenomenon of fractional quantum Hall effect (FQHE), a problem first explained by Laughlin.
Laughlin’s State at Filling Factor 1/3
The variables, zj are two-dimensional position vectors in complex notation (magnetic length is set to one).
(From Bell Labs Picture Gallery)
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POSTERS AND REPORTS
The parameter a breaks the rotational symmetry of the ground state wave function. Plane waves correspond to lowest-lying plane wave energy states of an ideal 2D spin-polarized Fermi gas.
Acknowledgements R&I’s Office of Undergraduate Research (OUR and Physics Department,, Prairie View A&M University.
