The problem of a single electron in a magnetic field is revisited from first principles. It is shown that the standard quantization, used by Landau, is inconsistent for this problem, whence Landau's wave functions spontaneously break the gauge symmetry of translations in the plane. Because of this Landau's (and Fock's) wave functions have a spurious second quantum number. The one-body wave function of the physical orbit, with only one quantum number, is derived, and expressed as a superposition of Landau's wave functions. Conversely, it is shown that Landau's wave functions are a limiting case of physical solutions of a different problem, where two quantum numbers naturally appear. When the translation gauge symmetry is respected, the degeneracy related to the choice of orbit center does not appear in the one-body problem.The problem of a single electron in a magnetic field is revisited from first principles. It is shown that the standard quantization, used by Landau, is inconsistent for this problem, whence Landau's wave functions spontaneously break the gauge symmetry of translations in the plane. Because of this Landau's (and Fock's) wave functions have a spurious second quantum number. The one-body wave function of the physical orbit, with only one quantum number, is derived, and expressed as a superposition of Landau's wave functions. Conversely, it is shown that Landau's wave functions are a limiting case of physical solutions of a different problem, where two quantum numbers naturally appear. When the translation gauge symmetry is respected, the degeneracy related to the choice of orbit center does not appear in the one-body problem.

arXiv:1111.6107Nov 25, 20110 citations

Electron wave functions in a given potential

Birkhäuser Basel eBooksJan 1, 20050 citations

Theory of the fractional quantum Hall effect

Zi-Zhao Gan, Zhao-Bin Su

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.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.

Physical review. B, Condensed matterApr 15, 19872 citations

Three-electron anisotropic quantum dots in variable magnetic fields: Exact results for excitation spectra, spin structures, and entanglement

Yuesong Li, Constantine Yannouleas, Uzi Landman

Exact-diagonalization calculations for $N=3$ electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection ${S}_{z}=1∕2$ are similar to the representative class of strongly entangled states referred to as the $W$ states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.Exact-diagonalization calculations for $N=3$ electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection ${S}_{z}=1∕2$ are similar to the representative class of strongly entangled states referred to as the $W$ states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.

arXiv:0710.4325Physical Review BDec 11, 200737 citations

On the Wave Functions of Many-Electron Systems

CRC Press eBooksMay 21, 20042 citations

Exact-diagonalization calculations for

N=3

electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection

{S}_{z}=1∕2

are similar to the representative class of strongly entangled states referred to as the

W

states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.