Recently, the authors discussed the possibility of the macroscopic quantum state similar to the Quantum Hall Effect in a semi-localized 2D electron system with a toroidal electron-wave amplitude in the absence of any magnetic field. In order to give the concrete statistical foundation of the study, the fermion-boson statistical transformation of the 2D electron system is made using a Chern–Simons gauge potential. Based on the solution of the resultant boson-type Hamiltonian, we construct the fermion-type solution via a unitary transformation. It is shown that the solution in the form of Laughlin function is stable when electrons form pairs. In the presence of hole doping, the pair Laughlin function leads to a representation of a superconducting state when the phase-coherence length λΘ exceeds the incompressibility length λQ, but when λΘ< λQ, it leads to a macroscopic quantum state characterized by particle-number definiteness.
Recently, the authors discussed the possibility of the macroscopic quantum state similar to the Quantum Hall Effect in a semi-localized 2D electron system with a toroidal electron-wave amplitude in the absence of any magnetic field. In order to give the concrete statistical foundation of the study, the fermion-boson statistical transformation of the 2D electron system is made using a Chern–Simons gauge potential. Based on the solution of the resultant boson-type Hamiltonian, we construct the fermion-type solution via a unitary transformation. It is shown that the solution in the form of Laughlin function is stable when electrons form pairs. In the presence of hole doping, the pair Laughlin function leads to a representation of a superconducting state when the phase-coherence length λΘ exceeds the incompressibility length λQ, but when λΘ< λQ, it leads to a macroscopic quantum state characterized by particle-number definiteness.
The derivation of the non-magnetic Laughlin state and other macroscopic quantum states in the semi-localized 2D electron system in the network of circular molecular orbits is made by the study of zero-point plasma oscillation. In the imaginary time representation, the electric field is transformed to the vector potential. After the cancellation of the mean-field component of the inter-electron repulsive field with the ion-lattice field, the boson Hamiltonian with respect to the phase-charge fluctuation is obtained using a Chern–Simons gauge field. Based on the resultant boson wave function, the macroscopic quantum state in hole doping is found to lead to a superfluidity that is described by a coherent function when λΘ > λQ, and to the particle-number-definite state described by a Laughlin function when λQ > λΘ, where λΘ is the phase-coherence length and λQ is the incompressibility length.
By using a bosonic Chern-Simons gauge theory the ground state of the planar electron system is analyzed in the presence of an external magnetic field. The distinctive feature of our theory is that a renormalization counter term (|α|/ M )|ψ| 4 which represents explicitly a hard-core condition plays important roles, where α and M are the statistics parameter and the mass of the electron. The following results are obtained systematically in the semiclassical approximation. When ν=1/ k ( k odd) with ν being the Landau level filling factor, the ground state is incompressible and described by the Laughlin wavefunction. On the other hand, when ν∼1/ k , the ground state is compressible and it is given by the Wigner crystal made of quasiholes (antivortices). The wavefunction of the Wigner crystal is also calculated, which is found to coincide with that of the Laughlin quasiholes. In these analyses the Coulomb interaction between electrons plays a crucial role.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2 corresponds to an effective time reversal transformation (i.e. {kj}→{−kj}) for composite fermions, and discuss how this connects to the absence of 2kF backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ levels (analogs of electronic Landau levels) we determine the Λ level spacing and find it to be very nearly independent of the Λ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.