An effort is made to review the Yang potential. The ground state and the wave function by using Bethe’s Ansatz are found. The Laughlin’s potential is studied and its various properties are investigated. The number of particles in cluster of electrons determines the angular momentum which is used to define the filling factor of the lowest Landau level. A flux factor is added to it so that it gives the desired denominators. The Laughlin state is obtained for any value of the filling factor, p. In all cases, the ground state belongs to nonphysical many-body Hamiltonians. The composite fermion wave function is not the ground state of any known single Hamiltonian. From the angular momentum, we construct the filling factor and then look for the wave function rather than for the Hamiltonian. In this approach the Hamiltonians found are unrealistic. It is possible to make the filling factors agree with the experimental values but the Hamiltonian are not the usual type. The relative angular momentum of two particles L2 is found to define a projection operator. The filling factor is defined for the state of two particles. The number of particles is g. If another particle approaches it to form a cluster, the modified filling factor is defined. In this methodology the flux quanta cannot be attached to the electron and the projection operator is not linked to the Coulomb Hamiltonian. A monolayer of carbon is called graphene. It exhibits unusual properties in the Hall effect and in the cyclotron resonance. Hence, the Kohn's theorem, which shows that the interactions do not play much role in determining the cyclotron resonance, becomes operative. The Hubbard model has been very successful in explaining the ground state of several electron systems. By using the spin in a particular way, we can obtain new features in the Hubbard model. There is a doubling in the Peierls-Luttinger phase factor and eigen values acquire higher multiplicities than are known for the usual treatment of spin. The flux is distributed on the area of the triangle. The graphene consists of hexagons of carbon atoms but the Hall effect shows that there are defects on which electrons form clusters so that there is spin wave type behaviour. The Hall resistivity as a function of magnetic field is not a linear function. Hence a suitable theory to understand the Hall effect is formulated. We find that there are phase transitions as a function of temperature. There are lots of fractions of charge which are explained on the basis of spin and orbital angular momentum of the electron. The nano meter size films of graphite also show that the Hall resistivity is non-linear and shows steps as a function of magnetic field. We make an effort to understand the steps in the Hall effect resistivity of graphite with quantum wells formed on the surface. The spin in the clusters is polarized so that it becomes NS which is not 1/2 but depends on the number N, of electrons in a cluster.
An effort is made to review the Yang potential. The ground state and the wave function by using Bethe’s Ansatz are found. The Laughlin’s potential is studied and its various properties are investigated. The number of particles in cluster of electrons determines the angular momentum which is used to define the filling factor of the lowest Landau level. A flux factor is added to it so that it gives the desired denominators. The Laughlin state is obtained for any value of the filling factor, p. In all cases, the ground state belongs to nonphysical many-body Hamiltonians. The composite fermion wave function is not the ground state of any known single Hamiltonian. From the angular momentum, we construct the filling factor and then look for the wave function rather than for the Hamiltonian. In this approach the Hamiltonians found are unrealistic. It is possible to make the filling factors agree with the experimental values but the Hamiltonian are not the usual type. The relative angular momentum of two particles L2 is found to define a projection operator. The filling factor is defined for the state of two particles. The number of particles is g. If another particle approaches it to form a cluster, the modified filling factor is defined. In this methodology the flux quanta cannot be attached to the electron and the projection operator is not linked to the Coulomb Hamiltonian. A monolayer of carbon is called graphene. It exhibits unusual properties in the Hall effect and in the cyclotron resonance. Hence, the Kohn's theorem, which shows that the interactions do not play much role in determining the cyclotron resonance, becomes operative. The Hubbard model has been very successful in explaining the ground state of several electron systems. By using the spin in a particular way, we can obtain new features in the Hubbard model. There is a doubling in the Peierls-Luttinger phase factor and eigen values acquire higher multiplicities than are known for the usual treatment of spin. The flux is distributed on the area of the triangle. The graphene consists of hexagons of carbon atoms but the Hall effect shows that there are defects on which electrons form clusters so that there is spin wave type behaviour. The Hall resistivity as a function of magnetic field is not a linear function. Hence a suitable theory to understand the Hall effect is formulated. We find that there are phase transitions as a function of temperature. There are lots of fractions of charge which are explained on the basis of spin and orbital angular momentum of the electron. The nano meter size films of graphite also show that the Hall resistivity is non-linear and shows steps as a function of magnetic field. We make an effort to understand the steps in the Hall effect resistivity of graphite with quantum wells formed on the surface. The spin in the clusters is polarized so that it becomes NS which is not 1/2 but depends on the number N, of electrons in a cluster.
The relative angular momentum of two particles, L2 is used to define a projection operator P2p which corresponds to the relative angular momentum L2+p the filling factor is defined by ν = 1/(L2min+p). The number of particles is g. If one more particle comes near it for the purpose of forming a cluster, the filling factor becomes ν = [(1/2)g(g+1)+p[−1. The g = 2, p = 4 is called Haffnian and g = 2, p = 3 is called Gaffnian. For g = 2, p = 3, v = 1/6. This state has a ground state of a special pseudopotential type Hamiltonian. Given a fraction, we find the ground state energy of a special Hamiltonian. The state wave function, the ground state energy and the special Hamiltonian are linked together. In this methodology, the flux quanta can not be attached to the electrons and the projection operator is not linked to the Coulomb Hamiltonian. Therefore, composite fermions (CF) with the flux quanta attached to the electrons, suggested by Jain cannot satisfy the wave functions, the Hamiltonian and the ground state requirements. The calculated ground state does not attach flux quanta.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
The Hall experiment can be explained by classical transport theory for lower magnetic fields (B ≲ 1 T). In Chapter 1 of this thesis, we review the Drude model to obtain cyclotron motion and to derive the conductivity tensor σ as a function of the magnetic field B. The quantum version of the classical Hall effect initiated a remarkable era in the search for new topological phases of two-dimensional materials and strongly correlated phases, treated via Landau quantization, which generates highly degenerate energy levels known as Landau levels (LLs). Strong, uniform magnetic fields lead to the quantization of the Hall resistivity ρ_H = h/(e^2*ν), where ν is the filling fraction of electrons, taking integer or fractional values depending on the system. This quantization reflects the presence of fully and partially filled LLs. Chapter 2 addresses the role of uniform magnetic and electric fields in a 2D system by choosing an appropriate vector potential A gauge. Also, we investigate the quantum mechanical current operator that allows us to quantize the conductivity tensor element σ_xy = e^2*ν/h. Subsequent sections present the derivations of the wave functions in both Landau and symmetric gauges and the corresponding eigenvalues. In the final sections of Chapter~2, we discuss the percolation concept, which explains Hall plateaux, and examine the role of edge states and random impurities in Hall samples by introducing confinement and random electrostatic potentials, which lead to backscattering effects at the boundaries and localized states in the bulk. Many-body physics plays a crucial role in the partially filled LL, highlighting the strong interactions between electrons in the context of the Fractional Quantum Hall Effect (FQHE). Chapter 3 begins with an exact solution for the two-body problem at filling fraction v = 1. Laughlin’s ansatz offers an elegant solution for the many-body problem at filling fraction v = 1/m, where m is even for bosons and odd for fermions. We review Laughlin’s incompressible wave function, which is an essentially exact solution for short-range interactions, and discuss the plasma analogy, which describes the Laughlin state as a quantum droplet with uniform density. The last section of Chapter 3 addresses the Haldane pseudopotential, which characterizes strong interactions in the FQHE regime. Rotating and trapped bosonic condensates exhibit unique properties for strongly correlated states in the FQHE regime, where the ground state lies in the lowest LL. The effective magnetic field generated by the Coriolis force mimics that of Quantum Hall (QH) systems, enabling the emergence of incompressible states for neutral bosons. In Chapter~4, we also study a three-dimensional rotating BEC in a harmonic trap, highlighting the connection between the FQHE and rapidly rotating BECs. In the final chapter, we investigate an equal-number binary bosonic gas with a fixed total angular momentum L in the z-direction. The tunable coupling ratio g_12/g allows us to identify a critical threshold beyond which the gas becomes unstable in the strongly attractive inter-species interaction regime in disc geometry. The collapse mode refers to the instability in which the binary gas undergoes spatial contraction. Overlap calculations for the ground state show that the Laughlin square wave function, which describes two species via separate Laughlin states, vanishes near the critical coupling ratio g_12/g ∼ -1, as expected. We introduce a molecular Laughlin ansatz that describes center-of-mass pairing between inter-species particles. Our molecular ansatz yields higher overlaps with both the ground state and low-lying excited states compared to the Laughlin square wave function. We also confirm that the ground state of the gas in this regime corresponds to a collapse mode, where the system rotates around its common center of mass. The dependence of the energy spectrum on g_12/g and the corresponding real-space density profiles are also presented.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta Nϕ is shown to be related to the number of electrons N by Nϕ=2N−(5−q) with q being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\ensuremath{\nu}=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\ensuremath{\mu}$ particles per lattice site, where $\ensuremath{\mu}$ may be different than $\ensuremath{\nu}$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of q and for different fillings $\ensuremath{\mu}$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\ensuremath{\le}4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\ensuremath{\nu}=1/q$, while for larger q there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\ensuremath{\mu}\ensuremath{\le}1/q$ or $\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q$ and when q=4 also at half filling.