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.
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.
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.
A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.
The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, V3. We here propose a closed-form ground-state wave function for V3 at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero V3 is essential together with V1 for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.
Composite fermion (CF) wave functions are used to describe both a two dimensional electron gas in a strong magnetic field (the quantum Hall effect system) and rotating two dimensional atomic gases that can be either bosons or fermions. In this thesis a new method for projecting fermionic CF wave functions (here called method 3) to the lowest Landau level is investigated. The new projection is based on attaching a Jastrow factor to a bosonic CF wave function projected in the standard way. This will be compared to two other projection methods: method 1 or Girvin-Jach projection and method 2 or Jain-Kamilla projection. I compare the minimal cyclotron energy compact CF ground state candidates for the ∇²δ interaction both to each other and to the exact ground state found by numerical diagonalization for up to 8 particles. The conclusion is that the CF candidates typically are good approximations of the exact ground state, and that method 3 is almost as good as the other two at this. An important motivation for method 3 is that it preserves linear dependencies from the bosonic states. The puzzle of linear dependencies is a part of CF theory that is not properly understood. Certain important results for low angular momentum states have been found for the bosonic case. Since method 3 works quite well in the cases tested here some of these results can be imported to the fermionic case. I have compared the linear dependencies among minimal cyclotron energy compact states for all three projection methods. If approximate linear dependencies are considered as proper linear dependencies then for all tested cases the same linear dependencies hold for all three projection methods, up to slightly different coefficients.
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.
It is commonly assumed in the studies of the fractional quantum Hall effect that the physics of a fractional quantum Hall state, in particular the character of its excitations, is invariant under a continuous deformation of the Hamiltonian during which the gap does not close. We show in this article that, at least for finite systems, as the interaction is changed from a model three body interaction to Coulomb, the ground state at filling factor $\ensuremath{\nu}=2/5$ evolves continuously from the so-called Gaffnian wave function to the composite fermion wave function, but the quasiholes alter their character in a nonperturbative manner. This is attributed to the fact that the Coulomb interaction opens a gap in the Gaffnian quasihole sector, pushing many of the states to very high energies. Interestingly, the states below the gap are found to have a one-to-one correspondence with the composite fermion theory, suggesting that the Gaffnian model contains composite fermions, and that the Gaffnian quasiholes are unstable to the formation of composite fermions when a two-body interaction term is switched on. General implications of this study are discussed.