Abstract The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu–Yang magnetic monopole at the center of the sphere. Wave functions are cross-section of a non-trivial U (1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah–Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions ν = 1 2 k + 1 are recovered.
Abstract The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu–Yang magnetic monopole at the center of the sphere. Wave functions are cross-section of a non-trivial U (1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah–Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions ν = 1 2 k + 1 are recovered.
Most analyses of the ground state wavefunctions of the QHE ignore electron spin and simply use anti-commuting scalars: either because strong magnetic fields split the spins and the ground state is completely spin polarised or because the ground state is effectively spin degenerate so the only effect of spin is to double the degeneracy of the ground state. A useful approach is that of the Haldane sphere where the system is put on a 2-dimensional sphere and a normal magnetic field is generated by placing a magnetic monopole at the centre of the sphere. The analysis of the fractional quantum Hall effect in this approach is somewhat subtle, as the ground state wavefunction of the unperturbed Hamiltonian is not unique in Haldane's approach. When the fermionic nature of the charged particles is fully incorporate into this picture the analysis is modified and we show that, incorporating Jain's composite fermion picture into the Haldane sphere leads to a unique unperturbed ground state for the fractional quantum Hall effect on a sphere. A mass gap then ensures stability under perturbations. An important tool in the analysis is the Atiyah-Singer index theorem.
The authors study a class of models for the fractional quantum Hall effect, where the system is mapped onto the surface of a sphere around a strong magnetic monopole. When the magnetic charge of the monopole is half-integer (in units of h(cross)/e), there exists an SU(2) quasispin group operating on the Fock space of these models. This group commutes with the group of rotations on the sphere, but not with the particle number operator N. This fact may be utilized to easily construct states of definite angular momentum at high particle numbers from states of the same angular momentum at lower particle numbers. Using the Wigner-Eckart theorem the matrix elements of the Hamiltonian between these states can also be constructed. For a large class of pair interactions this leads to a computational method of finding exact eigenstates and eigenvalues of the Hamiltonian.
This thesis explores the application of differential geometric and general relativistic techniques to deepen our understanding of quantum mechanical systems. We focus on three systems, employing these mathematical frameworks to uncover subtle features within each. First, we examine Unruh radiation in the context of an accelerated two-state atom, determining transition frequencies for a variety of accelerated trajectories via first-order perturbation theory. For harmonic motion of the atom in a vacuum, we derive transition rates with potential experimental realizations. Next, we investigate the quantum Hall effect in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field generated by a Wu-Yang monopole. The Atiyah-Singer index theorem constrains the degeneracy of the ground state, and the fractional quantum Hall effect is studied using the composite fermion model, where Dirac strings associated with the monopole field supply the statistical gauge field vortices. A unique, gapped ground state emerges, yielding fractions of the form ν=2k+11 for large particle numbers. Finally, we examine the AdS/CMT correspondence through a bulk fermionic field in an RN-AdS4 background (with a U(1) gauge field), dual to a boundary fermionic operator. Spherical and planar event horizon geometries are discussed, with the temperature of the RN black hole identified with that of the dual system on the boundary. By numerically solving for the spectral functions of the dual theory, for a spherical event horizon at zero temperature, we identify a shift in the Fermi surface from that which arises in the planar case. Preliminary evidence of a phase transition emerges upon examining these spectral functions, again for the spherical horizon, at non-zero temperature.
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.
With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.
The particle-hole (PH) symmetry at half-filled Landau level requires the relationship between the flux number N_phi and the particle number N on a sphere to be exactly N_phi - 2(N-1) = 1. The wave functions of composite fermions with 1/2 "orbital spin", which contributes to the shift "1" in the N_phi and N relationship, are proposed, shown to be PH symmetric, and validated with exact finite system results. It is shown the many-body composite electron and composite hole wave functions at half-filling can be formed from the two components of the same spinor wave function of a massless Dirac fermion at zero-magnetic field. It is further shown that away from half-filling, the many-body composite electron wave function at filling factor nu and its PH conjugated composite hole wave function at 1-nu can be formed from the two components of the very same spinor wave functions of a massless Dirac fermion at non-zero magnetic field. This relationship leads to the proposal of a very simple Dirac composite fermion effective field theory, where the two-component Dirac fermion field is a particle-hole spinor field coupled to the same emergent gauge field, with one field component describing the composite electrons and the other describing the PH conjugated composite holes. As such, the density of the Dirac spinor field is the density sum of the composite electron and hole field components, and therefore is equal to the degeneracy of the Lowest Landau level. On the other hand, the charge density coupled to the external magnetic field is the density difference between the composite electron and hole field components, and is therefore neutral at exactly half-filling. It is shown that the proposed particle-hole spinor effective field theory gives essentially the same electromagnetic responses as Son's Dirac composite fermion theory does.