We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton modes can be interpreted as fusion of charges in two-dimensional plasma picture, extending the plasma analogy to neutral excitations. A lattice diagrammatic method of representing these many-body wavefunctions and FQH elementary excitations is also presented.
We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton modes can be interpreted as fusion of charges in two-dimensional plasma picture, extending the plasma analogy to neutral excitations. A lattice diagrammatic method of representing these many-body wavefunctions and FQH elementary excitations is also presented.
We show the model wave functions for the neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of ``pair excitations''works for the magnetoroton modes of single-component Abelian and non-Abelian FQH states, as well as the neutral fermion mode for the Moore-Read state. The analytic wave functions allow us to compute the ``quadrupole gap'' of the magnetoroton mode in the thermodynamic limit, which was previously inaccessible to the numerics. The quadrupole gap is related to the fusion of the charges in the two-dimensional plasma picture, extending the plasma analogy to neutral excitations. A lattice diagrammatic method of representing these many-body wave functions and FQH elementary excitations is also presented.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
We study the neutral excitations in the bulk of the fractional quantum Hall (FQH) fluids generated by acting with the Girvin-MacDonald-Platzman (GMP) density operator on the uniform ground state. Creating these density modulations atop the ground state costs energy, since any density fluctuation in the FQH system has a gap stemming from underlying interparticle interactions. We calculate the GMP density-mode dispersion for various bosonic and fermionic FQH states on the Haldane sphere using the ground-state static structure factor computed on the same geometry. Previously, this computation was carried out on the plane. Analogous to the GMP algebra of the lowest Landau level (LLL) projected density operators in the plane, we derive the algebra for the LLL-projected density operators on the sphere, which facilitates the computation of the density-mode dispersion. Contrary to previous results on the plane, we find that, in the long-wavelength limit, the GMP mode accurately describes the dynamics of the primary Jain states.
Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this paper, we derive a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and the thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin $\ensuremath{\nu}=1/5$ and the Gaffnian $\ensuremath{\nu}=2/5$ phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a ``phase transition'' even when the ground state is invariant. We discuss the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the ``hollow-core'' modes, as well as their experimental consequences.
The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable. We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.
We study a model fractional quantum Hall (FQH) wave function called the Gaffnian state, which is believed to represent a gapless, strongly correlated state that is very different from conventional metals. To understand this exotic gapless state better, we provide a representation based on work of Halperin in which the pairing structure of the Gaffnian state becomes more explicit. We employ the single-mode approximation introduced by Girvin, MacDonald, and Platzman, here extended to three-body interactions, in order to treat a neutral collective excitation mode in order to clarify the physical origin of the gaplessness of the Gaffnian state. We discuss approaches to extract systematically the relevant physics in the long-distance, large-electron-number limit of FQH states using numerical calculations with relatively few electrons. In Appendices, we provide second-quantized expressions for many-body Haldane pseudopotentials in various geometries including the plane, sphere, cylinder, and torus based on the proper definition of the relative angular momentum.