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Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)

The fractional quantum Hall effect on a sphere and the Atiyah-Singer index theorem

Brian P. Dolan, Coauthor Hunter-McCabe Aonghus

Published Jul 31, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.22323/1.463.0263

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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.
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.
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.
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){U}_{q}{(su}_{2})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.
Strong correlation among electrons under high magnetic field gives rise to an entirely new arena of emergent physics, namely fractional quantum Hall effect. Such systems have entirely different elementary degrees of freedom and generally, demand non-perturbative approaches to develop a better understanding. In the literature, there are several analytical methodologies and numerical toolkits available to study such a system. Clustering of zeros, parent Hamiltonian, off-diagonal order parameter, parton construction, matrix product states are to be named among a few of those popular methodologies in the existing literature. Most of these methods work well in the lowest Landau level or holomorphic wavefunction framework. It is, however, imperative to develop such methodology to study systems with Landau levels mixing to study more exotic as well as experimentally relevant states. In this work, we have developed particular methodologies, which denounce the traditional importance of the analytic properties of first quantized model wavefunction thereby extend the existing parent Hamiltonian, topological order-parameter, matrix product states descriptions to mixed Landau level systems. Such extension produces a deeper, compact and holistic understanding of universal physics of exotic phases in strongly correlated systems from the microscopic viewpoint, as well as produces interesting new results. Our second quantized/ non-analytic approach allows us to construct the ``entangled Pauli principle", a guidebook to extract universal/topological properties such as braiding statistics, fractional charge quantization, topological degeneracy of the ground states starting from a relatively simple many-body wavefunction, ``root pattern" of fractional quantum Hall state. Such an entangled Pauli principle can be derived from a microscopic parent Hamiltonian setting, thereby provide us a potential tool to probe the non-universal physics in quantum Hall fluids as well. Essentially, entangled Pauli principle is the ``DNA" of fractional quantum Hall states. Using this guiding principle, we have shown ground states with non-abelian excitations, such as Majorana fermion or Fibonacci fermion can be stabilized for two-particle interaction. Fibonacci fermion supports universal quantum gates, thereby a potential candidate for the topologically protected universal quantum computer. Entangled Pauli principle, along with a recently developed topological order parameter for composite fermions, gives rise to Parent Hamiltonian description for composite fermions as well.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).