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arXiv:cond-mat/9803337 · Physical Review Letters

Pairing Effects at the Edge of Paired Quantum Hall States

Kazusumi Ino

Published Aug 1, 1998• 14 citations
arXivHTMLPDF

Abstract

We study pairing effects at the edge of paired fractional quantum Hall states using persistent edge currents as a probe. We give the grand partition functions for edge excitations of paired states (Haldane-Rezayi, Pfaffian, 331) coupled to an Aharanov-Bohm flux and derive the exact formulas of the persistent edge currents. At low temperatures, they exhibit anomalous oscillations in their flux dependence. They converge to the sawtooth function with period ${\ensuremath{\varphi}}_{0}/2$ at zero temperature, which indicates pair condensation. This phenomenon provides an interesting bridge between superconductivity in $2+1$ dimensions and superconductivity in $1+1$ dimensions.

DOI
10.1103/physrevlett.81.1078

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We study pairing effects at the edge of paired fractional quantum Hall states using persistent edge currents as a probe. We give the grand partition functions for edge excitations of paired states (Haldane-Rezayi, Pfaffian, 331) coupled to an Aharanov-Bohm flux and derive the exact formulas of the persistent edge currents. At low temperatures, they exhibit anomalous oscillations in their flux dependence. They converge to the sawtooth function with period \ensuremathφ0/2{\ensuremath{\varphi}}_{0}/2\ensuremathφ0​/2 at zero temperature, which indicates pair condensation. This phenomenon provides an interesting bridge between superconductivity in 2+12+12+1 dimensions and superconductivity in 1+11+11+1 dimensions.
The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely, the Pfaffian, Haldane-Rezayi, and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero-energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The wave functions for each case have a similar form, related to Slater determinants, and the edge states satisfy a ``projection rule,'' that the parity of the number of fermions added to the edge equals the parity of the charge added. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a noninteracting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system, we exhibit a chiral Lagrangian that has manifest SU(2) symmetry, but breaks Lorentz invariance, because of the breakdown of the spin-statistics connection implied by the scalar nature of the field and the positive-definite norm on the Hilbert space. Finally, we consider systems on a cylinder, where the fluid has two edges, and construct the sectors of zero-energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. The corresponding theory for the hierarchy and its generalizations is also given. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions. Two appendixes contain technical details. \textcopyright{} 1996 The American Physical Society.
We give a universal description of the mesoscopic effects occurring in fractional quantum Hall disks due to the Aharonov-Bohm flux threading the system. The analysis is based on the exact treatment of the flux within the conformal field theory framework and is relevant for all fractional quantum Hall states whose edge states CFTs are known. As an example we apply this scheme for the parafermion Hall states and extract the main characteristics of the low- and high- temperature asymptotic behavior of the persistent currents.
The quasihole states of several paired states, the Pfaffian, Haldane-Rezayi, and 331 states, which under certain conditions may describe electrons at filling factor \ensuremath{\nu}=1/2 or 5/2, are studied analytically and numerically in the spherical geometry, for the Hamiltonians for which the ground state are known exactly. We also find all the ground states (without quasiparticles) for these systems in the toroidal geometry. In each case, a complete set of linearly independent functions that are energy eigenstates of zero energy is found explicitly. For fixed positions of the quasiholes, the number of linearly independent states is ${2}^{\mathit{n}\mathrm{\ensuremath{-}}1}$ for the Pfaffian, and ${2}^{2\mathit{n}\mathrm{\ensuremath{-}}3}$ for the Haldane-Rezayi state; these degeneracies are needed if these systems are to possess non-Abelian statistics, and they agree with predictions based on conformal field theory. The dimensions of the spaces of states for each number of quasiholes agree with numerical results for moderate system sizes. The effects of tunneling and of the Zeeman term are discussed for the 331 and Haldane-Rezayi states, as well as the relation to Laughlin states of electron pairs. A model introduced by Ho, which was supposed to connect the 331 and Pfaffian states, is found to have the same degeneracies of zero-energy states as the 331 state, except at its Pfaffian point where it is much more highly degenerate than either the 331 or the Pfaffian. We introduce a modification of the model which has the degeneracies of the 331 state everywhere including the Pfaffian point; at the latter point, tunneling reduces the degeneracies to those of the Pfaffian state. An experimental difference is pointed out between the Laughlin states of electron pairs and the other paired states, in the current-voltage response when electrons tunnel into the edge. An appendix contains results for the permanent state, in which the zero modes can be occupied by composite bosons, rather than by composite fermions as in the other cases; the system is found to have an incipient instability toward a spin-polarized state. \textcopyright{} 1996 The American Physical Society.
We consider the edge states of the quantum Hall paired states, which were introduced as the ground-state wave function for $\ensuremath{\nu}=\frac{5}{2}.$ We propose that the paired Hall states such as the Haldane-Rezayi state and the Pfaffian state have the supersymmetry, and we clarify the exclusion statistics of the quasi-particle excitations. By use of an idea of the ``restricted partition function'' for the quasi-particle excitations and its relationship with the q-special functions, we explicitly compute the distribution functions for the quasi-particle. We also calculate the specific heat and show that our results coincide with previously reported results based on the conformal field theory. We further show that there is a duality between the Pfaffian state and the Haldane-Rezayi state.
Recent theoretical papers have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role in obtaining a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility.
Integrability of electron dynamics in one dimension is manifested by the nonequilibrium stationary states. They emerge near a point contact coupling two quantum Hall edges with different chemical potentials. I use the nonequilibrium bosonization technique to show that the effective temperature of such states at the fractional quantum Hall edges has a universal linear dependence on the current through the contact. In contrast, the temperature at eventual equilibrium scales as the square root of the power dissipating at the point contact. I propose to use this distinction to detect these intriguing nonequilibrium states.