Guillermo R. Zemba

Infinite Symmetry in the Quantum Hall Effect

Abstract Free planar electrons in a uniform magnetic field are shown to possess the dynamical symmetry of area-preserving diffeomorphisms (W-infinity algebra). Intuitively, this is a consequence of gauge invariance, which forces dynamics to depend only on the flux. The infinity of generators of this symmetry act within each Landau level, which is infinite dimensional in the thermodynamic limit. The incompressible ground states corresponding to completely filled Landau levels (integer quantum Hall effect) possess a dynamical symmetry, since they are left invariant by an infinite subset of generators. This geometrical characterization of incompressibility also holds for fractional fillings of the lowest level (simplest fractional Hall effect) in the presence of Haldanes effective two-body interactions. Although these modify the symmetry algebra, the corresponding incompressible ground states proposed by Laughlin are again symmetric with respect to the modified infinite algebra.

Conformal symmetry and universal properties of quantum Hall states

Abstract The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of (2 + 1)-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlins fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlins theory of incompressible quantum fluids.The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of

Stable hierarchical quantum Hall fluids as W1+∞ minimal models

(2+1)

Large N Limit in the Quantum Hall Effect

-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlins fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlins theory of incompressible quantum fluids.

Thermal Transport in Chiral Conformal Theories and Hierarchical Quantum Hall States

Abstract In this paper, we pursue our analysis of the W1+∞ symmetry of the low-energy edge excitations of incompressible quantum Hall fluids. These excitations are described by (1 + 1)-dimensional effective field theories, which are built by representations of the W1+∞ algebra. Generic W1+∞ theories predict many more fluids than the few, stable ones found in experiments. Here we identify a particular class of W1+∞ theories, the minimal models, which are made of degenerate representations only - a familiar construction in conformal field theory. The W1+∞ minimal models exist for specific values of the fractional Hall conductivity, which nicely fit the experimental data and match the results of the Jain hierarchy of quantum Hall fluids. We thus obtain a new hierarchical construction, which is based uniquely on the concept of quantum incompressible fluid and is independent of Jains approach and hypotheses. Furthermore, a surprising non-abelian structure is found in the W1+∞ minimal models: they possess neutral quark-like excitations with SU(m) quantum numbers and non-abelian fractional statistics. The physical electron is made of anyon and quark excitations. We discuss some properties of these neutral excitations which could be seen in experiments and numerical simulations.

Symmetry aspects and finite-size scaling of quantum Hall fluids

Abstract The Laughlin states for N interacting electrons at the plateaus of the fractional Hall effect are studied in the thermodynamic limit of large N . It is shown that this limit leads to the semiclassical regime for these states, thereby relating their stability to their semiclassical nature. The equivalent problem of two-dimensional plasmas is solved analytically, to leading order for N →∞, by the saddle-point approximation — a two-dimensional extension of the method used in random matrix models of quantum gravity and gauge theories. To leading order, the Laughlin states describe classical droplets of fluids with uniform density and sharp boundaries, as expected from the Laughlin “plasma analogy”. In this limit, the dynamical W ∞-symmetry of the quantum Hall states expresses the kinematics of the area-preserving deformations of incompressible liquid droplets.

Chiral partition functions of quantum Hall droplets

Chiral conformal field theories are characterized by a ground-state current at finite temperature, that could be observed, e.g. in the edge excitations of the quantum Hall effect. We show that the corresponding thermal conductance is directly proportional to the gravitational anomaly of the conformal theory, upon extending the well-known relation between specific heat and conformal anomaly. The thermal current could signal the elusive neutral edge modes that are expected in the hierarchical Hall states. We then compute the thermal conductance for the Abelian multi-component theory and the W1+∞ minimal model, two conformal theories that are good candidates for describing the hierarchical states. Their conductances agree to leading order but differ in the first, universal finite-size correction, that could be used as a selective experimental signature.

Coulomb blockade in hierarchical quantum Hall droplets

Abstract The exactness and universality observed in the quantum Hall effect suggests the existence of a symmetry principle underlying Laughlins theory. We review the role played by the infinite W ∞ and conformal algebras as dynamical symmetries of incompressible quantum fluids and show how they predict universal finite-size effects in the excitation spectrum.

W1+∞ minimal models and the hierarchy of the quantum Hall effect

Chiral partition functions of conformal field theory describe the edge excitations of isolated Hall droplets. They are characterized by an index specifying the quasiparticle sector and transform among themselves by a finite-dimensional representation of the modular group. The partition functions are derived and used to describe electron transitions leading to Coulomb blockade conductance peaks. We find the peak patterns for Abelian hierarchical states and non-Abelian Read-Rezayi states, and compare them. Experimental observation of these features can check the qualitative properties of the conformal field theory description, such as the decomposition of the Hilbert space into sectors, involving charged and neutral parts, and the fusion rules.

Effective Chern–Simons theories of pfaffian and parafermionic quantum Hall states, and orbifold conformal field theories

The degeneracy of energy levels in a quantum dot of Hall fluid, leading to conductance peaks, can be readily derived from the partition functions of conformal field theory. Their complete expressions can be found for Hall states with both Abelian and non-Abelian statistics, upon adapting known results for the annulus geometry. We analyze the Abelian states with hierarchical filling fractions, ν = m/(mp ± 1), and find a non-trivial pattern of conductance peaks. In particular, each one of them occurs with a characteristic multiplicity, which is due to the extended symmetry of the m-folded edge. Experimental tests of the multiplicity can shed more light on the dynamics of this composite edge.