The subject of this work is the application of conformal field theory in the fractional quantum Hall effect. The fractional quantum Hall effect is observed in a two-dimensional electron gas in a strong perpendicular magnetic field and at low temperature. The electron system forms a strongly correlated quantum liquid and exhibits the exact quantization of the Hall conductivity in to a fractional value vet/h, where v is the fractional filling fraction of Landau-levels in the system. In addition, the elementary excitations of the electron fluid have fractional charge and fractional statistics of anyons. First, we review the basic of two-dimensional conformal field theory and show how certain conformal quantum field theories can be solved non-perturbatively by using the representation theory of conformal symmetries. The low-energy effective field theory in the bulk of a fractional quantum Hall state is known to be a 2+1 dimensional topological field theory. The states of the topological quantum field theory are directly connected to the holomorphic correlation functions of an equivalent two-dimensional conformal field theory. It is known that the gapless excitations at the boundary of the quantum Hall state are described by the same conformal field theory. By using this connection, the so-called Laughlin and Moore-Read states are derived in detail using conformal field theory. The fractional charge and statistics of the quasiparticle excitations in these states are obtained from the conformal field theory description. Furthermore, the non-Abelian statistics of the Moore-Read states are verified explicitly from the quasiparticle wave functions. We argue how the different fractional quantum Hall phases of matter are described by the quantum numbers obtained using conformal field theory. Lastly, we study model interactions that have the Laughlin v = 1/3 and the Moore-Read v = 1/2 states as exact ground states. We compute the matrix elements of these interactions and diagonalize the Hamiltonian matrix in the disk geometry for both states. The low-energy edge excitations of these states are studied in the finite geometry and we verify that they are in accordance with the conformal field theory on the boundary. By creating a localized density disturbance in the system, we create the e/4 quasiparticle of the Moore-Read state, and by studying the properties of the edge theory it is seen that quasiparticle has the same properties as the non-Abelian quasihole in the conformal field theory. The obtained results verify the correspondence between the bulk state and the boundary conformal field theory at the edge in the fractional quantum Hall states in question.
The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.
We investigate the response to modular transformations and the fractional statistics of Abelian multicomponent fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using the anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the
K matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and
(2+1)D topological orders for multicomponent FQH states.