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arXiv:hep-th/0601075

Conformal field theory description of mesoscopic phenomena in the fractional quantum Hall effect

Lachezar S. Georgiev

Published Jan 12, 2006• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.hep-th/0601075

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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.
Using the effective conformal field theory for the quantum Hall edge states we propose a compact and convenient scheme for the computation of the periods, amplitudes, and temperature behavior of the chiral persistent currents and the magnetic susceptibilities in the mesoscopic disk version of the Zk{Z}_{k}Zk​ parafermion quantum Hall states in the second Landau level. Our numerical calculations show that the persistent currents are periodic in the Aharonov--Bohm flux with period exactly one flux quantum and have a diamagnetic nature. In the high-temperature regime their amplitudes decay exponentially with increasing the temperature and the corresponding exponents are universal characteristics of non-Fermi liquids. Our theoretical results for these exponents are in perfect agreement with those extracted from the numerical data and demonstrate that there is in general a nontrivial contribution coming from the neutral sector. We emphasize the crucial role of the nonholomorphic factors, first proposed by Cappelli and Zemba in the context of the conformal field theory partition functions for the quantum Hall states, which ensure the invariance of the annulus partition function under the Laughlin spectral flow.
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+12+12+1 dimensions and superconductivity in 1+11+11+1 dimensions.
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.
Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.
The authors consider here the behavior of quantum Hall edges away from the Luttinger liquid fixed point that describes the low energy, large system limit. Using the links between quantum Hall wave functions and conformal field theories, they construct effective Hamiltonians and constrain their forms by considering bulk symmetries. In examining the effect of bulk interactions on this edge, they find remarkable simplifications to these effective theories that allow a very accurate description of the low-energy physics of quantum Hall edges relatively far away from the Luttinger liquid fixed point, and that apply to small systems and higher energies.