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arXiv:0802.2208 · physica status solidi (b)

Laughlin's function on a cylinder: plasma analogy and representation as a quantum polymer

Sabine Jansen, Élliott H. Lieb, Ruedi Seiler

Published Jan 16, 2008• 8 citations
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Abstract

Abstract We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry of Laughlin's integer quantum Hall effect argument, at filling factor 1/3. We show that the plasma analogy leads to a periodic density, and that the wave function admits a representation as a “quantum polymer”, reminiscent of the quantum dimer model by Rokhsar and Kivelson. We explain how the representation can be exploited to compute the normalization and one‐particle density in the limit of infinitely many particles. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200743516

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Contributions to Plasma Physics•Jan 1, 2016•0 citations
Abstract We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry of Laughlin's integer quantum Hall effect argument, at filling factor 1/3. We show that the plasma analogy leads to a periodic density, and that the wave function admits a representation as a “quantum polymer”, reminiscent of the quantum dimer model by Rokhsar and Kivelson. We explain how the representation can be exploited to compute the normalization and one‐particle density in the limit of infinitely many particles. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract We investigate the nature of the plasma analogy for the Laughlin wave function on a torus describing the quantum Hall plateau at ν = 1 q . We first establish, as expected, that the plasma is screening if there are no short nontrivial paths around the torus. We also find that when one of the handles has a short circumference—i.e. the thin-torus limit—the plasma no longer screens. To quantify this we compute the normalization of the Laughlin state, both numerically and analytically. In the thin torus limit, the analytical form of the normalization simplify and we can reconstruct the normalization and analytically extend it back into the 2D regime. We find that there are geometry dependent corrections to the normalization, and this in turn implies that the plasma in the plasma analogy is not screening when in the thin torus limit. Despite the breaking of the plasma analogy in this limit, the analytical approximation is still a good description of the normalization for all tori, and also allows us to compute hall viscosity at intermediate thickness.
The most robust fractional quantum Hall states occur in the lowest Landau level at filling factors, 1/3 and 1/5. Such states are very well described by Laughlin's wave function. In this work, we have succeeded in calculating exactly the one-particle density function of the Laughlin states for some finite systems of particles in a disk geometry. The exact results we provide are not only important for the Laughlin states, but also for the general field of numerical calculations because they can serve as benchmarks to test the accuracy of various approaches, numerical schemes and computational methods used in studies of strongly correlated electronic systems.
We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\ensuremath{\nu}=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\ensuremath{\mu}$ particles per lattice site, where $\ensuremath{\mu}$ may be different than $\ensuremath{\nu}$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of qqq and for different fillings $\ensuremath{\mu}$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\ensuremath{\le}4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\ensuremath{\nu}=1/q$, while for larger qqq there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q1/q1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\ensuremath{\mu}\ensuremath{\le}1/q$ or $\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q$ and when q=4q=4q=4 also at half filling.
Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.