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Chinese Physics

Composite particle and field theory in atomic quantum Hall effect

Bo Zhao, Zeng‐Bing Chen

Published Jan 27, 2005• 0 citations
Publisher

Abstract

In this paper, we explore the composite particle description of the atomic quantum Hall (QH) effect. We further give the Chern–Simon–Gross–Pitaevskii (CSGP) effective theory for the atomic Hall liquid, which is the counterpart of Chern–Simon theory in electron Hall effect. What we obtained is equivalent to the Laughlin wavefunction approach. Our results show that in terms of composite particles, the atomic Hall effect is really the same as the electronic QH effect. The CSGP effective theory would shed new light on the atomic QH effect.

DOI
10.1088/1009-1963/14/2/027

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In this paper, we explore the composite particle description of the atomic quantum Hall (QH) effect. We further give the Chern–Simon–Gross–Pitaevskii (CSGP) effective theory for the atomic Hall liquid, which is the counterpart of Chern–Simon theory in electron Hall effect. What we obtained is equivalent to the Laughlin wavefunction approach. Our results show that in terms of composite particles, the atomic Hall effect is really the same as the electronic QH effect. The CSGP effective theory would shed new light on the atomic QH effect.
We exploit the analogy with the quantum Hall (QH) effect for electrons to study the possible atomic QH states of a rapidly-rotating Bose-Einstein condensate. Actually, there is a nearly perfect map of the present problem in the QH regime to the QH physics for electrons. The profound map enables one to give a physically appealing definitions of the filling fraction and the "atomic Hall conductance" that is quantized for atomic Laughlin states. This quantization might imply an exotic fractionalization of atomic mass. We also briefly discuss an effective Chern-Simons theory for describing the atomic QH liquids where a gravitational-like field naturally emerges.
This paper reviews progress on the Fractional Quantum Hall Effect (FQHE) based on what we term hamiltonian theories, i.e., theories that proceed from the microscopic electronic hamiltonian to the final solution via a sequence of transformations and approximations, either in the hamiltonian or path integral approach, as compared to theories based on exact diagonalization or trial wavefunctions. We focus on the Chern-Simons (CS) approach in which electrons are converted to CS fermions or bosons that carry along flux tubes and our Extended Hamiltonian Theory (EHT) in which electrons are paired with pseudo-vortices to form composite fermions (CF) whose properties are a lot closer to the ultimate low-energy quasiparticles. We address a variety of qualitative and quantitative questions: In what sense do electrons really bind to vortices? What is the internal structure of the Composite Fermion and what does it mean? What exactly is the dipole picture? How are (approximate) transport gaps, zero-temperature magnetic transitions, the temperature dependent polarizations of gapped and gapless states, the NMR relaxation rate 1/T11/T_11/T1​ in gapless states, and gaps in inhomogeneous states computed? It is seen that though the CS and EHT approaches agree whenever a comparison is possible, results that are transparent in one approach are typically opaque in the other, making them truly complementary.
The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain ‘composite fermion’ excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell–Chern–Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.