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arXiv:cond-mat/9601125 · International Journal of Modern Physics B

A FIELD THEORY FOR THE READ OPERATOR

RAMAMURTI RAJARAMAN, S. L. SONDHI

Published Mar 30, 1996• 37 citations
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Abstract

We introduce a new field theory for studying quantum Hall systems. The quantum field is a modified version of the bosonic operator introduced by Read. In contrast to Read's original work we do not work in the lowest Landau level alone, and this leads to a much simpler formalism. We identify an appropriate canonical conjugate field, and write a Hamiltonian that governs the exact dynamics of our bosonic field operators. We describe a Lagrangian formalism, derive the equations of motion for the fields and present a family of mean-field solutions. Finally, we show that these mean field solutions are precisely the Laughlin states. We do not, in this work, address the treatment of fluctuations.

DOI
10.1142/s0217979296000337

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We introduce a new field theory for studying quantum Hall systems. The quantum field is a modified version of the bosonic operator introduced by Read. In contrast to Read's original work we do not work in the lowest Landau level alone, and this leads to a much simpler formalism. We identify an appropriate canonical conjugate field, and write a Hamiltonian that governs the exact dynamics of our bosonic field operators. We describe a Lagrangian formalism, derive the equations of motion for the fields and present a family of mean-field solutions. Finally, we show that these mean field solutions are precisely the Laughlin states. We do not, in this work, address the treatment of fluctuations.
We introduce a new field theory for studying quantum Hall systems. The quantum field is a modified version of the bosonic operator introduced by Read. In contrast to Read’s original work we do not work in the lowest Landau level alone, and this leads to a much simpler formalism. We identify an appropriate canonical conjugate field, and write a Hamiltonian that governs the exact dynamics of our bosonic field operators. We describe a Lagrangian formalism, derive the equations of motion for the fields and present a family of mean-field solutions. Finally, we show that these mean field solutions are precisely the Laughlin states. We do not, in this work, address the treatment of fluctuations. 1 Much of our current understanding of the novel behavior in the quantum Hall (QH) regime relies upon microscopic, first quantized, wavefunctions for ideal QH states. The
Certain well known quantum Hall states---including the Laughlin states, the Moore-Read Pfaffian, and the Read-Rezayi Parafermion states---can be defined as the unique lowest degree symmetric analytic function that vanishes as at least ppp powers as some number (g+1)(g+1)(g+1) of particles approach the same point. Analogously, these same quantum Hall states can be generated as the exact highest density zero energy state of simple angular momentum projection operators. Following this theme we determine the highest density zero energy state for many other values of ppp and ggg.
The Hall experiment can be explained by classical transport theory for lower magnetic fields (B ≲ 1 T). In Chapter 1 of this thesis, we review the Drude model to obtain cyclotron motion and to derive the conductivity tensor σ as a function of the magnetic field B. The quantum version of the classical Hall effect initiated a remarkable era in the search for new topological phases of two-dimensional materials and strongly correlated phases, treated via Landau quantization, which generates highly degenerate energy levels known as Landau levels (LLs). Strong, uniform magnetic fields lead to the quantization of the Hall resistivity ρ_H = h/(e^2*ν), where ν is the filling fraction of electrons, taking integer or fractional values depending on the system. This quantization reflects the presence of fully and partially filled LLs. Chapter 2 addresses the role of uniform magnetic and electric fields in a 2D system by choosing an appropriate vector potential A gauge. Also, we investigate the quantum mechanical current operator that allows us to quantize the conductivity tensor element σ_xy = e^2*ν/h. Subsequent sections present the derivations of the wave functions in both Landau and symmetric gauges and the corresponding eigenvalues. In the final sections of Chapter~2, we discuss the percolation concept, which explains Hall plateaux, and examine the role of edge states and random impurities in Hall samples by introducing confinement and random electrostatic potentials, which lead to backscattering effects at the boundaries and localized states in the bulk. Many-body physics plays a crucial role in the partially filled LL, highlighting the strong interactions between electrons in the context of the Fractional Quantum Hall Effect (FQHE). Chapter 3 begins with an exact solution for the two-body problem at filling fraction v = 1. Laughlin’s ansatz offers an elegant solution for the many-body problem at filling fraction v = 1/m, where m is even for bosons and odd for fermions. We review Laughlin’s incompressible wave function, which is an essentially exact solution for short-range interactions, and discuss the plasma analogy, which describes the Laughlin state as a quantum droplet with uniform density. The last section of Chapter 3 addresses the Haldane pseudopotential, which characterizes strong interactions in the FQHE regime. Rotating and trapped bosonic condensates exhibit unique properties for strongly correlated states in the FQHE regime, where the ground state lies in the lowest LL. The effective magnetic field generated by the Coriolis force mimics that of Quantum Hall (QH) systems, enabling the emergence of incompressible states for neutral bosons. In Chapter~4, we also study a three-dimensional rotating BEC in a harmonic trap, highlighting the connection between the FQHE and rapidly rotating BECs. In the final chapter, we investigate an equal-number binary bosonic gas with a fixed total angular momentum L in the z-direction. The tunable coupling ratio g_12/g allows us to identify a critical threshold beyond which the gas becomes unstable in the strongly attractive inter-species interaction regime in disc geometry. The collapse mode refers to the instability in which the binary gas undergoes spatial contraction. Overlap calculations for the ground state show that the Laughlin square wave function, which describes two species via separate Laughlin states, vanishes near the critical coupling ratio g_12/g ∼ -1, as expected. We introduce a molecular Laughlin ansatz that describes center-of-mass pairing between inter-species particles. Our molecular ansatz yields higher overlaps with both the ground state and low-lying excited states compared to the Laughlin square wave function. We also confirm that the ground state of the gas in this regime corresponds to a collapse mode, where the system rotates around its common center of mass. The dependence of the energy spectrum on g_12/g and the corresponding real-space density profiles are also presented.
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.