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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Vortices of the Chern-Simons theory in the collective field approach

Ivan Andrić, Velimir Bardek

Published Jun 15, 1993• 2 citations
Publisher

Abstract

We formulate a collective field approach to a planar nonrelativistic system of bosons interacting via the Chern-Simons interaction. Adding the attractive $\ensuremath{\delta}$ potential this system can be solved and the vortex soliton appears. The ground state of this system is scale and inversion invariant. In the harmonic well this system is not stable against the creation of vortices. Leading and next to leading terms in the $\frac{1}{N}$ expansion of the ground-state energy are calculated.

DOI
10.1103/physrevd.47.5546

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We formulate a collective field approach to a planar nonrelativistic system of bosons interacting via the Chern-Simons interaction. Adding the attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ potential this system can be solved and the vortex soliton appears. The ground state of this system is scale and inversion invariant. In the harmonic well this system is not stable against the creation of vortices. Leading and next to leading terms in the 1N\frac{1}{N}N1​ expansion of the ground-state energy are calculated.
We consider the large-NNN Chern-Simons theory for attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian collective-field approach based on the 1/N1/N1/N expansion. We show that the dynamics of low-lying density excitations around the ground-state vortex configuration is equivalent to that of the Sutherland model. The correspondence between the Chern-Simons coupling constant $\ensuremath{\lambda}$ and the Calogero-Sutherland statistical parameter ${\ensuremath{\lambda}}_{s}$ signals some sort of statistical transmutation accompanying the dimensional reduction of the initial problem.
Based on the observation that a particle motion in one dimension maps to a two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimen- sional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In the case of well separated boundaries we derive the one-dimensional collective field Hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions.
I describe a condensed matter system in terms of interacting quantum fields. In this electronic system, gauge interactions of non electromagnetic origin are generated effectively with a topological term in the effective action and play a leading role in describing the physics. Using the renormalization group and large-N expansion technique, I analyze the low energy sector of a U (1) × U (1) model in three dimensional space-time with a potential of sixth order in the scalar fields coupled to mixed Chern-Simons gauge fields. From the quantum effective potential, I derive various renormalization group functions and the fixed points associated with a quantum phase transition.
Two-dimensional quantum antiferromagnets host rich physics, including long-range ordering, high-Tc{T}_{c}Tc​ superconductivity, quantum spin-liquid behavior, topological ordering, a variety of other exotic phases, and quantum criticalities. Frustrating perturbations in antiferromagnets may give rise to strong quantum fluctuations, challenging the theoretical understanding of the many-body ground state. Here we develop a method to describe the quantum antiferromagnets using fermionic degrees of freedom. The method is based on a formally exact mapping between spin-exchange models and theories describing fermionic matter with the emergent U(1)\text{U}(1)U(1) Chern-Simons gauge field. For the planar N\'eel state, this mapping self-consistently generates the Chern-Simons superconductor mean-field ground state of introduced spinless fermions. We systematically compare the Chern-Simons superconductor state with the planar N\'eel state at the level of collective modes as well as order parameters. We reveal qualitative and quantitative correspondences between these two states. We demonstrate that such a construction using the fractionalized excitations and Chern-Simons gauge field can not only describe the N\'eel order, but can also be applied to study quantum spin liquids. Furthermore, we show that the confinement-deconfinement transitions from the N\'eel order to quantum spin liquids are signaled and characterized by the instabilities of Chern-Simons superconductors, driven by strong frustration. The results suggest observing and classifying the descendants of antiferromagnets, including other ordered states and unconventional superconductors, as well as emergent quantum spin liquids.
Over the years, many theoretical frameworks have been developed to understand the remarkable physics of the quantum Hall system. In this work we discuss the interplay among quantum wires, Chern-Simons theory, bosonization, and particle-vortex duality, which enable a unified approach for the Laughlin states of the fractional quantum Hall system. Starting from the so-called quantum wires system, which is a semimicroscopic description in terms of (1+1)(1+1)(1+1)-dimensional theories, we discuss the emergence of (2+1)(2+1)(2+1)-dimensional low-energy effective field theories by using different maps connecting the microscopic degrees of freedom with the macroscopic ones. We show the origin of these maps by embedding the bosonized version of the original quantum wires model in a gauge-invariant theory and using particle vortex duality.