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Mean Field Description of the Fractional Quantum Hall Effect Near $\nu=1/(2k+1)$
High Energy Physics - Theory · Submitted 30 Apr 1995
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model.
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The abstract, sentence by sentence
The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model. This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. The vortex configurations do not depend on $\mu$ in a finite range about $\mu_0$ in this model. However when $\mu-\mu_0 < \mu_{cr}^-$ (or $\mu-\mu_0>\mu_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). We give numerical examples of vortex and antivortex configurations and discuss the range of $\mu$ and $\nu$ over which the system of vortices is dilute.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. | contribution 0.94 | 0.80 | 0.04 | unclear 0.74 |
| s1 | Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model. | result 0.86 | 0.91 | 0.50 | asserted 1.00 |
| s2 | This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. | result 1.00 | 0.86 | 0.18 | asserted 1.00 |
| s3 | The vortex configurations do not depend on $\mu$ in a finite range about $\mu_0$ in this model. | result 1.00 | 0.84 | 0.22 | asserted 1.00 |
| s4 | However when $\mu-\mu_0 < \mu_{cr}^-$ (or $\mu-\mu_0>\mu_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). | result 1.00 | 0.86 | 0.33 | asserted 1.00 |
| s5 | We give numerical examples of vortex and antivortex configurations and discuss the range of $\mu$ and $\nu$ over which the system of vortices is dilute. | contribution 0.88 | 0.93 | 0.06 | asserted 0.95 |
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Original abstract
The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model. This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. The vortex configurations do not depend on $\mu$ in a finite range about $\mu_0$ in this model. However when $\mu-\mu_0 < \mu_{cr}^-$ (or $\mu-\mu_0>\mu_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). We give numerical examples of vortex and antivortex configurations and discuss the range of $\mu$ and $\nu$ over which the system of vortices is dilute.
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