Download Conclusions Emergent geometry and Chern

Emergent geometry and Chern-Simons theory in the lowest
Landau level
Xi Luo1, Yong-shi Wu1,2,3, and Yue Yu1,2,4
1Department of Physics and Center for Field theory and Particle Physics,
Fudan University, Shanghai 200433, China
2Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, China
3Department of Physics and Astronomy , University of Utah, Salt Lake City, UT84112, USA
4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, P.O. Box 2735,Beijing 100190, China
Abstract
We relate the collective dynamic internal geometric degrees of freedom to gauge fluctuations in ν=1/m (m odd) fractional
quantum Hall effects. In this way, in the lowest Landau level, a highly nontrivial quantum geometry in two-dimensional
guiding center space emerges from these internal geometric modes. We propose that this quantum geometric field theory
is a topological non-commutative Chern-Simons theory. Topological indices, such as the guiding center angular
momentum (also called the shift) and the guiding center spin, which characterize the FQH states besides the filling factor,
are naturally defined.
.
Emergence of Noncommutative Chern-Simons Gauge Theory
Introduction to FQHE
Lagrangian
Moyal *-Product
Guiding Center
Zero Band Mass Limit
Second Class Constraint
and Dirac Bracket
The noncommutivity in the guiding center space suggests
that a noncommutative field theory will be more reasonable.
And if we separate the electron position into the guiding
center motion and the gauge fluctuation, we come up with
the noncommutative Chern-Simons gauge theory.
The gauge transformation
The constraint equation
( Borrowed from PRL 48.1559 by Tsui et al )
Emergent Geometry of Noncommutative Chern-Simons Theory
A Geometric Description
Then the Lagrangian turns out to be,
Define
And they form a SL(2,R) algebra under the
noncommutative bracket .
We relate the gauge fields with the
zweibein of the unimodular metric
through the following assumption,
Guiding Center Spin
With
A new topological number
Conclusions
We identified the electron position fluctuation around its guiding center in a given Laughlin state with the
collective dynamic internal geometric fluctuation. Furthermore, we see a quantum geometry emerging from the
Chern-Simons gauge fluctuations, whose dynamics are governed by the non-commutative Chern-Simons theory at
least to the leading order. The shift and guiding center spin were naturally defined. We have used the zero mass limit
to do the lowest Landau level projection. Therefore, the application to higher Landau level physics remains open.
The even denominator filling factor FQH states are beyond our reach at this moment.