Abstract
We show that a system of particles on the lowest Landau level can be coupled to a probe U(1) gauge field \( \mathcal{A} \)μ in such a way that the theory is invariant under a noncommutative U(1) gauge symmetry. While the temporal component \( \mathcal{A} \)0 of the probe field is coupled to the projected density operator, the spatial components \( \mathcal{A} \)i are best interpreted as quantum displacements, which distort the interaction potential between the particles. We develop a Seiberg-Witten-type map from the noncommutative U(1) gauge symmetry to a simpler version, which we call “baby noncommutative” gauge symmetry, where the Moyal brackets are replaced by the Poisson brackets. The latter symmetry group is isomorphic to the group of volume preserving diffeomorphisms. By using this map, we resolve the apparent contradiction between the noncommutative gauge symmetry, on the one hand, and the particle-hole symmetry of the half-filled Landau level and the presence of the mixed Chern-Simons terms in the effective Lagrangian of the fractional quantum Hall states, on the other hand. We outline the general procedure which can be used to write down effective field theories which respect the noncommutative U(1) symmetry.
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References
L. Susskind, The Quantum Hall fluid and noncommutative Chern-Simons theory, hep-th/0101029 [INSPIRE].
A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 04 (2001) 011 [hep-th/0103013] [INSPIRE].
S. Hellerman and M. Van Raamsdonk, Quantum Hall physics equals noncommutative field theory, JHEP 10 (2001) 039 [hep-th/0103179] [INSPIRE].
E. Fradkin, V. Jejjala and R.G. Leigh, Noncommutative Chern-Simons for the quantum Hall system and duality, Nucl. Phys. B 642 (2002) 483 [cond-mat/0205653] [INSPIRE].
A. Cappelli and I.D. Rodriguez, Matrix Effective Theories of the Fractional Quantum Hall effect, J. Phys. A 42 (2009) 304006 [arXiv:0902.0765] [INSPIRE].
Z. Dong and T. Senthil, Noncommutative field theory and composite Fermi liquids in some quantum Hall systems, Phys. Rev. B 102 (2020) 205126 [arXiv:2006.01282] [INSPIRE].
V. Pasquier and F.D.M. Haldane, A dipole interpretation of the ν = \( \frac{1}{2} \) state, Nucl. Phys. B 516 (1998) 719 [cond-mat/9712169] [INSPIRE].
N. Read, Lowest Landau level theory of the quantum Hall effect: The Fermi liquid-like state, Phys. Rev. B 58 (1998) 16262 [cond-mat/9804294] [INSPIRE].
X.G. Wen and A. Zee, A Classification of Abelian quantum Hall states and matrix formulation of topological fluids, Phys. Rev. B 46 (1992) 2290 [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
Y.-H. Du, U. Mehta, D.X. Nguyen and D.T. Son, Volume-preserving diffeomorphism as non-Abelian higher-rank gauge symmetry, SciPost Phys. 12 (2022) 050 [arXiv:2103.09826] [INSPIRE].
K. Yang, Geometry of compressible and incompressible quantum Hall states: Application to anisotropic composite-fermion liquids, Phys. Rev. B 88 (2013) 241105 [arXiv:1309.2830].
B.I. Halperin, P.A. Lee and N. Read, Theory of the half filled Landau level, Phys. Rev. B 47 (1993) 7312 [INSPIRE].
D.T. Son, Is the Composite Fermion a Dirac Particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].
D. Gočanin, S. Predin, M.D. Ćirić, V. Radovanović and M. Milovanović, Microscopic derivation of Dirac composite fermion theory: Aspects of noncommutativity and pairing instabilities, Phys. Rev. B 104 (2021) 115150 [arXiv:2102.11313] [INSPIRE].
H. Goldman and T. Senthil, Lowest Landau level theory of the bosonic Jain states, Phys. Rev. B 105 (2022) 075130 [arXiv:2110.03700] [INSPIRE].
D.X. Nguyen, Lowest Landau level limit, W∞ algebra, area-preserving diffeomorphism and non-commutative U(1) Chern-Simons, unpublished (2021).
S.M. Girvin, A.H. MacDonald and P.M. Platzman, Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33 (1986) 2481 [INSPIRE].
Acknowledgments
While this paper was being completed, the authors become aware of ref. [17], where, in particular, the problem with the mutual Chern-Simons term is discussed and partially solved. This paper is supported, in part, by the U.S. DOE grant No. DE-FG02-13ER41958, a Simons Investigator grant and by the Simons collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, DTS).
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Du, YH., Mehta, U. & Son, D.T. Noncommutative gauge symmetry in the fractional quantum Hall effect. J. High Energ. Phys. 2024, 125 (2024). https://doi.org/10.1007/JHEP08(2024)125
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DOI: https://doi.org/10.1007/JHEP08(2024)125