# The Laughlin liquid in an external potential

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## Abstract

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are *N*-particle wave functions that have the form of a product of Laughlin states and analytic functions of the *N* variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large *N* limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.

## Keywords

Fractional quantum hall effect Laughlin states Quasi-holes## Mathematics Subject Classification

81V70## Notes

### Acknowledgements

We thank Elliott H. Lieb for helpful remarks. N. Rougerie received financial support from the French ANR Project ANR-13-JS01-0005-01.

## References

- 1.Arovas, S., Schrieffer, J., Wilczek, F.: Fractional statistics and the quantum Hall effect. Phys. Rev. Lett.
**53**, 722–723 (1984)ADSCrossRefGoogle Scholar - 2.Chafaï, D., Gozlan, N., Zitt, P.-A.: First order asymptotics for confined particles with singular pair repulsions. Ann. Appl. Probab.
**24**, 2371–2413 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Ciftjá, O.: Monte Carlo study of Bose Laughlin wave function for filling factors \(1/2\), \(1/4\) and \(1/6\). Europhys. Lett.
**74**, 486–492 (2006)ADSCrossRefGoogle Scholar - 4.Frank, R.L., Lieb, E.H.: A liquid–solid phase transition in a simple model for swarming. Indiana Univ. J. Math. arXiv:1607.07971 (2017) (to appear)
- 5.Girvin, S.: Introduction to the fractional quantum Hall effect. Séminaire Poincaré
**2**, 54–74 (2004)Google Scholar - 6.Haldane, F.D.M.: Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett.
**51**, 605–608 (1983)ADSMathSciNetCrossRefGoogle Scholar - 7.Jain, J.K.: Composite Fermions. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
- 8.Laughlin, R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett.
**50**, 1395–1398 (1983)ADSCrossRefGoogle Scholar - 9.Laughlin, R.B.: Elementary Theory: the Incompressible Quantum Fluid. In: Prange, R.E., Girvin, S.E. (eds.) The Quantum Hall Effect. Springer, Heidelberg (1987)Google Scholar
- 10.Laughlin, R.B.: Nobel lecture: fractional quantization. Rev. Mod. Phys.
**71**, 863–874 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 11.Lieb, E.H., Lebowitz, J.L.: The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei. Adv. Math.
**9**, 316–398 (1972)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
- 13.Lieb, E.H., Rougerie, N., Yngvason, J.: Rigidity of the Laughlin liquid. arXiv:1609.03818 (2016)
- 14.Lieb, E.H., Rougerie, N., Yngvason, J.: Local incompressibility estimates for the Laughlin phase. arXiv:1701.09064 (2017)
- 15.Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
- 16.Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math.
**23**, 22–116 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Lundholm, D., Rougerie, N.: Emergence of fractional statistics for tracer particles in a Laughlin liquid. Phys. Rev. Lett.
**116**, 170401 (2016)ADSCrossRefGoogle Scholar - 18.Papenbrock, T., Bertsch, G.F.: Rotational spectra of weakly interacting Bose–Einstein condensates. Phys. Rev. A
**63**, 023616 (2001)ADSCrossRefGoogle Scholar - 19.Rougerie, N., Serfaty, S., Yngvason, J.: Quantum Hall states of bosons in rotating anharmonic traps. Phys. Rev. A
**87**, 023618 (2013)ADSCrossRefGoogle Scholar - 20.Rougerie, N., Serfaty, S., Yngvason, J.: Quantum Hall phases and plasma analogy in rotating trapped Bose gases. J. Stat. Phys.
**154**, 2–50 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Rougerie, N., Yngvason, J.: Incompressibility estimates for the Laughlin phase. Commun. Math. Phys.
**336**, 1109–1140 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 22.Rougerie, N., Yngvason, J.: Incompressibility estimates for the Laughlin phase, part II. Commun. Math. Phys.
**339**, 263–277 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 23.Störmer, H., Tsui, D., Gossard, A.: The fractional quantum Hall effect. Rev. Mod. Phys.
**71**, S298–S305 (1999)MathSciNetCrossRefGoogle Scholar - 24.Trugman, S., Kivelson, S.: Exact results for the fractional quantum Hall effect with general interactions. Phys. Rev. B
**31**, 5280 (1985)ADSCrossRefGoogle Scholar

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